Point Pattern Matching Algorithm for Planar Point Sets under Euclidean Transform

Efficient point pattern matching algorithm for planar point sets under transform of translation, rotation and scale

The problem of point pattern matching (PPM) is frequently encountered in computer vision, such as image registration and image matching. This paper investigates the manifold approaches to the problem of point pattern matching, and proposes a manifold correspondence based on Locally Linear Embedding (LLE). Our method operates on embeddings of the two data sets in the manifold space so as to get embedding features, which is invariance to rotation, scaling and translation (RST). By comparing the manifold embeddings of the points, we locate correspondences. We evaluate the method on both synthetic and real-world data, and experimental results demonstrate its high accuracy and robust to outliers.

This study examines various algorithms for computing the convex hull of a set of n points in a d-dimensional space. Convex hulls are fundamental in computational geometry and are applied in computer graphics, pattern recognition, and computational biology. Such convex hulls can also be useful in symmetry problems. For instance, when points are arranged symmetrically, the convex hull is also likely to be symmetrically shaped, which can be useful for object recognition in computer vision or pattern recognition. The focus is primarily on two-dimensional algorithms, including well-known methods like Gift Wrapping, Graham Scan, Divide and Conquer, QuickHull, TORCH, Kirkpatrick–Sediel, and Chan’s algorithms. These algorithms vary in terms of time complexity and scalability to higher dimensions. This study is extended to three-dimensional convex hull algorithms, such as NAW, randomized insertion, and parallelized versions, such as CudaHull and CudaChain. This study aimed to elucidate the operational principles, step-by-step procedures, and comparative time complexities of each algorithm. The implementation in Python facilitates a detailed comparison of the algorithmic performance through stepwise analysis and graphical outputs. The ultimate goal is to provide insights into the strengths and weaknesses of each algorithm under various scenarios, thereby offering a comprehensive guide for practical implementation.

In the paper, we present and prove Castle Theorem of Convex Hull, design and realize the Fast Rampart Searching Algorithm. The algorithm can be treated as the preprocess of Convex Hull calculation of very large planar point set. When calculating the Convex Hull of a very large planar point set, we can use the Fast Rampart Searching Algorithm to get a very small part points as candidate point set, and then we can get the Convex Hull of the whole planar point set from the candidate point set through other algorithms. Keywords-fast rampart searching algorithm; castle theorem; convex hull; computational geometry

This paper investigates spectral approaches to the problem of point pattern matching. Specifically, kernel principle component analysis (kernel PCA) methods are studied and compared with Shapiro and Brady’s approach and multidimensional scaling methods on both synthetic data and real world data. We demonstrate that kernel methods can be effectively used for solving the point correspondence matching problem with a performance that is comparable with other iterative-based algorithms in the literature under the existing of outliers and random position jitter. We also provide discussion of the theoretical support from kernel PCA to the earlier approach of Shapiro and Brady.KeywordsFeature PointPoint PatternPolynomial KernelKernel Principal Component AnalysisPoint CorrespondenceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

With the rapid development of computer, computer vision technology is also making rapid progress. In this paper, deep neural network algorithm is used to improve the technology of computer vision, improve the visual effect, and at the same time, innovative algorithm structure, improve the identification of pattern recognition. Computer vision and pattern recognition is a very cutting edge technology that has given people very advanced tools for vision and recognition. At present, the technology can control more accurate image resolution and improve the ability of pattern recognition. This article mainly explains the process of using deep neural network algorithm to improve computer vision and pattern recognition from the internal mechanism, and reveals the working principle and internal mechanism of computer vision and pattern recognition technology application. Data analysis proves that the pattern recognition application established by deep neural network algorithm performs very well in the field of vision and pattern recognition.KeywordsDeep neural network algorithmComputer visionPattern recognitionTechnical application

Traditional geometric algorithms are often presented as if input imprecision does not exist, even though it is often unavoidable. We argue that in some cases, it may be desirable for geometric algorithms to treat this imprecision as an explicit component of the input, and to reflect this imprecision in the output. Starting with three problems from computational geometry whose inputs are planar point sets (Voronoi diagrams, convex hulls, and smallest bounding discs), we recast these as problems where each input point’s location is imprecise, but known to lie within a particular region of uncertainty. Where algorithms to solve each of the original problems produce a single geometric object as output, the algorithms that we present typically produce either guaranteed or possible output objects. A guaranteed object represents qualities that can be guaranteed for every point set that is consistent with the uncertain regions, and a possible object represents qualities that exist for at least one such point set. By dealing with input imprecision explicitly, these guaranteed and possible objects can represent a more accurate view of what can be reliably inferred from the input data.

A non-parametric approach to shape reconstruction from planar point sets through Delaunay filtering

A chain of a set P of n points in the plane is a chain of the dominance order on P. A k-chain is a subset C of P that can be covered by k chains. A k-chain C is a maximum k-chain if no other k-chain contains more elements than C. This paper deals with the problem of finding a maximum k-chain of P in the cardinality and in the weighted case. Using the skeleton S(P) of a point set P introduced by Viennot we describe a fairly simple algorithm that computes maximum k-chains in time O(kn log n) and linear space. The basic idea is that the canonical chain partition of a maximum (k-1)-chain in the skeleton S(P) provides k regions in the plane such that a maximum k-chain for P can be obtained as the union of a maximal chain from each of these regions. By the symmetry between chains and antichains in the dominance order we may use the algorithm for maximum k-chains to compute maximum k-antichains for planar points in time O(kn log n). However, for large k one can do better. We describe an algorithm computing maximum k-antichains (and, by symmetry, k-chains) in time O((n2 k) log n) and linear space. Consequently, a maximum k-chain can be computed in time O(n3/2 log n) for arbitrary k. The background for the algorithms is a geometric approach to the Greene--Kleitman theory for permutations. We include a skeleton-based exposition of this theory and give some hints on connections with the theory of Young tableaux. The concept of the skeleton of a planar point set is extended to the case of a weighted point set. This extension allows to compute maximum weighted k-chains with an algorithm that is similar to the algorithm for the cardinality case. The time and space requirements of the algorithm for weighted k-chains are O(2kn log(2kn)) and O(2kn), respectively.

Given two sets P and Q of n and m (<or=n) labeled points, respectively, in plane, the matching problem considered here is to determine all occurrences of Q in P when Q is allowed to be translated. An occasional lower bound of Omega ((n-m)m+nlog n) is shown for this problem. For the matching the authors study five different algorithms based on the techniques of dynamic programming, point-line duality, instance matching, list traversal, and radix matching.

We show the existence of $\varepsilon$-nets of size $O\left(\frac{1}{\varepsilon}\log\log\frac{1}{\varepsilon}\right)$ for planar point sets and axis-parallel rectangular ranges. The same bound holds for points in the plane and “fat” triangular ranges and for point sets in $\boldsymbol{R}^3$ and axis-parallel boxes; these are the first known nontrivial bounds for these range spaces. Our technique also yields improved bounds on the size of $\varepsilon$-nets in the more general context considered by Clarkson and Varadarajan. For example, we show the existence of $\varepsilon$-nets of size $O\left(\frac{1}{\varepsilon}\log\log\log\frac{1}{\varepsilon}\right)$ for the dual range space of “fat” regions and planar point sets (where the regions are the ground objects and the ranges are subsets stabbed by points). Plugging our bounds into the technique of Brönnimann and Goodrich or of Even, Rawitz, and Shahar, we obtain improved approximation factors (computable in expected polynomial time by a randomized algorithm) for the hitting set or the set cover problems associated with the corresponding range spaces.

Matching objects is a fundamental problem for any object detection system. Feature-based methods in matching objects such as vehicles often encounter the problem of correspondences between features of two related patterns. The features may be points, lines, curves and regions. Point pattern matching (PPM) is a primary and essential approach for establishing a correspondence within two related patterns. Although some well-known Hausdorff distance measures work well for this task, they are very computational expensive and suffer from the noise of images. In this paper, we propose a novel similarity measure using Hubert curve named Hubert scanning distance (HSD) to resolve the problems. This method computes the distance measure in one-dimensional (1-D) sequence in stead of in two-dimensional (2-D) image space, which greatly reduce the computational complexity. By applying a threshold elimination function, extreme distances caused by noise and position errors (e.g. those that occur with feature or edge extraction) are removed. The experimental results show that HSD can provide sufficient information for matching vehicles within low computational complexity. We believe this point out a new direction for the research of PPM.

One of the fundamental problems of computer vision is to extract 3D shape and motion from images. This can be achieved when a scene or object is observed from different viewpoints or over a period of time. There is a wide range of applications, ranging from digitizing cultural heritage to vision-based autonomous robot navigation. This talk will present several approaches to solve this problem. First, we’ll discuss techniques for 3D shape recovery for static objects and scenes. One particular case is the 3D mapping and localization in large environments from images, e.g. urban 3D reconstruction from vehicleborne cameras or localization from cell-phone images. Next, we’ll shift our focus to modeling dynamic scenes, e.g. people who are moving around. In addition to explicitly 3D modeling an event, we'll consider the possibility to perform video-based rendering from casually captured videos. Speaker Bio-Sketch: Marc Pollefeys is a full professor in the Dept. of Computer Science of ETH Zurich since 2007 where he is the head of the Institute for Visual Computing and leads the Computer Vision and Geometry lab. He currently also remains associated with the Dept. of Computer Science of the University of North Carolina at Chapel Hill where he started as an assistant professor in 2002 and became an associate professor in 2005. Before this he was a postdoctoral researcher at the Katholieke Universiteit Leuven in Belgium, where he also received his M.S. and Ph.D. degrees in 1994 and 1999, respectively. His main area of research is computer vision. One of his main research goals is to develop flexible approaches to capture visual representations of real world objects, scenes and events. Dr. Pollefeys has received several prizes for his research, including a Marr prize, an NSF CAREER award, a Packard Fellowship and a European Research Council Starting Grant. He is the author or co-author of more than 130 peer-reviewed publications. He is the General Chair for the European Conference on Computer Vision 2014 (ECCV), was a Program Co-Chair for the IEEE Conference on Computer Vision and Pattern Recognition 2009 (CVPR), was general/program co-chair of the Third Symposium on 3D Data Processing, Visualization and Transmission and has organized workshops and courses at major vision and graphics conferences and has served on the program committees of many conferences. Prof. Pollefeys is/was on the Editorial Board of the IEEE Transactions on Pattern Analysis and Machine Intelligence and the International Journal of Computer Vision as well as several other journals in computer vision, graphics and robotics.

Hausdorff metrics are used in geometric settings for measuring the distance between sets of points. They have been used extensively in areas such as computer vision, pattern recognition and computational chemistry. While computing the distance between a single pair of sets under the Hausdorff metric has been well studied, no results are known for the nearest-neighbor problem under Hausdorff metrics. Indeed, no results were known for the nearest-neighbor problem for any metric without a norm structure, of which the Hausdorff is one. We present the first nearest-neighbor algorithm for the Hausdorff metric. We achieve our result by embedding Hausdorff metrics into l/sub /spl infin// and by using known nearest-neighbor algorithms for this target metric. We give upper and lower bounds on the number of dimensions needed for such an l/sub /spl infin// embedding. Our bounds require the introduction of new techniques based on superimposed codes and non-uniform sampling.

In the automation of proteome analysis, matching of two-dimensional (2-D) electropherograms represents a bottleneck in the process. Here we present a point pattern recognition approach to the matching of spots in 2-D electropherograms. The algorithm is based on a comparison of spot neighborhoods, converted to point patterns between reference and compared gels. The neighborhood was characterized by a syntactic descriptor which minimized the influence of spot displacements. A combined criterion utilizing the similarity of point patterns and a metric definition of position similarity was derived. The efficiency and accuracy of the algorithm was tested on a set of 69 gels with different levels of similarity. For a typical gel the accuracy of matching was higher than 98% and the number of correctly identified spots was higher than 95%.