A stable algorithm is proposed for image restoration based on the “mean curvature motion” equation. Existence and uniqueness of the “viscosity” solution of the equation are proved, a $L^\infty $ stable algorithm is given, experimental results are shown, and the subjacent vision model is compared with those introduced recently by several vision researchers. The algorithm presented appears to be the sharpest possible among the multiscale image smoothing methods preserving uniqueness and stability.

The authors define a new class of filters for noise elimination and edge enhancement by using shock filters and anisotropic diffusion. Some nonlinear partial differential equations used as models for these filters are studied The authors develop recursive and unconditional stable schemes which drastically reduce the computational effort of the algorithms. A new fast recursive approach to linear Gaussian filters is also shown by using the heat equation.

We introduce new results connecting differential and morphological operators that provide a formal and theoretically grounded approach for stable and fast contour evolution. Contour evolution algorithms have been extensively used for boundary detection and tracking in computer vision. The standard solution based on partial differential equations and level-sets requires the use of numerical methods of integration that are costly computationally and may have stability issues. We present a morphological approach to contour evolution based on a new curvature morphological operator valid for surfaces of any dimension. We approximate the numerical solution of the curve evolution PDE by the successive application of a set of morphological operators defined on a binary level-set and with equivalent infinitesimal behavior. These operators are very fast, do not suffer numerical stability issues, and do not degrade the level set function, so there is no need to reinitialize it. Moreover, their implementation is much easier since they do not require the use of sophisticated numerical algorithms. We validate the approach providing a morphological implementation of the geodesic active contours, the active contours without borders, and turbopixels. In the experiments conducted, the morphological implementations converge to solutions equivalent to those achieved by traditional numerical solutions, but with significant gains in simplicity, speed, and stability.