Notes on Geometric Measure Theory Applications to Image Processing; De-noising, Segmentation, Pattern, Texture, Lines, Gestalt and Occlusion

摘要

Regularization functionals that lower level set boundary length when usedwith L^1 fidelity functionals on signal de-noising on images create artifacts.These are (i) rounding of corners, (ii) shrinking of radii, (iii) shrinking ofcusps, and (iv) non-smoothing of staircasing. Regularity functionals based upontotal curvature of level set boundaries do not create artifacts (i) and (ii).An adjusted fidelity term based on the flat norm on the current (adistributional graph) representing the density of curvature of level setsboundaries can minimize (iii) by weighting the position of a cusp. A regularityterm to eliminate staircasing can be based upon the mass of the currentrepresenting the graph of an image function or its second derivatives.Densities on the Grassmann bundle of the Grassmann bundle of the ambient spaceof the graph can be used to identify patterns, textures, occlusion and lines.