Compensated convexity on bounded domains, mixed Moreau envelopes and computational methods

Compensated convex transforms have been introduced for extended real valued functions defined over Rn. In their application to image processing, interpolation and shape interrogation, where one deals with functions defined over a bounded domain, one was making the implicit assumption that the function coincides with its transform at the boundary of the data domain. In this paper, we introduce local compensated convex transforms for functions defined in bounded open convex subsets Ω of Rn by making specific extensions of the function to the whole space, and establish their relations to globally defined compensated convex transforms via the mixed critical Moreau envelopes. We find that the compensated convex transforms of such extensions coincide with the local compensated convex transforms in the closure of Ω. We also propose a numerical scheme for computing Moreau envelopes, establishing convergence of the scheme with the rate of convergence depending on the regularity of the original function. We give an estimate of the number of iterations needed for computing the discrete Moreau envelope. We then apply the local compensated convex transforms to image processing and shape interrogation. Our results are compared with those obtained by using schemes based on computing the convex envelope from the original definition of compensated convex transforms

Logarithmic Curvature Graph as a Shape Interrogation Tool

Abstract A compact formula for Logarithmic Curvature Graph(LCG) and its gradient for planar curves has been shown which can be used as shape interrogation tool. Using these entities, the mathematical definition for a curve to be aesthetic has been introduced to overcome the ambiguity that occurs in measuring the aesthetic value of a curve. Detailed examples are shown how LCG and its gradient can be used to identify curvature extrema and measure the aesthetic value of curves. Mathematics Subject Classification: 65D17, 68U0

Feature based volumes for implicit intersections.

The automatic generation of volumes bounding the intersection of two implicit surfaces (isosurfaces of real functions of 3D point coordinates) or feature based volumes (FBV) is presented. Such FBVs are defined by constructive operations, function normalization and offsetting. By applying various offset operations to the intersection of two surfaces, we can obtain variations in the shape of an FBV. The resulting volume can be used as a boundary for blending operations applied to two corresponding volumes, and also for visualization of feature curves and modeling of surface based structures including microstructures