[Paper Review] Two-Dimensional Adaptive Fourier Decomposition

Tao Qian|arXiv (Cornell University)|Jun 9, 2014

Image and Signal Denoising Methods18 references3 citations

TL;DR

This paper introduces a two-dimensional adaptive Fourier decomposition (2D-AFD) using multivariate complex Hardy spaces, proposing two approaches: Product-TM Systems and Product-Szegö Dictionaries. It introduces the Pre-Orthogonal Greedy Algorithm (P-OGA), proving its convergence and faster convergence rate than Orthogonal Greedy Algorithm (OGA), with theoretical guarantees for convergence in the $H^2$ space on the 2-torus using an induced complete dictionary.

ABSTRACT

One-dimensional adaptive Fourier decomposition, abbreviated as 1-D AFD, or AFD, is an adaptive representation of a physically realizable signal into a linear combination of parameterized Szegö and higher order Szegö kernels of the context. In the present paper we study multi-dimensional AFDs based on multivariate complex Hardy spaces theory. We proceed with two approaches of which one uses Product-TM Systems; and the other uses Product-Szegö Dictionaries. With the Product-TM Systems approach we prove that at each selection of a pair of parameters the maximal energy may be attained, and, accordingly, we prove the convergence. With the Product-Szegö dictionary approach we show that Pure Greedy Algorithm is applicable. We next introduce a new type of greedy algorithm, called Pre-Orthogonal Greedy Algorithm (P-OGA). We prove its convergence and convergence rate estimation, allowing a weak type version of P-OGA as well. The convergence rate estimation of the proposed P-OGA evidences its advantage over Orthogonal Greedy Algorithm (OGA). In the last part we analyze P-OGA in depth and introduce the concept P-OGA-Induced Complete Dictionary, abbreviated as Complete Dictionary . We show that with the Complete Dictionary P-OGA is applicable to the Hardy $H^2$ space on $2$-torus.

Motivation & Objective

• To extend one-dimensional adaptive Fourier decomposition (1-D AFD) to two dimensions using multivariate complex Hardy space theory.
• To develop a stable, adaptive signal representation method for 2D signals with positive instantaneous frequency and improved convergence.
• To introduce and analyze a new greedy algorithm, Pre-Orthogonal Greedy Algorithm (P-OGA), for efficient decomposition in multidimensional Hardy spaces.
• To establish a theoretical framework for convergence and convergence rate estimation in 2D settings, particularly in $H^2$ space on the 2-torus.

Proposed method

• Uses Product-TM Systems to construct a rational orthogonal system in the product space of two unit discs, enabling maximal energy selection at each step.
• Employs Product-Szegö Dictionaries based on parameterized Szegö kernels and their higher-order derivatives for greedy selection.
• Introduces the Pre-Orthogonal Greedy Algorithm (P-OGA), which selects atoms based on inner products with a pre-orthogonalized remainder, ensuring convergence.
• Defines the P-OGA-Induced Complete Dictionary as a system formed by orthogonalizing all finite linear combinations of directional derivatives of Szegö kernels, enabling P-OGA applicability in $H^2$ space.
• Applies the Plemelj formula and Hilbert transform theory in the context of complex Hardy spaces to derive boundary limit representations and inner product identities.
• Uses $L^2$-normalized kernels and reproducing kernel properties to derive convergence conditions, particularly as parameters approach the boundary ($|a| \to 1$ or $|b| \to 1$).

Experimental results

Research questions

• RQ1 Can 1-D AFD be generalized to two dimensions using multivariate complex Hardy space theory?
• RQ2 Does the Pre-Orthogonal Greedy Algorithm (P-OGA) achieve faster convergence than the Orthogonal Greedy Algorithm (OGA) in 2D settings?
• RQ3 Can a complete dictionary be induced from the Szegö dictionary to ensure P-OGA convergence in $H^2$ space on the 2-torus?
• RQ4 What is the theoretical convergence rate of P-OGA in two-dimensional Hardy spaces, and how does it compare to OGA?
• RQ5 How do Product-TM Systems and Product-Szegö Dictionaries compare in terms of energy maximization and convergence in 2D AFD?

Key findings

• The P-OGA converges in the $H^2$ space on the 2-torus when applied with the P-OGA-Induced Complete Dictionary, ensuring stable and adaptive decomposition.
• The convergence rate of P-OGA is faster than that of the Orthogonal Greedy Algorithm (OGA), as evidenced by theoretical estimation and comparison.
• The maximal energy is attained at each step in the Product-TM Systems approach, leading to guaranteed convergence of the decomposition process.
• The P-OGA-Induced Complete Dictionary is formed by orthogonalizing all finite linear combinations of directional derivatives of Szegö kernels, enabling systematic and stable greedy selection.
• As $|a| \to 1$ or $|b| \to 1$, the $L^2$-norm of the remainder after projection tends to zero, confirming the convergence of the greedy selection process.
• In the one-dimensional case, the P-OGA under the Complete Szegö Dictionary reduces exactly to classical 1-D AFD, validating consistency with the established theory.