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Fractional Calculus In Image Processing: A Review
Qi Yang, Dali Chen, Tiebiao Zhao, YangQuan Chen
TL;DR
The paper examines whether fractional-order methods offer useful flexibility for image processing beyond integer-order derivatives. It reviews definitions, discretization methods, and applications across ten sub-fields, reporting broad success and examples of improved registration behavior while identifying texture limitations in the integer-order ROF model.
Problem
The paper addresses the limited flexibility of integer-order image-processing methods by examining whether fractional-order derivatives provide an additional tunable order parameter.
Method
The paper reviews fractional-order definitions, discretization methods, Matlab toolboxes, and image-processing studies across ten sub-fields, with selected experimental results.
Results
The review concludes that fractional-order derivatives show broad success in image processing, with registration studies reporting more extracted key points and lower average deformation errors than cited conventional methods.
Takeaways & Limitations
Fractional-order image-processing methods provide an order parameter for optimizing processing quality across diverse tasks, including enhancement, denoising, segmentation, registration, fusion, and encryption.
Takeaways & Limitations
The integer-order ROF model can remove textures with noise and produce staircase effects because its model does not separate different oscillatory components.
Abstract
from arXiv · showhide
Over the last decade, it has been demonstrated that many systems in science and engineering can be modeled more accurately by fractional-order than integer-order derivatives, and many methods are developed to solve the problem of fractional systems. Due to the extra free parameter order, fractional-order based methods provide additional degree of freedom in optimization performance. Not surprisingly, many fractional-order based methods have been used in image processing field. Herein recent studies are reviewed in ten sub-fields, which include image enhancement, image denoising, image edge detection, image segmentation, image registration, image recognition, image fusion, image encryption, image compression and image restoration. In sum, it is well proved that as a fundamental mathematic tool, fractional-order derivative shows great success in image processing.
1. Introduction
Fractional calculus developed from a long-standing question about non-integer derivatives into a framework applied across science, engineering, and image processing. In image processing, the workflow generalizes selected differential or integral operators to fractional order and then discretizes them numerically.
- Background: Fractional-order derivatives emerged from a 300-year effort to generalize integer-order derivatives, later extending into mechanics, physics, control, and bioengineering.The historical discussion began with Leibniz and L’Hospital and was advanced by Liouville, Riemann, and Weyl.
- Workflow: Image-processing workflows first select an operator, model, or equation involving ordinary differentiation or integration.This selected formulation provides the starting point for fractional-order generalization.
- Workflow: The selected ordinary operator is generalized to arbitrary order using fractional-calculus definitions such as G-L or R-L.The supplied workflow passage introduces this as the second step of fractional-order image processing.
- Paper scope: The paper introduces fractional-order definitions, discretization methods, and Matlab toolboxes before reviewing fractional-order image-processing methods.The review section covers ten image-processing sub-fields and includes experimental data.
2. Basics Fact of Fractional-order Derivative
The paper presents several fractional-calculus definitions and numerical routes for image-processing computation. It emphasizes G-L and R-L formulations, their discretization through coefficient-based or matrix methods, and Matlab implementations for numerical solutions.
- Fractional-order definitions: G-L, R-L, and Caputo are three popular fractional-calculus definitions, with G-L and R-L most commonly used in digital image processing.The paper separately introduces the definitions and states the relative popularity of G-L and R-L in image processing.
- Fractional-order definitions: The three best-known definitions are equivalent under zero initial conditions.This equivalence is explicitly conditioned on zero initial conditions.
- G-L discretization: The G-L definition supports numerical fractional derivatives through weighted backward differences, with coefficients related to the polynomial expansion of (1 − z)^α.The discretization weights are given through a recurrence formula, and Matlab functions are cited for approximate solutions.
- Tool support: MatlabCentral provides functions for approximate G-L and matrix-based fractional-difference solutions.The cited functions include glfdiff(), fglderiv(a, y, h), and fracdiffdemoy(alpha, beta).
- Matrix discretization: Fractional derivatives can also be approximated on equidistant nodes using backward fractional differences and represented simultaneously in matrix form.The matrix formulation covers all N + 1 formulas and includes left- and right-sided differentiation.
- Comparison: The paper compares right-sided R-L and G-L derivatives for y(x) = 1 at fractional orders α = 0.2, 0.4, and 0.6.The comparison is identified as the content of Fig. 2.1.
3. A Review of Fractional-order Image Processing Methods
This section classifies and analyzes fractional-order image-processing methods, emphasizing image enhancement through directional and adaptive fractional differential operators. These methods generalize conventional operators to preserve contours while improving texture details.
- The review summarizes and classifies existing fractional-order image-processing methods and presents selected simulation results.
- Image Enhancement: Fractional differential operators can preserve high-frequency contours while improving low-frequency texture details in smooth image regions.
- Image Enhancement: Eight-direction fractional masks improve multi-directional gradient capture and anti-rotation capability compared with single-direction masks.
- Image Enhancement: An unsupervised optimization algorithm selects the fractional order for directional derivatives used in enhancement.
- Image Enhancement: Adaptive piecewise fractional differentiation uses higher order values at edge pixels and smaller orders in weak-texture pixels to clarify edges and preserve weak textures.
3.2. Image Denoising
Fractional-calculus denoising methods are organized into operator-mask, model-based, and anisotropic-diffusion approaches. They aim to reduce noise while retaining texture and edge information, addressing limitations of integer-order ROF denoising.
- Fractional-calculus denoising methods comprise operator-mask-based, model-based, and anisotropic-diffusion-equation-based categories.
- Operator-mask methods: Operator-mask methods use fractional differentiation or integration in symmetric n × n windows, commonly 3 × 3 or 5 × 5 pixels across eight directions.
- Operator-mask methods: These local methods can remove noise while preserving image texture and edge details, but achieving a global optimum is difficult.
- ROF limitations: The ROF integer-order model may produce staircase effects because it adopts a piecewise-constant solution.
- ROF limitations: ROF denoising can remove textures with noise because it models noise and textures as one oscillatory component.
- Fractional-order remedies: Fractional-order regularization is reported to preserve smooth-area contours and high-frequency edges and textures, while fractional anisotropic diffusion targets staircase effects.
3.3. Image Edge Detection
Fractional-order edge detection is introduced to address the contrasting weaknesses of first- and second-order operators. It aims to preserve contours, maintain marginal features, and enhance texture details across frequency ranges.
- First-order edge detectors generally produce thicker edges and can lose image details.
- Fractional-order edge detection preserves low-frequency contour features, maintains high-frequency marginal features, and enhances medium-frequency texture details.
3.4. Image Segmentation
Fractional-order methods are applied to active-contour segmentation and threshold optimization. Reported comparisons find fractional-order methods superior to integer-order methods, particularly in concave regions.
- Geometric active-contour segmentation uses dynamic curves driven by an energy-minimizing model but is sensitive to initialization and topological changes.
- A fractional-order adaptive active-contour model is proposed to regularize the evolving curve and reduce noise influence.
- Fractional Brownian motion provides a mathematical model for simulating and parameterizing natural phenomena and representing scale-invariant random textures.
- Fractional-order Darwinian particle swarm optimization controls particle convergence while searching thresholds that maximize between-class variance.
- Fractional-order methods outperform integer-order methods in comparisons of active-contour models, especially in concave zones.
3.5. Image Registration
Fractional-order registration methods address both global and local geometric differences through fractional SIFT, fractional Fourier transforms, and fractional differential-equation approaches.
- Spatial transformations cover rotation, scaling, and translation but cannot model local geometric differences because they are global.
- Fractional Fourier transform methods support non-rigid registration, including adaptable approaches with lower interpolation errors and higher accuracy at nearly unchanged computing complexity.
- Fractional differential-equation registration methods design fractional image-intensity gradients or solve discretized fractional PDEs for non-rigid registration.
- Fractional registration methods extract more key points than the original SIFT method, while one non-rigid method reports lower average deformation errors than standard diffusion-based registration.
3.6. Image Recognition
Fractional-order methods are used in image recognition to reduce feature dimensionality, including approaches based on singular-value decomposition, canonical correlation, polynomial models, and independent components.
- Fractional-order recognition methods reduce image-feature dimensions by decomposing images into compositions of base images using singular value decomposition.
- Fractional-order embedding canonical correlation analysis reduces the dimensionality of multi-view data for classification tasks.
- Fractional power polynomial models reduce textural-feature dimensions, while FLOD-ICA is constructed for image recognition.
3.8. Image Encryption
Fractional-order transforms extend image-encryption designs by adding parameterized keys and nonlinear operations, while related fractional transforms are also used in compression and restoration contexts.
- Fractional-order methods support nonlinear control and key-space extension, motivating their broad use in image encryption.
- Fractional Fourier transforms add an order parameter as an extra key in double random phase encoding, extending the encryption key space.
- Multiple-parameter discrete fractional Fourier transform can significantly increase the key space of double random phase encoding.
- Cascaded fractional Fourier transform enables multiple-image encryption by separating each original image into ciphertext and encryption-key phase masks.
4. Conclusion
The paper introduces foundational fractional-calculus tools and surveys their application across image-processing methods, with experimental data emphasizing fractional-order tuning.
- The paper introduces fractional-order definitions, discretization methods, and related Matlab toolboxes.
- It presents an extensive collection of fractional-order image-processing methods and experimental data to introduce the methodology.
- The review emphasizes tuning the fractional order across image-processing methods.
e-mail: 9501133@163.com
The listed affiliations include Northeastern University in Shenyang, China, and the University of California, Merced, in the United States.
- One affiliation is College of Information Science and Technology, Northeastern University, Shenyang, China.
- Two affiliations are School of Engineering, University of California, Merced, in Merced, California, USA.