Variational Methods in Image Segmentation

This book contains both a synthesis and mathematical analysis of a wide set of algorithms and theories whose aim is the automatic segmen tation of digital images as well as the understanding of visual perception. A common formalism for these theories and algorithms is obtained in a variational form. Thank to this formalization, mathematical questions about the soundness of algorithms can be raised and answered. Perception theory has to deal with the complex interaction between regions and "edges" (or boundaries) in an image: in the variational seg mentation energies, "edge" terms compete with "region" terms in a way which is supposed to impose regularity on both regions and boundaries. This fact was an experimental guess in perception phenomenology and computer vision until it was proposed as a mathematical conjecture by Mumford and Shah. The third part of the book presents a unified presentation of the evi dences in favour of the conjecture. It is proved that the competition of one-dimensional and two-dimensional energy terms in a variational for mulation cannot create fractal-like behaviour for the edges. The proof of regularity for the edges of a segmentation constantly involves con cepts from geometric measure theory, which proves to be central in im age processing theory. The second part of the book provides a fast and self-contained presentation of the classical theory of rectifiable sets (the "edges") and unrectifiable sets ("fractals").

Résumé

"Well-written and very user-friendly. Each chapter is introduced by a brief summary of its aims... A very readable introduction to the mathematical theory of image segmentation."

--ZAA

Contenu
I. Modelisation.- 1. Edge detection and segmentation.- 2. Linear and nonlinear multiscale filtering.- 3. Region and edge growing methods.- 4. Variational theories of segmentation.- 5. The piecewise constant Mumford-Shah model: mathematical analysis.- II. Elements Of Geometric Measure Theory.- 6. Hausdorff measure.- 7. Covering lemmas in a metric space.- 8. Density properties.- 9. Tangency properties of regular subsets of ?N.- 10. Semicontinuity properties of Hausdorff measure.- 11. Rectiflable sets.- 12. Properties of regular and rectifiable 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.- 14. Small oscillation coverings&and the excision method.- 15. Density properties and existence theory&for the Mumford-Shah minimizers.- 16. Further properties of the minimizers:&covering the edge set with a single curve.- 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.