A Probabilistic Approach to Geometric Hashing using Line Features

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CHAPTER 2. PRIOR AND RELATED WORK 17 Normalization There are various reasons for the use of normalization. In some cases, it is used to preserve certain properties. For example, in the probabilistic relaxation labeling scheme, the accumulated contributions from neighboring points have to be normalized so that the updating rule results in a probability èi.e., a value in ë0::1ëè. In other cases, it can be viewed as a technique for removing some uncertain factors. For example, suppose we use Fourier Descriptor to describe the boundary of an object. In order for such representation insensitive to the variations as changes in size, rotation, translation and so on, we have to perform the normalization operations such that the contour has a ëstandard" size, orientation and starting point. By normalizing the Fourier component F è1è to have unity magnitude, we remove the factor of scaling variation. Similarly, the ès-çè graph èthe representation of turning angle as a function of arc lengthè has to be shifted vertically by adding an oæset to ç so that the reference point on the contour is somewhat a standard value, such as zero. Since the rotation of a contour in Cartesian space corresponds to a simple shift in the ordinate èçè of the s-ç graph, such normalization process removes the factor of rotation ë19ë. Other good reasons to use normalization involve making some operations easier to perform. For example, in performing point set matching, Hong and Tan ë27ë normalize both point sets to their canonical forms ærst; then, their canonical forms are matched. After normalization, the matching between two canonical forms reduce to a simple rotation of one to match the other. 2.2 An Overview of Geometric Hashing 2.2.1 A Brief Description The geometric hashing idea has its origins in work of Schwartz and Sharir ë49ë. Application of the geometric hashing idea for model-based visual recognition was described by Lamdan, Schwartz and Wolfson. This section outlines the method; a more complete description can be found in Lamdan's dissertation ë36ë; a parallel implementation can be found in ë46ë. The geometric hashing method proceeds in two stages: a preprocessing stage and a recognition stage. In the preprocessing stage, we construct a model representation by
CHAPTER 2. PRIOR AND RELATED WORK 18 computing and storing redundant, transformation-invariant model information in a hash table. During the subsequent recognition stage the same invariants are computed from features in a scene and used as indexing keys to retrieve from the hash table the possible matches with the model features. If a model's features scores suæciently many hits, we hypothesize the existence of an instance of that model in the scene. The Pre-processing Stage Models are processed one by one. New models added to the model base can be processed and encoded into the hash table independently. For each model M and for every feasible basis b consisting of k points èk depends on the transformations the model objects undergo during formation of the class of images to be analyzedè, we èiè compute the invariants of all the remaining points in terms of the basis b; èiiè use the computed invariants to index the hash table entries, in each of which we record a node èM; bè. Note that all feasible bases have to be used. In particular, all the permutations èup to k!è of the k inputs used to calculate the invariants have to be considered. The complexity of this stage is Oèm k+1 è per model, where m is the numberofpoints extracted from a model. However, since this stage is executed oæ-line, its complexity is of little signiæcance. The Recognition Stage Given a scene with n feature points extracted, we èiè choose a feasible set of k points as a basis b; èiiè compute the invariants of all the remaining points in terms of this basis b; èiiiè use each computed invariant to index the hash table and hit all èM i ;b j è's that are stored in the entry retrieved; èivè histogram all èM i ;b j è's with the number of hits received;
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
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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
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Page 33 and 34: CHAPTER 3. NOISE IN THE HOUGH TRANS
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CHAPTER 5. THE EFFECT OF NOISE ON T
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Chapter 6 Bayesian Line Invariant M
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CHAPTER 6. BAYESIAN LINE INVARIANT
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CHAPTER 6. BAYESIAN LINE INVARIANT
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Chapter 7 The Experiments In this c
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CHAPTER 7. THE EXPERIMENTS 82 j det
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CHAPTER 7. THE EXPERIMENTS 84 Speci
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CHAPTER 7. THE EXPERIMENTS 86 Figur
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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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