[Paper Review] Aspects of 2D-Adaptive Fourier Decompositions

You Gao, Tao Qian+2 more|arXiv (Cornell University)|Oct 24, 2017

Image and Signal Denoising Methods31 references3 citations

TL;DR

This paper introduces and numerically validates 2D-Adaptive Fourier Decomposition (2D-AFD) methods for image representation, focusing on Product AFD and 2D-Pre-OGA algorithms. It demonstrates that these methods achieve superior image reconstruction accuracy and faster convergence than traditional Fourier, greedy, and orthogonal matching pursuit techniques, with 2D-Pre-OGA yielding the best PSNR (30 dB) and structural similarity (MSSIM) results after 256 terms.

ABSTRACT

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.

Motivation & Objective

• To develop and implement numerical algorithms for 2D-Adaptive Fourier Decomposition (2D-AFD) on the 2-torus for image representation.
• To compare the performance of five image reconstruction methods—Fourier Decomposition (FD), Greedy Algorithm (GA), Orthogonal Greedy Algorithm (OGA), Product AFD, and 2D-Pre-OGA—using PSNR and MSSIM metrics.
• To derive new, sharper a posteriori error bounds for the Weak Orthogonal Greedy Algorithm (WOGA) and Weak Pre-Orthogonal Greedy Algorithm (WPre-OGA) in general Hilbert spaces.
• To establish that 2D-Pre-OGA achieves optimal convergence and reconstruction quality without requiring Gram-Schmidt orthogonalization.

Proposed method

• Proposes a numerical realization of 2D-AFD via Product AFD and 2D-Pre-OGA, implemented through Algorithms 1 and 3, respectively, for image data on the 2-torus.
• Employs the generalized backward shift transform to construct orthonormal systems from non-orthogonal dictionary elements (Szegő kernels), avoiding numerical instability from Gram-Schmidt.
• Uses the maximal selection principle (MSP) to iteratively select parameters $ a_k otin ar{ ext{D}} $ that maximize energy reduction at each step, ensuring fast convergence.
• Applies the Takenaka-Malmquist (TM) system as a complete orthonormal basis in $ H^2( ext{D}) $, with $ B_k(z) = rac{ ho_k}{1 - ar{a}_k z} imes ext{product of Blaschke factors} $, enabling rational orthogonal decomposition.
• Employs the Bhattacharyya distance for histogram comparison and PSNR/MSSIM for quantitative image quality evaluation across 256-term approximations.
• Derives new a posteriori error bounds for WOGA and WPre-OGA, incorporating real-time iteration data to improve upon standard a priori bounds.

Experimental results

Research questions

• RQ1 How do 2D-AFD methods, particularly Product AFD and 2D-Pre-OGA, compare to traditional Fourier, greedy, and orthogonal matching pursuit methods in image reconstruction quality?
• RQ2 Can 2D-Pre-OGA achieve faster convergence and better image fidelity than existing greedy-type algorithms without requiring Gram-Schmidt orthogonalization?
• RQ3 What are the theoretical error bounds for the weak orthogonal and pre-orthogonal greedy algorithms, and can they be improved using a posteriori information from iterations?
• RQ4 To what extent does the 2D-AFD framework inherit the fast energy-convergence and non-negative frequency properties of 1D-AFD in image processing?
• RQ5 How do PSNR and MSSIM values evolve across different numbers of terms (N = 1, 16, 64, 256) in 2D-AFD-based image approximations?

Key findings

• 2D-Pre-OGA achieves the highest PSNR of 30 dB after 256-term approximation, outperforming all other methods including Product AFD and OGA.
• The 2D-Pre-OGA method maintains the best structural similarity (MSSIM), with QA scores reaching 0.8763 at N = 256, significantly higher than GA (0.8107) and FD (0.8516).
• Product AFD and 2D-Pre-OGA show the most consistent improvement in Bhattacharyya distance, increasing from 0.6376 at N=1 to 0.8456 and 0.8763 at N=256, respectively.
• The 2D-Pre-OGA algorithm achieves $ t = 1 $ in the weak type, indicating optimal convergence behavior, unlike GA and OGA, which fail to reach this threshold.
• The new a posteriori error bounds for WOGA and WPre-OGA are sharper and more accurate than standard a priori bounds, as they incorporate real-time iteration data.
• GA produces blurred reconstructions even at 256 terms, indicating poor convergence for signals with decentralized energy or singularities, while 2D-AFD methods preserve fine image details effectively.