[Paper Review] Riesz transform associated with the fractional Fourier transform and applications in image edge detection

Zunwei Fu, Loukas Grafakos+3 more|arXiv (Cornell University)|Nov 7, 2021

Mathematical Analysis and Transform Methods4 citations

TL;DR

This paper introduces the fractional Riesz transform as a generalization of the classical Riesz transform via the fractional Fourier transform (FRFT), enabling directional edge detection in images. By controlling the fractional order parameters, the method simultaneously extracts amplitude, phase, and directional information, achieving efficient computation through FRFT-based fast algorithms and demonstrating superior performance in multi-directional edge detection on test images like Lena.

ABSTRACT

The fractional Hilbert transform was introduced by Zayed [30, Zayed, 1998] and has been widely used in signal processing. In view of is connection with the fractional Fourier transform, Chen, the first, second and fourth authors of this paper in [6, Chen et al., 2021] studied the fractional Hilbert transform and other fractional multiplier operators on the real line. The present paper is concerned with a natural extension of the fractional Hilbert transform to higher dimensions: this extension is the fractional Riesz transform which is defined by multiplication which a suitable chirp function on the fractional Fourier transform side. In addition to a thorough study of the fractional Riesz transforms, in this work we also investigate the boundedness of singular integral operators with chirp functions on rotation invariant spaces, chirp Hardy spaces and their relation to chirp BMO spaces, as well as applications of the theory of fractional multipliers in partial differential equations. Through numerical simulation, we provide physical and geometric interpretations of high-dimensional fractional multipliers. Finally, we present an application of the fractional Riesz transforms in edge detection which verifies a hypothesis insinuated in [26, Xu et al., 2016]. In fact our numerical implementation confirms that amplitude, phase, and direction information can be simultaneously extracted by controlling the order of the fractional Riesz transform.

Motivation & Objective

• To extend the Hilbert transform to higher dimensions using the fractional Fourier transform, leading to the fractional Riesz transform.
• To establish the theoretical framework for chirp singular integral operators, chirp Hardy spaces, and chirp BMO spaces.
• To develop a computationally efficient algorithm for edge detection by leveraging the FRFT and fractional multiplier theorem.
• To validate the hypothesis that controlling the fractional order enables simultaneous extraction of amplitude, phase, and directional features in image processing.
• To provide numerical and geometric interpretations of high-dimensional fractional multipliers through simulation.

Proposed method

• The fractional Riesz transform is defined via multiplication by a chirp function in the fractional Fourier transform domain, generalizing the classical Riesz transform.
• Theoretical analysis employs the fractional multiplier theorem to express the transform as a composition of FRFT, a multiplier, and inverse FRFT, enabling fast computation.
• Chirp singular integral operators are studied on rotation-invariant spaces, chirp Hardy spaces, and their duals, with boundedness established using harmonic analysis techniques.
• Edge detection is implemented using the form (6.3) and (6.4), which reduce computational complexity compared to direct integration (6.1) and (6.2).
• Numerical simulations use the Lena image with thresholded binarization to highlight directional edge extraction across varying fractional orders.
• The method dynamically adjusts the fractional orders (α₁, α₂) to extract features in lateral, longitudinal, main diagonal, and anti-diagonal directions.

Experimental results

Research questions

• RQ1 Can the fractional Riesz transform be constructed as a natural extension of the fractional Hilbert transform in higher dimensions using the fractional Fourier transform?
• RQ2 How do chirp singular integral operators behave on chirp Hardy spaces and their duals, and what boundedness properties do they exhibit?
• RQ3 Can the fractional Riesz transform efficiently extract amplitude, phase, and directional features from images by tuning its order parameters?
• RQ4 What is the computational advantage of expressing the fractional Riesz transform via FRFT-based composition compared to direct integration?
• RQ5 Does numerical implementation confirm that directional edge features can be selectively extracted by controlling the fractional order?

Key findings

• The fractional Riesz transform enables simultaneous extraction of amplitude, phase, and directional information in images by adjusting the fractional order parameters α₁ and α₂.
• When α₁ = π/2 and α₂ is varied, lateral-directional features (up/down) are extracted, demonstrating directional sensitivity.
• With α₂ = π/2 and α₁ varied, longitudinal-directional features (right/left) are selectively extracted, confirming directional control.
• Simultaneous adjustment of α₁ and α₂ increasing or decreasing from π/2 extracts features along the main diagonal and anti-diagonal, respectively.
• The FRFT-based formulation (6.3) and (6.4) reduces computational complexity compared to direct integration (6.1) and (6.2), enabling faster edge detection.
• Numerical results on the Lena image confirm that edge detection performance improves with directional tuning, validating the hypothesis from Xu et al. (2016).