4. Curve evolution and level sets

4.1 From curve operators to function operators and vice versa 4.1.1 Signed distance function and supporting function4.1.2 Monotone and translation invariant operators4.1.3 Level sets and their properties4.1.4 Extension of sets operators to functions operators4.1.5 Characterization of monotone, contrast invariant operator4.1.6 Asymptotic behavior of morphological operators4.1.7 Morphological operators yield PDEs4.2 Curve evolution and Scale Space theory 4.2.1 Multiscale analysis are given by PDEs4.2.2 Morphological scale space4.3 Viscosity solutions 4.3.1 Definition of viscosity solution4.3.2 Proof of uniqueness: the maximum principle4.3.3 Existence of solution by Perron’s Method4.3.4 Contrast invariance of level sets flow4.3.5 Viscosity solutions shorten level lines4.4 Morphological operators and viscosity solution 4.4.1 Median filter and mean curvature motion4.4.2 Affine invariant schemes4.5 Conclusions4.6 Curvature thresholding 4.6.1 Viscosity approach4.6.2 Opening and closing4.7 Alternative weak solutions of curve evolution 4.7.1 Brakke’s varifold solution4.7.2 Reaction diffusion approximation4.7.3 Minimal barriers4.8 Bibliographical notes