Lecture 11 (Digital Image Processing)
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This document discusses modifications to B-spline interpolation formulations. It begins by describing B-splines and their use in interpolation. It notes that constant, linear, quadratic, and cubic interpolation correspond to B-spline orders of 1, 2, 3, and 4, respectively. It then presents a modified relation where a shifted B-spline is used, with the shift determined by the interpolation order. For example, with linear interpolation the shift is 1. Graphs demonstrate how this modified relation gives nearer points higher weight. The document concludes by discussing constant interpolation in 2D images.
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The document proposes an algorithm called Robust Tensor Decomposition (RTD) to recover a low-rank tensor from observations that are corrupted by block sparse errors. RTD iteratively estimates the low-rank and sparse components using hard thresholding and tensor projections. It compares RTD to existing approaches on video sequences, finding RTD yields more accurate recovery with fewer assumptions.
State equations model based on modulo 2 arithmetic and its applciation on rec...
1) The document discusses state equations that can model digital control systems based on modulo-2 arithmetic and their application to recursive convolutional coding.
2) State equations can be derived from the transfer function of a discrete-time controller and expressed using modulo-2 arithmetic.
3) A recursive convolutional encoder can be modeled by state equations in modulo-2 algebra, where the state at each time step is a function of the previous states and the input bit.
State Equations Model Based On Modulo 2 Arithmetic And Its Applciation On Rec...
1) The document discusses state equations that can model digital control systems based on modulo-2 arithmetic and their application to recursive convolutional coding.
2) Key concepts covered include cyclic groups, rings, state equations derived from transfer functions expressed in z-transform, and direct hardware realizations of state equations.
3) An example of a recursive convolutional encoder is described based on a set of recursive signal equations, and its corresponding algebraic state equations in modulo-2 arithmetic are provided.
Lecture 9 (Digital Image Processing)
This document discusses interpolation and resampling in image processing. It introduces interpolation as a way to assign intensity values to fractional coordinate locations that result from image transformations like rotation or scaling. Common interpolation methods are described, including constant, linear, and higher-order polynomial interpolation using B-spline basis functions. Resampling is needed when transforming digital images between discrete coordinate grids, and interpolation helps estimate values at non-integer coordinates. The document outlines the use of B-splines for smooth interpolation and local curve approximation in images.
Programming project
This document describes the implementation of an Extended Kalman Filter (EKF) to estimate the state (position and heading angle) of a bicycle model. The EKF was able to provide reasonably accurate estimates of position over time based on position measurements and steering/velocity inputs, but struggled to accurately estimate the heading angle due to a lack of direct measurements. Histograms of the final state errors across many test cases showed normally distributed position errors and a uniformly distributed random heading angle error. While the EKF provided an approximation, a more advanced filter may have yielded better heading angle estimates.
B spline
The document describes the DeBoor-Cox calculation, which relates the analytical and geometric definitions of B-spline curves. It begins by defining a B-spline curve analytically as a weighted sum of normalized B-spline blending functions and control points. The blending functions are defined recursively. DeBoor and Cox showed that starting from this analytical definition, one can derive the geometric definition of a B-spline curve as a pyramid of control points. Their calculation demonstrated the relationship between the two common definitions of B-splines.
Unit 2.1
This document provides an overview of linear and quadratic functions and modeling. It defines polynomial functions and discusses linear functions, their graphs, and applications in business and economics. Quadratic functions and their graphs are also introduced, including how to find the vertex and axis of a quadratic function. Examples are provided to illustrate related concepts like finding equations of linear functions, average rate of change, and describing quadratic graphs algebraically.
論文紹介：Towards Robust Adaptive Object Detection Under Noisy Annotations
Xinyu Liu, Wuyang Li, Qiushi Yang, Baopu Li, Yixuan Yuan, "Towards Robust Adaptive Object Detection Under Noisy Annotations" CVPR2022
https://openaccess.thecvf.com/content/CVPR2022/html/Liu_Towards_Robust_Adaptive_Object_Detection_Under_Noisy_Annotations_CVPR_2022_paper.html
Computer vision 3 4
This document discusses image transforms and processing techniques in computer vision. It introduces important 2D linear transforms like the discrete cosine transform. It explains how transforms allow recovering the original image using the inverse transform. Examples are provided on image denoising using DCT. Probabilistic methods for analyzing images like calculating mean, variance and moments are also described. The document contrasts processing images in the spatial vs transform domains. Several basic intensity transformation functions are illustrated including negatives, log transforms, and piecewise linear transformations. Histogram processing techniques like equalization and matching are explained in detail with examples.
Quantum logic synthesis (srikanth)
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MVPA with SpaceNet: sparse structured priors
The GraphNet (aka S-Lasso), as well as other “sparsity + structure” priors like TV (Total-Variation), TV-L1, etc., are not easily applicable to brain data because of technical problems
relating to the selection of the regularization parameters. Also, in
their own right, such models lead to challenging high-dimensional optimization problems. In this manuscript, we present some heuristics for speeding up the overall optimization process: (a) Early-stopping, whereby one halts the optimization process when the test score (performance on leftout data) for the internal cross-validation for model-selection stops improving, and (b) univariate feature-screening, whereby irrelevant (non-predictive) voxels are detected and eliminated before the optimization problem is entered, thus reducing the size of the problem. Empirical results with GraphNet on real MRI (Magnetic Resonance Imaging) datasets indicate that these heuristics are a win-win strategy, as they add speed without sacrificing the quality of the predictions. We expect the proposed heuristics to work on other models like TV-L1, etc.
Determinants
This document provides information about determinants of square matrices:
- It defines the determinant of a matrix as a scalar value associated with the matrix. Determinants are computed using minors and cofactors.
- Properties of determinants are described, such as how determinants change with row/column operations or identical rows/columns.
- Examples are provided to demonstrate computing determinants by expanding along rows or columns and using cofactors and minors.
- Applications of determinants include finding the area of triangles and solving systems of linear equations.
A Novel Cosine Approximation for High-Speed Evaluation of DCT
This article presents a novel cosine approximation for high-speed evaluation of DCT (Discrete Cosine Transform) using Ramanujan Ordered Numbers. The proposed method uses the Ramanujan ordered number to convert the angles of the cosine function to integers. Evaluation of these angles is by using a 4th degree Polynomial that approximates the cosine function with error of approximation in the order of 10^-3. The evaluation of the cosine function is explained through the computation of the DCT coefficients. High-speed evaluation at the algorithmic level is measured in terms of the computational complexity of the algorithm. The proposed algorithm of cosine approximation increases the overhead on the number of adders by 13.6%. This algorithm avoids floating-point multipliers and requires N/2log2N shifts and (3N/2 log2 N)- N + 1 addition operations to evaluate an N-point DCT coefficients thereby improving the speed of computation of the coefficients .
Modeling quadratic fxns
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Geneva Plan of Action
The Geneva Plan of Action outlines the World Summit on the Information Society (WSIS) process which took place in two phases. The first phase was held in Geneva in 2003 and resulted in a plan of action. The second phase was held in Tunis in 2005 and focused on implementing the Geneva plan and reaching agreements on internet governance and financing mechanisms. Key outcomes included connecting villages, schools, and other institutions to ICTs and ensuring over half the world's population has access to ICTs. UNESCO played a prominent role in both phases and the follow up processes by focusing on themes like education, cultural diversity, and access to information.
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needed blood group during the time of the emergency.
This is an android application developed in Java and XML with the connectivity of
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Lecture 11 (Digital Image Processing)
- 1. Modified Interpolation Formulation Through B-Spline Function Subject: Image Procesing & Computer Vision Dr. Varun Kumar Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 11 1 / 11
- 2. Outlines 1 Problem with existing interpolation through B-spline function 2 Modified Relation 3 References Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 11 2 / 11
- 3. B-spline function: It is a piece wise polynomial function. It is useful for local approximation of a curve. Mathematical representation: x(t) = n i=0 pi Bi,k(t) (1) pi → control point how the B-spline function should be guided for smooth curve. Bi,k(t) → Normalized B-spline of order k Bi,1(t) = 1 ∀ ti < t < ti + 1 = 0 otherwise (2) Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 11 3 / 11
- 4. Different interpolation From (2), for different value of k, the type of interpolation has been given below. Constant interpolation k = 1 Linear interpolation k = 2 Quadratic interpolation k = 3 Cubic interpolation k = 4 1. Constant Interpolation k = 1 x(t) = n i=0 pi Bi,1(t) ⇒ Control point pi remain same, what ever the value observed at fs(ti ) ∀ ti ≤ t < ti + 1. Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 11 4 / 11
- 5. Modified Relation ⇒ Nearer integer point should have higher weight compare to farther one. Modified interpolation relation x(t) = n i=0 pi Bi−s,k(t) (3) ⇒ For constant interpolation s = 0.5 at k = 1 ⇒ For linear interpolation s = 1 at k = 2 ⇒ For cubic interpolation s = 2 at k = 4 Note: For quadratic interpolation modified relation is not available, because there is no point of symmetry. Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 11 5 / 11
- 6. Graphical representation of B-spline function 0 5 10 0 0.5 1 1.5 B0,1 (t) 0 5 10 0 0.5 1 1.5 B0,2 (t) 0 5 10 0 0.5 1 1.5 B0,3 (t) 0 5 10 0 0.5 1 1.5 B0,4 (t) Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 11 6 / 11
- 7. Modified constant interpolation 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 Constant Interpolation B0,1 (t) -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 Constant Interpolation B-0.5,1 (t), B0.5,1 (t) p0 ' p1 ' x(t) = p0 B-0.5,1 (t) + p1 B0.5,1 (t) 0≤ t<1 p0 = p0 'fs (0) and p1 = p1 'fs (1) Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 11 7 / 11
- 8. Modified constant interpolation 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.2 0.4 0.6 0.8 1 Linear Interpolation B0,2 (t) -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 Linear Interpolation B-1,2 (t), B0,2 (t) x(t) = p0 B-1,2 (t) + p1 B0,2 (t) 0≤ t< 1 p0 =p0 ' fs (0) and p1 = p1 ' fs (1) p0 ' p1 ' Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 11 8 / 11
- 9. Modified Linear interpolation 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 B0,4 (t) -2 -1 0 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 Cubic Interpolation B-2,4 (t), B-1,4 (t), B0,4 (t) p1 ' p0 =p0 ' fs (0) p1 =p1 ' fs (1) p2 =p2 ' fs (2) p2 ' p0 ' x(t)=p0 B-2,4 (t) + p1 B-1,4 (t) + p2 B0,4 (t) 0≤ t<1 Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 11 9 / 11
- 10. Constant interpolation in 2D plane k nearest neighbor Q. Let a gray scale image having pixel location (2, 3) and amplitude level is 234. What is the amplitude level of new pixel location having co-ordinate is (2.2, 3.3). Ans. 234, when constant interpolation is done. Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 11 10 / 11
- 11. References M. Sonka, V. Hlavac, and R. Boyle, Image processing, analysis, and machine vision. Cengage Learning, 2014. D. A. Forsyth and J. Ponce, “A modern approach,” Computer vision: a modern approach, vol. 17, pp. 21–48, 2003. L. Shapiro and G. Stockman, “Computer vision prentice hall,” Inc., New Jersey, 2001. R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital image processing using MATLAB. Pearson Education India, 2004. Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 11 11 / 11