Variational Restoration of Nonflat Image Features: Models and Algorithms

Abstract

We develop both mathematical models and computational algorithms for variational denoising and restoration of nonflat image features. Nonflat image features are those that live on Riemannian manifolds, instead of on the Euclidean spaces. Familiar examples include the orientation feature (from optical flows or gradient flows) that lives on the unit circle S1 , the alignment feature (from fingerprint waves or certain texture images) that lives on the real projective line $\mathbb{RP}^1$, and the chromaticity feature (from color images) that lives on the unit sphere S2 . In this paper, we apply the variational method to denoise and restore general nonflat image features. Mathematical models for both continuous image domains and discrete domains (or graphs) are constructed. Riemannian objects such as metric, distance and Levi--Civita connection play important roles in the models. Computational algorithms are also developed for the resulting nonlinear equations. The mathematical framework can be applied to restoring general nonflat data outside the scope of image processing and computer vision.