Numerical Analysis in Visual Computing What we can Learn from each Other

Visual computing is a wide area that includes computer graphics and image processing, where the “eye-ball norm” rules. This paper discusses two case studies involving numerical methods and analysis applied to this wide domain. The focus is on highlighting relative strengths in those intersection areas where there are many problems of interest both to numerical analysts and to researchers in the visual computing community. The first involves motion simulation and calibration of soft objects such as cloth, plants, and skin. The governing elastodynamics PDE system, discretized in space already at the variational level using co-rotated FEM, leads to a large, expensive to assemble, dynamical system in time, where the damped motion may mask highly oscillatory stiffness. Geometric integration and exponential time differencing ideas are making their way into visual computing research these days in search for more quantitative computations that are required for applications such as control and 3D printing. The other case study involves some image processing problems where there is a premium for local approaches that do not necessarily use underlying PDEs. The popular paradigm of solving a data fitting problem with a penalty term involving the gradient or higher derivatives of a function approximating the sought surface is employed in the visual community much less often than in the numerical analysis community. We explain why. Concepts are demonstrated and discussed.