A Novel Two-Dimensional Unwinding Decomposition for Image Signals

Abstract

This paper presents a new two-dimensional (2D) signal analysis method, namely partial unwinding adaptive Fourier decomposition. Because of its efficiency and accuracy it has great ability in practice. Besides theory and algorithm we provide the application in image reconstruction. One-dimensional (1D) unwinding adaptive Fourier decomposition has been studied. It depends on Nevanlinna factorization and a maximal selection. The 1D method leads to a fast convergence nonlinear phase decomposition. However, for a multi-dimensional ( $m\text{D}$ ) case, the usual factorization fails. Consequently, $m\text{D}$ signals have no genuine unwinding decomposition. This work aims to propose a 2D partial unwinding adaptive Fourier decomposition that depends on the basic mathematical operations reducing the 2D case to the 1D case. Its decomposing terms are orthogonal to each other and result in 2D-nonlinear phase decomposition. Its fast convergence gives rise to efficient image reconstruction. A number of experiments illustrate that our proposed 2D-PUAFD exhibits the best reconstruction performance among all the tested methods.

Keywords

• Fourier transform
• unwinding adaptive Fourier decomposition
• maximal selection
• Nevanlinna factorization
• Blaschke product
• image reconstruction