Numerical Geometry of Images: Theory, Algorithms, and ApplicationsNumerical Geometry of Images examines computational methods and algorithms in image processing. It explores applications like shape from shading, color-image enhancement and segmentation, edge integration, offset curve computation, symmetry axis computation, path planning, minimal geodesic computation, and invariant signature calculation. In addition, it describes and utilizes tools from mathematical morphology, differential geometry, numerical analysis, and calculus of variations. Graduate students, professionals, and researchers with interests in computational geometry, image processing, computer graphics, and algorithms will find this new text / reference an indispensable source of insight of instruction. |
Contents
Introduction | 1 |
11 Mathematical Tools and Machinery | 2 |
12 Applications | 4 |
13 Exercises | 13 |
Basic Differential Geometry | 17 |
22 Invariant Signatures | 20 |
23 Calculus of Variations in Parametric Form | 26 |
24 Geometry of Surfaces | 27 |
73 Fast Marching on Triangulated Manifolds | 98 |
74 Applications of Fast Marching on Surfaces | 102 |
75 Exercises | 107 |
Shape from Shading | 109 |
81 Problem Formulation | 110 |
82 Horn Characteristic Strip Expansion Method | 111 |
83 Brucksteins EqualHeight Contours Expansion Method | 113 |
84 Tracking Level Sets by Level Sets | 114 |
25 A Brief Introduction to Intrinsic Geometry | 31 |
26 Exercises | 33 |
Curve and Surface Evolution | 35 |
32 Properties of Curve Evolution | 40 |
33 Surface Evolution | 45 |
34 Exercises | 48 |
The OsherSethian Level Set Method | 50 |
41 The Eulerian Formulation | 51 |
42 From Curve to Image Evolution | 55 |
43 Exercises | 59 |
The Level Set Method Numerical Considerations | 61 |
52 Conservation Laws and HamiltonJacobi Equations | 63 |
53 Entropy Condition and Vanishing Viscosity | 64 |
54 Numerical Methodologies | 65 |
55 The CFL Condition | 67 |
56 OneDimensional Example of a Differential Conservation Law | 68 |
57 TwoDimensional Example of the CFL Condition | 70 |
58 Viscosity Solutions | 72 |
59 Summary | 73 |
510 Exercises | 74 |
Mathematical Morphology and Distance Maps | 75 |
61 Continuous Morphology by Curve Evolution | 77 |
62 ContinuousScale Morphology | 78 |
63 Distance Maps | 80 |
64 Skeletons | 82 |
65 Exercises | 86 |
Fast Marching Methods | 87 |
71 The OneDimensional Eikonal Equation | 88 |
72 Fast Marching on TwoDimensional Rectangular Grids | 91 |
85 Extracting the Surface Topography | 116 |
86 Oblique Light Source | 118 |
87 Summary | 122 |
2D and 3D Image Segmentation | 123 |
91 The Level Set Geometric Model | 124 |
93 Relation to Image Enhancement Methods | 126 |
94 Nongeometric Measures and the Maupertuis Principle of Least Action | 128 |
95 Edge Integration | 130 |
96 Geometric Segmentation in 3D | 132 |
97 Efficient Numerical Schemes | 133 |
98 Exercises | 136 |
Geometric Framework in Image Processing | 141 |
101 Images as Surfaces | 144 |
102 The Geometric Framework | 145 |
103 Movies and Volumetric Medical Images | 149 |
105 The Metric as a Structure Tensor | 156 |
106 Inverse Diffusion Across the Edge | 158 |
107 Summary | 161 |
108 Exercises | 162 |
Texture Mapping Matching Isometric Surfaces and 3D Face Recognition | 163 |
111 Flat Embedding | 164 |
112 Texture Mapping | 168 |
113 Isometric Signatures for Surfaces | 171 |
114 Face Recognition | 173 |
115 Exercises | 178 |
Solutions to Selected Problems | 179 |
Bibliography | 192 |
206 | |
Other editions - View all
Numerical Geometry of Images: Theory, Algorithms, and Applications Ron Kimmel No preview available - 2012 |
Numerical Geometry of Images: Theory, Algorithms, and Applications Ron Kimmel No preview available - 2012 |
Common terms and phrases
A. M. Bruckstein active contours affine algorithm apply approximation boundary C₁ color images Computer Vision conservation law convex coordinates curvature flow curve evolution defined derivative diffusion distance map domain ds² dx² dy² edge eigenvalues eikonal equation embedding equal-height contour Euclidean space evolution equation example fast marching method Figure function geodesic active contour geodesic curvature geodesic distance geometric framework given gradient gray-level image grid point heat equation IEEE Trans image processing integration introduced invariant J. A. Sethian Kimmel level set formulation level set method light source manifold mathematical matrix mean curvature measure metric minimal geodesic normal numerical scheme offset parameterization path pixels planar curve plane problem propagating scale space segmentation Sethian shading image shape from shading skeleton smooth Solution to Ex structuring element surface T₁ tangent texture mapping theorem triangle update variation vector viscosity solution Voronoi diagrams მ მ
Popular passages
Page 205 - SD Yanowitz and AM Bruckstein. A new method for image segmentation.