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 OsherSethian 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. Language: English