Numerical Geometry of Images Theory, Algorithms, and Applications

Numerical 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.

1 Introduction.- 1.1 Mathematical Tools and Machinery.- 1.2 Applications.- 1.3 Exercises.- 2 Basic Differential Geometry.- 2.1 Introduction to Differential Geometry in the Plane.- 2.2 Invariant Signatures.- 2.3 Calculus of Variations in Parametric Form.- 2.4 Geometry of Surfaces.- 2.5 A Brief Introduction to Intrinsic Geometry.- 2.6 Exercises.- 3 Curve and Surface Evolution.- 3.1 Evolution.- 3.2 Properties of Curve Evolution.- 3.3 Surface Evolution.- 3.4 Exercises.- 4 The Osher—Sethian Level Set Method.- 4.1 The Eulerian Formulation.- 4.2 From Curve to Image Evolution.- 4.3 Exercises.- 5 The Level Set Method: Numerical Considerations.- 5.1 Finite Difference Approximation.- 5.2 Conservation Laws and Hamilton-Jacobi Equations.- 5.3 Entropy Condition and Vanishing Viscosity.- 5.4 Numerical Methodologies.- 5.5 The CFL Condition.- 5.6 One-Dimensional Example of a Differential Conservation Law.- 5.7 Two-Dimensional Example of the CFL Condition.- 5.8 Viscosity Solutions.- 5.9 Summary.- 5.10 Exercises.- 6 Mathematical Morphology and Distance Maps.- 6.1 Continuous Morphology by Curve Evolution.- 6.2 Continuous-Scale Morphology.- 6.3 Distance Maps.- 6.4 Skeletons.- 6.5 Exercises.- 7 Fast Marching Methods.- 7.1 The One-Dimensional Eikonal Equation.- 7.2 Fast Marching on Two-Dimensional Rectangular Grids.- 7.3 Fast Marching on Triangulated Manifolds.- 7.4 Applications of Fast Marching on Surfaces.- 7.5 Exercises.- 8 Shape from Shading.- 8.1 Problem Formulation.- 8.2 Horn Characteristic Strip Expansion Method.- 8.3 Bruckstein’s Equal-Height Contours Expansion Method.- 8.4 Tracking Level Sets by Level Sets.- 8.5 Extracting the Surface Topography.- 8.6 Oblique Light Source.- 8.7 Summary.- 8.8 Exercises.- 9 2D and 3D Image Segmentation.- 9.1 The Level Set Geometric Model.- 9.2Geodesic Active Contours.- 9.3 Relation to Image Enhancement Methods.- 9.4 Nongeometric Measures and the Maupertuis Principle of Least Action.- 9.5 Edge Integration.- 9.6 Geometric Segmentation in 3D.- 9.7 Efficient Numerical Schemes.- 9.8 Exercises.- 10 Geometric Framework in Image Processing.- 10.1 Images as Surfaces.- 10.2 The Geometric Framework.- 10.3 Movies and Volumetric Medical Images.- 10.4 The Image Area as a Measure for Color Processing.- 10.5 The Metric as a Structure Tensor.- 10.6 Inverse Diffusion Across the Edge.- 10.7 Summary.- 10.8 Exercises.- 11 Texture Mapping, Matching Isometric Surfaces, and 3D Face Recognition.- 11.1 Flat Embedding.- 11.2 Texture Mapping.- 11.3 Isometric Signatures for Surfaces.- 11.4 Face Recognition.- 11.5 Exercises.- 12 Solutions to Selected Problems.