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

TL;DR

This paper explores advanced geometric measure theory techniques to improve image processing tasks like de-noising and segmentation, aiming to reduce artifacts and better identify patterns, textures, and occlusions.

Contribution

It introduces new regularization and fidelity functionals based on total curvature and currents, reducing artifacts and enhancing pattern detection in images.

Findings

Total curvature-based regularity avoids corner rounding and shrinking artifacts.

Adjusted fidelity with flat norm minimizes cusp shrinking.

Currents on Grassmann bundles help identify textures and occlusions.

Abstract

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