This document describes a narrow band region approach for 2D and 3D image segmentation using deformable curves and surfaces. Specifically, it develops a region energy term involving a fixed-width band around the evolving curve or surface. This energy achieves a balance between local gradient features and global region statistics. The region energy is formulated to allow efficient computation on explicit parametric models and implicit level set models. Two different region terms are introduced, each suited to different configurations of the target object and its surroundings. The document derives the mathematical framework for computing the region energies and their gradients to allow minimization via gradient descent. It then discusses numerical implementations and provides experiments segmenting medical and natural images.
The document describes a method for multiresolution image fusion using nonsubsampled contourlet transform (NSCT). NSCT provides multiresolution analysis with shift invariance and directionality. The method uses NSCT to learn edges from a high-resolution panchromatic image and fuse it with low-resolution multispectral images. Maximum a posteriori estimation with a Markov random field prior is used to optimize the fused image. Experimental results on fusing QuickBird satellite images show the proposed method produces higher quality fused images compared to other fusion methods.
This document presents a method for interactive image segmentation using constrained active contours. It begins with an overview of existing interactive segmentation techniques, including boundary-based methods like active contours/snakes and region-based methods like random walks and graph cuts. The proposed method initializes a contour using region-based segmentation then refines it using a convex active contour model that incorporates both regional information from seed pixels and boundary smoothness. This allows the contour to globally evolve to object boundaries while handling topology changes.
The development of multimedia system technology in Content based Image Retrieval (CBIR) System is
one in every of the outstanding area to retrieve the images from an oversized collection of database. The feature
vectors of the query image are compared with feature vectors of the database images to get matching images.It is
much observed that anyone algorithm isn't beneficial in extracting all differing kinds of natural images. Thus an
intensive analysis of certain color, texture and shape extraction techniques are allotted to spot an efficient CBIR
technique that suits for a selected sort of images. The Extraction of an image includes feature description and
feature extraction. During this paper, we tend to projected Color Layout Descriptor (CLD), grey Level Co-
Occurrences Matrix (GLCM), Marker-Controlled Watershed Segmentation feature extraction technique that
extract the matching image based on the similarity of Color, Texture and shape within the database. For
performance analysis, the image retrieval timing results of the projected technique is calculated and compared
with every of the individual feature.
REGION CLASSIFICATION BASED IMAGE DENOISING USING SHEARLET AND WAVELET TRANSF...csandit
This paper proposes a neural network based region classification technique that classifies
regions in an image into two classes: textures and homogenous regions. The classification is
based on training a neural network with statistical parameters belonging to the regions of
interest. An application of this classification method is applied in image denoising by applying
different transforms to the two different classes. Texture is denoised by shearlets while
homogenous regions are denoised by wavelets. The denoised results show better performance
than either of the transforms applied independently. The proposed algorithm successfully
reduces the mean square error of the denoised result and provides perceptually good results.
A New Approach for Segmentation of Fused Images using Cluster based ThresholdingIDES Editor
This paper proposes the new segmentation technique
with cluster based method. In this, the multi source medical
images like MRI (Magnetic Resonance Imaging), CT
(computed tomography) & PET (positron emission
tomography) are fused and then segmented using cluster based
thresholding approach. The edge details of an image have
become an essential technique in clinical and researchoriented
applications. The more edge details of the fused image
have obtainable with this method. The objective of the
clustering process is to partition a fused image coefficients
into a number of clusters having similar features. These
features are useful to generate the threshold value for further
segmentation of fused image. Finally the segmented output
is compared with standard FCM method and modified Otsu
method. Experimental results have shown that the proposed
cluster based thresholding method is able to effectively extract
important edge details of fused image.
This document discusses Gabor convolutional networks (GCNs), which incorporate Gabor filters into deep convolutional neural networks (DCNNs) to improve robustness to orientation and scale changes. GCNs modulate the learned convolution filters with Gabor filters, generating convolutional Gabor orientation filters (GoFs) that endow the ability to capture spatial localization, orientation selectivity, and spatial frequency selectivity. This allows GCNs to learn more compact yet enhanced representations with improved generalization to image transformations, outperforming conventional DCNN architectures. The incorporation of Gabor filters into the convolution operation can be integrated into any deep learning model to enhance robustness to scale and rotation variations.
The document summarizes research on mesh representations in computer graphics. It discusses Greg Turk's introduction of "mutual tessellation" to represent objects at different levels of detail. It also covers Hugues Hoppe's work on "mesh optimization" to minimize triangles in dense meshes and "progressive meshes" to preserve overall appearance while simplifying. The document outlines challenges of mesh simplification, level-of-detail approximations, and compression. It describes representing meshes as sets of vertices, connectivity, and attributes. An energy function is defined to optimize meshes by minimizing distance and spring energies while preserving scalar attributes. Applications in medical imaging and reduced manual work in graphics are mentioned.
This document summarizes an approach for single image depth estimation that first performs semantic segmentation to predict semantic labels for each pixel and then uses the predicted semantic labels to guide 3D reconstruction and estimate depth. It presents advantages of this approach over direct depth prediction, including enabling depth and geometry constraints based on semantic class and conditioning depth prediction on appearance differences for a given class. The approach achieves state-of-the-art depth perception results with a simpler model than previous work by leveraging semantic context.
The document describes a method for multiresolution image fusion using nonsubsampled contourlet transform (NSCT). NSCT provides multiresolution analysis with shift invariance and directionality. The method uses NSCT to learn edges from a high-resolution panchromatic image and fuse it with low-resolution multispectral images. Maximum a posteriori estimation with a Markov random field prior is used to optimize the fused image. Experimental results on fusing QuickBird satellite images show the proposed method produces higher quality fused images compared to other fusion methods.
This document presents a method for interactive image segmentation using constrained active contours. It begins with an overview of existing interactive segmentation techniques, including boundary-based methods like active contours/snakes and region-based methods like random walks and graph cuts. The proposed method initializes a contour using region-based segmentation then refines it using a convex active contour model that incorporates both regional information from seed pixels and boundary smoothness. This allows the contour to globally evolve to object boundaries while handling topology changes.
The development of multimedia system technology in Content based Image Retrieval (CBIR) System is
one in every of the outstanding area to retrieve the images from an oversized collection of database. The feature
vectors of the query image are compared with feature vectors of the database images to get matching images.It is
much observed that anyone algorithm isn't beneficial in extracting all differing kinds of natural images. Thus an
intensive analysis of certain color, texture and shape extraction techniques are allotted to spot an efficient CBIR
technique that suits for a selected sort of images. The Extraction of an image includes feature description and
feature extraction. During this paper, we tend to projected Color Layout Descriptor (CLD), grey Level Co-
Occurrences Matrix (GLCM), Marker-Controlled Watershed Segmentation feature extraction technique that
extract the matching image based on the similarity of Color, Texture and shape within the database. For
performance analysis, the image retrieval timing results of the projected technique is calculated and compared
with every of the individual feature.
REGION CLASSIFICATION BASED IMAGE DENOISING USING SHEARLET AND WAVELET TRANSF...csandit
This paper proposes a neural network based region classification technique that classifies
regions in an image into two classes: textures and homogenous regions. The classification is
based on training a neural network with statistical parameters belonging to the regions of
interest. An application of this classification method is applied in image denoising by applying
different transforms to the two different classes. Texture is denoised by shearlets while
homogenous regions are denoised by wavelets. The denoised results show better performance
than either of the transforms applied independently. The proposed algorithm successfully
reduces the mean square error of the denoised result and provides perceptually good results.
A New Approach for Segmentation of Fused Images using Cluster based ThresholdingIDES Editor
This paper proposes the new segmentation technique
with cluster based method. In this, the multi source medical
images like MRI (Magnetic Resonance Imaging), CT
(computed tomography) & PET (positron emission
tomography) are fused and then segmented using cluster based
thresholding approach. The edge details of an image have
become an essential technique in clinical and researchoriented
applications. The more edge details of the fused image
have obtainable with this method. The objective of the
clustering process is to partition a fused image coefficients
into a number of clusters having similar features. These
features are useful to generate the threshold value for further
segmentation of fused image. Finally the segmented output
is compared with standard FCM method and modified Otsu
method. Experimental results have shown that the proposed
cluster based thresholding method is able to effectively extract
important edge details of fused image.
This document discusses Gabor convolutional networks (GCNs), which incorporate Gabor filters into deep convolutional neural networks (DCNNs) to improve robustness to orientation and scale changes. GCNs modulate the learned convolution filters with Gabor filters, generating convolutional Gabor orientation filters (GoFs) that endow the ability to capture spatial localization, orientation selectivity, and spatial frequency selectivity. This allows GCNs to learn more compact yet enhanced representations with improved generalization to image transformations, outperforming conventional DCNN architectures. The incorporation of Gabor filters into the convolution operation can be integrated into any deep learning model to enhance robustness to scale and rotation variations.
The document summarizes research on mesh representations in computer graphics. It discusses Greg Turk's introduction of "mutual tessellation" to represent objects at different levels of detail. It also covers Hugues Hoppe's work on "mesh optimization" to minimize triangles in dense meshes and "progressive meshes" to preserve overall appearance while simplifying. The document outlines challenges of mesh simplification, level-of-detail approximations, and compression. It describes representing meshes as sets of vertices, connectivity, and attributes. An energy function is defined to optimize meshes by minimizing distance and spring energies while preserving scalar attributes. Applications in medical imaging and reduced manual work in graphics are mentioned.
This document summarizes an approach for single image depth estimation that first performs semantic segmentation to predict semantic labels for each pixel and then uses the predicted semantic labels to guide 3D reconstruction and estimate depth. It presents advantages of this approach over direct depth prediction, including enabling depth and geometry constraints based on semantic class and conditioning depth prediction on appearance differences for a given class. The approach achieves state-of-the-art depth perception results with a simpler model than previous work by leveraging semantic context.
A Comparative Study of Wavelet and Curvelet Transform for Image DenoisingIOSR Journals
Abstract : This paper describes a comparison of the discriminating power of the various multiresolution based thresholding techniques i.e., Wavelet, curve let for image denoising.Curvelet transform offer exact reconstruction, stability against perturbation, ease of implementation and low computational complexity. We propose to employ curve let for facial feature extraction and perform a thorough comparison against wavelet transform; especially, the orientation of curve let is analysed. Experiments show that for expression changes, the small scale coefficients of curve let transform are robust, though the large scale coefficients of both transform are likely influenced. The reason behind the advantages of curvelet lies in its abilities of sparse representation that are critical for compression, estimation of images which are denoised and its inverse problems, thus the experiments and theoretical analysis coincide . Keywords: Curvelet transform, Face recognition, Feature extraction, Sparse representation Thresholding rules,Wavelet transform..
Automated Medical image segmentation for detection of abnormal masses using W...IDES Editor
Research in the segmentation of medical images
will strive towards improving the accuracy, precision, and
computational speed of segmentation methods, as well as
reducing the amount of manual interaction. Accuracy and
precision can be improved by incorporating prior information
from atlases and by combining discrete and continuous-based
segmentation methods. The medical images consists a general
framework to segment curvilinear 2D objects The proposed
method based on Watershed algorithm is traditionally applied
on image domain. Then model uses a Markov chain in scale
to model the class labels that form the segmentation, but
augments this Markov chain structure by incorporating tree
based classifiers to model the transition probabilities between
adjacent scales. Experiments with real medical images show
that the proposed segmentation framework is efficient.
Abstract Image Segmentation plays a vital role in image processing. The research in this area is still relevant due to its wide applications. Image segmentation is a process of assigning a label to every pixel in an image such that pixels with same label share certain visual characteristics. Sometimes it becomes necessary to calculate the total number of colors from the given RGB image to quantize the image, to detect cancer and brain tumour. The goal of this paper is to provide the best algorithm for image segmentation. Keywords: Image segmentation, RGB
Performance of Efficient Closed-Form Solution to Comprehensive Frontier Exposureiosrjce
This document discusses boundary detection techniques for images. It proposes a generalized boundary detection method (Gb) that combines low-level and mid-level image representations in a single eigenvalue problem to detect boundaries. Gb achieves state-of-the-art results at low computational cost. Soft segmentation and contour grouping methods are also introduced to further improve boundary detection accuracy with minimal extra computation. The document presents outputs of Gb on sample images and concludes that Gb effectively detects boundaries in a principled manner by jointly resolving constraints from multiple image interpretation layers in closed form.
This document summarizes an automatic left ventricle segmentation technique using iterative thresholding and an active contour model adapted for short-axis cardiac MRI images. It begins with background on image segmentation and its applications. Then, it reviews related work on cardiac segmentation techniques and their limitations. The proposed method segments the endocardium using iterative thresholding and the epicardium using an active contour model. It estimates blood and myocardial intensities, applies region growing to segment the endocardium in each slice, and propagates the segmentation to remaining slices. Finally, it measures left ventricle volume and compares the results to manual segmentation.
The document describes a new method called Cluster Sensing Superpixel (CSS) for generating superpixels. CSS identifies ideal cluster centers using pixel density, which reflects pixel concentration. CSS searches for local optimal cluster centers that have high pixel density (representativeness) and are isolated from other high density pixels. It is approximately 5 times faster than state-of-the-art methods while maintaining comparable performance. When applied to microscopy image segmentation, CSS benefits from its efficient implementation.
A version of watershed algorithm for color image segmentationHabibur Rahman
The document summarizes a master's thesis presentation on a new watershed algorithm for color image segmentation. The thesis addresses issues with existing watershed algorithms like over-segmentation and sensitivity to noise. The contributions of the thesis include an adaptive masking and thresholding mechanism to overcome over-segmentation and perform well on noisy images. The thesis is evaluated using five image quality assessment metrics on 20 classes of images, showing the proposed method performs better and has lower computational complexity than other algorithms. In conclusions, the adaptive watershed algorithm ensures accurate segmentation and is suitable for real-time applications.
A HYBRID MORPHOLOGICAL ACTIVE CONTOUR FOR NATURAL IMAGESIJCSEA Journal
Morphological active contours for image segmentation have become popular due to their low computational complexity coupled with their accurate approximation of the partial differential equations
involved in the energy minimization of the segmentation process. In this paper, a morphological active contour which mimics the energy minimization of the popular Chan-Vese Active Contour without Edges is coupled with a morphological edge-driven segmentation term to accurately segment natural images. By using morphological approximations of the energy minimization steps, the algorithm has a low computational complexity. Additionally, the coupling of the edge-based and region-based segmentation techniques allows the proposed method to be robust and accurate. We will demonstrate the accuracy and robustness of the algorithm using images from the Weizmann Segmentation Evaluation Database and report on the segmentation results using the Sorensen-Dice similarity coefficient.
A HYBRID MORPHOLOGICAL ACTIVE CONTOUR FOR NATURAL IMAGESIJCSEA Journal
Morphological active contours for image segmentation have become popular due to their low computational complexity coupled with their accurate approximation of the partial differential equations involved in the energy minimization of the segmentation process. In this paper, a morphological active contour which mimics the energy minimization of the popular Chan-Vese Active Contour without Edges is coupled with a morphological edge-driven segmentation term to accurately segment natural images. By using morphological approximations of the energy minimization steps, the algorithm has a low computational complexity. Additionally, the coupling of the edge-based and region-based segmentation techniques allows the proposed method to be robust and accurate. We will demonstrate the accuracy and robustness of the algorithm using images from the Weizmann Segmentation Evaluation Database and report on the segmentation results using the Sorensen-Dice similarity coefficient.
An efficient image segmentation approach through enhanced watershed algorithmAlexander Decker
This document proposes an efficient image segmentation approach combining an enhanced watershed algorithm and color histogram analysis. The watershed algorithm is applied to preprocessed images after merging the results with an enhanced edge detection. Over-segmentation issues are addressed through a post-processing step applying color histogram analysis to each segmented region, improving overall performance. The document provides background on image segmentation techniques, reviews related work applying watershed algorithms, and discusses challenges like over-segmentation that watershed approaches can face.
ROLE OF HYBRID LEVEL SET IN FETAL CONTOUR EXTRACTIONsipij
Image processing technologies may be employed for quicker and accurate diagnosis in analysis and
feature extraction of medical images. Here, existing level set algorithm is modified and it is employed for
extracting contour of fetus in an image. In traditional approach, fetal parameters are extracted manually
from ultrasound images. An automatic technique is highly desirable to obtain fetal biometric measurements
due to some problems in traditional approach such as lack of consistency and accuracy. The proposed
approach utilizes global & local region information for fetal contour extraction from ultrasonic images.
The main goal of this research is to develop a new methodology to aid the analysis and feature extraction.
IRJET- Lunar Image Fusion based on DT-CWT, Curvelet Transform and NSCTIRJET Journal
This document compares three image fusion techniques - Dual Tree Complex Wavelet Transform (DT-CWT), Curvelet transform, and Nonsubsampled Contourlet Transform (NSCT) - for fusing high spectral resolution lunar images from the HySI instrument and high spatial resolution images from the TMC instrument on the Chandrayaan-1 satellite. Statistical analysis of the fused images shows that the NSCT technique best preserves both the spectral information of the HySI image and the spatial information of the TMC image, with correlation coefficients over 0.99, higher entropy and average gradients, and lower root mean square errors than the other techniques. Therefore, NSCT produces the highest quality fused lunar images according to these evaluation metrics.
In this paper, we analyze and compare the performance of fusion methods based on four different
transforms: i) wavelet transform, ii) curvelet transform, iii) contourlet transform and iv) nonsubsampled
contourlet transform. Fusion framework and scheme are explained in detail, and two different sets of
images are used in our experiments. Furthermore, eight different performancemetrics are adopted to
comparatively analyze the fusion results. The comparison results show that the nonsubsampled contourlet
transform method performs better than the other three methods, both spatially and spectrally. We also
observed from additional experiments that the decomposition level of 3 offered the best fusion performance,
anddecomposition levels beyond level-3 did not significantly improve the fusion results.
Presented by Adrien Depeursinge, PhD, at MICCAI 2015 Tutorial on Biomedical Texture Analysis (BTA), Munich, Oct 5 2015.
Texture-based imaging biomarkers complement focal, invasive biopsy based biomarkers by providing information on tissue structure over broad regions, non-invasively, and repeatedly across multiple time points. Texture has been used to predict patient survival, tissue function, disease subtypes and genomics (imagenomics and radiogenomics). Nevertheless, several challenges remain, such as: the lack of an appropriate framework for multi-scale, multi-spectral analysis in 2D and 3D; localization uncertainty of texture operators; validation; and, translation to routine clinical applications.
Performance analysis of contourlet based hyperspectral image fusion methodijitjournal
Recently, contourlet transform has been widely used in hyperspectral image fusion due to its advantages,
such as high directionality and anisotropy; and studies show that the contourlet-based fusion methods
perform better than the existing conventional methods including wavelet-based fusion methods. Few studies
have been done to comparatively analyze the performance of contourlet-based fusion methods;
furthermore, no research has been done to analyze the contourlet-based fusion methods by focusing on
their unique transform mechanisms. In addition, no research has focused on the original contourlet
transform and its upgraded versions. In this paper, we investigate three different kinds of contourlet
transform: i) original contourlet transform, ii) nonsubsampled contourlet transform, iii) contourlet
transform with sharp frequency localization. The latter two transforms were developed to overcome the
major drawbacks of the original contourlet transform; so it is necessary and beneficial to see how they
perform in the context of hyperspectral image fusion. The results of our comparative analysis show that the
latter two transforms perform better than the original contourlet transform in terms of increasing spatial
resolution and preserving spectral information.
We propose a novel imaging biomarker of lung cancer relapse from 3-D texture analysis of CT images. Three-dimensional morphological nodular tissue properties are described in terms of 3-D Riesz-wavelets. The responses of the latter are aggregated within nodular regions by means of feature covariances, which leverage rich intra- and inter-variations of the feature space dimensions. The obtained Riesz-covariance descriptors lie on a manifold governed by Riemannian geometry requiring specific geodesic metrics to locally approximate scalar products. The latter are used to construct a kernel for support vector machines (SVM). The effectiveness of the presented models is evaluated on a dataset of 92 patients with non-small cell lung carcinoma (NSCLC) and cancer recurrence information. Disease recurrence within a timeframe of 12 months could be predicted with an accuracy above 80, and highlighted the importance of covariance-based texture aggregation. At the end of the talk, computer tools will be presented to easily extract 3D radiomics quantitative features from PET-CT images.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Research on Image Classification Model of Probability Fusion Spectrum-Spatial...CSCJournals
For insufficient information of imaging spectrum with high spatial resolution, detailed imaging information, reduction of mixed pixels, increase of pure pixels and problems of image characteristic extraction and model classification produced from this, we provide a classifier model of a united spectrum-spatial multi-characteristic based on SVM, and use this model to finish the image classification. The model completely uses the multi-characteristic information, and overcomes the over-fitting problems produced by accumulating high-dimensional characteristics. The model includes three classifications of spectrum-spatial characteristics, namely spectral characteristics-spectral characteristic of multi-scale morphology, spectral characteristics-physical characteristics of underlaying surfaces of multi-scale morphology and spectral characteristics-features spatial extension characteristics of multi-scale morphology. Firstly the three classifications of spectrum-spatial characteristics are classified through SVM, then carries out the probability fusion for the classification results based on the pixels to obtain the final image classification results. This article respectively uses WorldView-2 image and ROSIS image to experiment, and the results show that the model has better classification effect compared with VS-SVM algorithm.
Dokumen tersebut merupakan ringkasan implementasi pendekatan brain based teaching dalam pembelajaran tema "menyelamatkan air" di SD. Guru membagi siswa menjadi kelompok dan memberikan tugas untuk membuat kliping dan gambar tentang dampak aktivitas manusia terhadap siklus air serta cara menghemat air. Pendekatan ini dimaksudkan untuk melibatkan emosi, sosial, kognitif, dan fisik siswa dalam pembelajaran.
The document introduces several of Pinki's classmates and friends, providing brief descriptions of each person including their name, age or class level, positive traits like being intelligent or hardworking, and sometimes hobbies or activities they enjoy like badminton, watching television or cricket, driving, or studying.
El documento proporciona información sobre el software libre. Explica que el software libre da a los usuarios la libertad de ejecutar, copiar, distribuir, estudiar, cambiar y mejorar el software. También describe algunas ventajas del software libre como la economía, libertad de uso y modificación. Finalmente, resume brevemente el decreto 3.390 de Venezuela que promueve el uso de software libre en la administración pública para fortalecer la industria nacional de software.
A Comparative Study of Wavelet and Curvelet Transform for Image DenoisingIOSR Journals
Abstract : This paper describes a comparison of the discriminating power of the various multiresolution based thresholding techniques i.e., Wavelet, curve let for image denoising.Curvelet transform offer exact reconstruction, stability against perturbation, ease of implementation and low computational complexity. We propose to employ curve let for facial feature extraction and perform a thorough comparison against wavelet transform; especially, the orientation of curve let is analysed. Experiments show that for expression changes, the small scale coefficients of curve let transform are robust, though the large scale coefficients of both transform are likely influenced. The reason behind the advantages of curvelet lies in its abilities of sparse representation that are critical for compression, estimation of images which are denoised and its inverse problems, thus the experiments and theoretical analysis coincide . Keywords: Curvelet transform, Face recognition, Feature extraction, Sparse representation Thresholding rules,Wavelet transform..
Automated Medical image segmentation for detection of abnormal masses using W...IDES Editor
Research in the segmentation of medical images
will strive towards improving the accuracy, precision, and
computational speed of segmentation methods, as well as
reducing the amount of manual interaction. Accuracy and
precision can be improved by incorporating prior information
from atlases and by combining discrete and continuous-based
segmentation methods. The medical images consists a general
framework to segment curvilinear 2D objects The proposed
method based on Watershed algorithm is traditionally applied
on image domain. Then model uses a Markov chain in scale
to model the class labels that form the segmentation, but
augments this Markov chain structure by incorporating tree
based classifiers to model the transition probabilities between
adjacent scales. Experiments with real medical images show
that the proposed segmentation framework is efficient.
Abstract Image Segmentation plays a vital role in image processing. The research in this area is still relevant due to its wide applications. Image segmentation is a process of assigning a label to every pixel in an image such that pixels with same label share certain visual characteristics. Sometimes it becomes necessary to calculate the total number of colors from the given RGB image to quantize the image, to detect cancer and brain tumour. The goal of this paper is to provide the best algorithm for image segmentation. Keywords: Image segmentation, RGB
Performance of Efficient Closed-Form Solution to Comprehensive Frontier Exposureiosrjce
This document discusses boundary detection techniques for images. It proposes a generalized boundary detection method (Gb) that combines low-level and mid-level image representations in a single eigenvalue problem to detect boundaries. Gb achieves state-of-the-art results at low computational cost. Soft segmentation and contour grouping methods are also introduced to further improve boundary detection accuracy with minimal extra computation. The document presents outputs of Gb on sample images and concludes that Gb effectively detects boundaries in a principled manner by jointly resolving constraints from multiple image interpretation layers in closed form.
This document summarizes an automatic left ventricle segmentation technique using iterative thresholding and an active contour model adapted for short-axis cardiac MRI images. It begins with background on image segmentation and its applications. Then, it reviews related work on cardiac segmentation techniques and their limitations. The proposed method segments the endocardium using iterative thresholding and the epicardium using an active contour model. It estimates blood and myocardial intensities, applies region growing to segment the endocardium in each slice, and propagates the segmentation to remaining slices. Finally, it measures left ventricle volume and compares the results to manual segmentation.
The document describes a new method called Cluster Sensing Superpixel (CSS) for generating superpixels. CSS identifies ideal cluster centers using pixel density, which reflects pixel concentration. CSS searches for local optimal cluster centers that have high pixel density (representativeness) and are isolated from other high density pixels. It is approximately 5 times faster than state-of-the-art methods while maintaining comparable performance. When applied to microscopy image segmentation, CSS benefits from its efficient implementation.
A version of watershed algorithm for color image segmentationHabibur Rahman
The document summarizes a master's thesis presentation on a new watershed algorithm for color image segmentation. The thesis addresses issues with existing watershed algorithms like over-segmentation and sensitivity to noise. The contributions of the thesis include an adaptive masking and thresholding mechanism to overcome over-segmentation and perform well on noisy images. The thesis is evaluated using five image quality assessment metrics on 20 classes of images, showing the proposed method performs better and has lower computational complexity than other algorithms. In conclusions, the adaptive watershed algorithm ensures accurate segmentation and is suitable for real-time applications.
A HYBRID MORPHOLOGICAL ACTIVE CONTOUR FOR NATURAL IMAGESIJCSEA Journal
Morphological active contours for image segmentation have become popular due to their low computational complexity coupled with their accurate approximation of the partial differential equations
involved in the energy minimization of the segmentation process. In this paper, a morphological active contour which mimics the energy minimization of the popular Chan-Vese Active Contour without Edges is coupled with a morphological edge-driven segmentation term to accurately segment natural images. By using morphological approximations of the energy minimization steps, the algorithm has a low computational complexity. Additionally, the coupling of the edge-based and region-based segmentation techniques allows the proposed method to be robust and accurate. We will demonstrate the accuracy and robustness of the algorithm using images from the Weizmann Segmentation Evaluation Database and report on the segmentation results using the Sorensen-Dice similarity coefficient.
A HYBRID MORPHOLOGICAL ACTIVE CONTOUR FOR NATURAL IMAGESIJCSEA Journal
Morphological active contours for image segmentation have become popular due to their low computational complexity coupled with their accurate approximation of the partial differential equations involved in the energy minimization of the segmentation process. In this paper, a morphological active contour which mimics the energy minimization of the popular Chan-Vese Active Contour without Edges is coupled with a morphological edge-driven segmentation term to accurately segment natural images. By using morphological approximations of the energy minimization steps, the algorithm has a low computational complexity. Additionally, the coupling of the edge-based and region-based segmentation techniques allows the proposed method to be robust and accurate. We will demonstrate the accuracy and robustness of the algorithm using images from the Weizmann Segmentation Evaluation Database and report on the segmentation results using the Sorensen-Dice similarity coefficient.
An efficient image segmentation approach through enhanced watershed algorithmAlexander Decker
This document proposes an efficient image segmentation approach combining an enhanced watershed algorithm and color histogram analysis. The watershed algorithm is applied to preprocessed images after merging the results with an enhanced edge detection. Over-segmentation issues are addressed through a post-processing step applying color histogram analysis to each segmented region, improving overall performance. The document provides background on image segmentation techniques, reviews related work applying watershed algorithms, and discusses challenges like over-segmentation that watershed approaches can face.
ROLE OF HYBRID LEVEL SET IN FETAL CONTOUR EXTRACTIONsipij
Image processing technologies may be employed for quicker and accurate diagnosis in analysis and
feature extraction of medical images. Here, existing level set algorithm is modified and it is employed for
extracting contour of fetus in an image. In traditional approach, fetal parameters are extracted manually
from ultrasound images. An automatic technique is highly desirable to obtain fetal biometric measurements
due to some problems in traditional approach such as lack of consistency and accuracy. The proposed
approach utilizes global & local region information for fetal contour extraction from ultrasonic images.
The main goal of this research is to develop a new methodology to aid the analysis and feature extraction.
IRJET- Lunar Image Fusion based on DT-CWT, Curvelet Transform and NSCTIRJET Journal
This document compares three image fusion techniques - Dual Tree Complex Wavelet Transform (DT-CWT), Curvelet transform, and Nonsubsampled Contourlet Transform (NSCT) - for fusing high spectral resolution lunar images from the HySI instrument and high spatial resolution images from the TMC instrument on the Chandrayaan-1 satellite. Statistical analysis of the fused images shows that the NSCT technique best preserves both the spectral information of the HySI image and the spatial information of the TMC image, with correlation coefficients over 0.99, higher entropy and average gradients, and lower root mean square errors than the other techniques. Therefore, NSCT produces the highest quality fused lunar images according to these evaluation metrics.
In this paper, we analyze and compare the performance of fusion methods based on four different
transforms: i) wavelet transform, ii) curvelet transform, iii) contourlet transform and iv) nonsubsampled
contourlet transform. Fusion framework and scheme are explained in detail, and two different sets of
images are used in our experiments. Furthermore, eight different performancemetrics are adopted to
comparatively analyze the fusion results. The comparison results show that the nonsubsampled contourlet
transform method performs better than the other three methods, both spatially and spectrally. We also
observed from additional experiments that the decomposition level of 3 offered the best fusion performance,
anddecomposition levels beyond level-3 did not significantly improve the fusion results.
Presented by Adrien Depeursinge, PhD, at MICCAI 2015 Tutorial on Biomedical Texture Analysis (BTA), Munich, Oct 5 2015.
Texture-based imaging biomarkers complement focal, invasive biopsy based biomarkers by providing information on tissue structure over broad regions, non-invasively, and repeatedly across multiple time points. Texture has been used to predict patient survival, tissue function, disease subtypes and genomics (imagenomics and radiogenomics). Nevertheless, several challenges remain, such as: the lack of an appropriate framework for multi-scale, multi-spectral analysis in 2D and 3D; localization uncertainty of texture operators; validation; and, translation to routine clinical applications.
Performance analysis of contourlet based hyperspectral image fusion methodijitjournal
Recently, contourlet transform has been widely used in hyperspectral image fusion due to its advantages,
such as high directionality and anisotropy; and studies show that the contourlet-based fusion methods
perform better than the existing conventional methods including wavelet-based fusion methods. Few studies
have been done to comparatively analyze the performance of contourlet-based fusion methods;
furthermore, no research has been done to analyze the contourlet-based fusion methods by focusing on
their unique transform mechanisms. In addition, no research has focused on the original contourlet
transform and its upgraded versions. In this paper, we investigate three different kinds of contourlet
transform: i) original contourlet transform, ii) nonsubsampled contourlet transform, iii) contourlet
transform with sharp frequency localization. The latter two transforms were developed to overcome the
major drawbacks of the original contourlet transform; so it is necessary and beneficial to see how they
perform in the context of hyperspectral image fusion. The results of our comparative analysis show that the
latter two transforms perform better than the original contourlet transform in terms of increasing spatial
resolution and preserving spectral information.
We propose a novel imaging biomarker of lung cancer relapse from 3-D texture analysis of CT images. Three-dimensional morphological nodular tissue properties are described in terms of 3-D Riesz-wavelets. The responses of the latter are aggregated within nodular regions by means of feature covariances, which leverage rich intra- and inter-variations of the feature space dimensions. The obtained Riesz-covariance descriptors lie on a manifold governed by Riemannian geometry requiring specific geodesic metrics to locally approximate scalar products. The latter are used to construct a kernel for support vector machines (SVM). The effectiveness of the presented models is evaluated on a dataset of 92 patients with non-small cell lung carcinoma (NSCLC) and cancer recurrence information. Disease recurrence within a timeframe of 12 months could be predicted with an accuracy above 80, and highlighted the importance of covariance-based texture aggregation. At the end of the talk, computer tools will be presented to easily extract 3D radiomics quantitative features from PET-CT images.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Research on Image Classification Model of Probability Fusion Spectrum-Spatial...CSCJournals
For insufficient information of imaging spectrum with high spatial resolution, detailed imaging information, reduction of mixed pixels, increase of pure pixels and problems of image characteristic extraction and model classification produced from this, we provide a classifier model of a united spectrum-spatial multi-characteristic based on SVM, and use this model to finish the image classification. The model completely uses the multi-characteristic information, and overcomes the over-fitting problems produced by accumulating high-dimensional characteristics. The model includes three classifications of spectrum-spatial characteristics, namely spectral characteristics-spectral characteristic of multi-scale morphology, spectral characteristics-physical characteristics of underlaying surfaces of multi-scale morphology and spectral characteristics-features spatial extension characteristics of multi-scale morphology. Firstly the three classifications of spectrum-spatial characteristics are classified through SVM, then carries out the probability fusion for the classification results based on the pixels to obtain the final image classification results. This article respectively uses WorldView-2 image and ROSIS image to experiment, and the results show that the model has better classification effect compared with VS-SVM algorithm.
Dokumen tersebut merupakan ringkasan implementasi pendekatan brain based teaching dalam pembelajaran tema "menyelamatkan air" di SD. Guru membagi siswa menjadi kelompok dan memberikan tugas untuk membuat kliping dan gambar tentang dampak aktivitas manusia terhadap siklus air serta cara menghemat air. Pendekatan ini dimaksudkan untuk melibatkan emosi, sosial, kognitif, dan fisik siswa dalam pembelajaran.
The document introduces several of Pinki's classmates and friends, providing brief descriptions of each person including their name, age or class level, positive traits like being intelligent or hardworking, and sometimes hobbies or activities they enjoy like badminton, watching television or cricket, driving, or studying.
El documento proporciona información sobre el software libre. Explica que el software libre da a los usuarios la libertad de ejecutar, copiar, distribuir, estudiar, cambiar y mejorar el software. También describe algunas ventajas del software libre como la economía, libertad de uso y modificación. Finalmente, resume brevemente el decreto 3.390 de Venezuela que promueve el uso de software libre en la administración pública para fortalecer la industria nacional de software.
The document discusses abuse victims and potential interventions. It asks the reader to brainstorm problems faced by victims and possible solutions. It provides an example two-part question format that could help structure an intervention - asking both what the issue is and how to help move a victim from abuse to helping others in similar situations.
Orientasi baru pendidikan terhadap perubahan sosial Rasmitadila Mita
1) Dokumen tersebut membahas tentang orientasi baru pendidikan terhadap perubahan sosial budaya, meliputi hubungan antara pendidikan dan budaya, penemuan, difusi budaya, akulturasi, asimilasi, dan inovasi.
2) Pendidikan memainkan peran penting dalam pengajaran nilai-nilai budaya secara kreatif melalui proses interaksi antara individu dan budaya.
3) Perkembangan teknologi dan komunikasi mempercepat proses difusi, akultur
Perkembangan kemampuan matematika anak SD dipengaruhi oleh perkembangan kognitif menurut teori-teori seperti Piaget, Bruner, dan Vygotsky. Kemampuan dasar matematika seperti hitung, bilangan, dan aritmatika berkembang sesuai dengan tahapan kognitif anak. Geometri dan hubungan spasial juga berkembang dari pengenalan bentuk dasar hingga pemahaman yang lebih luas. Prinsip pengembangan kemampuan mate
Dokumen tersebut membahas tentang proses pemerolehan bahasa anak, mulai dari tahap awal mengenal bahasa hingga perkembangan bahasa di sekolah. Proses pemerolehan bahasa pertama dan kedua juga dibahas, termasuk dampak bahasa ibu terhadap pemerolehan bahasa kedua.
Ornament is a decoration used to embellish parts of a building or object, excluding large figurative elements like sculpture. Architectural ornament can be carved from stone, wood, or metal, formed with plaster or clay, or painted onto surfaces. Common types of ornament include carved stone, wood, metal, cast iron, plaster, painted, ceiling, dome, floor, columns, and exterior house ornamentation.
This document discusses the differences between direct and indirect speech. It notes that when changing from direct to indirect speech, pronouns may change, tenses usually change, and contextual details like temporal words and locations are adjusted. Questions require different transformations than statements. Indirect speech uses conjunctions like "that" and "to" and maintains the meaning while adjusting grammar based on the rules discussed.
Este documento describe los procedimientos y cuidados necesarios para el traslado seguro de recién nacidos enfermos o prematuros. Explica la importancia de mantener la temperatura corporal, estabilizar la función respiratoria y asegurar la oxigenación antes y durante el traslado. También cubre el equipamiento requerido como incubadoras portátiles y el rol del personal médico para monitorear signos vitales y prevenir complicaciones durante el proceso de traslado.
Pakistani architecture has developed over various periods, starting from ancient Indus Valley civilizations to modern times. The Indus Valley civilization included well-planned cities with public buildings and drainage systems. Buddhist architecture later flourished with structures borrowing from Greek styles. Islamic architecture rose with the arrival of Islam, shown through early mosques oriented towards Arab styles. Under the Mughal Empire, architecture blended Persian and Indian influences. During British rule, a mixture of European and Islamic styles emerged. After independence, Pakistan strived to express its national identity through modern mosques, monuments, and buildings.
Environmental scanning involves collecting, analyzing, and distributing information for strategic purposes. It allows businesses to understand what opportunities exist in their environment and what threats might challenge their position. When conducting an environmental scan, businesses should research factors like technology, the economy, politics, competitors, and their own community. This helps them identify appropriate business ideas and develop strategies to get ahead of competitors.
Role of Hybrid Level Set in Fetal Contour Extractionsipij
Image processing technologies may be employed for quicker and accurate diagnosis in analysis and
feature extraction of medical images. Here, existing level set algorithm is modified and it is employed for
extracting contour of fetus in an image. In traditional approach, fetal parameters are extracted manually
from ultrasound images. An automatic technique is highly desirable to obtain fetal biometric measurements
due to some problems in traditional approach such as lack of consistency and accuracy. The proposed
approach utilizes global & local region information for fetal contour extraction from ultrasonic images.
The main goal of this research is to develop a new methodology to aid the analysis and feature extraction.
This document presents a method for interactive image segmentation using constrained active contours. It begins with an overview of existing interactive segmentation techniques, including boundary-based and region-based approaches. It then describes initializing a contour using region-based segmentation. The main contribution is a method that uses a convex active contour model incorporating both regional and boundary information to find a globally optimal segmentation. The model includes terms for regional color distributions estimated from user seeds and a boundary term to regularize the contour shape.
A MORPHOLOGICAL MULTIPHASE ACTIVE CONTOUR FOR VASCULAR SEGMENTATIONijbbjournal
This paper presents a morphological active contour ideal for vascular segmentation in biomedical images.
The unenhanced images of vessels and background are successfully segmented using a two-step
morphological active contour based upon Chan and Vese’s Active Contour without Edges. Using dilation
and erosion as an approximation of curve evolution, the contour provides an efficient, simple, and robust
alternative to solving partial differential equations used by traditional level-set Active Contour models. The
proposed method is demonstrated with segmented data set images and compared to results garnered from
multiphase Active Contour without Edges, morphological watershed, and Fuzzy C-means segmentations.
A HYBRID MORPHOLOGICAL ACTIVE CONTOUR FOR NATURAL IMAGESIJCSEA Journal
Morphological active contours for image segmentation have become popular due to their low
computational complexity coupled with their accurate approximation of the partial differential equations
involved in the energy minimization of the segmentation process. In this paper, a morphological active
contour which mimics the energy minimization of the popular Chan-Vese Active Contour without Edges is
coupled with a morphological edge-driven segmentation term to accurately segment natural images. By
using morphological approximations of the energy minimization steps, the algorithm has a low
computational complexity. Additionally, the coupling of the edge-based and region-based segmentation
techniques allows the proposed method to be robust and accurate. We will demonstrate the accuracy and
robustness of the algorithm using images from the Weizmann Segmentation Evaluation Database and
report on the segmentation results using the Sorensen-Dice similarity coefficient
Geospatial Data Acquisition Using Unmanned Aerial SystemsIEREK Press
The Rivers State University campus in Portharcourt is one of the university campuses in the city of Portharcourt,
Nigeria covering over 21 square kilometers and housing a variety of academic, residential, administrative and other
support buildings. The University Campus has seen significant transformation in recent years, including the
rehabilitation of old facilities, the construction of new academic facilities and the most recent update on the creation
of new collages, faculties and departments. The current view of the transformations done within the University
Campus is missing from several available maps of the university. Numerous facilities have been constructed on the
University Campus that are not represented on these maps as well as the qualities associated with these facilities.
Existing information on the various landscapes on the map is outdated and it needs to be streamlined in light of
recent changes to the University's facilities and departments. This research article aims to demonstrate the
effectiveness of unmanned aerial systems (UAS) in geospatial data collection for physical planning and mapping of
infrastructures at the Rivers State University Port Harcourt campus by developing a UAS-based digital map and
tour guide for RSU's main campus covering all collages, faculties and departments and this offers visitors, staff and
students with location and attribute information within the campus.
Methodologically, Unmanned Aerial Vehicles were deployed to obtain current visible images of the campus
following the growth and increasing infrastructural development. At a flying height of 76.2m (250 ft), a DJI
Phantom 4 Pro UAS equipped with a 20-megapixel visible camera was flown around the campus, generating
imagery with 1.69cm spatial resolution per pixel. To obtain 3D modeling capabilities, visible imagery was acquired
using the flight-planning software DroneDeploy with a near nadir angle and 75 percent front and side overlap.
Vertical positions were linked to the World Geodetic System 1984 and horizontal positions to the 1984 World
Geodetic Datum universal transverse Mercator (UTM) (WGS 84). To match the UAS data, GCPs were transformed
to UTM zone 32 north.
Finally, dense point clouds, DSM, and an orthomosaic which is a geometrically corrected aerial image that provides
an accurate representation of an area and can be used to determine true distances, were among the UAS-derived
deliverables.
Image registration using a weighted region adjacency graph. The document presents an image registration technique that uses weighted region adjacency graphs (RAGs). RAGs are constructed from medical images segmented using watershed transformation. Graph matching is performed using a multi-spectral technique based on singular value decomposition to find correspondences between weighted RAG vertices. The method is shown to successfully co-register 2D MRI brain images with errors between corresponding region centroids typically less than 7.5%.
Boosting ced using robust orientation estimationijma
In this paper, Coherence Enhancement Diffusion (CED) is boosted feeding external orientation using new
robust orientation estimation. In CED, proper scale selection is very important as the gradient vector at
that scale reflects the orientation of local ridge. For this purpose a new scheme is proposed in which pre
calculated orientation, by using local and integration scales. From the experiments it is found the proposed
scheme is working much better in noisy environment as compared to the traditional Coherence
Enhancement Diffusion
The mean shift procedure is a general nonparametric technique for analyzing complex multimodal feature spaces and delineating arbitrarily shaped clusters. It works by recursively finding the nearest stationary point of the underlying density function, which corresponds to the mode of the density. The mean shift procedure relates to kernel density estimation and robust M-estimators of location. It provides a versatile tool for feature space analysis that can solve many low-level computer vision tasks with few parameters.
OBIA on Coastal Landform Based on Structure Tensor csandit
This paper presents the OBIA method based on structure tensor to identify complex coastal
landforms. That is, develop Hessian matrix by Gabor filtering and calculate multiscale structure
tensor. Extract edge information of image from the trace of structure tensor and conduct
watershed segment of the image. Then, develop texons and create texton histogram. Finally,
obtain the final results by means of maximum likelihood classification with KL divergence as
the similarity measurement. The study findings show that structure tensor could obtain
multiscale and all-direction information with small data redundancy. Moreover, the method
described in the current paper has high classification accuracy
Study on Reconstruction Accuracy using shapiness index of morphological trans...ijcseit
Basin, lakes, and pore-grain space are important geophysical shapes, which can fit with the several
classical and fractal binary shapes, are processed by employing morphological transformations, and
methods. The decomposition of skeleton network (minimum morphological information) using various
classical structures like square, octagon and rhombus. Then derive the dilated subsets respective degree by
the structures for reconstruct the original image. Through shapiness index of pattern spectrum procedure,
we try test the reconstruction accuracy in a quantitative manner. It gives some general procedure to
characterise the shape-size complexity of surface water body. The reconstruction accuracy is against the
size of water bodies with which we produce the some example of different shapiness index for different
structuring element of shapes. In which quantitative manner approach yields better reconstruction level.
The complexity of water bodies are compared with the surfaces.
This document proposes a new model for segmenting overlapping, near-circular objects in images. It combines a multi-layer nonlocal phase field prior model that favors configurations of possibly overlapping near-circular shapes, with a new additive image likelihood model that accounts for intensities summing in overlapping regions. The full posterior energy is minimized using gradient descent to obtain a maximum a posteriori estimate of the object instances. The model is tested on synthetic and fluorescence microscopy cell nucleus image data.
REGION CLASSIFICATION BASED IMAGE DENOISING USING SHEARLET AND WAVELET TRANSF...cscpconf
This paper proposes a neural network based region classification technique that classifies
regions in an image into two classes: textures and homogenous regions. The classification is
based on training a neural network with statistical parameters belonging to the regions of
interest. An application of this classification method is applied in image denoising by applying
different transforms to the two different classes. Texture is denoised by shearlets while
homogenous regions are denoised by wavelets. The denoised results show better performance
than either of the transforms applied independently. The proposed algorithm successfully
reduces the mean square error of the denoised result and provides perceptually good results.
Boosting CED Using Robust Orientation Estimationijma
n this paper, Coherence Enhancement Diffusion (CED) is boosted feeding external orientation using new
robust orientation estimation. In CED, proper scale selection is very important as the gradient vector at
that scale reflects the orientation of local ridge. For this purpose a new scheme is proposed in which pre
calculated orientation, by using local and integration scales. From the experiments it is found the proposed
scheme is working much better in noisy environment as compared to the traditional Coherence
Enhancement Diffusion
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
An Efficient K-Nearest Neighbors Based Approach for Classifying Land Cover Re...IDES Editor
In recent times, researchers in the remote
sensing community have been greatly interested in
utilizing hyperspectral data for in-depth analysis of
Earth’s surface. In general, hyperspectral imaging comes
with high dimensional data, which necessitates a pressing
need for efficient approaches that can effectively process
on these high dimensional data. In this paper, we present
an efficient approach for the analysis of hyperspectral
data by incorporating the concepts of Non-linear manifold
learning and k-nearest neighbor (k-NN). Instead of
dealing with the high dimensional feature space directly,
the proposed approach employs Non-linear manifold
learning that determines a low-dimensional embedding of
the original high dimensional data by computing the
geometric distances between the samples. Initially, the
dimensionality of the hyperspectral data is reduced to a
pairwise distance matrix by making use of the Johnson's
shortest path algorithm and Multidimensional scaling
(MDS). Subsequently, based on the k-nearest neighbors,
the classification of the land cover regions in the
hyperspectral data is achieved. The proposed k-NN based
approach is evaluated using the hyperspectral data
collected by the NASA’s (National Aeronautics and Space
Administration) AVIRIS (Airborne Visible/Infrared
Imaging Spectrometer) from Kennedy Space Center,
Florida. The classification accuracies of the proposed k-
NN based approach demonstrate its effectiveness in land
cover classification of hyperspectral data.
Self-organizing maps (SOMs) can be used to classify seismic attributes in an unsupervised manner and reveal geological patterns that improve seismic interpretation. SOMs reduce a large set of seismic attributes into a smaller set of clusters that relate to geologic features of interest. The classified seismic data can then be analyzed using computer vision algorithms like convolutional neural networks to automatically identify depositional sequences, seismic facies, play types, leads, and prospects. This facilitates a more robust and timely seismic interpretation that can help identify hydrocarbon traps and reduce exploration risk and costs.
Radar reflectance model for the extraction of height from shape from shading ...eSAT Journals
Abstract
The shape-from-shading (SFS) technique deals with the recovery of shape of an object through a gradual variation of shading
encoded in the image. Most SFS approaches have assumed Lambertian surface to extract DEM from individual images. The
quality of the derived DEM from radar SFS in particular, depends on the appropriate radar reflectance model, which relates the
radar backscatter to the surface normal.This paper will focus on a new reflectance model for relatingthe radar SAR backscatter
coefficient values to surface normal orientation. An iterative minimization SFS algorithm was implemented using this radar
reflectance model to derive the height measurements.The most important key of derivation of the surface height using this model is
forward and inverse Fast Fourier Transform (FFT). The model performance was evaluated on RADARSAT-1 image using both
graphical and statistical analysis. Root mean square error (RMSE) and coefficient of determination (R2) were used as evaluation
criteria for the model performance. The model has shown good performance in reconstructing surface heights from RADARSAT-1
imagery. It gave 17.47m and 97.2% for RMSE and R2, respectively.
Keywords:3-D, SFS, Remote Sensing, Radar Remote Sensing, Satellite Images, SAR Imageries.
A Survey Of Deformable Modeling In Computer GraphicsMartha Brown
This document summarizes research on deformable modeling techniques in computer graphics. It organizes the diverse research approaches by technique rather than application. The summary focuses on physically-based approaches including mass-spring models, finite element models, and continuum models. Mass-spring models represent objects as connected point masses and springs and have been widely used for facial animation. Finite element models offer the greatest accuracy but have seen limited use in computer graphics.
Shape analysis using conformal mapping is a technique that analyzes geometric shapes by mapping their connections and distributions into computer programs. It is useful in fields like archaeology, architecture, medical imaging, and computer design. Conformal mapping preserves angles between shapes and can map 2D and 3D shapes onto each other. For brain mapping specifically, spherical harmonics equations are used to compress brain surface data and compare different brain scans. Conformal mapping proves useful for medical applications due to its ability to uniformly map points on brain surfaces without distortion.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Narrow Band Active Contour
1. Narrow band region-based active contours and surfaces for 2D and 3D segmentation
Julien Mille
Universit´ Fran¸ois Rabelais de Tours, Laboratoire Informatique (EA2101)
e c
64 avenue Jean Portalis, 37200 Tours, France
Abstract
We describe a narrow band region approach for deformable curves and surfaces in the perspective of 2D and 3D image
segmentation. Basically, we develop a region energy involving a fixed-width band around the curve or surface. Classical
region-based methods, like the Chan-Vese model, often make strong assumptions on the intensity distributions of the
searched object and background. In order to be less restrictive, our energy achieves a trade-off between local features
of gradient-like terms and global region features. Relying on the theory of parallel curves and surfaces, we perform a
mathematical derivation to express the region energy in a curvature-based form allowing efficient computation on explicit
models. We introduce two different region terms, each one being dedicated to a particular configuration of the target
object. Evolution of deformable models is performed by means of energy minimization using gradient descent. We
provide both explicit and implicit implementations. The explicit models are a parametric snake in 2D and a triangular
mesh in 3D, whereas the implicit models are based on the level set framework, regardless of the dimension. Experiments
are carried out on MRI and CT medical images, in 2D and 3D, as well as 2D color photographs.
Key words: Segmentation, narrow band region energy, deformable model, active contour, active surface, level sets
1. Introduction evolution algorithm. Conversely, implicit implementa-
tions, based on the level set framework [4], handle the
Segmentation by means of deformable models has been evolving boundary as the zero level of a hypersurface,
a widely studied aspect of computer vision over the last defined on the same domain as the image. They are
two decades. Since their introduction by Kass et al. [1], often chosen for their natural handling of topological
deformable models have found many applications in im- changes and intuitive extensibility to higher dimensions.
age segmentation and tracking. From an initial location, Their algorithmic complexity is a function of the image
which may be manually or automatically provided, these resolution, making them time-consuming. Despite the
models deform according to an iterative evolution algo- development of accelerating methods, like the narrow
rithm until they fit one or more structures of interest. The band technique [4] or the fast marching method [5], their
evolution method is usually derived from the minimization computational cost remains higher than their explicit
of some energy functional, including regularizing terms counterparts.
for geometrical smoothness and external terms relating
the model to the data. They are powerful tools thanks Deformable models, whether they are explicit or
to their ability to adapt their geometry and incorporate implicit, are attached to the image by means of a local
prior knowledge about the structure of interest. edge-based energy or force. Since they consider only local
boundaries, classical snakes are relatively blind, in the
Several implementations of these active models were sense they are unable to reach boundaries if their initial
developed. Explicit deformable models represent the location is far from them. The increasing use of region
evolving boundary as a set of interconnected control terms inspired by the Mumford-Shah functional [6, 7] has
points or vertices. Among these, the original 2D paramet- proven to overcome the limitations of uniquely gradient-
ric contour and the 3D triangular mesh [2, 3] are intuitive based models, especially when dealing with data sets
implementations, in which the boundary is deformed by suffering from noise and lack of contrast. Indeed, many
direct modifications of vertices coordinates. The main anatomical structures encountered in medical imaging
drawback is that polygon and meshes do not modify lend themselves to region-based segmentation. Global
their topology naturally, i.e. techniques for detection statistical data computed over the entire region of interest
of topological changes must be implemented beside the is a well established technique to improve the behaviour
of snakes. Early work, including the anticipating snake by
Ronfard [8] and the active region model by Ivins and Por-
Email address: julien.mille@univ-tours.fr (Julien Mille)
rill [9], introduced the use of region terms in the evolution
Preprint submitted to Computer Vision and Image Understanding May 18, 2009
2. of parametric snakes. The region competition method
by Zhu and Yuille [10] was developed later, combining
aspects of snakes and region growing techniques. Many
papers have dealt with region-based approaches using the
level set framework, including the Chan-Vese model [11],
(a) (b) (c)
the deformable regions by Jehan-Besson et al. [12] and
the geodesic active regions by Paragios and Deriche [13]. Figure 1: Different object configurations for different region energies
These implementations have the advantage of adaptive
topology at the expense of computational cost. In the
context of 3D segmentation, a deformable mesh endowed of the image in medical data, the background contains
with a Chan-Vese region energy was presented in [14], various anatomical structures, which differ in their overall
whereas Dufour et al. [15] used an implicit active surface intensities and textures. In this context, the use of local
to perform segmentation and tracking of cells, where features was already addressed in the literature. For
computations are particularly time-expensive. other work dealing with local statistics in region-based
segmentation, the reader may refer to [18, 19, 20, 21].
Most existing region-based deformable models segment For the same purpose, active contours embedded with
images according to statistical data computed over the en- both edge and region terms were studied in [22, 23, 24]
tire regions, i.e. the object of interest and the background. and extended to textured region segmentation [25]. In
These approaches have an underlying notion of homogene- cases (b) and (c), the background, made up of the floor
ity, in the sense that image partitions should be uniform in and the plate, is now piecewise uniform. Case (b) depicts
terms of intensity, whether prior knowledge on the distri- a particular situation where the background is uniform in
bution of pixel intensities is available [16] or not. Instead a small band around the cup boundaries. We believe that
of raw pixel intensity, higher level features like texture de- many objects can be discriminated from the background
scriptors may also be considered [17]. We now focus on according to intensity features only in the vicinity of their
the region energy of the Chan-Vese model [11]. Let Rin be boundaries, which leads to the development of our first
the region enclosed by deformable curve Γ, and Rout its narrow band region energy. Extending the work in [26],
complement. The energy penalizes the curve splitting the we formulate our energy as the intensity variance over an
image into heterogeneous regions, using intensity devia- inner and an outer band around the evolving boundary.
tions. In addition to length and area terms, the Chan-Vese Case (c) represents an even more general case, where
model has the following global data term: the outer band around the target object is piecewise
C-V
constant. Indeed, the cup is surrounded by the floor in
Eregion [Γ] = λin (I(x)−kin )2 dx the upper half and the plate in the bottom half. The
Rin (1) role of our second narrow band region energy is to handle
+λout (I(x)−kout )2 dx configurations in which the outer neighborhood of the
Rout target object presents several distinct areas.
where kin and kout are intensity descriptors inside and In the paper, we first describe the theoretical frame-
outside the curve, respectively. By gradient descent, these work of the narrow band energy. This includes mathe-
descriptors are assigned to average intensity values [11]. matical derivations to yield a suitable form for the region
At the end of the segmentation process, region Rin is term, i.e. a formulation enabling natural implementation.
expected to coincide with the target object. Hence, Our mathematical development is based on the theory of
although constraints on intensity deviations can be parallel curves and surfaces [27, 28]. We endeavour to de-
adjusted by tuning parameters λin and λout , the global velop a framework which is applicable both to 2D and 3D
region term is by definition devoted to segment uniform segmentation. Indeed, after describing our region terms
objects and backgrounds. Let us consider the images on a planar curve, we extend them to a deformable sur-
depicted in fig. 1, in which the object of interest is the face model. Then, in order to allow gradient descent af-
white cup. Ignoring the influence of illumination changes, terwards, we determine the variational derivatives of the
case (a) is the typical configuration which the Chan-Vese region energies with respect to the curve, thanks to calcu-
model aims at, since the cup and the floor areas are nearly lus of variations, and extend them to the surface model as
constant with respect to color. well. Then, we deal with numerical implementation issues,
including model structure and energy minimization. We
Uniformity of intensity over regions is a rather first present the explicit implementation, which lies in a 2D
strong assumption. However, strict homogeneity is polygonal contour and a 3D triangular mesh. These mod-
not necessarily a desirable property, especially for the els are able to perform resampling, in order to overcome
background. The ideal case (a) is rarely encountered the lack of geometrical flexibility of traditional snakes and
in most of computer vision applications. For instance, meshes. We also provide a level-set implementation, which
when one wants to isolate a single structure from the rest
2
3. offers the advantage of a common mathematical descrip- Γ[−B]
tion in 2D and 3D, in addition to the topological adapt- Γ
ability. Finally, experiments are carried out on medical
data and natural color images. For both explicit and im- Γ[B]
plicit implementations, the tests discuss the advantages of
our narrow band terms over other data terms including
edge energies and global region energies.
2. Active contour model
Bin
2.1. Energies Bout
The continuous active contour model is represented as
a parameterized curve Γ with position vector c:
Figure 2: Inner and outer bands for narrow band region energy
Γ: Ω −→ R2
(2)
u −→ c(u) = [x(u) y(u)]T
Let Bin be the inner band domain and Bout the outer
where x and y are continuously differentiable with respect band domain (see fig. 2), and B the band thickness, which
to parameter u. The parameter domain is normalized: is constant as we move along Γ. We propose two different
Ω = [0, 1]. We assume that the curve is simple, i.e. non- region energies. Thus, in eq. (3), the region term will be
intersecting, and closed: c(0) = c(1). Segmentation of either Eregion 1 or Eregion 2 . To obtain Eregion 1 , we consider
an object of interest is performed by finding the curve Γ eq. (1) and we replace Rin with Bin and Rout with Bout ,
minimizing the following energy functional: which yields:
E[Γ] = ωEsmooth [Γ] + (1 − ω)Eregion [Γ] (3)
Eregion 1 [Γ]= (I(x)−kin )2 dx + (I(x)−kout )2 dx (5)
where Esmooth and Eregion are respectively the smooth- Bin Bout
ness and region energies. The user-provided coefficient
ω weights the significance of the smoothness term. We Increased flexibility is achieved thanks to the narrow band
express the smoothness energy in terms of first-order principle, since it does not convey a strict homogeneity
derivative, as it appears in the original snake model by condition like classical region-based approaches. The sec-
Kass et al. [1]: ond energy is a generalization of the first one. Its purpose
is to handle cases where the background is locally homo-
dc
2 geneous in the vicinity around the object (see fig. 1c). For
Esmooth [Γ] = du (4) now, we express it using a local outer descriptor hout de-
Ω du
pending on current position x:
The first-order regularization term usually prevents the
contour to undergo large variations of its area. In our Eregion 2 [Γ]= (I(x)−kin )2 dx + (I(x)−hout (x))2 dx (6)
case, it is a non desirable property, since the contour will Bin Bout
be initialized as a small shape inside a target object and
inflated afterwards. Once discretized as a polygon, the In eq. (1), one may note that the Chan-Vese region
contour is periodically reparameterized to keep control term is asymmetric, as region integrals are independently
points practically equidistant and to allow inflation. In weighted in order to favour minimization of intensity devi-
this context, resampling and remeshing techniques are ation inside or outside. However, we use symmetric terms
discussed in section 5. in our approach, as it is the most common case with region-
based active contours. In what follows, we show in what
Curve Γ splits the image domain D into an inner extent the narrow band principle allows easier implemen-
region Rin and an outer region Rout , over which the tation than classical region-based approaches.
homogeneity criterion is usually expressed. The narrow
band principle, which has proven its efficiency in the 2.2. Parallel curves
evolution of level sets [4], is used in our approach to The theoretical background of our narrow band frame-
formulate a new region term. Instead of dealing with the work is based on parallel curves, also known as ”offset
entire domains delineated by the evolving curve, we only curves” [27, 28]. The curve Γ[B] is called a parallel curve
consider an inner and outer band both sides apart from of Γ if its position vector c[B] verifies
the curve, as depicted in fig. 2. One may note that the c[B] (u) = c(u) + Bn(u) (7)
bands are not limited to the snake’s initial location and
are updated during curve evolution. where B is a real constant, corresponding to the amount
of translation, and n in the inward unit normal of Γ.
3
4. An important property resulting from the definition in
eq. (7) is that the velocity vector of parallel curves de-
pends on the curvature of Γ. The velocity vector of curve
Γ[B] is expressed as a function of the velocity vector of Γ,
as well as its curvature and normal. Using the identity
nu = − κcu , we have:
c[B] u = cu + Bnu = (1 − Bκ)cu (11)
which yields, for the length element of inner parallel curve:
Γ[B1 ] Γ[B2 ]
Γ ℓ[B] = c[B] u = ℓ |1 − Bκ|
The same development is valid for Γ[−B] , replacing B
with −B. This is a known result in parallel curve the-
Figure 3: Main curve (black) and two parallel curves. Small trans- ory [32, 33]. The expressions of ℓ[B] and ℓ[−B] suggest the
lation B1 yields regular curve (blue) whereas large translation B2 smoothness condition of curves Γ[B] ans Γ[−B] . Indeed,
yields a curve with singulaties (red). Corresponding points on par- their length elements should remain strictly positive. This
allel curves are linked with dashed lines.
implies a constraint on the maximal curvature of curve Γ,
i.e. the band width should not exceed the radius of cur-
vature. We should assume that Γ is smooth enough such
Hereafter, we will use the index [B] to denote all quantities that:
related to the parallel curve. The definition in eq. (7) 1 1
− < κ(u) < , ∀u ∈ Ω (12)
is suitable to our narrow band formulation, in the sense B B
that bands Bin and Bout are bounded by parallel curves If condition 12 is well verified, curves Γ[B] and Γ[−B] are
of Γ, respectively Γ[B] and Γ[−B] . This implies that both simple and regular. The impact of this assumption on
curves are continuously differentiable and do not exhibit numerical implementation is discussed in section 5.
singularities. Fig. 3 depicts a case where width B2 ,
unlike B1 , is larger than the curve’s radius of curvature, 2.3. Transformation of area integral
yielding singularities (also known as cusps). Afterwards, In this section, we show that the domain integrals ap-
we refer to the eroded inner region by Rin[B] , bounded pearing in eq. (5) can be expressed in terms of c and B.
by Γ[B] , and the dilated inner region by Rin [−B] bounded This conversion is mandatory for the calculation of the
by Γ[−B] . variational derivative of Eregion with respect to c. More-
over, it brings a formulation suitable for implementa-
Before introducing our simplification, let us recall the tion on explicit models. The proof is based on Green-
notion of line integral. Given a real-valued function f de- Riemann theorem, stating that for every region R, if
fined over R2 and a domain D ⊂ R2 , we introduce the [P (x, y) Q(x, y)]T is a continuously differentiable R2 → R2
general notation J(f, D) representing the integral of f over vector field, then:
domain D. If D is a region R, J(f, R) is an area integral ∂Q ∂P
whereas if D is a curve Γ, J(f, Γ) is written as a line inte- − dxdy = P dx + Qdy
∂x ∂y ∂R
gral: R
dc
J(f, Γ) = f (c(u)) du (8) In order to apply the theorem on J(f, R), where f is a
Ω du
real-valued function defined on the image domain D, one
where the length element (or velocity) should determine vector field [P Q] such that
dc ∂Q ∂P
ℓ= (9) − = kf (x, y)
du ∂x ∂y
where k is a real constant. By choosing P and Q as follows,
makes J(f, Γ) intrinsic, i.e. independent of the param-
the previous condition is satisfied:
eterization. This idea was first introduced in deformable
x
models with the geodesic active contour model [29, 30, 31]. 1
Q(x, y) = f (t, y)dt
From now on, we will use indexed notations for derivatives: 2 −∞
1 y (13)
dc d2 c P (x, y) = − f (x, t)dt
cu = , cuu = 2 ... (10) 2 −∞
du du
Hereafter, we will rely on the following equation to trans-
The curvature of Γ is: form region integrals:
xu yuu − xuu yu xu yuu − xuu yu
κ(u) = = J(f, R) = P dx + Qdy (14)
(x2
u + 2 3
yu ) 2 ℓ3 ∂R
4
5. Γ1 Γ1 2.4. Region energies
B Thanks to the previous result, we now express our two
Γ2 Γ2 narrow band region energies in terms of contour, curvature
and band thickness. The first narrow band region energy,
which aims at minimizing the intensity deviation in the
two bands, is rewritten:
B
(a) (b)
Eregion 1 [Γ] = (I(c+bn) − kin )2 ℓ(1−bκ)dbdu
Figure 4: Region enclosed by two simple closed curves Γ1 and Γ2 (a) B Ω 0 (18)
split into infinitesimal quadrilaterals (b) + (I(c−bn) − kout )2 ℓ(1+bκ)dbdu
Ω 0
For the second narrow band region energy, intensity devi-
Let us consider the more general case of a band B ation should be minimized in the outer band locally along
bounded by curves Γ1 and Γ2 , as depicted in fig. 4a. Rely- the curve. Hence, we replace global descriptor kout of
ing on eq. (14), the integral of f over B is expressed using eq. (18) by a local counterpart, which is now a function of
Green’s theorem: the position on the curve:
J(f, B) = J(f, R1 ) − J(f, R2 ) B
Eregion 2 [Γ] = (I(c+bn) − kin )2 ℓ(1−bκ)dbdu
= x1u P (c1 ) + y1 u Q(c1 )du (15) B Ω 0 (19)
Ω
+ (I(c−bn) − hout (u))2 ℓ(1+bκ)dbdu
− x2u P (c2 ) + y2 u Q(c2 )du Ω 0
Ω
Up to now, we have used intensity descriptors without ex-
˜
We introduce a family of curves {Γ(α)}α∈[0,1] interpolating plicitly providing their expressions. They may be consid-
˜
from Γ1 to Γ2 . The position vector of Γ is ered as unknowns which will be determined during energy
minimization. Their values will be determined by calculus
c(α, u) = (1 − α)c2 (u) + αc1 (u)
˜ of variations of the energies, described in section 4.
Relying on the following equality,
1 3. Active surface model
d
c1 − c2 = (1 − α)c2 + αc1 dα
0 dα 3.1. Energies
and using integration by parts, we transform eq. (15) and The active contour method approach naturally extends
show that J(f, B) can be directly expressed as a function to a three dimensional segmentation problem. In a con-
of f , c1 and c2 (a detailed derivation is provided in ap- tinuous space, a deformable model is represented by a pa-
pendix A.1). rameterized surface Γ.
1 Γ : Ω2 −→ R3
J(f, B) = f (˜)(c1 − c2 ) × cu dαdu
c ˜ (16) (u, v) −→ s(u, v) = [x(u, v) y(u, v) z(u, v)]T
Ω 0
This expression is intuitively understood since (1 − α)c2 + In all subsequent derivations, we will assume a closed sur-
αc1 sweeps all curves between Γ1 and Γ2 as α varies from 0 face with a parameterization homeomorphic to a torus:
to 1. The cross product corresponds to the area of the in-
finitesimal quadrilaterals spanned by (c1− 2 ) and cu , as de-
c ˜ s(0, v) = s(1, v) ∀v ∈ Ω
(20)
picted in fig. 4b. Relying on parallel curves, the mathemat- s(u, 0) = s(u, 1) ∀u ∈ Ω
ical definition of bands Bin allows us to express J(f, Bin )
or a sphere:
in a convenient form. We apply the general result in
eq. (16) on inner band Bin , considering curves Γ and Γ[B] s(0, v) = s(1, v) ∀v ∈ Ω
instead of Γ1 and Γ2 . Using a variable thickness b (see s(u1 , 0) = s(u2 , 0) ∀(u1 , u2 ) ∈ Ω2 (21)
appendix A.2 for more details), we finally obtain: s(u1 , 1) = s(u2 , 1) ∀(u1 , u2 ) ∈ Ω2
B
J(f, Bin ) = f (c + bn)ℓ(1 − bκ)dbdu (17) Note that these parameterizations are given only for math-
Ω 0 ematical transformation purpose and do not generate any
The formulation for J(f, Bout ) is easily obtained by re- constraint on the topology of the surface once this last
placing b with −b in eq. (17). This expression is especially one is discretized. Hence, they do not restrict numerical
useful when the curve is discretized as a polygonal line, as implementation. The surface is endowed with the energy
described in section 5.
5
6. functional E. Replacing c by s in eq. (3), we obtain the The gaussian curvature κG and mean curvature κM may
surface energy to be minimized. The smoothness term is: be expressed in terms of coefficients of the fundamental
2 2 forms:
∂s ∂s LN − M 2
Esmooth [Γ] = + dudv (22) κG =
∂u ∂v EG − F 2
Ω2
GL − 2F M + EN
κM =
Considering now that image I is a R3 → R function, the 2(EG − F 2 )
narrow band region energy is a function of the volume Normal derivatives nu and nv are orthogonal to n. In the
integrals over the two bands Bin and Bout : tangential plane at point s(u, v), they can be expressed as
combinations of basis vectors su and sv according to the
Eregion 1 [Γ] = (I(x) − kin )2 dx
Weingarten equations [34]:
Bin (23)
+ (I(x) − kout )2 dx F M − GL F L − EM
nu = su + sv
Bout EG − F 2 EG − F 2 (26)
F N − GM F M − EN
and similarly for Eregion 2 . As in the two dimensional case, nv = su + sv
EG − F 2 EG − F 2
terms defined over bands are not computed as is. They
should undergo some mathematical transformation in or- which lead to the following combinations, holding mean
der to be differentiated and implemented. This is done and gaussian curvatures:
through the framework of parallel surfaces described in nu ×sv + su ×nv = −2κM su ×sv
the next section. (27)
nu ×nv = κG su ×sv
3.2. Parallel surfaces An important property, resulting from eq. (27), is that the
In three dimensions, regions Bin and Bout are bounded normal vector of parallel surface Γ[B] is colinear to the
by Γ and its parallel surfaces Γ[B] and Γ[−B] , respectively. normal vector of Γ. Its magnitude is a function of the
As an example, if Γ describes a sphere, Bin and Bout may mean and gaussian curvatures of Γ:
be viewed as two empty balls with thickness B. s[B] u ×s[B] v = (su + Bnu ) × (sv + Bnv )
(28)
s[B] (u, v) = s(u, v) + Bn(u, v) (24) = (1 − 2BκM + B 2 κG )su ×sv
and similarly for s[−B] . As previous, B is the constant Considering the magnitude of the previous vector, we ob-
band thickness and n(u, v) is the unit inward normal: tain the area element of the parallel surface:
su × sv a[B] = s[B] u ×s[B] v
n(u, v) = (29)
su × sv = a 1 − 2BκM + B 2 κG
A surface integral of f over Γ is which will be useful for expressing the simplified form of
∂s ∂s the narrow band region energy described below.
J(f, Γ) = f (s(u, v)) × dudv
∂u ∂v
Ω2 3.3. Transformation of volume integral
Volume integrals can be converted to surface integrals
where the area element
thanks to the divergence theorem, also known as Green-
∂s ∂s Ostrogradski’s theorem. For every volumic region R, given
a(u, v) = × (25)
∂u ∂v F(x) = [P (x) Q(x) R(x)]T a continuously differentiable
makes the surface integral J(f, Γ) independent of the pa- R3 → R3 vector field, we have:
rameterization. In accordance with our mathematical
derivations in the previous section, we demonstrate how div(F) dV = F, N dA (30)
the normal vector of parallel surface can be expressed as R ∂R
a function of su ×sv . Moreover, we show how the various where dA and dV are the differential area and volume
surface curvatures intervene in this expression. To express elements, respectively. N is the surface outward normal.
surface curvature, we introduce basic elements of differen- The divergence of vector field F is:
tial geometry [34, 33]. E, F and G are the coefficients of
∂P ∂Q ∂R
the first fundamental form, whereas L, M and N are the div(F) = + + (31)
coefficients of the second fundamental form. At a given ∂x ∂y ∂z
surface point s(u, v), we have If the boundary ∂R is parameterized by s(u, v), the surface
integral can be written:
E = su , su F = su , sv G = sv , sv
L = − nu , su = n, suu ∂s ∂s
F, N dA = − F(s(u, v)), × dudv
M = − nu , sv = − nv , su = n, suv ∂u ∂u
∂R Ω2
N = − nv , sv = n, svv (32)
6
7. where the negative sign appears since n(u, v) is the unit 4. Calculus of variations
inward normal. To convert the volume integral of f into
a surface integral, one should find F such that div(F) = Image segmentation is performed through numerical
f . This condition is verified by choosing P , Q and R as minimization of the energy functional using gradient de-
follows: scent. The negative discretized variational derivative of
1 x the energy term is usually considered for the descent direc-
P (x, y, z) = f (t, y, z)dt tion. In this section, we express the variational derivatives
3 0
1 y of the energies, especially focusing on the region terms, for
Q(x, y, z) = f (x, t, z)dt (33) both contour and surface.
3 0
z
1
R(x, y, z) = f (x, y, t)dt 4.1. Active contour
3 0
Let us consider a general energy term E, depending on
In what follows, we demonstrate how we can express the
the curve position c and its successive derivatives:
3D region energy in terms of surface integrals. The deriva-
tion is similar in philosophy to the 2D case, since our 3D
scheme is also based on curvature. As in the 2D section, E[Γ] = L(c, cu , cuu ) du
Ω
we consider a general case of a volumic band B bounded
by surfaces Γ1 and Γ2 . The variational derivative of the energy with respect to the
curve can be computed thanks to calculus of variations [1]:
J(f, B) = J(f, R1 ) − J(f, R2 )
˜
This theorem is based on a family of surfaces Γ(α) δE ∂L d ∂L d2 ∂L
0≤α≤1 = − + 2 ∂c (37)
δΓ ∂c du ∂cu du uu
with position vector:
˜(α, u, v) = (1 − α)s1 (u, v) + αs2 (u, v)
s According to the Euler-Lagrange equation, if curve Γ is a
local minimizer of E, the previous variational derivative
Using the divergence theorem in eq. (32), the volume in- vanishes. Curve evolution is achieved by iterative solving
tegral over the region bounded by two surfaces Γ1 and Γ2 of the Euler-Lagrange equation, by means of gradient de-
can be expressed as follows (details of the proof are given scent. It is more convenient to calculate the variational
in appendix A.3): derivative of each energy. From eq. (3), we have:
1
J(f, B) = f (˜) s2 − s1 , ˜u טv dαdudv
s s s (34) δE δEsmooth δEregion
0 =ω + (1 − ω)
Ω2 δΓ δΓ δΓ
In the previous expression, the scalar triple product is the The derivative of the smoothness term is well known [1],
volume of the parallelepiped spanned by vectors (s2 − s1 ), since eq. (37) is easily applicable on Esmooth :
˜u and ˜v . We apply this general result in our case, where
s s
Γ1 = Γ and Γ2 = Γ[B] . Given the area element of parallel δEsmooth d2 c
surface in eq. (29), we write the final approximation of the = −2 2 (38)
δΓ du
volume integral:
As regards the first narrow band region energy, it is more
J(f, Bin ) = conveniently differentiated when expressed with integrals
B
f (s+bn) su ×sv (1−2bκM +b2 κG )dbdudv (35) over Rin and its related regions, rather than over bands.
0 Therefore, the inner band term is split between Rin and the
Ω2
eroded inner region Rin [B] , whereas the outer band term
The transformation from eq. (34) to eq. (35) is detailed is split between Rin and the dilated inner region Rin[−B] ,
in appendix A.4. Again, the volume integral over outer which leads to the following variational derivative:
band Bout is obtained by replacing b with −b. The first
narrow band region energy is found by replacing adequates δEregion 1
=
quantities in eq. (18): δΓ
B δJ (I−kin )2 , Rin δJ (I−kin )2 , Rin [B] (39)
Eregion 1 [Γ] = a[b] (I(s[b] )−kin )2 dbdudv + −
δΓ δΓ
Ω2
0 δJ (I−kout )2 , Rin[−B] δJ (I−kout )2 , Rin
B (36) + −
δΓ δΓ
+ a[−b] (I(s[−b] )−kout )2 dbdudv
0 In this way, region terms are transformed using Green’s
Ω2
theorem and subsequently derived. From the appendix
where area elements a[b] and a[−b] should be expanded ac-
in [10], we have:
cording to eq. (29). The explicit form of the second energy
may be obtained by replacing kout with surface-dependent δJ(f, Rin )
local descriptor hout (u, v). = −ℓf (c)n (40)
δΓ
7
8. In appendix B, we develop the calculation of the varia- The local outer descriptor function hout is determined by
tional derivative of the general term J(f, Rin[B] ), which solving another Euler-Lagrange equation:
results in:
δEregion 2
δJ(f, Rin[B] ) =0
= −ℓ(1 − Bκ)f (c[B] )n δhout
δΓ
which yields the average weighted intensity along the out-
Its counterpart on the dilated region Rin [−B] is obtained ward normal line of length B, at a given contour point:
by replacing B with −B in eq. (67). This eventually leads
B
to:
ℓ(1 + bκ)I(c − bn)db
0
δEregion 1 hout (u) = B
= ℓ[
δΓ ℓ(1 + bκ)db
−(I(c)−kin )2 + (1−Bκ)(I(c[B] )−kin )2 (41) 0
−(1+Bκ)(I(c[−B] )−kout )2 + (I(c)−kout )2 n According to the previous definition of hout , we assume
that piecewise constancy over the outer band is verified if
The energy should also be minimized with respect to inten- intensity is uniform along finite length lines in the direc-
sity descriptors kin and kout . These are found by solving tion normal to the object boundary. For a given point on
the contour, the length element is constant and may be
∂Eregion 1 ∂Eregion 1 omitted, which reduces the mean intensity to:
=0 and =0
∂kin ∂kout B
2
which yield average intensities on the inner and outer hout (u) = (1 + bκ)I(c − bn)db (45)
B(2 + Bκ) 0
bands:
B 4.2. Extension to the surface
1
kin = I(c+bn)ℓ(1−bκ)dbdu The general energy term depending on surface position
|Bin | Ω 0
B (42) as well as its u and v-derivatives,
1
kout = I(c−bn)ℓ(1+bκ)dbdu
|Bout | Ω 0
E[Γ] = L(s, su , sv , suu , suv , svv ) dudv
Band areas |Bin | and |Bout | are expressed by considering Ω2
eq. (17) with f (x) = 1:
has the following variational derivative [35]:
2
B
|Bin | = ℓ B− κ du δE ∂L d ∂L d ∂L
Ω 2 = − −
2 (43) δΓ ∂s du ∂su dv ∂sv
B
|Bout | = ℓ B+ κ du d2 ∂L d2 ∂L d2 ∂L
Ω 2 + 2 ∂s + + 2
du uu dudv ∂suv dv ∂svv
The derivative in eq. (41) holds the term
The variation of the smoothness term is straightforward
(I(c) − kout )2 − (I(c) − kin )2 , which is clearly in ac-
to calculate and is a function of the laplacian:
cordance with the region-based segmentation principle.
Indeed, the sign of the above quantity depends on the δEsmooth ∂2s ∂2s
likeness of the current point’s intensity with respect to kin = −2 + 2 (46)
δΓ ∂u2 ∂v
or kout . If I(c) is closer to kin than kout , the contour
will locally expand, as it would be the case with a region As in the 2D case, it is practical to differentiate the region
growing approach. Moreover, one may note that this term term when formulated in terms of integrals over Rin , Rin [B]
is also found in the Chan-Vese region-based method. The and Rin[−B] . Hence, a similar derivation as in eq. (39) is
derivative holds additional curvature-dependent terms performed. A detailed calculation of the variational deriva-
which are addressed in section 5. tive of a general term J(f, Rin ) may be found in the ap-
pendix of [36]. It gives:
We now deal with the second region energy. From
eq. (41), we extrapolate a consistent variational derivative δJ(f, Rin )
= −af (s)n
of the second narrow band region term. We obtain: δΓ
δEregion 2 The variation of the term expressed on the eroded inner
≈ ℓ[ region is extended from appendix B. In particular, the
δΓ
(44) final result of eq. (67) gives:
−(I(c)−kin )2 + (1−Bκ)(I(c[B] )−kin )2
δJ(f, Rin [B] )
−(1+Bκ)(I(c[−B] )−hout )2 + (I(c)−hout )2 n = −a(1 − 2BκM + BκG )f (s[B] )n
δΓ
8
9. and similarly on the dilated inner region. Replacing the pj pj
curvature-dependent terms of eq. (41) and (44), this even- αij
tually leads to: θij
δEregion 1 βij
= a −(I(s)−kin )2 + (I(s)−kout )2 pi pi
δΓ
+(1−2BκM +B 2 κG )(I(s[B] )−kin )2 (47)
−(1+2BκM +B 2 κG )(I(s[−B] )−kout )2 n
Minimizing Eregion 1 with respect to intensity descrip- Figure 5: Angles in the neighborhood of pi for discrete mean and
gaussian curvature estimation
tors kin and kout , we end up with average intensities:
B
1
kin = a[b] I(s[b] ) dbdudv where ptk , k = 1, 2, 3 are the vertices of triangle t. In
|Bin | 0 a given triangle, vertex indices should be sorted so that
Ω2
1 B the normal vector points towards the inside of the sur-
kout = a[−b] I(s[−b] ) dbdudv face. Since the iterative evolution algorithm described
|Bout | 0
Ω2 below modifies vertex coordinates, all normals should be
updated after each iteration (when all vertices have been
The derivative of Eregion 2 may be obtained from eq. (47), moved). For the contour, the discretized length element ℓ
replacing kout with local descriptor hout (u, v). As in the 2D associated to pi is
case, minimizing Eregion 2 with respect to hout , we obtain
the average intensity along outer normal line segment at pi − pi−1 + pi − pi+1
surface point s (u, v): ℓi =
2
hout (u, v) = For the mesh, to compute the area element associated to
B pi , we use the sum of areas of its neighboring triangles:
3 (48)
I(s[−b] )(1+2bκM +b2 κG )db
3B(1 + B) + B 3 0 |Ni |
(pi − pNi [k] ) × (pi − pNi [k+1] )
Once the first variations of the smoothness and region Ai =
2
terms are known, we are able to perform gradient descent k=1
of the discretized energy over explicit implementations of The area element is simply ai = Ai /3. The sum of area
the curve and surface. elements equal to the sum of triangle areas, which is itself
the total mesh area. To estimate the mean and gaussian
5. Implementation on explicit models curvatures, we use the discrete operators described in [38]
and [39].
5.1. Polygon and mesh
To describe the discrete forms of active 2D contour and 1
3D surface models simultaneously, we introduce a general κM i = (cot αij + cot βij )(pi − pj )
4Ai
framework. The contour is a discrete closed curve, whereas j∈Ni
the surface model is a triangular mesh built by subdividing 1
an icosahedron [37]. The models have a constant global κG i = 2π − θij
Ai
topology, their initial shape being circular and spherical, j∈Ni
respectively. Both are made up of a set of n vertices,
denoted pi = [xi yi ]T in 2D and pi = [xi yi zi ]T in 3D. where αij , βij and θij are the angles formed by pi , pj and
Each vertex pi has a set of neighboring vertices, denoted their common neighboring vertices, as shown in fig. 5.
Ni . In the 2D contour, index i is the discrete equivalent
of the curve parameter, hence Ni = {i − 1, i + 1}. For 5.2. Reparameterization
the mesh, Ni is sorted in such a way that the k th and To maintain a stable vertex distribution along the 2D
(k + 1)th neighbors of pi are also neighbors between them. contour (or 3D surface), adaptive resampling (or remesh-
While the computation of tangent and normal vectors is ing) is performed [3]. The contour is allowed to add or
straightforward on the polygon, computing normals on the delete vertices to keep the distance between neighboring
mesh needs some explanation. The normal of vertex pi is vertices homogeneous. It insures that every couple of
the mean computed over the normals of the neighboring neighbors (pi , pj ) satisfies the constraint:
triangles [3]. The normal nt of a given triangle is the
normalized cross product between two of its edges. w ≤ pi − pj ≤ 2w (50)
(pt2 − pt1 ) × (pt3 − pt1 ) where w is the sampling between consecutive vertices.
nt = (49) Resampling the 2D contour is simple: when the distance
(pt2 − pt1 ) × (pt3 − pt1 )
9
10. step ∆t:
(t+1) (t)
pj pj pi = pi + ∆tf (pi ) (52)
where f (pi ) is the force vector, expressed in terms of the
pa pb pa pn+1 pb pa pb discretization of the energy derivative at a given vertex pi :
pi
pi pi δE
f (pi ) = −
δΓ c=pi
= ωfsmooth (pi ) + (1 − ω)fregion (pi )
Figure 6: Remeshing operations on the triangular mesh: vertex in-
serting (middle) and deleting (left) We first consider the region force fregion (the smoothness
force is studied in the next section). To compute band
areas and means on the polygonal contour, we apply the
discretization templates in eq. (51) on expressions of areas
between pi − pi+1 exceeds 2w, the line segment is split in eq. (43) and intensity means in eq. (42). Similarly, quan-
by creating a new vertex at coordinates (pi + pi+1 )/2. tities in the 3D region energy in eq. (36) are implemented.
When pi − pi+1 gets below w, vertex pi+1 is deleted For the first narrow band region energy, the corresponding
and pi is connected to pi+2 . force on the contour is:
Active surface remeshing is described in [3] and [14]. fregion1 (pi ) = (I(pi )−kin )2 − (I(pi )−kout )2 ni (53)
While resampling the 2D contour is rather straightforward,
remeshing the 3D active surface is carefully performed. In addition to the squared differences between I(c) and
Adding or deleting vertices modifies local topology, thus the average band intensities, the variational derivative in
topological constraints should be verified. Let us consider eq. (41) also contains curvature-based terms depending on
the couple of neighbors (pi ,pj ). To perform vertex adding the intensity at points c[B] and c[−B] . Actually, these
or deleting, pi and pj should share exactly two common terms turn out to go against the region growing or shrink-
neighbors, denoted pa and pb . When pi − pj > 2w, a ing principle, as they oppose the other terms depending
new vertex is created at the middle of line segment pi pj on I(c). As stated in [40], the usual energy gradient may
and connected to pa and pb (see middle part of fig. 6). not be consistently the best direction to take, which jus-
When pi − pj < w, pj is deleted and pi is translated to tifies our choice to remove side effect terms. Moreover,
the middle location (see right part of fig. 6). Vertex merg- by doing so, we keep the same evolution principle as the
ing prevents neighboring vertices from getting too close, Chan-Vese region term. In a similar way, the force result-
which might result in vertex overlapping and intersections ing from the second narrow band region energy is:
between triangles. Adding vertices allows the contour and
fregion2 (pi ) = (I(pi )−kin )2 − (I(pi )−µNL (pi ))2 ni
surface to inflate significantly while keeping a sufficient
(54)
vertex density.
with
5.3. Energy minimization κi (B + 1)
b=B
We give the discrete forms of quantities used in the µNL (pi ) = B 1 + (1 + bκi )I(pi − bni )
2
region energies. Over Bin , area integrals are computed b=1
according to the following template formula, which is a
5.4. Gaussian filtering
discrete implementation of eq. (17):
Minimizing the regularization term boils down to apply
n B−1 laplacian smoothing of the contour - see eqs. (38) and (46)
J(f, Bin ) ≈ f (pi + bni )ℓi (1 − bκi ) (51) - which is formalized by the following PDE:
i=1 b=0
∂c ∂2c
where ℓi , ni and κi are the discretized length element, nor- =α 2
∂t ∂u
mal and curvature at vertex pi , using finite differences.
A similar computation is performed over the outer band, where α ∈ [0, 1/2]. When discretized, laplacian smoothing
in which b varies from −B to −1. There are two com- only intervenes in the direct neighborhood of vertices and
plementary techniques to address the regularity condition is consequently limited in space. However, highly noise-
in eq. (12). The first one is to prevent each vertex from corrupted data require very strong regularity constraint
making a sharp angle with its neighbors, so that its cur- on the contour. Should the regularization be insufficient,
vature κi is well bounded. Moreover, the case of a neg- the contour is exposed to unstable behaviour. Hence, to
ative length element can be handled. Hence, in eq. (51), achieve more diffuse regularization, the contour is con-
ℓi (1−bκi ) is actually computed as max(0, ℓi (1−bκi )) and volved with a gaussian kernel of zero mean and standard
similarly on the outer band. Vertex coordinates are itera- deviation σ. Regularization often comes with curve shrink-
tively modified using gradient descent of eq. (3) with time age, as studied in [41]. To avoid this unwanted effect, we
10
11. use the two-pass method of Taubin [42]. This consists in 5.6. Implementation of Green’s and divergence theorems
performing the gaussian smoothing twice, firstly with a Our experiments, described in section 7, include a
positive weight and secondly with a negative one. The comparison between segmentation results obtained with
force, resulting from the first pass, applied on a given ver- our narrow band region terms and the ones obtained
tex is with the Chan-Vese region energy in eq. (1). The imple-
k=η
mentation of the latter on the explicit polygon and mesh
1 k2 raises the difficulty of computing region integrals. The
fsmooth (pi ) = √ exp − 2 pi+k − pi
σ 2π k=−η 2σ direct solution consists in using region filling algorithms
(55) to determine inner pixels, as in [9] and [14], which would
where η is the rank of neighborhood. One usually admits be computationally expensive if performed after each
that the value of the gaussian distribution is nearly zero deformation step. Another solution, which we chose,
beyond 3σ, we choose η = 3σ. This force is used instead is based on an discretization of Green-Riemann and
of a discretization of the variational derivative in eq. (38). Green-Ostrogradski theorems in 2D and 3D, respectively.
A similar smoothing is performed on the deformable mesh
when working with 3D data. In eq. (55), the k th neigh- In the 2D case, we consider Green’s theorem as stated
bors of pi should then be replaced with successive rings of in eqs. (13) and (14). For instance, a brute-force imple-
neighbors - the first one being Ni - around pi . The choice mentation of the integral J(I, Rin ) on the polygon would
of standard deviation σ has a substantial impact on the yield:
final segmentation result and is discussed in section 7. n xi
1 yi+1 − yi−1
J(I, Rin ) ≈ I(k, yi )
5.5. Bias force 2 i=1
2
k=0
yi
In a particular case, the formulation of fregion presents xi+1 − xi−1
− I(xi , l)
a shortcoming. Indeed, the magnitude of fregion is low 2
l=0
when kin and kout are similar, since in eq. (53), the term
(I(pi )−kout )2 − (I(pi )−kin )2 tends to 0. This situation Computing this term in this way may be time-consuming,
arise when the contour, including the bands, is initialized since intensities should be summed horizontally and
inside a uniform area. However, we expect the contour vertically at each vertex position. Nevertheless, it is
to grow if the intensity at the current vertex matches the possible to compute and store the summed intensities
inner band features, whatever the value of kout . Thus, we only once, before polygon deformation is performed. This
introduce a bias fbias expanding the boundary if I(pi ) is reduces the algorithmic complexity to O(n), whereas
near kin : narrow band region energies induce a O(nB) complexity.
Note that the additonal memory cost imputed to the
fbias (pi ) = − 1 − (I(pi ) − kin )2 ni 2D arrays, storing summed intensities in the x and y
directions for each pixel, is insignificant.
This bias acts like the balloon force described in [35]. Its
influence should decrease as kin gets far from kout . To do On the other hand, for volumetric images, the extra
so, we weight fbias with a negative exponential-like coeffi- memory burden caused by a similar implementation of the
cient γ. divergence theorem would be problematic. In this case, in
addition to the initial image, we store a unique 3D array S
1 − (kin − kout )2
γ = holding summed intensities in the x, y and z dimensions.
1 + ρ(kin − kout )2 (56) Its corresponding continuous expression is:
fregion+bias (pi ) = γfbias (pi ) + (1 − γ)fregion (pi )
z y x
where ρ should be high enough to ensure a quickly de- S(x, y, z) = I(x′ , y ′ , z ′ )dx′ dy ′ dz ′
−∞ −∞ −∞
creasing slope (any value above 50 turns out to be suit-
able). Hence, fbias is predominant when mean intensities which is actually computed according to the following re-
are close. Its influence decreases to the advantage of fregion cursive scheme:
as inner and outer mean intensities become significantly
S(x, y, z) = S(x−1, y, z) + S(x, y−1, z)
different. One may note that the bias force is also ap-
plied when using the second region force in eq. (54). Con- + S(x, y, z−1) − S(x−1, y−1, z)
sequently, our region terms maintain the same ability to − S(x−1, y, z−1) − S(x, y−1, z−1)
grow or retract than classical region-based active contours. + S(x−1, y−1, z−1) + I(x, y, z)
The region bias guarantees the contour has a similar cap-
ture range as other region-based models. As in 2D, S needs to be computed only once at the be-
ginning of the segmentation process. Then, during surface
11
12. evolution, image primitives P , Q and R are determined by If it is assumed that ψ remains a signed euclidean distance
finite differences. We provide the details for P : function, the narrow band region energy may be written
as:
1 ∂2S
P (x, y, z) = Eregion 1 [ψ] =
3 ∂y∂z
1
≈ (S(x, y, z) − S(x, y−1, z) H(ψ(x)+B)(1−H(ψ(x)))(I(x)−kin )2 dx
3
−S(x, y, z−1) + S(x, y−1, z−1)) D
+ H(ψ(x))(1−H(ψ(x)−B))(I(x)−kout )2 dx
For an implementation of the divergence theorem in the
D
context of 3D segmentation, the reader may refer to [43].
where the Heaviside step function H is used in a similar
6. Level set implementation manner as in [45] or [46]. Unlike in the explicit case, the
regularity condition in eq. (12) has no impact on the im-
We provide an implicit implementation of our narrow plementation of band integrals in the implicit case. Due
band energies as well. In addition to topological flexibility, to the geometric nature of level sets, there is no need to
the level set formulation presents the advantage of a com- compute any length element and the curvature does not
mon formulation for both 2D and 3D models. We consider intervene in the computation of band integrals. Indeed,
the level set function ψ : Rd → R, where d is the image considering the sign of ψ, pixels belonging to Bin or Bout
dimension. The contour or surface is the zero level set of are easily determined by dilating the front with the cir-
ψ. We define the region enclosed by the contour or sur- cular window WB . Regarding the average intensity along
face by Rin = {x|ψ(x) ≤ 0}. Instead of forces applied outward normal lines, we rely on the curvature-based for-
on vertices, we now deal with speeds applied to function mulation of the explicit curve:
samples. Function ψ deforms according to the evolution
equation: hout (x) =
B
2
∂ψ I(x + bnψ (x))(1 + bκψ (x))db
= F (x) ∇ψ(x) ∀x ∈ Rd (57) B(2 + Bκψ (x)) 0
∂t
where the unit outward normal to the front at x is:
where speed function F is to some extent the level set-
equivalent of the explicit energy in eq. (3), i.e. a weighted ∇ψ(x)
nψ (x) =
sum of smoothness and region terms: ∇ψ(x)
F (x) = ωFsmooth (x) + (1 − ω)Fregion (x) This last expression is only adequate when x is located
on the zero-level front. Computed as is, to maintain nψ
In the level set framework, regularization is usually per- normal to the front, ψ should remain a distance function.
formed with a curvature-dependent term. With this tech- This implies to update ψ as a signed euclidean distance
nique, for the same reasons as explained in section 5.4, the in the neighborhood of the front before estimating normal
effect is limited to the direct neighborhood of pixels. In vectors. Regardless of the dimension of ψ, the curvature
order to achieve a regularization as diffuse as in eq. (55), is expressed in terms of divergence:
we replace the usual curvature term with a gaussian con-
volution, as in [44]: ∇ψ(x)
κψ (x) = div
∇ψ(x)
2
1 x′ −x
Fsmooth (x)= √ exp − ψ(x′ ) −ψ(x)
which allows to write the level-set formulation of the sec-
σ 2πx′ ∈W (x) 2σ 2 ond region term. From eq. (19), it follows:
3σ
where W is a circular window of a given radius around x: Eregion 2 [ψ] =
Wη (x) = {x′ | x′ −x ≤ η} H(ψ(x)+B)(1−H(ψ(x)))(I(x)−kin )2 dx +
D
Parameters ω and σ play the same role as in the explicit B
2
implementation described in section 5. Areas, volumes and δ(ψ(x)) (I(x+bnψ (x))−hout (x)) (1+bκψ (x))dbdx
average intensities upon inner and outer bands are easily 0
D
computed on the level set implementation, since a circular
window of radius B may be considered around each pixel For a point x located on the front, the speeds correspond-
located on the front. ing to the narrow band region terms are:
Bin = {x|ψ(x) ≤ 0 and ∃x′ ∈ WB (x) s.t. ψ(x′ ) = 0} Fregion 1 (x) = (I(x)−kout )2 − (I(x)−kin )2
Bout = {x|ψ(x) ≥ 0 and ∃x′ ∈ WB (x) s.t. ψ(x′ ) = 0} Fregion 2 (x) = (I(x)−hout (x))2 − (I(x)−kin )2
12
13. On the implicit 2D contour, the curvature may be ex- to the data term of the Chan-Vese model [11] described in
panded as eq. (1):
2 2
ψxx ψy − 2ψx ψy ψxy + ψx ψyy
κψ = Eglobal [Γ] = λ (I(x)−kin )2 dx
2
(ψx + ψy )3/2
2
Rin
+(2 − λ) (I(x)−kout )2 dx
According to [47], the mean and gaussian curvatures of the
implicit surface are: Rout
2 2 2 2 2 2
ψxx (ψy + ψz ) + ψyy (ψx + ψz ) + ψzz (ψx + ψy ) Weights on inner and outer integrals are expressed in terms
κM ψ = of a single parameter λ, since the full data term is already
(ψx + ψy + ψz )3/2
2 2 2
weighted by (1−ω). In subsequent experiments, the asym-
ψxy ψx ψy + ψxz ψx ψz + ψyz ψy ψz metric configuration (λ = 1) will be explicitly indicated.
−2
(ψx + ψy + ψz )3/2
2 2 2
Otherwise, the symmetric configuration is used. The data
2 2 term of the combined model by Kimmel [23] holds a robust
ψi (ψjj ψkk − ψjk ) + 2ψ
(i,j,k)∈C alignment term, encouraging intensity variations normal to
κGψ = the curve, and a symmetric global region term:
(ψx + ψy + ψz )2
2 2 2
where C = {(x, y, z), (y, z, x), (z, x, y)} is the set of circular Ecombined [Γ] = − | ∇I(c), n | du
shifts of (x, y, z). Eventually, the reader may note that the Ω
bias technique used in the explicit implementation is also
+β (I(x)−kin )2 dx + (I(x)−kout )2 dx
applied in the level set model. The level set function ψ
evolves according to the narrow band technique [4], so that Rin Rout
only pixels located in the neighborhood of the front are For the edge and combined data terms, image gradient ∇I
treated. is computed on data convolved with first-order derivative
of gaussian, where scale s is empirically chosen to yield the
7. Results and discussion most significant edges. The choice of s is a tradeoff be-
tween noise removal and edge sharpness. The variational
Regarding the results, we should first point out that the derivatives of the previous terms are:
goal of our experiments is not to compare explicit and im-
plicit implementations, since it is well accepted that both δEedge
= −∇ ∇I(c) − αn
exhibit their own advantages. These ones are typically δΓ
topological and geometrical freedom for level sets. On the δEglobal
= [−λ(I(c)−kin )2 + (2−λ)(I(x)−kout )2 ]n
other hand, explicit approaches with polygonal snakes and δΓ
δEcombined
triangular meshes yield less computational cost than level = −sign( ∇I(c), n )∇2 I(c)n
set and allow more control. The purpose of our tests is δΓ
+β[−(I(c)−kin )2 + (I(c)−kout )2 ]n
to compare the behavior of active contours and surfaces
endowed with different data terms, on both explicit and We perform segmentation of visually uniform structures.
implicit implementations. We intend to show the interest Since we are looking for perceptually homogeneous ob-
of the narrow band approach regardless of the implemen- jects, segmentation quality can be assessed visually. One
tation. To do so, the narrow band region energies are can reasonably admit that the target object corresponds
compared with an edge term, the global region term of to the area containing the major part of the initial region.
the Chan-Vese model [11] as well as the combined term For all 2D datasets, the model, whether it is our explicit
by Kimmel [23]. For every data term involved, we used contour or the level set, is initialized as a small circle
the same smoothness term. Hence, the full energy is ob- fully or partially inside the area of interest, far from
tained by replacing Eregion with the following data terms the target boundaries. Similarly, for 3D images, our
in eq. (3). The edge term is based on the image gradient active surface and the 3D level set are initialized as small
magnitude: spheres. On a given image, a common initialization is
used for all models. The initial curve is depicted for some
Eedge [Γ] = − ∇I(c) du + α dx experiments. In all subsequent figures, explicit contours
Ω and surfaces are drawn in red whereas implicit ones
Rin
appear in blue.
where α weights an additional balloon force [35] increasing
the capture range. It allows the contour to be initialized For all experiments, the regularization weight ω is set
far from the target boundaries, similarly to a region-based to 0.5. The standard deviation of the gaussian smooth σ
contour. The following global region energy is equivalent takes its values between 1 and 2, which achieves the best
regularization for all tested images. On noisy data, we
13
14. Global Narrow band 1
Figure 8: Left ventricle in MRI with level set
found that contours and surfaces with lower σ are prone to
boundary leaking. In addition, insufficient regularization
makes level set implementations leave spurious isolated
pixels inside and outside the inner region. Conversely,
values above 2 might prevent the surface from propagating
Global Narrow band 1
into narrow structures, like thin blood vessels. Moreover,
since the width of the gaussian mask is proportional to σ, Figure 9: Kidney in abdominal CT with parametric contour (top)
large standard deviations lead to significant increase of and level set (bottom)
computational cost, especially for 3D segmentation.
Fig. 7 shows segmentation results of the brain ventricle
in a 2D axial MRI (Magnetic Resonance Imaging) slice. only the implicit method was tested on this dataset. In
As it is the case with many medical datasets, the partially order to keep a critical eye on our approach, we draw the
blurred boundaries between ventricle and gray matter attention on its equivalence with the Chan-Vese model on
prevent the extraction of reliable edges in places. For the this particular image. The background is not uniform but
edge-based model, we could not find a suitable balloon still significantly darker than the target object. Thus, the
weight α and gradient scale s preventing the contour from classical region speed manages to make the front stabilize
being trapped in spurious noisy edges inside the shape on the actual boundaries. Fig. 8 depicts the typical case
while stopping on the actual boundaries. As regards in which there is no particular benefit in using the narrow
combined and symmetric global approaches (see columns band region energy.
2 and 4), the contour did not manage to grow, as inner and
outer average intensities were not sufficiently different. In fig. 9 and 10, we illustrate the recovery of the kidney
Setting λ to 0.9, we decrease the significance of inner and aorta inner walls, respectively, in 2D CT (Computed
deviation to the benefit of outer deviation, consequently Tomography) data. The global region energy makes the
allowing the contour to grow. As depicted in column 3, contour leak outside the target object, into neighboring
the contour undesirably flows into the gray matter part structures of rather light gray intensity. Indeed, since
as soon as this area is reached, which is inconsistent with the black background occupies a large area of the image,
the initialization in our context. In this column, one may every bright grayscale is considered as a part of inner
note that the drawn curves are not the final ones, but statistics. Conversely, the narrow band energy, which
intermediate states to illustrate the leaking effect. Indeed, ignores the major part of the background, has a more
on this particular dataset, gray/white separation is the local view and manages to keep the contour inside the
most probable partition with respect to a global two-class kidney and the aorta. One may note that all segmented
segmentation. objects of interest are homogeneous. It has no incidence
if we consider the inner band or the whole inner region
We also tested the snake on slices of the human for the homogeneity criterion in Eregion 1 . Indeed, in the
heart in short-axis view, where the shape of the endo- region force of eq. (53), descriptor kin may be equally the
cardium, i.e. the inner wall of the left ventricle, should average intensity on Rin or Bin .
be recovered. The slices were extracted from 3D+T
MRI sequences. Finding the endocardium boundary is The band thickness B is an important parameter of
made more complex by the presence of papillary muscles, our method and should be discussed. Apart from its
appearing as small dark areas inside the blood pool, as impact on the algorithmic complexity - computing average
shown in fig. 8. This requires topological changes, hence intensities µ(Bin ) and µ(Bout ) takes at least O(nB) oper-
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15. Edge Global Global (asymmetric) Combined Narrow band 1
Figure 7: Brain ventricle in 2D axial MRI, with parametric contour (top) and level set (bottom). The initial curve is drawn in black dashed
line
Global Narrow band 1
Figure 11: Succesive gaussian smoothings of an axial slice of 3D CT
Figure 10: Aorta in abdominal CT with parametric contour (top) data
and level set (bottom)
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