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.

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∧ 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.