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Refining Image Edge Detection via Linear Canonical Riesz Transforms

Shuhui Yang, Zunwei Fu, Dachun Yang, Yan Lin, Zhen Li

TL;DR

This work introduces the linear canonical Riesz transform (LCRT), a linear canonical multiplier that combines the linear canonical transform (LCT) with a Riesz-type kernel to analyze non-stationary images. It proves the LCRT decomposes as the LCT, a multipliers in the frequency domain, and the inverse LCT, and establishes its $L^p$-boundedness. A new edge-sharpness measure $R^{\rm E}_{\rm sc}$ and the LCRT-based edge-detection method (LCRT-IED) are developed, with theoretical support (Theorem p-sharp) and demonstrations showing controllable edge strength/continuity and improved local feature preservation in grayscale and RGB images. The results indicate that LCRT offers reduced computational complexity, directionally flexible edge processing, and enhanced feature extraction, making it a promising tool for image matching and refinement tasks.

Abstract

Combining the linear canonical transform and the Riesz transform, we introduce the linear canonical Riesz transform (for short, LCRT), which is further proved to be a linear canonical multiplier. Using this LCRT multiplier, we conduct numerical simulations on images. Notably, the LCRT multiplier significantly reduces the complexity of the algorithm. Based on these we introduce the new concept of the sharpness $R^{\rm E}_{\rm sc}$ of the edge strength and continuity of images associated with the LCRT and, using it, we propose a new LCRT image edge detection method (for short, LCRT-IED method) and provide its mathematical foundation. Our experiments indicate that this sharpness $R^{\rm E}_{\rm sc}$ characterizes the macroscopic trend of edge variations of the image under consideration, while this new LCRT-IED method not only controls the overall edge strength and continuity of the image, but also excels in feature extraction in some local regions. These highlight the fundamental differences between the LCRT and the Riesz transform, which are precisely due to the multiparameter of the former. This new LCRT-IED method might be of significant importance for image feature extraction, image matching, and image refinement.

Refining Image Edge Detection via Linear Canonical Riesz Transforms

TL;DR

This work introduces the linear canonical Riesz transform (LCRT), a linear canonical multiplier that combines the linear canonical transform (LCT) with a Riesz-type kernel to analyze non-stationary images. It proves the LCRT decomposes as the LCT, a multipliers in the frequency domain, and the inverse LCT, and establishes its -boundedness. A new edge-sharpness measure and the LCRT-based edge-detection method (LCRT-IED) are developed, with theoretical support (Theorem p-sharp) and demonstrations showing controllable edge strength/continuity and improved local feature preservation in grayscale and RGB images. The results indicate that LCRT offers reduced computational complexity, directionally flexible edge processing, and enhanced feature extraction, making it a promising tool for image matching and refinement tasks.

Abstract

Combining the linear canonical transform and the Riesz transform, we introduce the linear canonical Riesz transform (for short, LCRT), which is further proved to be a linear canonical multiplier. Using this LCRT multiplier, we conduct numerical simulations on images. Notably, the LCRT multiplier significantly reduces the complexity of the algorithm. Based on these we introduce the new concept of the sharpness of the edge strength and continuity of images associated with the LCRT and, using it, we propose a new LCRT image edge detection method (for short, LCRT-IED method) and provide its mathematical foundation. Our experiments indicate that this sharpness characterizes the macroscopic trend of edge variations of the image under consideration, while this new LCRT-IED method not only controls the overall edge strength and continuity of the image, but also excels in feature extraction in some local regions. These highlight the fundamental differences between the LCRT and the Riesz transform, which are precisely due to the multiparameter of the former. This new LCRT-IED method might be of significant importance for image feature extraction, image matching, and image refinement.

Paper Structure

This paper contains 7 sections, 4 theorems, 47 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Linear Canonical Riesz Transforms
  3. Numerical Simulations
  4. Applications
  5. Refining Image Edge Detection by Using $R^{\rm E}_{\rm sc}$
  6. Advantages and Features
  7. Conclusions

Key Result

Theorem 2.7

Let $k,j\in\{1,\ldots,n\}$ and $\boldsymbol A:=(A_1,\ldots ,A_n)$ with $A_k:=\in{M_{2\times2}}(\mathbb{R})$ and both $b_k\ne0$ and $a_k=d_k$ for all $k$, and let $\boldsymbol{a}:=(a_1,\ldots,a_n)$, $\boldsymbol{b}:=(b_1,\ldots,b_n)$, and $\boldsymbol{d}:=(d_1,\ldots,d_n)$. From the perspective of th where $\frac{\boldsymbol{\omega}}{\boldsymbol b}:= (\frac{\omega_1}{b_1},\ldots,\frac{\omega_n}{b_n

Figures (13)

  • Figure 1: The decomposition of the LCT $\mathscr{L}_{\boldsymbol A}$.
  • Figure 2: The decomposition of the $j$th LCRT $R_j^{\boldsymbol A}f$.
  • Figure 3: Gaussian function as a test image.
  • Figure 4: 2D and 3D images of the test image after applying LCRT with different parameters are shown. Graphs (a) and (b), (c) and (d), and (d) and (e) correspond to LCRT parameters ${\boldsymbol A}$, ${\boldsymbol B}$, and ${\boldsymbol C}$, respectively.
  • Figure 5: The amplitude, the real part, and the imaginary part of the original test image in the LCT domain, and those of the original image after applying LCRTs in the LCT domain with the corresponding parameter ${\boldsymbol A}$.
  • ...and 8 more figures

Theorems & Definitions (24)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Remark 1.6
  • Definition 1.7
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • ...and 14 more