\contentsline {chapter}{Introduction.}{1}
\contentsline {chapter}{\numberline {1}Mathematical formalization and fundamental equations of the visual pyramid.}{3}
\contentsline {section}{\numberline {1.1}A short story of the subject. }{6}
\contentsline {section}{\numberline {1.2}The visual pyramid.}{6}
\contentsline {subsection}{\numberline {1.2.1}What is the {\pit input} of the visual pyramid ?}{7}
\contentsline {subsection}{\numberline {1.2.2}What is the {\pit output} of the visual pyramid ?}{8}
\contentsline {subsection}{\numberline {1.2.3}What basic principles must obey the visual pyramid ?}{8}
\contentsline {subsubsection}{The causality.}{8}
\contentsline {subsubsection}{Morphological and affine invariance.}{10}
\contentsline {section}{\numberline {1.3}Axiomatization of multiscale shape and image analysis and classification of the models depending on their fundamental equation.}{11}
\contentsline {subsection}{\numberline {1.3.1}Multiscale image analysis.}{11}
\contentsline {subsection}{\numberline {1.3.2}Axiomatization of multiscale shape analysis.}{13}
\contentsline {section}{\numberline {1.4}Multiscale segmentation.}{15}
\contentsline {section}{\numberline {1.5}An example : the discussion of a psychophysical theory, the Texton Theory.}{16}
\contentsline {chapter}{A. The ``fundamental theorem" of image analysis.}{}
\contentsline {chapter}{B. Affine Invariant Mathematical Morphology.}{}
\contentsline {chapter}{\numberline {2} A multiscale algorithm for image segmentation by variational method.}{31}
\contentsline {paragraph}{Abstract.}{31}
\contentsline {section}{\numberline {2.1}Introduction.}{31}
\contentsline {section}{\numberline {2.2}General principles of segmentation devices.}{32}
\contentsline {subsection}{\numberline {2.2.1}Formalization.}{32}
\contentsline {subsection}{\numberline {2.2.2}Choice of properties and algorithms for segmentation.}{34}
\contentsline {section}{\numberline {2.3}Compactness properties of segmentations obtained by ``merging''.}{36}
\contentsline {subsection}{\numberline {2.3.1}Topological properties of affine 1-normal segmentations.}{38}
\contentsline {subsection}{\numberline {2.3.2}Estimates on segmentations obtained by merging.}{39}
\contentsline {section}{\numberline {2.4}A pyramidal algorithm constructing 2-normal affine segmentations.}{42}
\contentsline {subsection}{\numberline {2.4.1}Description of the algorithm.}{43}
\contentsline {paragraph}{The criterion.}{43}
\contentsline {paragraph}{The algorithm.}{44}
\contentsline {paragraph}{Experimental results.}{44}
\contentsline {section}{\numberline {2.5}Application to texture discrimination.}{47}
\contentsline {section}{\numberline {2.6}Conclusion}{48}
\contentsline {chapter}{\numberline {3}Axiomatization of shape analysis and application to texture hyperdiscrimination.}{57}
\contentsline {section}{\numberline {3.1}Introduction.}{57}
\contentsline {section}{\numberline {3.2}The four principles of Multiscale Shape Analysis.}{58}
\contentsline {section}{\numberline {3.3}The fundamental equation of Shape Analysis.}{61}
\contentsline {section}{\numberline {3.4}The fundamental equation of Image Processing.}{63}
\contentsline {section}{\numberline {3.5}The "Texton Density" Doctrine of Julesz.}{64}
\contentsline {section}{\numberline {3.6}How to define Texton Densities. An Axiomatic Approach.}{66}
\contentsline {section}{\numberline {3.7}Experimental verification of Julesz Theory.}{67}
\contentsline {section}{\numberline {3.8}Conclusion.}{68}
\contentsline {chapter}{A. Proof of the renormalization lemma.}{}
\contentsline {chapter}{\numberline {4}Axiomatization of shape analysis and application to texture hyperdiscrimination.}{83}
\contentsline {section}{\numberline {4.1}Julesz axioms.}{83}
\contentsline {section}{\numberline {4.2}Texton formalization}{84}
\contentsline {section}{\numberline {4.3}The fundamental equations of shape and image analysis.}{86}
\contentsline {section}{\numberline {4.4}Experimental confirmation of Julesz Theory.}{87}
\contentsline {chapter}{\numberline {5}Numerical Analysis of the fundamental equation of image processing.}{91}
\contentsline {section}{\numberline {5.1}Introduction.}{92}
\contentsline {section}{\numberline {5.2}The four principles of Multiscale Shape Analysis and the fundamental equation.}{95}
\contentsline {subsection}{\numberline {5.2.1}The fundamental equation of Shape Analysis.}{97}
\contentsline {subsection}{\numberline {5.2.2}The fundamental equation of Image Processing.}{97}
\contentsline {subsection}{\numberline {5.2.3}Invariance and stability requirements for numerical schemes of the fundamental equation.}{98}
\contentsline {section}{\numberline {5.3}Quasilinear finite difference schemes.}{99}
\contentsline {subsection}{\numberline {5.3.1}Luis Alvarez's scheme.}{100}
\contentsline {subsection}{\numberline {5.3.2}Luis Alvarez, Frederic Guichard's scheme.}{100}
\contentsline {section}{\numberline {5.4}Morphological (set evolution) schemes.}{102}
\contentsline {subsection}{\numberline {5.4.1}A theoretical scheme.}{102}
\contentsline {subsection}{\numberline {5.4.2}Iterated median filters and curvature motion.}{104}
\contentsline {section}{\numberline {5.5}Purely morphological (curve evolution) schemes.}{105}
\contentsline {subsection}{\numberline {5.5.1}Introduction.}{105}
\contentsline {subsection}{\numberline {5.5.2}Quasilinear intrinsic diffusion schemes.}{105}
\contentsline {subsection}{\numberline {5.5.3}A geometrical curve evolution scheme.}{107}
\contentsline {subsubsection}{Curve sampling problems.}{107}
\contentsline {subsubsection}{A geometrical scheme.}{107}
\contentsline {section}{\numberline {5.6}Experimental results.}{109}
\contentsline {subsection}{\numberline {5.6.1} The Alvarez anisotropic diffusion scheme.}{109}
\contentsline {subsection}{\numberline {5.6.2}The Alvarez-Guichard rotationally invariant finite difference scheme.}{110}
\contentsline {subsection}{\numberline {5.6.3}"Morphologization" of the Alvarez-Guichard scheme.}{111}
\contentsline {subsection}{\numberline {5.6.4}Set morphology on a grid.}{111}
\contentsline {subsection}{\numberline {5.6.5}Purely morphological (curve evolution) scheme.}{112}
\contentsline {section}{\numberline {5.7}Conclusion and conjectures.}{113}
\contentsline {chapter}{A. Appendix.}{}
