Nonlinear total variation based noise removal algorithms
Proceedings of the eleventh annual international conference of the Center for Nonlinear Studies on Experimental mathematics : computational issues in nonlinear science: computational issues in nonlinear science
Invariant geometric evolutions of surfaces and volumetric smoothing
SIAM Journal on Applied Mathematics
Curvelets and curvilinear integrals
Journal of Approximation Theory
Particle-based anisotropic surface meshing
ACM Transactions on Graphics (TOG) - SIGGRAPH 2013 Conference Proceedings
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We propose and study quantitative measures of smoothness f 驴 A(f) which are adapted to anisotropic features such as edges in images or shocks in PDE's. These quantities govern the rate of approximation by adaptive finite elements, when no constraint is imposed on the aspect ratio of the triangles, the simplest example being $A_{p}(f)=\|\sqrt{|\mathrm{det}(d^{2}f)|}\|_{L^{\tau}}$ which appears when approximating in the L p norm by piecewise linear elements when $\frac{1}{\tau}=\frac{1}{p}+1$ . The quantities A(f) are not semi-norms, and therefore cannot be used to define linear function spaces. We show that these quantities can be well defined by mollification when f has jump discontinuities along piecewise smooth curves. This motivates for using them in image processing as an alternative to the frequently used total variation semi-norm which does not account for the smoothness of the edges.