This document provides a summary of corner detection techniques in digital images. It discusses the local structure matrix, which is used in two popular corner detectors: the Kanade-Lucas-Tomasi (KLT) corner detector and the Harris corner detector. Both detectors identify corners as locations where variations of image intensity are high in two orthogonal directions, as measured by the eigenvalues of the local structure matrix. The KLT detector uses an explicit threshold on the smaller eigenvalue, while the Harris detector uses an implicit threshold through a corner response measure. Examples are given to compare the outputs of the two detectors.
a way to specify points on a curve or surface (or part of one) using only the control points. The curve or surface can then be rendered at any precision. In addition, normal vectors can be calculated for surfaces automatically. You can use the points generated by an evaluator in many ways - to draw dots where the surface would be, to draw a wireframe version of the surface, or to draw a fully lighted, shaded, and even textured version.
In Computer Graphics, Hidden surface determination also known as Visible Surface determination or hidden surface removal is the process used to determine which surfaces
of a particular object are not visible from a particular angle or particular viewpoint. In this scribe we will describe the object-space method and image space method. We
will also discuss Algorithm based on Z-buffer method, A-buffer method, and Scan-Line Method.
Polygon is a figure having many slides. It may be represented as a number of line segments end to end to form a closed figure.
The line segments which form the boundary of the polygon are called edges or slides of the polygon.
The end of the side is called the polygon vertices.
Triangle is the most simple form of polygon having three side and three vertices.
The polygon may be of any shape.
Identify those parts of a scene that are visible from a chosen viewing position.
Visible-surface detection algorithms are broadly classified according to whether
they deal with object definitions directly or with their projected images.
These two approaches are called object-space methods and image-space methods, respectively
An object-space method compares
objects and parts of objects to each other within the scene definition to determine which surfaces, as a whole, we should label as visible.
In an image-space algorithm, visibility is decided point by point at each pixel position on the projection plane.
a way to specify points on a curve or surface (or part of one) using only the control points. The curve or surface can then be rendered at any precision. In addition, normal vectors can be calculated for surfaces automatically. You can use the points generated by an evaluator in many ways - to draw dots where the surface would be, to draw a wireframe version of the surface, or to draw a fully lighted, shaded, and even textured version.
In Computer Graphics, Hidden surface determination also known as Visible Surface determination or hidden surface removal is the process used to determine which surfaces
of a particular object are not visible from a particular angle or particular viewpoint. In this scribe we will describe the object-space method and image space method. We
will also discuss Algorithm based on Z-buffer method, A-buffer method, and Scan-Line Method.
Polygon is a figure having many slides. It may be represented as a number of line segments end to end to form a closed figure.
The line segments which form the boundary of the polygon are called edges or slides of the polygon.
The end of the side is called the polygon vertices.
Triangle is the most simple form of polygon having three side and three vertices.
The polygon may be of any shape.
Identify those parts of a scene that are visible from a chosen viewing position.
Visible-surface detection algorithms are broadly classified according to whether
they deal with object definitions directly or with their projected images.
These two approaches are called object-space methods and image-space methods, respectively
An object-space method compares
objects and parts of objects to each other within the scene definition to determine which surfaces, as a whole, we should label as visible.
In an image-space algorithm, visibility is decided point by point at each pixel position on the projection plane.
IVR - Chapter 2 - Basics of filtering I: Spatial filters (25Mb) Charles Deledalle
Moving averages. Finite differences and edge detectors. Gradient, Sobel and Laplacian. Linear translations invariant filters, cross-correlation and convolution. Adaptive and non-linear filters. Median filters. Morphological filters. Local versus global filters. Sigma filter. Bilateral filter. Patches and non-local means. Applications to image denoising.
Limits of Wiener filtering. Non-linear shrinkage functions. Limits of Fourier representation. Continuous and discrete wavelet transforms. Sparsity and shrinkage in wavelet domain. Undecimated wavelet transforms, a trous algorithm. Regularization. Sparse regression, combinatorial optimization and matching pursuit. LASSO, non-smooth optimization, and proximal minimization. Link with implcit Euler scheme. ISTA algorithm. Syntehsis versus analysis regularized models. Applications to image deconvolution.
Image segmentation is a computer vision task that involves dividing an image into multiple segments or regions, where each segment corresponds to a distinct object, region, or feature within the image. The goal of image segmentation is to simplify and analyze an image by partitioning it into meaningful and semantically relevant parts. This is a crucial step in various applications, including object recognition, medical imaging, autonomous driving, and more.
Key points about image segmentation:
Semantic Segmentation: This type of segmentation assigns each pixel in an image to a specific class, essentially labeling each pixel with the object or region it belongs to. It's commonly used for object detection and scene understanding.
Instance Segmentation: Here, individual instances of objects are separated and labeled separately. This is especially useful when multiple objects of the same class are present in the image.
Boundary Detection: Some segmentation methods focus on identifying the boundaries that separate different objects or regions in an image.
Methods: Image segmentation can be achieved through various techniques, including traditional methods like thresholding, clustering, and region growing, as well as more advanced techniques involving deep learning, such as using convolutional neural networks (CNNs) and fully convolutional networks (FCNs).
Challenges: Image segmentation can be challenging due to variations in lighting, color, texture, and object shape. Overlapping objects and unclear boundaries further complicate the task.
Applications: Image segmentation is used in diverse fields. For example, in medical imaging, it helps identify organs or abnormalities. In autonomous vehicles, it aids in identifying pedestrians, other vehicles, and obstacles.
Evaluation: Measuring the accuracy of segmentation methods can be complex. Metrics like Intersection over Union (IoU) and Dice coefficient are often used to compare segmented results to ground truth.
Data Annotation: Creating ground truth annotations for segmentation can be labor-intensive, as each pixel must be labeled. This has led to the development of datasets and tools to facilitate annotation.
Semantic Segmentation Networks: Deep learning architectures like U-Net, Mask R-CNN, and Deeplab have significantly improved the accuracy of image segmentation by effectively learning complex patterns and features.
Image segmentation plays a fundamental role in understanding and processing images, enabling computers to "see" and interpret visual information in ways that mimic human perception.
Image segmentation is a computer vision task that involves dividing an image into meaningful and distinct segments or regions. The goal is to partition an image into segments that represent different objects or areas of interest within the image. Image segmentation plays a crucial role in various applications, such as object detection, medical imaging, autonomous vehicles, and more.
Understanding High-dimensional Networks for Continuous Variables Using ECLHPCC Systems
Syed Rahman & Kshitij Khare, University of Florida, present at the 2016 HPCC Systems Engineering Summit Community Day.
The availability of high dimensional data (or “big data”) has touched almost every field of science and industry. Such data, where the number of variables (features) is often much higher than the number of samples, is now more pervasive than it has ever been. Discovering meaningful relationships between the variables in such data is one of the major challenges that modern day data scientists have to contend with.
The covariance matrix of the variables is the most fundamental quantity that can help us understand the complex multivariate relationships in the data. In addition to estimating the inverse covariance matrix, CSCS can be used to detect the edges in a directed acyclic graph, as opposed to the edges an undirected graph, which CONCORD (presented at the 2015 summit) was used for.
Similar to the CONCORD algorithm, the CSCS algorithm works by minimizing a convex objective function through a cyclic coordinate minimization approach. In addition, it is theoretically guaranteed to converge to a global minimum of the objective function. One of the main advantage of CSCS is that each row can be calculated independently of the other rows, and thus we are able to harness the power of distributed computing.
Syed Rahman
Syed Rahman is a PhD student in the Statistics department at the University of Florida working under the supervision of Dr. Kshitij Khare. He is interested in high-dimensional covariance estimation. In 2015, Syed programmed the CONCORD algorithm in ECL and presented this at the HPCC Systems Engineering Summit.
Kshitij Khare
Kshitij Khare is an Associate Professor of Statistics at the University of Florida. He earned his Ph.D. in Statistics from Stanford University in 2009. He has a variety of interests, which include covariance/network estimation in high-dimensional datasets, and Bayesian inference using Markov chain Monte Carlo methods. One of Dr. Khare's major research focus is development of novel statistical methods and algorithms for "big data" or high-dimensional data.
Presented at #H2OWorld 2017 in Mountain View, CA.
Enjoy the video: https://youtu.be/bas3-Ue2qxc.
Learn more about H2O.ai: https://www.h2o.ai/.
Follow @h2oai: https://twitter.com/h2oai.
- - -
Abstract:
Auto Visualization involves the problem of producing meaningful graphics when presented with data. Relevant to this task are the strategies that expert statisticians and data analysts use to gain insights through visualization, as well as the portfolio of diagnostic methods devised by statisticians in the last 50 years. While some researchers and companies may claim to do automatic visualization, the problem is much deeper than simply producing collections of histograms, bar charts, and scatterplots. The deeper problem is what subset of these graphics is critical to recognizing anomalies, outliers, unusual distributions, missing values, and so on. This talk will cover aspects of this deeper problem and will introduce H2O software that implements some of these algorithms.
Leland Wilkinson is Chief Scientist at H2O.ai and Adjunct Professor of Computer Science at the University of Illinois Chicago. He received an A.B. degree from Harvard in 1966, an S.T.B. degree from Harvard Divinity School in 1969, and a Ph.D. from Yale in 1975. Wilkinson wrote the SYSTAT statistical package and founded SYSTAT Inc. in 1984. After the company grew to 50 employees, he sold SYSTAT to SPSS in 1994 and worked there for ten years on research and development of visualization systems. Wilkinson subsequently worked at Skytree and Tableau before joining H2O.ai. Wilkinson is a Fellow of the American Statistical Association, an elected member of the International Statistical Institute, and a Fellow of the American Association for the Advancement of Science. He has won best speaker award at the National Computer Graphics Association and the Youden prize for best expository paper in the statistics journal Technometrics. He has served on the Committee on Applied and Theoretical Statistics of the National Research Council and is a member of the Boards of the National Institute of Statistical Sciences (NISS) and the Institute for Pure and Applied Mathematics (IPAM). In addition to authoring journal articles, the original SYSTAT computer program and manuals, and patents in visualization and distributed analytic computing, Wilkinson is the author (with Grant Blank and Chris Gruber) of Desktop Data Analysis with SYSTAT. He is also the author of The Grammar of Graphics, the foundation for several commercial and opensource visualization systems (IBMRAVE, Tableau, Rggplot2, and PythonBokeh).
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
Water billing management system project report.pdfKamal Acharya
Our project entitled “Water Billing Management System” aims is to generate Water bill with all the charges and penalty. Manual system that is employed is extremely laborious and quite inadequate. It only makes the process more difficult and hard.
The aim of our project is to develop a system that is meant to partially computerize the work performed in the Water Board like generating monthly Water bill, record of consuming unit of water, store record of the customer and previous unpaid record.
We used HTML/PHP as front end and MYSQL as back end for developing our project. HTML is primarily a visual design environment. We can create a android application by designing the form and that make up the user interface. Adding android application code to the form and the objects such as buttons and text boxes on them and adding any required support code in additional modular.
MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software. It is a stable ,reliable and the powerful solution with the advanced features and advantages which are as follows: Data Security.MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
For more technical information, visit our website https://intellaparts.com
NUMERICAL SIMULATIONS OF HEAT AND MASS TRANSFER IN CONDENSING HEAT EXCHANGERS...ssuser7dcef0
Power plants release a large amount of water vapor into the
atmosphere through the stack. The flue gas can be a potential
source for obtaining much needed cooling water for a power
plant. If a power plant could recover and reuse a portion of this
moisture, it could reduce its total cooling water intake
requirement. One of the most practical way to recover water
from flue gas is to use a condensing heat exchanger. The power
plant could also recover latent heat due to condensation as well
as sensible heat due to lowering the flue gas exit temperature.
Additionally, harmful acids released from the stack can be
reduced in a condensing heat exchanger by acid condensation. reduced in a condensing heat exchanger by acid condensation.
Condensation of vapors in flue gas is a complicated
phenomenon since heat and mass transfer of water vapor and
various acids simultaneously occur in the presence of noncondensable
gases such as nitrogen and oxygen. Design of a
condenser depends on the knowledge and understanding of the
heat and mass transfer processes. A computer program for
numerical simulations of water (H2O) and sulfuric acid (H2SO4)
condensation in a flue gas condensing heat exchanger was
developed using MATLAB. Governing equations based on
mass and energy balances for the system were derived to
predict variables such as flue gas exit temperature, cooling
water outlet temperature, mole fraction and condensation rates
of water and sulfuric acid vapors. The equations were solved
using an iterative solution technique with calculations of heat
and mass transfer coefficients and physical properties.
The Internet of Things (IoT) is a revolutionary concept that connects everyday objects and devices to the internet, enabling them to communicate, collect, and exchange data. Imagine a world where your refrigerator notifies you when you’re running low on groceries, or streetlights adjust their brightness based on traffic patterns – that’s the power of IoT. In essence, IoT transforms ordinary objects into smart, interconnected devices, creating a network of endless possibilities.
Here is a blog on the role of electrical and electronics engineers in IOT. Let's dig in!!!!
For more such content visit: https://nttftrg.com/
1. Institute of Informatics
E¨otv¨os Lor´and University
Budapest, Hungary
Basic Algorithms for Digital Image Analysis:
a course
Dmitrij Csetverikov
with help of Attila Lerch, Judit Verest´oy, Zolt´an Megyesi, Zsolt Jank´o
and Levente Hajder
http://visual.ipan.sztaki.hu
2. Lecture 8: Corner detection
• Zero-crossing edge detector
• Summary of edge detection
• Corner detection in greyscale images
• The local structure matrix
• The KLT corner detector
• The Harris corner detector
• Comparison of the two corner detectors
2
3. Zero-crossing edge detector
2
f(x)
___
Double edge
Zero crossing
Blurred edgeSteep edge
__
dx
df
_
2
d f
dx
-
00
00
+ +
+ +
-
--
Left: Principles of zero-crossing edge detector.
Right: Simple masks for detection of zero-crossings.
3
4. Implementation of the zero crossing filter: Gaussian smoothing followed by
Laplacian filtering.
• Using commutativity and associativity of linear filters and rotation symmetry of
Gaussian filter, we obtain the convolution mask of the zero-crossing operator,
called Laplacian-of-Gaussian (LoG):
wZ(r) = C
r2
σ2
− 1 exp
−r2
2σ2
◦ C: normalisation constant
◦ r2
= x2
+ y2
: square distance from centre of mask
◦ σ is scale parameter: the smaller the σ the finer the edges obtained
• Discrete zero-crossing mask: Threshold wZ(r) at a small level.
⇒ Larger mask obtained for larger σ: For example, when σ = 4 the size of the
mask is about 40 pixels.
4
5. • Another, more efficient but approximate, implementation of the zero-crossing
filter is the difference of two separable Gaussian filters, called DoG.
• Localising the zero-crossings corresponds to edge localisation in gradient-type
edge detectors.
◦ For more precise localisation, one can locally approximate output of LoG filter
by facets (flat patches), then find zero-crossings analytically.
LoG absolute LoG zero DoG zero
Examples of edge detection by 15 × 15 LoG and DoG operators. ‘LoG absolute’ is
absolute value of filter output: dark lines are contours. ‘LoG zero’ was obtained
with removal of weak edges, ‘DoG zero’ without removal.
5
6. Properties of zero crossing edge detector
• The continuous zero-crossing edge detector always gives closed contours.
◦ Reason: Cross-sections of continuous surface at zero level
◦ In principle, this may help in contour following
◦ In practice, many spurious loops appear
• Controlled operator size σ ⇒ Natural edge hierarchy within a scale-space.
◦ Edges may only merge or disappear at rougher scales (larger σ)
◦ This tree-like data structure facilitates structural analysis of image
• Does not provide edge orientation.
◦ Non-maxima suppression and hysteresis threshoding are not applicable
◦ Other ways of post-processing to remove unrealiable edges can be used
6
7. Prewitt 3 × 3 M´er˝o-Vassy 7 × 7 LoG 21 × 21
Canny 3 × 3 Canny 7 × 7 Canny 25 × 25
Examples of edge detection by different operators. The LoG result was obtained
with removal of weak edges. M´er˝o-Vassy is a non-gradient edge detector.
7
8. Summary of edge detection
• 3 × 3 gradient operators (Prewitt, Sobel) are simple and fast. Used when
◦ Fine edges are only needed
◦ Noise level is low
• By varying the σ parameter, the Canny operator can be used
◦ to detect fine as well as rough edges
◦ at different noise levels
• All gradient operators
◦ Provide edge orientation
◦ Need localisation: non-maxima suppression, hysteresis threshoding
• The zero-crossing edge detector
◦ Is supported by neurophysiological experiments
◦ Was popular in the 1980’s
◦ Today, less frequently used in practice
8
9. Corner detection in greyscale images
A reminder:
• Corners are used in shape analysis and motion analysis
◦ Motion is ambiguous at an edge, unambiguous at a corner
◦ Shapes can be approximately reconstructed from their corners
• Two different operations although related operations are called corner detection:
◦ Detection of corners in greyscale images
∗ does not assume extracted contours
◦ Detection of corners in digital curves
∗ assumes extracted contours
This lecture deals with corner detection in greyscale images. Corner detection in
contours will be discussed later.
9
10. Corners, edges, and derivatives of intensity function
Diference between greyscale corners and edges:
• Corners are local image features characterised by locations where variations of
intensity function f(x, y) in both X and Y directions are high.
⇒ Both partial derivatives fx and fy are large
• Edges are locations where the variation of f(x, y) in a certain direction is high,
while the variation in the orthogonal direction is low.
⇒ In an edge oriented along the Y axis, fx is large, while is fy small
Y
X
Y
X
EdgeCorner
A corner and an edge.
10
11. Two selected corner detectors
Different corner detectors exist, but we will only consider two of them:
• The Kanade-Lucas-Tomasi (KLT) operator
• The Harris operator
Reasons:
• Most frequently used: Harris in Europe, KLT in US.
• Can select corners and other interest points.
• Have many application areas, for example:
◦ motion tracking, stereo matching, image database retrieval
• Are relatively simple but still efficient and reliable.
The two operators are closely related and based on the local structure matrix.
11
12. The local structure matrix Cstr
Definition of the local structure matrix (tensor):
Cstr = wG(r; σ) ∗
f2
x fxfy
fxfy f2
y
(1)
Explanation of the definition:
• The derivatives of the intensity function f(x, y) are first calculated in each point.
◦ If necessary, the image is smoothed before taking the derivatives
• Then, the entries of the matrix (f2
x, etc.) are obtained.
• Finally, each of the entries is smoothed (integrated) by Gaussian filter wG(r; σ)
of selected size σ.
◦ Often, a simple box (averaging) filter is used instead of the Gaussian.
12
13. Properties of the local structure matrix
Denoting in (1) the smoothing by ff, we have
Cstr =
f2
x fxfy
fxfy f2
y
The local structure matrix Cstr is
• Symmetric
⇒ It can be diagonalised by rotation of the coordinate axes. The diagonal entries
will be the two eigenvalues λ1 and λ2:
Cstr =
λ1 0
0 λ2
• Positive definite
⇒ The eigenvalues are nonnegative. Assume λ1 ≥ λ2 ≥ 0.
13
14. The meaning of the eigenvalues of Cstr
The geometric interpretation of λ1 and λ2:
• For a perfectly uniform image: Cstr = 0 and λ1 = λ2 = 0.
• For a perfectly black-and-white step edge: λ1 > 0, λ2 = 0, where the eigenvector
associated with λ1 is orthogonal to the edge.
• For a corner of black square against a white background: λ1 ≥ λ2 > 0.
◦ The higher the contrast in that direction, the larger the eigenvalue
Basic observations:
• The eigenvectors encode edge directions, the eigenvectors edge magnitudes.
• A corner is identified by two strong edges ⇒ A corner is a location where the
smaller eigenvalue, λ2, is large enough.
14
15. The KLT corner detector has two parameters: the threshold on λ2, denoted by
λthr, and the linear size of a square window (neighbourhood) D.
Algorithm 1: The KLT Corner Detector
1. Compute fx and fy over the entire image f(x, y).
2. For each image point p:
(a) form the matrix Cstr over a D × D neighbourhood of p;
(b) compute λ2, the smaller eigenvalue of Cstr;
(c) if λ2 > λthr, save the p into a list, L.
3. Sort L in decreasing order of λ2.
4. Scan the sorted list from top to bottom. For each current point, p, delete all
points apperaring further in the list which belong to the neighbourhood of p.
15
16. The output is a list of feature points with the following properties:
• In these points, λ2 > λthr.
• The D-neighbourhoods of these points do not overlap.
Selection of the parameters λthr and D:
• The threshold λthr can be estimated from the histogram of λ2: usually, there is
an obvious valley near zero.
◦ Unfortunately, such valley is not always present
• There is no simple criterion for the window size D. Values between 2 and 10
are adequate in most practical cases.
◦ For large D, the detected corner tends to move away from its actual position
◦ Some corners which are close to each other may be lost
16
18. The Harris corner detector
The Harris corner detector (1988) appeared earlier than KLT. KLT is a different
interpretation of the original Harris idea.
Harris defined a measure of corner strength:
H(x, y) = det Cstr − α (trace Cstr)
2
,
where α is a parameter and H ≥ 0 if 0 ≤ α ≤ 0.25.
A corner is detected when
H(x, y) > Hthr,
where Hthr is another parameter, a threshold on corner strength.
Similar to the KLT, the Harris corner detector uses D-neighbourhoods to discard
weak corners in the neighbourhood of a strong corner.
18
19. Parameter of Harris operator and relation to KLT
Assume as before that λ1 ≥ λ2 ≥ 0. Introduce λ1 = λ, λ2 = κλ, 0 ≤ κ ≤ 1.
Using the relations between eigenvalues, determinant and trace of a matrix A
det A =
i
λi
trace A =
i
λi,
we obtain that
H = λ1λ2 − α(λ1 + λ2)2
= λ2
κ − α(1 + κ)2
Assuming that H ≥ 0, we have
0 ≤ α ≤
k
(1 + κ)2
≤ 0.25 and, for small κ, H ≈ λ2
(κ − α) , α κ
19
20. In the Harris operator, α plays a role similar to that of λthr in the KLT operator.
• Larger α ⇒ smaller H ⇒ less sensitive detector: less corners detected.
• Smaller α ⇒ larger H ⇒ more sensitive detector: more corners detected.
Usually, Hthr is set close to zero and fixed, while α is a variable parameter.
original image α = 0.05 α = 0.10
α = 0.20 α = 0.22 α = 0.24
Corner detection by Harris operator: influence of α. (Hthr = 0.)
20
22. KLT 40 corners Harris α = 0.2
Comparison of the two operators.
22
23. Summary of corner detection
• The KLT and the Harris corner detectors are conceptually related.
◦ Based on local structure matrix Cstr
◦ Search for points where variations in two orthogonal directions are large
• Difference between the two detectors:
◦ KLT sets explicit threshold on the diagonalised Cstr
◦ Harris sets implicit threshold via corner magnitude H(x, y)
• The KLT detector
◦ usually gives results which are closer to human perception of corners;
◦ is often used for motion tracking in the wide-spread KLT Tracker.
• The Harris detector
◦ provides good repeatability under varying rotation and illumination;
◦ if often used in stereo matching and image database retrieval.
• Both operators may detect interest points other than corners.
23