Algorithmic Efficiency in Convex Hull Computation: Insights from 2D and 3D Implementations

Finding efficient and powerful techniques for representing 2D and 3D entities is central in many areas, such as computer vision, graphics, CAD or data visualization. The large amount of data often involved and the small processing time required place strong demands on modeling techniques. This chapter gives an overview of the possibilities wavelets offer for object modeling, specially from the point of view of object recognition in computer vision environments. For the 2D case, techniques based on segmented contours or raw grey level information are described. For 3D data, both the cases of dense, regularly sampled volumetric information and sparse, irregularly sampled geometric surface information are considered. Ideas on how Bayesian methods may enhance wavelet based techniques for object representation and recognition are outlined.

Chapter 7 A survey of computational geometry

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.

A new algorithm for the determination of the relative convex hull in the plane of a simple polygon A with respect to another simple polygon B which contains A, is proposed. The relative convex hull is also known as geodesic convex hull, and the problem of its determination in the plane is equivalent to find the shortest curve among all Jordan curves lying in the difference set of B and A and encircling A. Algorithms solving this problem known from Computational Geometry are based on the triangulation or similar decomposition of that difference set. The algorithm presented here does not use such decomposition, but it supposes that A and B are given as ordered sequences of vertices. The algorithm is based on convex hull calculations of A and B and of smaller polygons and polylines, it produces the output list of vertices of the relative convex hull from the sequence of vertices of the convex hull of A.

An approximate algorithm for computing multidimensional convex hulls

Polyhedral and Algebraic Methods in Computational Geometry provides a thorough introduction into algorithmic geometry and its applications. It presents its primary topics from the viewpoints of discrete, convex and elementary algebraic geometry. The first part of the book studies classical problems and techniques that refer to polyhedral structures. The authors include a study on algorithms for computing convex hulls as well as the construction of Voronoi diagrams and Delone triangulations. The second part of the book develops the primary concepts of (non-linear) computational algebraic geometry. Here, the book looks at Grbner bases and solving systems of polynomial equations. The theory is illustrated by applications in computer graphics, curve reconstruction and robotics. Throughout the book, interconnections between computational geometry and other disciplines (such as algebraic geometry, optimization and numerical mathematics) are established. Polyhedral and Algebraic Methods in Computational Geometry is directed towards advanced undergraduates in mathematics and computer science, as well as towards engineering students who are interested in the applications of computational geometry.

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

Passive polarization based imaging is a useful tool in computer vision and pattern recognition. A passive polarization imaging system forms a polarimetric image from the reflection of ambient light that contains useful information for computer vision tasks such as object detection (classification) and recognition. Applications of polarization based pattern recognition include material classification and automatic shape recognition. In this paper, we present two target detection algorithms for images captured by a passive polarimetric imaging system. The proposed detection algorithms are based on Bayesian decision theory. In these approaches, an object can belong to one of any given number classes and classification involves making decisions that minimize the average probability of making incorrect decisions. This minimum is achieved by assigning an object to the class that maximizes the a posteriori probability. Computing a posteriori probabilities requires estimates of class conditional probability density functions (likelihoods) and prior probabilities. A Probabilistic neural network (PNN), which is a nonparametric method that can compute Bayes optimal boundaries, and a -nearest neighbor (KNN) classifier, is used for density estimation and classification. The proposed algorithms are applied to polarimetric image data gathered in the laboratory with a liquid crystal-based system. The experimental results validate the effectiveness of the above algorithms for target detection from polarimetric data.

3D registration based on the direction sensor measurements

Computer Graphics is a part of computer science that deals with the creation and manipulation of (visual) images digitally. A simple form of computer graphics is 2D computer graphics which later evolved into 3D computer graphics, image-based (image processing), and pattern recognition (pattern recognition). Computer graphics are often also known as data visualization. Graphics include images and other imagery generated by the computer in the form of lines. OpenGL (Open Graphics Library) is an API standard that can be used to create graphics-based applications, both two-dimensional (2D) and three-dimensional (3D). OpenGL is cross-platform, meaning that it can be run on various operating system platforms that exist today. using OpenGL, we first need a conception of the interface in its implementation of object protection. One method that is commonly used is to create a window-based OpenGL. To be able to conceptualize windowing in OpenGL, we need a certain tool. What we are using this time is GLUT (OpenGL Utility Toolkit).

This paper is concerned with the problem of feature point registration and scene recognition from images under weak perspective transformations which are well approximated by affine transformations and under possible occlusion and/or appearance of new objects. It presents a set of local absolute affine invariants derived from the convex hull of scattered feature points (e.g., fiducial or marking points, corner points, inflection points, etc.) extracted from the image. The affine invariants are constructed from the areas of the triangles formed by connecting three vertices among a set of four consecutive vertices (quadruplets) of the convex hull, and hence do make direct use of the area invariance property associated with the affine transformation. Because they are locally constructed, they are very well suited to handle the occlusion and/or appearance of new objects. These invariants are used to establish the correspondences between the convex hull vertices of a test image with a reference image in order to undo the affine transformation between them. A point matching approach for recognition follows this. The time complexity for registering L feature points on the test image with N feature points of the reference image is of order O(N x L). The method has been tested on real indoor and outdoor images and performs well.

A comparative molecular field analysis study on several bioactive peptides using the alignment rules derived from identification of commonly exposed groups

This paper describes an algorithm for computing three-dimensional convex hulls. First, we introduce a simple algorithm to compute two-dimensional convex hulls using graphics hardware. This algorithm computes the edges and the facets of the convex hulls by analyzing bitmap images drawn using given points. Graphics hardware can generate bitmap images of three-dimensional objects very rapidly. We then expand the algorithm to compute three-dimensional convex hulls. Finally, we apply the algorithm to compute Delaunay diagrams. The algorithm is very simple. Hoff et al. reported an algorithm to draw Voronoi diagrams using graphics hardware [3]. Our algorithms are based on the same technique used by [3]. However, our algorithm employs an analyzing phase of the images and obtains topological information from the image.

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

Machine learning algorithm based on convex hull analysis