CoMeDiC

Discrete exterior calculus has emerged in the last decade as a powerful framework for solving discrete variational problems in image and geometry processing. It simplifies both the formulation of variational problems and their numerical resolution, and is able to extract global optima in many cases. However nothing guarantees that, on digital data like 2D or 3D images, digital curves and surfaces, it approaches the expected result of standard calculus, even when refining the discrete domain toward the limit continuous domain. The CoMeDiC project aims at filling the gap between discrete calculus and standard calculus for subsets of the digital space Z^n. The general idea is to define well-chosen metrics for discrete calculus that make it converge toward continuous values. This approach is now possible due to recent advances in digital geometry on multigrid convergent estimators. Digital calculus then addresses variational problems involving domains such as digital surfaces, curves, graphs living in a higher dimensional ambient space, as well as problems involving discontinuities or subtle boundary conditions. This project addresses theoretical problems like the definition of a sound digital calculus, the study of appropriate estimators for metrics, the statement of convergence properties. It is also concerned with its efficient numerical implementation. It studies also variational problems that present difficulties to standard numerical methods, like problems with discontinuities or free boundaries, or problems involving domains of codimension greater or equal to one as surfaces or curves. Last, this project focuses on three domains of application for digital calculus --- image analysis, digital geometry processing and shape optimisation --- both to guide and nourish theoretical developments, as well as to serve as testbed for digital calculus.