Variational Methods in Image Segmentation

I. Modelisation.- 1. Edge detection and segmentation.- 1. Is there a general theory for segmentation?.- 2. Principles of multiscale analysis.- 2. Linear and nonlinear multiscale filtering.- 1. The linear model (Hildreth, Marr, Witkin, Koenderink,...).- 2. The Perona and Malik theory.- 3. Anisotropic diffusion and "mean curvature motion".- 4. Image restoration by mean curvature motion.- 3. Region and edge growing methods.- 1. Formalization of region growing methods. (Brice and Fennema, Pavlidis, ...).- 2. Edge growing methods.- 3. Hybrid methods.- 4. How to translate a heuristic algorithm into a variational theory.- 4. Variational theories of segmentation.- 1. The Nitzberg and Mumford 2.1-D sketch.- 2. The "structural saliency" of Sha'ashua and Ullman.- 3. The "minimum description length" method of Yvon Leclerc.- 4. Algorithms for minimizing the general Mumford and Shah energy.- 5. Affine invariant segmentation.- 5. The piecewise constant Mumford-Shah model: mathematical analysis.- 1. Main theorem.- 2. Definition and elementary properties of segmentations.- 3. Properties of segmentations obtained by merging. Proof of the main theorem.- 4. A pyramidal algorithm computing 2-normal segmentations.- 5. Examples.- II. Elements Of Geometric Measure Theory.- 6. Hausdorff measure.- 1. Hausdorff outer measure.- 2. Outer measures in topological spaces.- 3. Connected sets of finite 1-dimensional Hausdorff measure.- 4. A suggesting example of Hausdorff measurable set with finite length.- 7. Covering lemmas in a metric space.- 1. Elementary properties of generic coverings.- 2. Vitali coverings.- 3. Hausdorff and Lebesgue measure.- 8. Density properties.- 1. Upper and lower density.- 2. Regular sets.- 9. Tangency properties of regular subsets of ?N.- 1. Reflection lemmas.- 2. Approximate tangent space.- 3. Existence of a tangent space.- 10. Semicontinuity properties of Hausdorff measure.- 1. Hausdorff distance.- 2. Counterexamples to the semicontinuity of the Hausdorff measure for the Hausdorff metric.- 3. Upper semicontinuity in [0,1].- 4. Lower semicontinuity of the Hausdorff measure on uniformly concentrated sets.- 5. Sequences of compact connected 1-sets.- 11. Rectiflable sets.- 1. Rectifiable surfaces and rectifiable sets.- 2. Regularity of rectifiable sets.- 3. Rectifiable curves.- 12. Properties of regular and rectifiable sets.- 1. Geometric lemmas.- 2. Equivalence of the notions of regularity and rectifiability.- 3. Projection properties of irregular sets.- III. Existence and Structural Properties of the Minimal Segmentations ror the Mumford-Shah Model.- 13. Properties of the approximating image&&&in the Mumford-Shah model.- 1. Notation and preliminary lemmas.- 2. Green Formulas.- 3. Interior regularity estimates.- 14. Small oscillation coverings&&∧ the excision method.- 1. The excision method.- 2. Small oscillation coverings.- 3. The excision lemmas.- 4. The Interpolation Lemma.- 5. Energy jump estimates.- 6. Proof of the excision lemmas.- 15. Density properties and existence theory&&&for the Mumford-Shah minimizers.- 1. The Essentiality Property.- 2. The Uniform Density Property.- 3. The Concentration Property.- 4. The Projection Properties.- 5. Existence theory for the Mumford-Shah minimizers.- 6. Nonuniqueness of the Mumford-Shah minimizers.- 16. Further properties of the minimizers:&&&covering the edge set with a single curve.- 1. The David-Semmes Concentrated Rectifiability Property.- 2. Dual curve lemmas.- 3. Existence of a rectifiable curve containing the edge set.- 4. Existence of a regular curve containing the edge set.- Bibliographical notes.- References References of Part I.- I-A) Image segmentation and edge detection, surveys and monographs..- I-B) Articles proposing algorithms for edge detection and image segmentation..- I-C) Scale space theory..- I-D) Mathematical analysis related to scale space theory..- I-E) Monographs in Image Processing..- I-F) Articles on texture analysis and segmentation..- I-G) Wavelets, theory and relation to image processing and scale space..- I-H) Related topics in psychophysics, neurobiology and gestalt theory..- References of Part II 232.- II-A) References on geometric measure theory and rectifiability.- II-B) Monographs on mathematics and geometric measure theory..- References of Part III:Mathematical analysis of the Mumford-Shah model.- Index of segmentation algorithms.- Notation.