2D Partial Unwinding—A Novel Non-Linear Phase Decomposition of Images

Abstract

This paper aims at proposing a novel 2D non-linear phase decomposition of images, which performs the image processing tasks better than the traditional Fourier transformation (linear phase decomposition), but further, it has additional mathematical properties allowing more effective image analysis, including adaptive decomposition components and positive instantaneous phase derivatives. 1D unwinding Blaschke decomposition has recently been proposed and studied. Through factorization it expresses arbitrary 1D signal into an infinite linear combination of Blaschke products. It offers fast converging positive frequency decomposition in the form of rational approximation. However, in the multi-dimensional cases, the usual factorization mechanism does not work. As a consequence, there is no genuine unwinding decomposition for multi-dimensions. In this paper, a 2D partial unwinding decomposition based on algebraic transforms reducing multi-dimensions to the 1D case is proposed and analyzed. The result shows that the fast convergence offers efficient image reconstruction. The tensor type decomposing terms are mutually orthogonal, giving rise to 2D positive frequency decomposition. The comparison results show that the proposed method outperforms the standard greedy algorithm and the most commonly used methods in the Fourier category. An application in watermarking is presented to demonstrate its potential in applications.