Mathematical Optimization in Graphics and Vision - Luiz Velho - Impa

CONTENTS
vii
5.2.1 Solving least squares problems . . . . . . . . . . . . . . . 81
5.3 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3.1 Newton’s method . . . . . . . . . . . . . . . . . . . . . . 83
5.3.2 Unidimensional search algorithms . . . . . . . . . . . . . 84
5.3.3 Conjugate gradient . . . . . . . . . . . . . . . . . . . . . 86
5.3.4 Quasi-Newton algorithms . . . . . . . . . . . . . . . . . 87
5.4 Constrained optimization . . . . . . . . . . . . . . . . . . . . . . 88
5.4.1 Optimality conditions . . . . . . . . . . . . . . . . . . . 88
5.4.2 Least squares with linear constraints . . . . . . . . . . . . 91
5.4.3 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.5 Linear programming . . . . . . . . . . . . . . . . . . . . . . . . 96
5.5.1 Simplex algorithm for linear programs . . . . . . . . . . . 99
5.5.2 The complexity of the simplex method . . . . . . . . . . . 100
5.6 Applications in Graphics . . . . . . . . . . . . . . . . . . . . . . 101
5.6.1 Camera calibration . . . . . . . . . . . . . . . . . . . . . 101
5.6.2 Registration and color correction for a sequence of images 105
5.6.3 Visibility for real time walktrough . . . . . . . . . . . . . 109
6 Combinatorial optimization 113
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.1.1 Solution methods in combinatorial optimization . . . . . . 114
6.1.2 Computational complexity . . . . . . . . . . . . . . . . . 116
6.2 Description of combinatorial problems . . . . . . . . . . . . . . . 116
6.2.1 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.2.2 Adjacency matrix . . . . . . . . . . . . . . . . . . . . . . 117
6.2.3 Incidence matrix . . . . . . . . . . . . . . . . . . . . . . 118
6.3 Dynamic programming . . . . . . . . . . . . . . . . . . . . . . . 119
6.3.1 Constructing the state of the problem . . . . . . . . . . . 122
6.3.2 Remarks about general problems . . . . . . . . . . . . . . 124
6.3.3 Problems of resource allocation . . . . . . . . . . . . . . 124
6.4 Shortest paths in graphs . . . . . . . . . . . . . . . . . . . . . . . 125
6.4.1 Shortest paths in acyclic oriented graphs . . . . . . . . . . 125
6.4.2 Directed graphs possibly with cycles . . . . . . . . . . . . 126
6.4.3 Dijkstra algorithm . . . . . . . . . . . . . . . . . . . . . 128
6.5 Integer programming . . . . . . . . . . . . . . . . . . . . . . . . 128