A Probabilistic Approach to Geometric Hashing using Line Features

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Chapter 4 Invariant Matching Using Line Features The idea behind the geometric hashing method is to encode local geometric features in a manner which isinvariant under the geometric transformation that model objects undergo during formation of the class of images being analyzed. This encoding can then be used in a hash function that makes possible fast retrieval of model features from the hash table, which can accordingly be viewed as an encoded model base. The technique can be applied directly to line features, without resorting to point features indirectly derived from lines, providing that we use some method of encoding line features in a way invariant under transformations considered. Potentially relevant transformation groups include rigid transformations, similarity transformations, aæne transformations and projective transformations, depending on the manner in which an image is formed. Each of these classes of transformations is a subgroup of the full group of projective transformations. Since a projective transformation is a collineation èreferring to p. 89 of ë35ëè, all these transformations preserve lines. This allows us to use line features as inputs to the recognition procedures. 4.1 Change of Coordinates in Various Spaces Since we will represent line features by their normal parameterization èç; rè, we show in this section how the change of coordinates in image space by a linear transformation acts 39
CHAPTER 4. INVARIANT MATCHING USING LINE FEATURES 40 to change of coordinates in èç; rè-space. This is preliminary to the invariant line encodings exploited in its following sections. 4.1.1 Change of Coordinates in Image Space It is often easiest to work in homogeneous coordinates, since this allows projective transformations to be represented by matrices. Achange of coordinates in homogeneous coordinates is given by w 0 = Tw; where w = èw 1 ;w 2 ;w 3 è t is the homogeneous coordinate before transformation, w 0 = èw 0 1 ;w0 2 3è t is the homogeneous coordinate after transformation and T is a non-singular ;w0 3 æ 3 matrix. Note that T and çT, for any ç 6= 0, deæne the same transformation, since we are dealing with homogeneous coordinates. A point èx; yè t in image space is represented by a non-zero 3-D point w =èw 1 ;w 2 ;w 3 è t in homogeneous coordinate systems, such that x = w 1 w 3 and y = w 2 w 3 ; where w 3 6=0. This representation is not unique, since çw, for any ç 6= 0 is also a homogeneous representation of èx; yè t . Every non-singular 3æ3 matrix deænes a 2-D projective transformation of homogeneous coordinates. Various subgroups of the projective transformation group are deæned by diæerent restrictions on the form of the matrix T. 4.1.2 Change of Coordinates in Line Parameter Space A line in a 2-D plane can be represented by its parameter vector and is usually parameterized as a 1 w 1 + a 2 w 2 + a 3 w 3 =0 or a t w =0; where a =èa 1 ;a 2 ;a 3 è t is the parameter vector of the line ènote that a 1 and a 2 are not both equal to 0è and w =èw 1 ;w 2 ;w 3 è t is a homogeneous coordinate of any point on the line.
Page 1 and 2: A Probabilistic Approach to Geometr
Page 3 and 4: Dedication This dissertation is ded
Page 5 and 6: Abstract One of the most important
Page 7 and 8: 2.2.1 A Brief Description : : : : :
Page 9 and 10: List of Figures 3.1 A comparison of
Page 11 and 12: CHAPTER 1. INTRODUCTION 2 1.1 Model
Page 13 and 14: CHAPTER 1. INTRODUCTION 4 texture a
Page 15 and 16: CHAPTER 1. INTRODUCTION 6 reconstru
Page 17 and 18: CHAPTER 1. INTRODUCTION 8 Both synt
Page 19 and 20: CHAPTER 2. PRIOR AND RELATED WORK 1
Page 33 and 34: CHAPTER 3. NOISE IN THE HOUGH TRANS
Page 47: CHAPTER 3. NOISE IN THE HOUGH TRANS
Page 51 and 52: CHAPTER 4. INVARIANT MATCHING USING
Page 69 and 70: Chapter 5 The Eæect of Noise on th
Page 71 and 72: CHAPTER 5. THE EFFECT OF NOISE ON T
Page 83 and 84: Chapter 6 Bayesian Line Invariant M
Page 85 and 86: CHAPTER 6. BAYESIAN LINE INVARIANT
Page 87 and 88: CHAPTER 6. BAYESIAN LINE INVARIANT
Page 89 and 90: Chapter 7 The Experiments In this c
Page 91 and 92: CHAPTER 7. THE EXPERIMENTS 82 j det
Page 93 and 94: CHAPTER 7. THE EXPERIMENTS 84 Speci
Page 95 and 96: CHAPTER 7. THE EXPERIMENTS 86 Figur
Page 97 and 98: CHAPTER 7. THE EXPERIMENTS 88 èaè
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CHAPTER 7. THE EXPERIMENTS 90 èaè
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CHAPTER 7. THE EXPERIMENTS 92 èeè
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CHAPTER 7. THE EXPERIMENTS 94 of li
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CHAPTER 8. CONCLUSIONS 96 We have a
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Appendix A Given a correspondence b
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Appendix B The components of the ma
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Bibliography ë1ë A. Aho, J. Hopcr
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BIBLIOGRAPHY 104 ë23ë W. E. L. Gr
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BIBLIOGRAPHY 106 ë47ë I. Rigoutso
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A Probabilistic Approach to Geometric Hashing using Line Features

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