Aspects of 2D-Adaptive Fourier Decompositions
For researchers in signal and image processing, this work provides numerical algorithms and convergence rate estimates for 2D-AFDs, showing they outperform other Fourier-based methods in image reconstruction.
This paper studies numerical aspects of 2D-Adaptive Fourier Decompositions (AFDs) for image representation, developing algorithms and comparing five reconstruction methods. The 2D-AFD methods achieve optimal results among the Fourier category methods.
As a new type of series expansion, the so-called one-dimensional adaptive Fourier decomposition (AFD) and its variations (1D-AFDs) have effective applications in signal analysis and system identification. The 1D-AFDs have considerable influence to the rational approximation of one complex variable and phase retrieving problems, etc. In a recent paper, Qian developed 2D-AFDs for treating square images as the essential boundary of the 2-torus embedded into the space of two complex variables. This paper studies the numerical aspects of multi-dimensional AFDs, and in particular 2D-AFDs, which mainly include (i) Numerical algorithms of several types of 2D-AFDs in relation to image representation; (ii) Perform experiments for the algorithms with comparisons between 5 types of image reconstruction methods in the Fourier category; and (iii) New and sharper estimations for convergence rates of orthogonal greedy algorithm and pre-orthogonal greedy algorithm. The comparison shows that the 2D-AFD methods achieve optimal results among the others.