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Two Results in Computer Vision Using Projective Geometry University of Florence Faculty of Engineering Dipartimento di Sistemi e Informatica Dottorato di ricerca in informatica, multimedialita` e telecomunicazioni Two Results in Computer Vision using Projective Geometry by Federico Pernici Doctoral committee: Prof. Alberto Del Bimbo Prof. Giacomo Bucci Year 2005 This thesis is submitted to the Dipartimento di Sistemi e Informatica, University of Florence, for the degree of Doctor of Philosophy. This thesis is entirely my own work, and, except where otherwise indicated, describes my own research. To my family and myself Abstract In the ¯rst part of this thesis the problem of metric reconstruction and texture acquisition from a single uncalibrated view of a surface of revolution (SOR) is addressed. Geometric constraints induced in the image by the symmetry properties of the SOR structure are exploited to perform self-calibration of a natural camera, 3D metric reconstruction, texture acquisition & mosaicing and automatic reconstruction. By exploiting the analogy with the geometry of single axis motion, i demonstrate that the imaged apparent contour and the visible segments of two imaged cross sections in a single SOR view provide enough information for these tasks. In the second part of this thesis the problem of scheduling an active observer to visit as many1 targets in an area of surveillance as possible is proposed. I show how it is possible to plan a sequence of decisions regarding what target to look at through such a foveal-sensing action. I propose a framework in which a pan/tilt/zoom camera executes saccades in order to visit, and acquire high resolution images (at least one) of, as many moving targets as possible before they leave the scene. The whole problem is casted into a dynamic discrete optimization framework. In particular, we will show that the problem can be solved by modeling the attentional gaze control as a kinetic traveling salesperson problem whose solution is approximated by iteratively solving time dependent orienteering problems 1An intelligent choice of the order of sensing the targets can signi¯cantly reduce the total dead-time wasted by the active camera and, consequently, its cycle time. Acknowledgement I would like to acknowledge the e®orts and input of my supervisor, Professor Alberto Del Bimbo, and my colleagues of the VipLab, who were invaluable in shaping the course of this research. Preface This thesis address two problems in Computer Vision in which the proposed methods are related by the use of Projective Geometry. Projective Geometry is the important tool which forms the basis for the formulation of the two main frameworks here proposed. The geometry of single views is explored and monocular vision is shown to be su±cient to obtain a partial/complete three-dimensional reconstruction or measurements of a scene. To achieve this the properties of planar homographies, planar homologies and conics are extensively exploited. The ¯rst part of this thesis regards about 3d reconstruction through cam- era (auto) calibration. A camera is a remarkably useful measuring device - it not only produces a realistic picture of a scene, but also provides informa- tion from which geometric properties of the scene can be measured and so reconstructed. Reconstructing scene geometry from images is one of the most active areas in computer vision. This task has proved to be very challenging. Still, with the almost exponential increase in computation resources that the last decade has seen, and with the application of principled mathematical methods, much progress has been made in the last decade especially in the geometry of multiple images. In this part the author propose a new method for the 3D reconstruction of surfaces of revolution from only a single uncali- brated view. Uncalibrated means that nothing is known about the geometry ii of the camera taking the picture. In particular no knowledge of the camera internal parameters (such as focal length and aspect ratio) or its pose (po- sition and orientation) is required at any time. In this minimal setting i'll show mathematically how to recover the global 3D structure (up to an over- all scaling factor) of a SOR and how to calibrate the camera from the solely image content. The novel contributions of the thesis will be addressed more precisely in the conclusion. In broad terms, however, the methods described integrate constraints on calibration and reconstruction of surfaces of revo- lution into a single framework extending previous approaches dealing with piecewise planar scenes. In the second part of the thesis a very novel problem is proposed in which an active observer (i.e. an active zooming camera) plans a sequence of decisions regarding what target to look at, through a foveal sensing action aimed to obtain high resolution images of such targets. The motivation is to collect images at some minimal ground resolution. This task might be de¯ned to support gait recognition, or the acquisition of an appearance model that could be used to subsequently identify the person. I propose a optimization framework in which Projective Geometry of monocular vision is used for evaluating the cost in the optimization. This work is motivated by the goal of reproducing the ability of humans to recognize a person in a crowd of moving people for surveillance purposes. The overall layout of subsequent chapters is as follows. Chapter 1 discuss some background and the single view geometry of surfaces of revolution. Chapter 2 propose a surface mosaicing algorithm for registering into a global images single views of SOR. Chapter 3 address the problem of giving an automatization to the whole process of SOR reconstruction from a single image. In chapter 4 (the second part of this thesis) present an approach iii for planning an active zooming camera for the task of high resolution image sensing. Chapter 5 draws the thesis to a close, summing up the contributions made in the research and outlining areas of further investigation. Contents Abstract i Preface i Contents iv I Single view geometry and Surfaces of Revolution 1 1 Single view Reconstruction 2 1.1 Introduction . 3 1.2 Related Work . 5 1.3 Background . 10 1.3.1 Basic terminology . 10 1.3.2 Basic imaged SOR properties . 12 1.3.3 The analogy between SOR geometry and single axis motion . 14 1.4 The Approach . 16 1.4.1 Derivation of the ¯xed entities . 16 1.4.2 Camera calibration . 20 1.4.3 3D Metric Reconstruction . 22 Contents v 1.4.4 Texture acquisition . 28 1.5 Experimental Results . 30 1.5.1 Accuracy evaluation . 30 1.5.2 3D SOR reconstruction . 36 1.5.3 Texture acquisition . 38 1.6 Summary . 39 2 Mosaicing 50 2.1 Introduction and Related Work . 50 2.2 The Approach . 53 2.2.1 Imaged SOR Geometry and Camera Calibration . 53 2.2.2 Imaged SOR Parameterization . 55 2.2.3 Texture Alignment . 62 2.3 Experimental Results . 62 2.4 Summary . 66 3 Automatic Reconstruction 68 3.1 Introduction and Related Work . 69 3.2 Problem Statement and Overview of the Approach . 70 3.3 Homology Estimation and Curve Grouping . 73 3.3.1 RANSAC-based initialization. 73 3.3.2 Robust homology estimation. 74 3.3.3 Curve Grouping. 75 3.3.4 The algorithm . 79 3.4 Curve Classi¯cation . 80 3.5 Experimental Results . 83 3.6 Summary . 84 Contents vi II Video Surveillance and Active Zooming Camera 89 4 Saccade planning with PTZ camera 90 4.1 Introduction . 91 4.2 Related Work . 94 4.3 Problem Formulation . 97 4.3.1 Kinetic Travelling Salesperson Problem . 97 4.3.2 Time Dependent Orienteering (TDO) . 99 4.3.3 Deadlines . 101 4.4 Saccades Planning Geometry . 103 4.5 Simulation Results . 109 4.5.1 Estimating Camera Speeds . 109 4.5.2 Congestion Analysis . 110 4.6 Summary . 117 5 Conclusion 119 5.1 Novel Contribution . 119 5.2 Future Research . 121 A Appendix of the Part A 124 A.1 Computing the harmonic homology from the vertices of the complete quadrangle . 124 A.2 Parametrizing the image of the absolute conic . 125 A.3 Computing the point of an imaged SOR parallel for a given Euclidean angle . 127 A.4 Linear Constraints on the IAC . 129 A.4.1 The constraint of the circular points . 129 A.4.2 The constraint of the pole-polar relationship . 130 Contents vii A.4.3 The constraint of the zero skew and unit aspect ratio . 131 A.5 Closed form solution for the intersection of two ellipses . 132 B Appendix of the Part B 136 B.1 Absolute Conic and rotating cameras . 136 B.2 Computing the Slave internal internal camera parameter . 137 Bibliography 139 Part I Single view geometry and Surfaces of Revolution Chapter 1 Single view Reconstruction Image analysis and computer vision can be e®ectively employed to recover the three-dimensional structure of imaged objects, together with their surface properties. In this chapter, we address the problem of metric reconstruction and texture acquisition from a single uncalibrated view of a surface of revo- lution (SOR). Geometric constraints induced in the image by the symmetry properties of the SOR structure are exploited to perform self-calibration of a natural camera, 3D metric reconstruction, and texture acquisition. By exploiting the analogy with the geometry of single axis motion, we demon- strate that the imaged apparent contour and the visible segments of two imaged cross sections in a single SOR view provide enough information for these tasks. Original contributions of this part of the thesis are: single view self-calibration and reconstruction based on planar recti¯cation, previously developed for planar surfaces, has been extended to deal also with the SOR class of curved surfaces; self-calibration is obtained by estimating both cam- era focal length (one parameter) and principal point (two parameters) from three independent linear constraints for the SOR ¯xed entities; the invariant- based description of the SOR scaling function has been extended from a±ne Chapter 1.
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