Fractional calculus in image processing: a review
---
paper_title: Modeling the Cardiac Tissue Electrode Interface Using Fractional Calculus

paper_content:

The tissue electrode interface is common to all forms of biopotential recording (e.g., ECG, EMG, EEG) and functional electrical stimulation (e.g., pacemaker, cochlear implant, deep brain stimulation). Conventional lumped element circuit models of electrodes can be extended by generalization of the order of differentiation through modification of the defining current-voltage relationships. Such fractional order models provide an improved description of observed bioelectrode behaviour, but recent experimental studies of cardiac tissue suggest that additional mathematical tools may be needed to describe this complex system.
---
paper_title: Frequency-band complex noninteger differentiator: characterization and synthesis

paper_content:

The state-of-the-art on generalized (or any order) derivatives in physics and engineering sciences, is outlined for justifying the interest of the noninteger differentiation. The problems subsequent to its use in real-time operations are then set out so as to motivate the idea of synthesizing it by a recursive distribution of zeros and poles. An analysis of the existing work is also proposed to support this idea. A comprehensive study is given of the synthesis of differentiators with integer, noninteger, real or complex orders, and whose action is limited to any given frequency bandwidth. First, a definition, in the operational and frequency domains, of a frequency-band complex noninteger order differentiator, is given in a mathematical space with four dimensions which is a Banach algebra. Then, the determination of its synthesized form, by a recursive distribution of complex zeros and poles characterized by complex recursive factors, is presented. The complex noninteger differentiation order is expressed as a function of these recursive factors. The number of zeros and poles is calculated to be as low as possible while still ensuring the stability of the synthesized differentiator to be synthesized. A time validation is presented. Finally, guidelines are proposed for the conception of the synthesized differentiator.
---
paper_title: Discretization schemes for fractional-order differentiators and integrators

paper_content:

For fractional-order differentiator s/sup r/ where r is a real number, its discretization is a key step in digital implementation. Two discretization methods are presented. The first scheme is a direct recursive discretization of the Tustin operator. The second one is a direct discretization method using the Al-Alaoui operator via continued fraction expansion (CFE). The approximate discretization is minimum phase and stable. Detailed discretization procedures and short MATLAB scripts are given. Examples are included for illustration.
---
paper_title: Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation.

paper_content:

Diffusion weighted MRI is used clinically to detect and characterize neurodegenerative, malignant and ischemic diseases. The correlation between developing pathology and localized diffusion relies on diffusion-weighted pulse sequences to probe biophysical models of molecular diffusion-typically exp[-(bD)]-where D is the apparent diffusion coefficient (mm(2)/s) and b depends on the specific gradient pulse sequence parameters. Several recent studies have investigated the so-called anomalous diffusion stretched exponential model-exp[-(bD)(alpha)], where alpha is a measure of tissue complexity that can be derived from fractal models of tissue structure. In this paper we propose an alternative derivation for the stretched exponential model using fractional order space and time derivatives. First, we consider the case where the spatial Laplacian in the Bloch-Torrey equation is generalized to incorporate a fractional order Brownian model of diffusivity. Second, we consider the case where the time derivative in the Bloch-Torrey equation is replaced by a Riemann-Liouville fractional order time derivative expressed in the Caputo form. Both cases revert to the classical results for integer order operations. Fractional order dynamics derived for the first case were observed to fit the signal attenuation in diffusion-weighted images obtained from Sephadex gels, human articular cartilage and human brain. Future developments of this approach may be useful for classifying anomalous diffusion in tissues with developing pathology.
---
paper_title: Matrix approach to discrete fractional calculus II: partial fractional differential equations

paper_content:

A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubny's matrix approach (Fractional Calculus and Applied Analysis, vol. 3, no. 4, 2000, 359--386). Four examples of numerical solution of fractional diffusion equation with various combinations of time/space fractional derivatives (integer/integer, fractional/integer, integer/fractional, and fractional/fractional) with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach. The fifth example illustrates that the method can be equally simply used for fractional differential equations with delays. A set of MATLAB routines for the implementation of the method as well as sample code used to solve the examples have been developed.
---
paper_title: Fractional Differential Mask: A Fractional Differential-Based Approach for Multiscale Texture Enhancement

paper_content:

In this paper, we intend to implement a class of fractional differential masks with high-precision. Thanks to two commonly used definitions of fractional differential for what are known as Grumwald-Letnikov and Riemann-Liouville, we propose six fractional differential masks and present the structures and parameters of each mask respectively on the direction of negative x-coordinate, positive x-coordinate, negative y-coordinate, positive y-coordinate, left downward diagonal, left upward diagonal, right downward diagonal, and right upward diagonal. Moreover, by theoretical and experimental analyzing, we demonstrate the second is the best performance fractional differential mask of the proposed six ones. Finally, we discuss further the capability of multiscale fractional differential masks for texture enhancement. Experiments show that, for rich-grained digital image, the capability of nonlinearly enhancing complex texture details in smooth area by fractional differential-based approach appears obvious better than by traditional integral-based algorithms.
---
paper_title: Fractional poisson enhancement model for text detection and recognition in video frames

paper_content:

Performing Laplacian operation on video images is a common technique to improve image contrast to achieve good text detection and recognition accuracies. However, it is a fact that when Laplacian operation enhances contrast, at the same time it introduces too many noises. To alleviate this, the existing methods propose different enhancement methods and filters. In this paper, we propose a generalized enhancement model based on fractional calculus to increase the quality of images obtained by Laplacian operation. The proposed method considers edges and their neighbor information to derive a mathematical model for enhancing low contrast information in video as well as in scene images. Experimental results of text detection and recognition methods on different databases show that the proposed enhancement model improves their accuracies significantly. The enhancement model is compared with standard enhancement models to show that the proposed model outperforms the existing models in terms of quality measures. The usefulness of the proposed model is validated through text detection and recognition experiments. A new Fractional Poisson enhancement model for noise removal in video.The model can be used for reducing distortion effects in text detection and recognition.The model enhances the results of existing text detection and recognition methods.
---
paper_title: Image enhancement in the fractional Fourier domain

paper_content:

Image enhancement in the transform domain is an indispensable technique in the image processing area. In this paper, a novel method of image enhancement with respect to the fractional Fourier transform is presented. This method filters image in the fractional Fourier domain instead of the Fourier domain which is usually applied to the classical image enhancement. The fractional Fourier transform has a rotation angle a, characters of image thus change in different transform domain. In a proper fractional Fourier domain with angle a, ideal low-pass filter makes image smoother and ideal high-pass filter loses less information of image than in the traditional Fourier domain, which provides an alternative way to enhance image with proper filter designing. The validity of this method is demonstrated via simulations.
---
paper_title: The fractional differential enhancement of image texture features and its parallel processing optimization

paper_content:

This paper mainly discusses the applications of the fractional differential in image enhancement and its parallel processing optimization. Firstly, it makes the principle analysis of fractional calculus antagonistic properties and texture extraction of the digital image. Then it constructs the fractional differential mask and determines its operation rules. The simulation result shows that the mask enhances the digital images texture features effectively with the operation rules. As the mask processing costs large amount of calculation, which is time-consuming, therefore, the parallel algorithm of digital image processing is introduced, so that the large amount calculation problem is better resolved, and satisfactory results are obtained.
---
paper_title: Fractional differential and variational method for image fusion and super-resolution

paper_content:

This paper introduces a novel fractional differential and variational model that includes the terms of fusion and super-resolution, edge enhancement and noise suppression. In image fusion and super-resolution term, the structure tensor is employed to describe the geometry of all the input images. According to the fact that the fused image and the source inputs should have the same or similar structure tensor, the energy functional of the image fusion and super-resolution is established combining with the down-sampling operator. For edge enhancement, the bidirectional diffusion term is incorporated into the image fusion and super-resolution model to enhance the visualization of the fused image. In the noise suppression term, a new variational model is developed based on the fractional differential and fractional total variation. Thanks to the above three terms, the proposed model can realize the image fusion, super-resolution, and the edge information enhancement simultaneously. To search for the optimal solution, a gradient descent iteration scheme derived from the Euler-Lagrange equation of the proposed model is employed. The numerical results indicate that the proposed method is feasible and effective.
---
paper_title: Texture enhancement algorithm based on fractional differential mask of adaptive non-integral step

paper_content:

Image texture enhancement is an important topic in computer graphics, computer vision and pattern recognition. By applying the fractional differential principle to analyze texture characteristics, a new fractional differential mask with adaptive non-integral step is proposed in this paper to enhance texture images. A non-regular self-similar support region is constructed based on a local texture similarity measure, which can exclude low-correlated pixels and noise. Then, through applying sub-pixel division and introducing a local linear piecewise model to estimate the gray value in between the pixels, the resulting nonintegral steps can improve the characterization of self-similarity that is inherent in digital images. Finally, the non-regular fractional differential mask which incorporates adaptive nonintegral step is constructed. Experimental results show that, for rich-grained digital images, the capability of improved self-similarity and texture characterization based on our proposed approach leads to improved image enhancement results when compared with conventional approaches.
---
paper_title: Adaptive fractional differential approach and its application to medical image enhancement

paper_content:

Display Omitted We use the improved Otsu algorithm to segment medical images.The function of order v is constructed based on the features of image areas.The fractional differential order of each pixel is obtained by the function.Each pixel of medical image is processed by adaptive fractional differential mask.The edge of medical image is enhanced while the weak texture is preserved. This paper presents a new medical image enhancement method that adjusts the fractional order according to the dynamic gradient feature of the entire image. The presented method can extract the edges of an image accurately and enhance them while preserving smooth areas and weak textures; these improvements can be particularly helpful to doctors' diagnoses. The primary contribution of this paper is the Adaptive Fractional Differential Algorithm (AFDA), which uses the improved Otsu algorithm to segment the edges, textures and smooth areas of images. This algorithm allows the optimal fractional order of each pixel to be obtained using an adaptive fractional differential function constructed based on the area feature of image. As a result, the image can be enhanced adaptively. Experimental results show that for medical images, AFDA shows better image enhancement than other methods by making edges clearer and textures richer.
---
paper_title: Image denoising and enhancement based on adaptive fractional calculus of small probability strategy

paper_content:

This paper presents two methods to deal with the problem that traditional image denoising algorithms may easily neglect image texture details. The first one is global adaptive fractional integral algorithm (GAFIA) which deals with common noises. It selects the optimal integral order of each pixel based on the local average gradient. The second is image denoising and enhancement algorithm based on adaptive fractional calculus of small probability strategy (AFC-SPS) which deals with salt & pepper noise. It regards the appearance of noise points as small probability events, divides them, and segments the image edges and weak textures by the improved two-dimensional Otsu algorithm. Then, the function of adaptive fractional order is constructed. Experimental results show that, both of the methods have good image denoising effect, and the AFC-SPS algorithm has a better effect than other methods in enhancing the edge and preserving the texture. The optimal order of each pixel is selected based on the local average gradient.The appearance of noise points is regarded as small probability events.We use the improved two-dimensional Otsu algorithm to segment images.The function of order v is constructed based on the features of image areas.AFC-SPS algorithm can remove noise and enhance image edge simultaneously.
---
paper_title: An Iterative Lagrange Multiplier Method for Constrained Total-Variation-Based Image Denoising

paper_content:

Various effective algorithms have been proposed in the past two decades for nonlinear PDEs arising from the unconstrained total-variation-based image denoising problem regularizing the total variation constrained minimization model. Such algorithms can be used to obtain a satisfactory result as long as a suitable regularization parameter balancing the trade-off between a good fit to the data and a regular solution is given. However, it is generally difficult to obtain a suitable regularization parameter without which restored images can be unsatisfactory: if it is too large, then the resulting solution is still contaminated by noise, while if too small, the solution is a poor approximation of the true noise-free solution. To provide an automatic method for the regularization parameter when the noise level is known a priori, one way is to address the coupled Karush–Kuhn–Tucker (KKT) systems from the constrained total variation optimization problem. So far much less work has been done on this problem. This ...
---
paper_title: Fractional-order total variation image denoising based on proximity algorithm

paper_content:

The fractional-order total variation(TV) image denoising model has been proved to be able to avoid the "blocky effect". However, it is difficult to be solved due to the non-differentiability of the fractional-order TV regularization term. In this paper, the proximity algorithm is used to solve the fractional-order TV optimization problem, which provides an effective tool for the study of the fractional-order TV denoising model. In this method, the complex fractional-order TV optimization problem is solved by using a sequence of simpler proximity operators, and therefore it is effective to deal with the problem of algorithm implementation. The final numerical procedure is given for image denoising, and the experimental results verify the effectiveness of the algorithm.
---
paper_title: Spatial fractional telegraph equation for image structure preserving denoising

paper_content:

In this paper, we propose a spatial fractional telegraph equation which could be applied to image denoising. The proposed equation interpolates between the second and the fourth order anisotropic diffusion equation by the use of spatial fractional derivatives. On the other hand, the telegraph equation interpolates between diffusion equation and wave equation, which leads to a mixed behavior of diffusion and wave propagation and thus it can preserve edges in the highly oscillatory regions. The existence, uniqueness and stability of the solution of our model are proved in this paper. The experimental results indicate superiority of the proposed model over the existing methods. HighlightsFractional derivatives lead to a interpolation between second and fourth order models.Telegraph equation interpolates between a diffusion equation and a wave equation.Telegraph equation is beneficial to enhance edges.We have proved that the proposed model is well-posed.The stable and convergent numerical scheme is given.
---
paper_title: An Optimization-Based Multilevel Algorithm for Total Variation Image Denoising

paper_content:

This paper proposes a fast multilevel method using primal relaxations for the total variation image denoising and analyzes its convergence. The basic primal relaxation is known to get stuck at a nonstationary point (nearly a local minimum) of the minimization, whose solution is known to be “nonsmooth” in the space of functions with bounded variation. Our idea is to use coarse level corrections, overcoming the deadlock in a basic primal relaxation scheme and achieving much improvement over relaxation. Moreover, to reach a global minimizer, further refinement of the multilevel method is needed, and we propose a nonregular coarse level based on a patch‐detection idea (relating to hemivariateness) to correct and improve the standard multilevel method. Both algorithmic and analytical results together with numerical experiments on both one‐ and two‐dimensional images are presented.
---
paper_title: Statistics of Natural Stochastic Textures and Their Application in Image Denoising.

paper_content:

Natural stochastic textures (NSTs), characterized by their fine details, are prone to corruption by artifacts, introduced during the image acquisition process by the combined effect of blur and noise. While many successful algorithms exist for image restoration and enhancement, the restoration of natural textures and textured images based on suitable statistical models has yet to be further improved. We examine the statistical properties of NST using three image databases. We show that the Gaussian distribution is suitable for many NST, while other natural textures can be properly represented by a model that separates the image into two layers; one of these layers contains the structural elements of smooth areas and edges, while the other contains the statistically Gaussian textural details. Based on these statistical properties, an algorithm for the denoising of natural images containing NST is proposed, using patch-based fractional Brownian motion model and regularization by means of anisotropic diffusion. It is illustrated that this algorithm successfully recovers both missing textural details and structural attributes that characterize natural images. The algorithm is compared with classical as well as the state-of-the-art denoising algorithms.
---
paper_title: A fractional-order adaptive regularization primal-dual algorithm for image denoising

paper_content:

This paper aims to develop a fractional-order model and a primal-dual algorithm for image denoising, where a regularization parameter can be adjusted adaptively according to Morozov discrepancy principle at each iteration to ensure that the denoised image retains in a specific set. In the light of saddle-point theory, the convergence of our proposed algorithm is guaranteed. Simulations with comparisons are carried out to demonstrate the effectiveness of our proposed algorithm for image denoising.
---
paper_title: An improved fractional-order differentiation model for image denoising

paper_content:

This paper investigates fractional order differentiation and its applications in digital image processing. We propose an improved model based on the Grunwald-Letnikov (G-L) fractional differential operator. Our improved denoising operator mask is based on G-L fractional order differentiation. The total coefficient of this mask is not equal to zero, which means that its response value is not zero in flat areas of the image. This nonlinear filter mask enhances and preserves detailed features while effectively denoising the image. Our experiments on texture-rich digital images demonstrated the capabilities of the filter. We used the information entropy and average gradient to quantitatively compare our method to existing techniques. Additionally, we have successfully used it to denoise three-dimensional magnetic resonance images. An improved fractional differential operator for image denoising is proposed.Our denoising filter operator mask based on G-L fractional is constructed.The total coefficient of filter mask is not equal to zero.This nonlinear filter mask can preserve the detail features while denoising.
---
paper_title: Properties of $BV-G$ Structures $+$ Textures Decomposition Models. Application to Road Detection in Satellite Images

paper_content:

In this paper, we present some theoretical results about a structures-textures image decomposition model which was proposed by the second author. We prove a theorem which gives the behavior of this model in different cases. Finally, as a consequence of the theorem we derive an algorithm for the detection of long and thin objects applied to a road networks detection application in aerial or satellite images.
---
paper_title: A fractional diffusion-wave equation with non-local regularization for image denoising

paper_content:

This paper introduces a novel fractional diffusion-wave equation with non-local regularization for noise removal. Using the fractional time derivative, the model interpolates between the heat diffusion equation and the wave equation, which leads to a mixed behavior of diffusion and wave propagation and thus it can preserve edges in a highly oscillatory region. On the other hand, the usual diffusion is used to reduce the noise whereas the non-local term which exhibits an anti-diffusion effect is used to enhance the image structure. We prove that the proposed model is well-posed, and the stable and convergent numerical scheme is also given in this paper. The experimental results indicate superiority of the proposed model over the baseline diffusion models. HighlightsThe fractional PDE interpolates between a diffusion equation and a wave equation.The anti-diffusion effect of the non-local term results in edge enhancement.We have proved that the proposed model is well-posed.The stable and convergent numerical scheme is given.We have discussed the choice of the parameters in our model.
---
paper_title: A Total Fractional-Order Variation Model for Image Restoration with Non-homogeneous Boundary Conditions and its Numerical Solution

paper_content:

To overcome the weakness of a total variation based model for image restoration, various high order (typically second order) regularization models have been proposed and studied recently. In this paper we analyze and test a fractional-order derivative based total $\alpha$-order variation model which can outperform the currently popular high order regularization models. There exist several previous works using total $\alpha$-order variations for image restoration; however, first, no analysis has been done yet, and second, all tested formulations, differing from each other, utilize the zero Dirichlet boundary conditions which are not realistic (while nonzero boundary conditions violate definitions of fractional-order derivatives). This paper first reviews some results of fractional-order derivatives and then analyzes the theoretical properties of the proposed total $\alpha$-order variational model rigorously. It then develops four algorithms for solving the variational problem---one based on the variational...
---
paper_title: Fractional Calculus Approach to Texture of Digital Image

paper_content:

In this paper, I put forward and discuss multiform covering templates and algorithms of fractional derivative for digital image. The computer experiment shows the textural detail enhancing capability of fractional derivative-based texture operator is much better than integral derivative-based one for rich-grained digital image
---
paper_title: A variational framework for image denoising based on fractional-order derivatives

paper_content:

In this paper, we propose a variational framework for noise removal by combining different fractional order derivatives. In smooth regions, we use the regularization with the fractional order greater than 2 to effectively remove the noise and avoid the staircase effect; in the region of image edges, we use the regularization with the fractional order that lies in (0,1] to better preserve them. A main advantage of this framework is the superiority in eliminating the staircase effect and dealing with better textures and repetitive structures. A set of experiments will be given to demonstrate the advantages of the proposed method.
---
paper_title: A new variational model for image denoising based on fractional-order derivative

paper_content:

This paper presents a new fractional variational model for removing multiplicative noise in ultrasound images. This model not only inherits the perfect denoising effects of total variational models, but also retains texture details of smooth regions owing to the characteristics of fractional calculus. The experimental results demonstrate that the quality of images denoised by the proposed model is quite ideal and excellent.
---
paper_title: Multigrid Algorithm for High Order Denoising

paper_content:

Image denoising has been a research topic deeply investigated within the last two decades. Excellent results have been obtained by using such models as the total variation (TV) minimization by Rudin, Osher, and Fatemi [Phys. D, 60 (1992), pp. 259-268], which involves solving a second order PDE. In more recent years some effort has been made [Y.-L. You and M. Kaveh, IEEE Trans. Image Process., 9 (2000), pp. 1723-1730; M. Lysaker, S. Osher, and X.-C. Tai, IEEE Trans. Image Process., 13 (2004), pp. 1345-1357; M. Lysaker, A. Lundervold, and X.-C. Tai, IEEE Trans. Image Process., 12 (2003), pp. 1579-1590; Y. Chen, S. Levine, and M. Rao, SIAM J. Appl. Math., 66 (2006), pp. 1383-1406] in improving these results by using higher order models, particularly to avoid the staircase effect inherent to the solution of the TV model. However, the construction of stable numerical schemes for the resulting PDEs arising from the minimization of such high order models has proved to be very difficult due to high nonlinearity and stiffness. In this paper, we study a curvature-based energy minimizing model [W. Zhu and T. F. Chan, Image Denoising Using Mean Curvature, preprint, http://www.math.nyu.edu/ wzhu/], for which one has to solve a fourth order PDE. For this model we develop two new algorithms: a stabilized fixed point method and, based upon this, an efficient nonlinear multigrid (MG) algorithm. We will show numerical experiments to demonstrate the very good performance of our MG algorithm.
---
paper_title: Fractional Differential Analysis for Texture of Digital Image

paper_content:

This paper discusses the capabilities of the fractional differential approach for the detection of textural features in two-dimensional digital images and the involved Lateral Inhibition Principle, and fractional differential masks and algorithms of digital image. Firstly, the kinetic physical meaning of fractional differential and the relationship between fractional calculus and classical time-frequency analysis and the separability of two-dimensional fractional calculus on certain conditions are deduced. Secondly, the difference between two Gaussians receptive fields for fractional differential of digital image involved signal processing and biologic vision nerve model is discussed. An analysis of its Mach band is also included. Thirdly, the implements and parameters of eight n × n fractional differential masks, which are mutual central symmetric, on negative x-coordinate, positive x-coordinate, negative y-coordinate, positive y-coordinate, left lower diagonal, left upper diagonal, right lower diagonal,...
---
paper_title: A MULTILEVEL ALGORITHM FOR SIMULTANEOUSLY DENOISING AND DEBLURRING IMAGES ∗

paper_content:

In this paper, we develop a fast multilevel algorithm for simultaneously denoising and deblurring images under the total variation regularization. Although much effort has been devoted to developing fast algorithms for the numerical solution and the denoising problem was satisfactorily solved, fast algorithms for the combined denoising and deblurring model remain to be a challenge. Recently several successful studies of approximating this model and subsequently finding fast algorithms were conducted which have partially solved this problem. The aim of this paper is to generalize a fast multilevel denoising method to solving the minimization model for simultaneously denoising and deblurring. Our new idea is to overcome the complexity issue by a detailed study of the structured matrices that are associated with the blurring operator. A fast algorithm can then be obtained for directly solving the variational model. Supporting numerical experiments on gray scale images are presented.
---
paper_title: Robust Kronecker product video denoising based on fractional-order total variation model

paper_content:

Abstract Most existing sparse representation-based video denoising algorithms assume video noise is additive Gaussian white noise, which is often violated in practice. In this paper, a robust Kronecker product video denoising (RKPVD) algorithm based on fractional-order total variation model is proposed to remove serious mixed Gaussian-impulse noises from the video data. Using the temporal and spatial correlations of videos, the problem of denoising mixed noises is formulated as a robust low rank video recovery minimization problem based on fractional-order total variation (FTV) model. The resulting under-determined minimization problem, which consists of nuclear norm, Kronecker product sparse l 1 norm and FTV, can be efficiently solved by a two-stage algorithm combined with alternating direction method (ADM). The robustness and effectiveness of the proposed RKPVD denoising algorithm on removing mixed Gaussian impulsive noise are validated in the experiments. Compared with several state-of-the-art algorithms, such as total variation (TV), sparse and redundant representation (SARR), video block matching and 3D filtering (VBM3D), robust principal component analysis (RPCA) and robust temporal spatial decomposition (RTSD), intensive experiments show that the proposed RKPVD method has a higher PSNR (peak signal-to-noise ratio) and a better visual detail preservation.
---
paper_title: A novel fractional-order signal processing based edge detection method

paper_content:

Image edge detection is a classic problem of machine vision and image processing. Edge detection often uses an integer-order differential operation. The paper adopts fractional differentiation and integration to obtain a new edge detection operator. The performances in terms of detection accuracy and noise immunity of the new operator are compared with those of the traditional operators through examples. The comparison shows that the new operator is promising.
---
paper_title: Edge detection of colour image based on quaternion fractional differential

paper_content:

According to the development of the real fractional differential and its applications in the modern signal processing, the authors extend it to quaternion body and put forward a new concept: quaternion fractional differential (QFD), and apply it to edge detection of colour image. This method is called edge detection based on QFD. Simulation experiments indicate that this method has special advantages. Furthermore, the authors give an indicator to evaluate the effectiveness of different edge filters. Comparing with Sobel and mix edges of real fractional differential to every channels of colour image, they discover that QFD has fewer false negatives in the textured regions and is also better at detecting edges that are partially defined by texture, which means the authors can obtain much better results in the interesting regions using QFD and is more consistent with the characteristics of human visual system.
---
paper_title: Fractional differentiation for edge detection

paper_content:

In image processing, edge detection often makes use of integer-order differentiation operators, especially order 1 used by the gradient and order 2 by the Laplacian. This paper demonstrates how introducing an edge detector based on noninteger (fractional) differentiation can improve the criterion of thin detection, or detection selectivity in the case of parabolic luminance transitions, and the criterion of immunity to noise, which can be interpreted in term of robustness to noise in general.
---
paper_title: A fractional-order edge detection operator for medical image structure feature extraction

paper_content:

This paper introduces a novel fractional-order gradient operator for medical image structure feature extraction. The proposed operator can be seen as generalization of the first-order Sobel operator based on the GL fractional derivative definition. The generalization goal is to utilize the frequency characteristic of the fractional derivative for extracting more structure feature details. A thresholding is set based on the average fractional-order gradient for marking the edge points, and then the image structure can be extracted. Experiments show that the proposed fractional-order operator yields good visual effects.
---
paper_title: Edge Detection Based on the Newton Interpolation's Fractional Differentiation

paper_content:

In this paper, according to the development of the fractional differentiation and its applications in the modern signal processing, we improve the numerical calculation of fractional differentiation by Newton interpolation equation, and propose a new mask, the Newton Interpolation's Fractional Differentiation (NIFD). Then, we apply this new mask to image edge detection and can obtain the better edge information image. In order to get continuous and thin edges, we synthesize a new gradient and adopt the non-maxima suppression method. For a comparison, we consider the edge map yielded by the sobel operator and canny operator. By contrast, we discover that the edge image obtained by NIFD operator is better than those of sobel and canny operators, and specially for a noisy image, NIFD operator has the best anti-noise ability.
---
paper_title: Adaptive active contour model driven by fractional order fitting energy

paper_content:

In this paper, a new adaptive active contour model is proposed for image segmentation, which is built based on fractional order differentiation, level set method and curve evolution. The energy functional for the proposed model consists of three terms: fitting term, regularization tern and penalty term. By incorporating the fractional order fitting term, the novel fitting term can describe the original image more accurately, and be robustness to noise. In order to ensure stable evolution of the level set function, a penalty tern is added into the proposed model. The results evolution of the level set function is the gradient flow that minimizes the overall energy functional. Experimental results for both synthetic and real image show desirable performance of our method. We proposed an adaptive active contour model based on fractional differentiation.The new fitting term can describe the original image more accurately.Adaptive length regularization is used to smooth the level set function.
---
paper_title: Fractional discrimination for texture image segmentation

paper_content:

Texture image segmentation plays an important role in texture analysis. This paper presents an approach to image segmentation by texture classification based on fractional discrimination functions. The idea behind this method is to enhance the texture edge points by means of image decomposition and contextual filtering in terms of the proposed fractional function. In addition, the function is described in a unified form with three-parameters. The parameters determine the global scale in conjunction with local scales for feature identification. Our experimental results show that texture features can be effectively extracted on the basis of the selective fractional discrimination function.
---
paper_title: A local information based variational model for selective image segmentation

paper_content:

Many effective models are available for ::: segmentation of an image to extract all ::: homogenous objects within it. ::: For applications where segmentation of a single object identifiable by ::: geometric constraints within an image is desired, much less work has been done for this purpose. ::: This paper presents an improved selective segmentation model, without ::: `balloon' force, combining geometrical constraints and local image intensity ::: information around zero level set, aiming to overcome ::: the weakness of getting spurious solutions by Badshah and Chen's model [8]. ::: A key step in our new strategy is an adaptive local band selection algorithm. ::: Numerical experiments show that the new model appears to be able to detect an object ::: possessing highly complex and nonconvex features, and to ::: produce desirable results in terms of segmentation quality and robustness.
---
paper_title: Forest Fire Image Segmentation Based on Contourlet Transform and Fractional Brownian Motion

paper_content:

A new segmentation algorithm for the forest fire image based on Fractional Brownian motion and Contour let transform is proposed. Firstly, the algorithm decomposes the image into several sub-bands of multi-scale, location and multi-direction by Contour let transform. Fractional Brownian motion model is performed for low frequency sub-band, and then estimate the Hurst coefficient of each frequency coefficient in the sub-band. Hurst coefficient used as the characteristics of flame parameters for image segmentation. Experimental results show that the algorithm can segment flame effectively, and improve the efficiency of segmentation for forest fire image.
---
paper_title: Segmentation of textures with different roughness using the model of isotropic two-dimensional fractional Brownian motion

paper_content:

An attempt is made to apply a novel multidirectional resolution analysis to obtain boundaries between areas with different grades of 'roughness'. To this end, the image texture is parametrized with respect to direction, based on the model of isotropic 2-D fractional Brownian motion (2-D FBM). Some theoretical aspects of this model have not been clarified yet. The authors first provide some of the crucial missing links in the theory of 2-D FBM and present a concrete realization of this stochastic process. They then describe the segmentation procedure in detail. First, they present a suboptimal but fast estimation algorithm to obtain angle-dependent fractal parameter subimages, and compare its performance with the Cramer-Rao bound. They then apply matched edge-detection procedures to each subimage separately, and combine the angle-dependent boundary pixels to obtain the texture boundaries of the entire scene. >
---
paper_title: Multilevel Image Segmentation Based on Fractional-Order Darwinian Particle Swarm Optimization

paper_content:

Hyperspectral remote sensing images contain hundreds of data channels. Due to the high dimensionality of the hyperspectral data, it is difficult to design accurate and efficient image segmentation algorithms for such imagery. In this paper, a new multilevel thresholding method is introduced for the segmentation of hyperspectral and multispectral images. The new method is based on fractional-order Darwinian particle swarm optimization (FODPSO) which exploits the many swarms of test solutions that may exist at any time. In addition, the concept of fractional derivative is used to control the convergence rate of particles. In this paper, the so-called Otsu problem is solved for each channel of the multispectral and hyperspectral data. Therefore, the problem of n-level thresholding is reduced to an optimization problem in order to search for the thresholds that maximize the between-class variance. Experimental results are favorable for the FODPSO when compared to other bioinspired methods for multilevel segmentation of multispectral and hyperspectral images. The FODPSO presents a statistically significant improvement in terms of both CPU time and fitness value, i.e., the approach is able to find the optimal set of thresholds with a larger between-class variance in less computational time than the other approaches. In addition, a new classification approach based on support vector machine (SVM) and FODPSO is introduced in this paper. Results confirm that the new segmentation method is able to improve upon results obtained with the standard SVM in terms of classification accuracies.
---
paper_title: A fractional-order regulatory CV model for brain MR image segmentation

paper_content:

In this paper, we introduce fractional derivative into CV level set model for image segmentation. Specifically, the first-order gradient operator in the CV level set model is generalized to fractional-order gradient by energy formulation regulation, which considers the nonlinear protecting capability of fractional-order derivative for texture and lower frequency features of images. The corresponding fractional Euler-Lagrange equation is given for level set evolution, and then the numerical algorithm is analyzed. The novel model has been validated on real and simulated brain MR images, with desirable performance in the presence of intensity inhomogeneity, compared with the traditional CV level set model.
---
paper_title: Fractional Brownian motion models for synthetic aperture radar imagery scene segmentation

paper_content:

The application of fractal random process models and their related scaling parameters as features in the analysis and segmentation of clutter in high-resolution, polarimetric synthetic aperture radar (SAR) imagery is demonstrated. Specifically, the fractal dimension of natural clutter sources, such as grass and trees, is computed and used as a texture feature for a Bayesian classifier. The SAR shadows are segmented in a separate manner using the original backscatter power as a discriminant. The proposed segmentation process yields a three-class segmentation map for the scenes considered in this study (with three clutter types: shadows, trees, and grass). The difficulty of computing texture metrics in high-speckle SAR imagery is addressed. In particular, a two-step preprocessing approach consisting of polarimetric minimum speckle filtering followed by noncoherent spatial averaging is used. The relevance of the resulting segmentation maps to constant-false-alarm-rate (CFAR) radar target detection techniques is discussed. >
---
paper_title: Multilayer-Pseudopolar Fractional Fourier Transform Approach for Image Registration

paper_content:

In this paper, a method is proposed to estimate the log-polar coordinates coefficients using Multilayer-peudopolar Fractional Fourier transform (MPFFT). The MPFFT grid is dense at low coefficients and sparse at high coefficients. This character is geometrically similar to the log-polar grid. Large translations, rotations and scaling in images can be estimated. Experimental results show that the proposed method is more efficient for image registration.
---
paper_title: Research on Image Matching Combining on Fractional Differential With Scale Invariant Feature Transform

paper_content:

Image matching is one of the most important problems in computer vision. Scale Invariant Feature Transform (SIFT) algorithm has been proved to be effective for detecting features for image matching. However SIFT algorithm has limitation to extract features in textile image or self-similar construction image. Fortunately fractional differentiation has advantage to strengthen and extract textural features of digital images. Aiming at the problem, this paper proposes a new method for image matching based on fractional differentiation and SIFT. The method calculates the image pyramid combining the Riemann-Liouville (R-L) fractional differentiation and the derivative of the Gaussian function. Thus image feature has been enhanced, and more feature points can be extracted. As a result the matching accuracy is improved. Additionally, a new feature detection mask based on fractional differential is constructed. The proposed method is a significant extension of SIFT algorithm. The experiments carried out with images in database and real images indicate that the proposed method can obtain good matching results. It can be used for matching textile image or some self-similar construct image.Copyright © 2015 by ASME
---
paper_title: Security of registration data of fingerprint image with a server by use of the fractional Fourier transform

paper_content:

Recently, the personal identities by the fingerprint authentication have been increasing everywhere, for example, in the automatic logging into a PC, the access control and the diligence & indolence management in an office, and so on. In the fingerprint authentication, the fingerprint does not have to be remembered and there is no worry to be lost like a password, if once the information is registered with a server. The fingerprint, however, cannot be changed like a password if the information leaks out from the server. Therefore, the data processing is necessary not to be able to be decoded by another person. In this study, we use the fractional Fourier transform (FRT) as the method of data processing and analyze the fundamental properties of the decoded image in comparison with the original image.
---
paper_title: Medical image registration in fractional Fourier transform domain

paper_content:

Abstract The fractional Fourier transform as a powerful signal analysis and processing technique, the studies on using it for image registration are very rewarding and promising. This paper focuses on the medical image registration in the domain of fractional Fourier transform. The Powell algorithm is implemented to optimize the parameters by evaluating the minimum MSE between the magnitude of images. Two kinds of fractional Fourier transform are used to construct the algorithms. And they are tested on two sets of medica images and the experimental results were compared and analyzed.
---
paper_title: An Adaptable-Multilayer Fractional Fourier Transform Approach for Image Registration

paper_content:

A novel adaptable accurate way for calculating polar FFT and log-polar FFT is developed in this paper, named multilayer fractional Fourier transform (MLFFT). MLFFT is a necessary addition to the pseudo-polar FFT for the following reasons: It has lower interpolation errors in both polar and log-polar Fourier transforms; it reaches better accuracy with the nearly same computing complexity as the pseudo-polar FFT; it provides a mechanism to increase the accuracy by increasing the user-defined computing level. This paper demonstrates both MLFFT itself and its advantages theoretically and experimentally. By emphasizing applications of MLFFT in image registration with rotation and scaling, our experiments suggest two major advantages of MLFFT: 1) scaling up to 5 and arbitrary rotation angles, or scales up to 10 without rotation can be recovered by MLFFT while currently the result recovered by the state-of-the-art algorithms is the maximum scaling of 4; 2) No iteration is needed to obtain large rotation and scaling values of images by MLFFT, hence it is more efficient than the pseudopolar-based FFT methods for image registration.
---
paper_title: Using fractional gradient information in non-rigid image registration: application to breast MRI

paper_content:

This work applies fractional differentiation (differentiation to non-integer order) to the gradients determined ::: from image intensities for enhanced image registration. The technique is used to correct known simulated ::: deformations of volumetric breast MR data using two algorithms: direct registration of gradient magnitude ::: images and an extension of a previously published method that incorporates both image intensity and image ::: gradient information to enhance registration performance. Better recovery of known deformations are seen ::: when using non-integer order derivatives: half-derivative breast images are better registered when these ::: methods are incorporated into a standard diffusion-based registration algorithm.
---
paper_title: Image registration based on Fractional Fourier Transform

paper_content:

Abstract Image registration (IR) has been a fundamental task in many image processing applications. Conventional FFT-based methods ignored the spatial information of images, which is also important for IR. So, the Fractional Fourier Transform (FRFT) which contains both spatial and frequency information, has been taken granted in applying to IR scheme. In this paper, a novel method based on the combination of FRFT and conventional phase correlation technique is proposed. Simulation results of the proposed technique prove its better noise immunity than FFT-based method and its superiority than existing methods based on FRFT.
---
paper_title: Image Registration using Fractional Fourier Transform

paper_content:

This image registration (IR) is a fundamental task in many image processing applications such as medical diagnosis, satellite imaging, super-resolution image reconstruction etc. Fourier transform based methods have also been in use for IR purposes for both the shifted and rotated images using the phase correlation method (Brown, 1992). As fractional Fourier transform (FRFT) is a generalization of the conventional Fourier transform, it is natural to extend its use in IR applications. In this paper, the authors propose the use of the FRFT in IR problems involving either linear shifts or pure rotations in the given images to be registered. Simulation results of the proposed techniques are also presented
---
paper_title: Variational image registration by a total fractional-order variation model

paper_content:

In this paper, a new framework of nonlocal deformation in non-rigid image registration is presented. It is well known that many non-rigid image registration techniques may lead to unsteady deformation (e.g. not one to one) if the dissimilarity between the reference and template images is too large. We present a novel variational framework of the total fractional-order variation to derive the underlying fractional Euler-Lagrange equations and a numerical implementation combining the semi-implicit update and conjugate gradients (CG) solution to solve the nonlinear systems. Numerical experiments show that the new registration not only produces accurate and smooth solutions but also allows for a large rigid alignment, the evaluations of the new model demonstrate substantial improvements in accuracy and robustness over the conventional image registration approaches.
---
paper_title: Nonrigid image registration with two-sided space-fractional partial differential equations

paper_content:

Fractional partial differential equations can be used to model super-diffusive processes. Recent research into fractional PDEs has explored their impact on nonrigid image registration, which typically uses differential operators of integer order. In this paper, we propose a nonrigid registration algorithm that involves directly and rapidly solving a discretized fractional PDE at each time step that is subject to physically realistic image boundary conditions. The proposed algorithm is validated through registration experiments on breast MR imagery with simulated biomechanical deformations, indicating that the super-diffusive model yields lower average deformation errors than standard diffusion-based registration.
---
paper_title: Fractional-order embedding canonical correlation analysis and its applications to multi-view dimensionality reduction and recognition

paper_content:

Due to the noise disturbance and limited number of training samples, within-set and between-set sample covariance matrices in canonical correlation analysis (CCA) usually deviate from the true ones. In this paper, we re-estimate within-set and between-set covariance matrices to reduce the negative effect of this deviation. Specifically, we use the idea of fractional order to respectively correct the eigenvalues and singular values in the corresponding sample covariance matrices, and then construct fractional-order within-set and between-set scatter matrices which can obviously alleviate the problem of the deviation. On this basis, a new approach is proposed to reduce the dimensionality of multi-view data for classification tasks, called fractional-order embedding canonical correlation analysis (FECCA). The proposed method is evaluated on various handwritten numeral, face and object recognition problems. Extensive experimental results on the CENPARMI, UCI, AT&T, AR, and COIL-20 databases show that FECCA is very effective and obviously outperforms the existing joint dimensionality reduction or feature extraction methods in terms of classification accuracy. Moreover, its improvements for recognition rates are statistically significant on most cases below the significance level 0.05. A new method, namely FECCA, is presented for multi-view dimensionality reduction.Fractional-order within-set and between-set scatter matrices are constructed by sample spectrum modeling.The extracted features by FECCA have strong discriminant power for recognition.Experimental results show FECCA has better recognition rates than the existing joint feature extraction methods.
---
paper_title: Gabor filter based optical image recognition using Fractional Power Polynomial model based common discriminant locality preserving projection with kernels

paper_content:

Abstract This paper presents Gabor filter based optical image recognition using Fractional Power Polynomial model based Common Kernel Discriminant Locality Preserving Projection. This method tends to solve the nonlinear classification problem endured by optical image recognition owing to the complex illumination condition in practical applications, such as face recognition. The first step is to apply Gabor filter to extract desirable textural features characterized by spatial frequency, spatial locality and orientation selectivity to cope with the variations in illumination. In the second step we propose Class-wise Locality Preserving Projection through creating the nearest neighbor graph guided by the class labels for the textural features reduction. Finally we present Common Kernel Discriminant Vector with Fractional Power Polynomial model to reduce the dimensions of the textural features for recognition. For the performance evaluation on optical image recognition, we test the proposed method on a challenging optical image recognition problem, face recognition.
---
paper_title: Fractional low-order independent component analysis for face recognition robust to partial occlusion

paper_content:

As biometric vector may not follow the Gaussian distribution under complex light, pose and accessories, systems often yield unacceptable performance when subjected to impulsive, non-Gaussian noise. This paper adopts signal symmetric alpha stable distribution theory to construct fractional low-order independent component analysis algorithm (FLOD-ICA) and applied FLOD-ICA to solve the partial occlusion face recognition problem. Experiments verified that our proposed scheme is effective for face recognition under partial occlusion.
---
paper_title: Three-dimensional object recognition using joint fractional Fourier transform correlators with the help of digital Fresnel holography

paper_content:

Abstract Digital holography (DH) helps record true three-dimensional (3D) perspective of a 3D object. The technique of DH can be used for various applications, including 3D object recognition, which can be implemented optically. In this paper, 3D object recognition with joint fractional Fourier transform (JFRT) correlators is proposed in which the technique of digital Fresnel holography is used. Digital Fresnel hologram of a 3D object is synthesized and reconstructed digitally. After removal of the dc and twin image terms from the reconstructed image, two different target objects have been compared using linear and nonlinear JFRT correlators. The results are also verified using an optically recorded digital Fresnel hologram. The performance measure parameters, such as, discrimination ratio, peak-to-correlation energy, and peak-to-sidelobe ratio have been computed to compare the two proposed correlation schemes.
---
paper_title: Radar/SAR Image Resolution Enhancement via Unifying Descriptive Experiment Design Regularization and Wavelet-Domain Processing

paper_content:

Modern approaches for resolution enhancement (RE) and superresolution (SR) of coherent remote sensing (RS) imagery suggest to exploit the sparsity of the desired image representations in some appropriately chosen overcomplete dictionaries and treat the related RE/SR imaging inverse problems in descriptive settings imposing some structured regularization constraints. However, such approaches are not properly adapted to the SR recovery of the speckle-corrupted low resolution (LR) coherent radar imagery with preservation of salient image features. In this letter, we address a new multistage iterative SR technique for feature-enhanced radar/fractional synthetic aperture radar computational imaging. First, the despeckled high-resolution image is recovered from the LR speckle-corrupted radar image applying the descriptive-experiment-design-regularization-based reconstructive processing. Next, the multistage RE is consequently performed in each nested refined SR frame via the iterative reconstruction of the upscaled radar images, followed by the discrete-wavelet-transform-based sparsity-promoting denoising with guaranteed consistency preservation in each resolution frame.
---
paper_title: Fractional differential and variational method for image fusion and super-resolution

paper_content:

This paper introduces a novel fractional differential and variational model that includes the terms of fusion and super-resolution, edge enhancement and noise suppression. In image fusion and super-resolution term, the structure tensor is employed to describe the geometry of all the input images. According to the fact that the fused image and the source inputs should have the same or similar structure tensor, the energy functional of the image fusion and super-resolution is established combining with the down-sampling operator. For edge enhancement, the bidirectional diffusion term is incorporated into the image fusion and super-resolution model to enhance the visualization of the fused image. In the noise suppression term, a new variational model is developed based on the fractional differential and fractional total variation. Thanks to the above three terms, the proposed model can realize the image fusion, super-resolution, and the edge information enhancement simultaneously. To search for the optimal solution, a gradient descent iteration scheme derived from the Euler-Lagrange equation of the proposed model is employed. The numerical results indicate that the proposed method is feasible and effective.
---
paper_title: Novel image fusion method based on adaptive pulse coupled neural network and discrete multi-parameter fractional random transform

paper_content:

Abstract In this paper, we first propose the discrete multi-parameter fractional random transform (DMPFRNT), which can make the spectrum distributed randomly and uniformly. Then we introduce this new spectrum transform into the image fusion field and present a new approach for the remote sensing image fusion, which utilizes both adaptive pulse coupled neural network (PCNN) and the discrete multi-parameter fractional random transform in order to meet the requirements of both high spatial resolution and low spectral distortion. In the proposed scheme, the multi-spectral (MS) and panchromatic (Pan) images are converted into the discrete multi-parameter fractional random transform domains, respectively. In DMPFRNT spectrum domain, high amplitude spectrum (HAS) and low amplitude spectrum (LAS) components carry different informations of original images. We take full advantage of the synchronization pulse issuance characteristics of PCNN to extract the HAS and LAS components properly, and give us the PCNN ignition mapping images which can be used to determine the fusion parameters. In the fusion process, local standard deviation of the amplitude spectrum is chosen as the link strength of pulse coupled neural network. Numerical simulations are performed to demonstrate that the proposed method is more reliable and superior than several existing methods based on Hue Saturation Intensity representation, Principal Component Analysis, the discrete fractional random transform etc.
---
paper_title: A Novel Image Fusion Method Based on FRFT-NSCT

paper_content:

Nonsubsampled Contourlet transform (NSCT) has properties such as multiscale, localization, multidirection, and shift invariance, but only limits the signal analysis to the time frequency domain. Fractional Fourier transform (FRFT) develops the signal analysis to fractional domain, has many super performances, but is unable to attribute the signal partial characteristic. A novel image fusion algorithm ::: based on FRFT and NSCT is proposed and demonstrated in this paper. Firstly, take FRFT on the two source images to obtain fractional domain matrices. Secondly, the NSCT is performed on the aforementioned ::: matrices to acquire multiscale and multidirection images. Thirdly, take fusion rule for low-frequency subband coefficients and directional bandpass subband coefficients to get the fused coefficients. Finally, the ::: fused image is obtained by performing the inverse NSCT and inverse FRFT on the combined coefficients. Three modes images and three fusion rules are demonstrated in the proposed algorithm test. The simulation results show that the proposed fusion approach is better than the methods based on NSCT at the same parameters.
---
paper_title: Image fusion method based on adaptive pulse coupled neural network in the discrete fractional random transform domain

paper_content:

Abstract In this paper, we present a new approach for the remote sensing image fusion, which utilizes both adaptive pulse coupled neural network (PCNN) and the discrete fractional random transform in order to meet the requirements of both high spatial resolution and low spectral distortion. In the proposed scheme, the multi-spectral (MS) and panchromatic (Pan) images are converted to the discrete fractional random transform domains, respectively, which can make the spectrum distribute randomly and uniformly. In DFRNT spectrum domain, high amplitude spectrum (HAS) and low amplitude spectrum (LAS) components carry different information of original images. We take full advantage of pulse coupled neural network synchronization pulse issuance characteristics of PCNN to extract the HAS and LAS components properly, and give us the PCNN ignition mapping images which can be used to confirm the fusion parameters. In the fusion process, local standard deviation of amplitude spectrum is chosen as the link strength of pulse coupled neural network. Numerical simulations are performed to demonstrate that the proposed method is more reliable and superior than several existing methods based on Hue Saturation Intensity representation, Principal Component Analysis, the discrete fractional random transform, etc.
---
paper_title: Modified computational integral imaging-based double image encryption using fractional Fourier transform

paper_content:

Abstract In this paper, we propose an image encryption technique to simultaneously encrypt double or multiple images into one encrypted image using computational integral imaging (CII) and fractional Fourier transform (FrFT). In the encryption, each of the input plane images are located at different positions along a pickup plane, and simultaneously recorded in the form of an elemental image array (EIA) through a lenslet array. The recorded EIA to be encrypted is multiplied by FrFT with two different fractional orders. In order to mitigate the drawbacks of occlusion noise in computational integral imaging reconstruction (CIIR), the plane images can be reconstructed using a modified CIIR technique. To further improve the solution of the reconstructed plane images, a block matching algorithm is also introduced. Numerical simulation results verify the feasibility and effectiveness of the proposed method.
---
paper_title: Double-image encryption based on discrete multiple-parameter fractional angular transform and two-coupled logistic maps

paper_content:

A new discrete fractional transform defined by the fractional order, periodicity and vector parameters is presented, which is named as the discrete multiple-parameter fractional angular transform. Based on this transform and two-coupled logistic map, a double-image encryption scheme is proposed. First, an enlarged image is obtained by connecting two plaintext images sequentially and scrambled by using a chaotic permutation process, in which the sequences of chaotic pairs generated by using the two-coupled logistic map. Then, the scrambled enlarged image is decomposed into two new components. Second, a chaotic random phase mask is generated based on the logistic map, with which one of two components is converted to the modulation phase mask. Another component is encoded into an interim matrix with the help of the modulation phase mask. Finally, the two-dimensional discrete multiple-parameter fractional angular transform is performed on the interim matrix to obtain the ciphertext with stationary white noise distribution. The proposed encryption scheme has an obvious advantage that no phase keys are used in the encryption and decryption process, which is convenient to key management. Moreover, the security of the cryptosystem can be enhanced by using extra parameters such as initial values of chaos functions, fractional orders and vector parameters of transform. Simulation results and security analysis verify the feasibility and effectiveness of the proposed scheme.
---
paper_title: Image encryption based on nonseparable fractional Fourier transform and chaotic map

paper_content:

Abstract In this paper an image cryptosystem is constructed by using double random phase masks and a chaotic map together with a novel transform which is similar to fractional Fourier transform and gyrator transform to some extent. The new transform is not periodic with respect to the transform order and cannot be expressed as a tensor product of two one-dimensional transforms neither in the space domain nor in the Wigner space-frequency domain. In the cryptosystem, the parameters of Arnold map, transform orders of the proposed transform and phase information serve as the main keys. The numerical simulations have demonstrated the validity and high security level of the image cryptosystem based on the proposed transform.
---
paper_title: The multiple-parameter discrete fractional Fourier transform

paper_content:

The discrete fractional Fourier transform (DFRFT) is a generalization of the discrete Fourier transform (DFT) with one additional order parameter. In this letter, we extend the DFRFT to have N order parameters, where N is the number of the input data points. The proposed multiple-parameter discrete fractional Fourier transform (MPDFRFT) is shown to have all of the desired properties for fractional transforms. In fact, the MPDFRFT reduces to the DFRFT when all of its order parameters are the same. To show an application example of the MPDFRFT, we exploit its multiple-parameter feature and propose the double random phase encoding in the MPDFRFT domain for encrypting digital data. The proposed encoding scheme in the MPDFRFT domain significantly enhances data security.
---
paper_title: An Enhanced Image Encryption Algorithm Using Fractional Chaotic Systems

paper_content:

Abstract The attractors in many fractional-order chaotic systems have more complex dynamics than their integer-order systems. Consequently, there is a recent trend of exploring fractional chaotic systems for the design of image cryptosystems. The fractional derivative orders can act as part of secret key. Recently, an image encryption algorithm is suggested based on synchronized fractional-order chaotic systems. In this paper, a security appraisal is actioned to evince that the encryption algorithm is completely breakable under chosen-plaintext attack, has no mandatory plaintext sensitivity, and unsafe. Moreover, an enhanced version of encryption scheme is accounted to improve its attack defensibleness, plaintext sensitivity and inherent encryption strength. Experimental analyses are exercised and statistical results are catered to manifest the improved performance. The anticipated enhanced version has high security, consistency and practicability for encryption of digital images.
---
paper_title: Image Encryption With Multiorders of Fractional Fourier Transforms

paper_content:

The original information in the existing security system based on the fractional Fourier transform (FRFT) is essentially protected by only a certain order of FRFT. In this paper, we propose a novel method to encrypt an image by multiorders of FRFT. In the image encryption, the encrypted image is obtained by the summation of different orders inverse discrete FRFT of the interpolated subimages. And the original image can be perfectly recovered using the linear system constructed by the fractional Fourier domain analysis of the interpolation. The proposed method can be applied to the double or more image encryptions. Applying the transform orders of the utilized FRFT as secret keys, the proposed method is with a larger key space than the existing security systems based on the FRFT. Additionally, the encryption scheme can be realized by the fast-Fourier-transform-based algorithm and the computation burden shows a linear increase with the extension of the key space. It is verified by the experimental results that the image decryption is highly sensitive to the deviations in the transform orders.
---
paper_title: A non-linear preprocessing for opto-digital image encryption using multiple-parameter discrete fractional Fourier transform

paper_content:

Abstract In this paper, a novel opto-digital image encryption technique is proposed by introducing a new non-linear preprocessing and using the multiple-parameter discrete fractional Fourier transform (MPDFrFT). The non-linear preprocessing is performed digitally on the input image in the spatial domain using a piecewise linear chaotic map (PLCM) coupled with the bitwise exclusive OR (XOR). The resulting image is multiplied by a random phase mask before applying the MPDFrFT to whiten the image. Then, a chaotic permutation is performed on the output of the MPDFrFT using another PLCM different from the one used in the spatial domain. Finally, another MPDFrFT is applied to obtain the encrypted image. The parameters of the PLCMs together with the multiple fractional orders of the MPDFrFTs constitute the secret key for the proposed cryptosystem. Computer simulation results and security analysis are presented to show the robustness of the proposed opto-digital image encryption technique and the great importance of the new non-linear preprocessing introduced to enhance the security of the cryptosystem and overcome the problem of linearity encountered in the existing permutation-based opto-digital image encryption schemes.
---
paper_title: The Effect of Fractional Fourier Transform Angle in Encryption Quality for Digital Images

paper_content:

This paper investigates the effect of the angle of the Fractional Fourier transform (FrFT) on encryption quality in a chaotic image cryptosystem. The adopted approach for encryption is 2D chaotic Baker map. This map is applied to the transformed images using FrFT with different angles. We inspect its application to digital images and its exhaustive security investigation. The security investigation contains statistical tests like histogram and correlation coefficient, the maximum and the irregular deviation metrics, and the processing time. Experimental results prove that the rotation angle of the FrFT does not affect the encryption quality. This indicates a wide range of angles that can be used in this transform domain for encryption in addition to increasing the scheme sensitivity and making it hard for the attacker to exactly determine the right angle used in the transform domain.
---
paper_title: Color image encryption based on color blend and chaos permutation in the reality-preserving multiple-parameter fractional Fourier transform domain

paper_content:

Abstract In this paper, we propose a novel color image encryption method by using Color Blend (CB) and Chaos Permutation (CP) operations in the reality-preserving multiple-parameter fractional Fourier transform (RPMPFRFT) domain. The original color image is first exchanged and mixed randomly from the standard red–green–blue (RGB) color space to R ′ G ′ B ′ color space by rotating the color cube with a random angle matrix. Then RPMPFRFT is employed for changing the pixel values of color image, three components of the scrambled RGB color space are converted by RPMPFRFT with three different transform pairs, respectively. Comparing to the complex output transform, the RPMPFRFT transform ensures that the output is real which can save storage space of image and convenient for transmission in practical applications. To further enhance the security of the encryption system, the output of the former steps is scrambled by juxtaposition of sections of the image in the reality-preserving multiple-parameter fractional Fourier domains and the alignment of sections is determined by two coupled chaotic logistic maps. The parameters in the Color Blend, Chaos Permutation and the RPMPFRFT transform are regarded as the key in the encryption algorithm. The proposed color image encryption can also be applied to encrypt three gray images by transforming the gray images into three RGB color components of a specially constructed color image. Numerical simulations are performed to demonstrate that the proposed algorithm is feasible, secure, sensitive to keys and robust to noise attack and data loss.
---
paper_title: New encryption method of 2D image by use of the fractional Fourier transform

paper_content:

Recently, secure transmission of image information has been very significant especially with the development in the Internet. In general, encryption of image has widely been known as one of the effective methods to realize the secure transmission. In this study, we propose a new encryption method of 2D image by use of the fractional Fourier transform (FRT) which is the generalization of the conventional Fourier transform (FT). Specifically the FRT with different order of the FT is conducted to the grayscale distribution in each line of the original image. The validity of our proposed encryption method is shown by analyzing the cross-correlation properties between the original image and the encrypted image.
---
paper_title: Image encryption based on the multiple-parameter discrete fractional Fourier transform and chaos function

paper_content:

In recent years, the chaos-based cryptographic algorithms have suggested some new and efficient ways to develop secure image encryption techniques. In this paper, we propose a new approach for image encryption based on the multiple-parameter discrete fractional Fourier transform and chaotic logistic maps in order to meet the requirements of the secure image transmission. In the proposed image encryption scheme, the image is encrypted by juxtaposition of sections of the image in the multiple-parameter discrete fractional Fourier domains and the alignment of sections is determined by chaotic logistic maps. This method does not require the use of phase keys. The new method has been compared with several existing methods and shows comparable or superior robustness to blind decryption.
---
paper_title: Vulnerability to known-plaintext attack of optical encryption schemes based on two fractional Fourier transform order keys and double random phase keys

paper_content:

From the perspective of optical information security, we demonstrate that conventional phase retrieval algorithms, such as the Gerchberg–Saxton algorithm, can be still valid when extended from the Fourier transform domain into the fractional Fourier transform (FRFT) domain. Based on this extension, we propose a method of known-plaintext attack (KPA) on double random phase encoding in the fractional Fourier transform domain (DRPE-FRFT). With this attack, an opponent can access two encryption phase keys with the help of the phase retrieval algorithm in the FRFT domain, and meanwhile obtain the two FRFT order keys by using an exhaustive search. This indicates that the optical encryption scheme based on the DRPE-FRFT scheme is vulnerable to KPA due to the nature of the linearity of FRFT. The validity of this attack is verified by numerical simulations.
---
paper_title: Double-image encryption scheme combining DWT-based compressive sensing with discrete fractional random transform

paper_content:

Abstract A new discrete fractional random transform based on two circular matrices is designed and a novel double-image encryption–compression scheme is proposed by combining compressive sensing with discrete fractional random transform. The two random circular matrices and the measurement matrix utilized in compressive sensing are constructed by using a two-dimensional sine Logistic modulation map. Two original images can be compressed, encrypted with compressive sensing and connected into one image. The resulting image is re-encrypted by Arnold transform and the discrete fractional random transform. Simulation results and security analysis demonstrate the validity and security of the scheme.
---
paper_title: Image encryption based on the fractional Fourier transform over finite fields

paper_content:

In this paper, we introduce fractional Fourier transforms over finite fields GF(p), where p=1(mod4). Such a definition is based on the approach given in [1], which corresponds to a finite field extension of the commuting matrix method for defining discrete fractional Fourier transforms [2]. The transforms we have constructed are then used as the basis of a novel image encryption scheme. Security aspects of such a scheme are analyzed through computer simulations and specific metrics.
---
paper_title: Security of image encryption scheme based on multi-parameter fractional Fourier transform

paper_content:

Abstract Recently, multi-parameter fractional Fourier transform (MPFRFT) has been widely applied in the optics cryptosystem, which has attracted more and more researchers’ attention. However, in further study we find a serious security problem on the MPFRFT which is the multi-choice of decryption key corresponding to an encryption key. The existence of multi-decryption-key hinders the application of this algorithm. We present a new generalized fractional Fourier transform, which can overcome the problem and enlarge the key space. The simulation results show that the proposed algorithm has higher security and key sensitivity.
---
paper_title: Image encryption using block based transformation with fractional Fourier transform

paper_content:

In order to transmit image data in open network, a novel image encryption algorithm based on fractional Fourier transform and block-based transformation is proposed in this paper. The image encryption process includes two steps: the original image was divided into blocks, which were rearranged into a transformed image using a transformation algorithm, and then the transformed image was encrypted using the fractional Fourier transform (FRFT) algorithm. The security of the proposed algorithm depends on the transformation algorithm, sensitivity to the randomness of phase mask and the orders of FRFT. Theoretical analysis and experimental results demonstrate that the algorithm is favorable.
---
paper_title: Asymmetric multiple-image encryption based on the cascaded fractional Fourier transform

paper_content:

Abstract A multiple-image cryptosystem is proposed based on the cascaded fractional Fourier transform. During an encryption procedure, each of the original images is directly separated into two phase masks. A portion of the masks is subsequently modulated into an interim mask, which is encrypted into the ciphertext image; the others are used as the encryption keys. Using phase truncation in the fractional Fourier domain, one can use an asymmetric cryptosystem to produce a real-valued noise-like ciphertext, while a legal user can reconstruct all of the original images using a different group of phase masks. The encryption key is an indivisible part of the corresponding original image and is still useful during decryption. The proposed system has high resistance to various potential attacks, including the chosen-plaintext attack. Numerical simulations also demonstrate the security and feasibility of the proposed scheme.
---
paper_title: Image compression and encryption scheme based on 2D compressive sensing and fractional Mellin transform

paper_content:

Abstract Most of the existing image encryption techniques bear security risks for taking linear transform or suffer encryption data expansion for adopting nonlinear transformation directly. To overcome these difficulties, a novel image compression–encryption scheme is proposed by combining 2D compressive sensing with nonlinear fractional Mellin transform. In this scheme, the original image is measured by measurement matrices in two directions to achieve compression and encryption simultaneously, and then the resulting image is re-encrypted by the nonlinear fractional Mellin transform. The measurement matrices are controlled by chaos map. The Newton Smoothed l 0 Norm ( NSL 0 ) algorithm is adopted to obtain the decryption image. Simulation results verify the validity and the reliability of this scheme.
---
paper_title: A secure fast 2D — Discrete fractional fourier transform based medical image compression using hybrid encoding technique

paper_content:

The arrival of CT and MRI imaging in the last two decades used for analysis of various diseases in medical field. Magnetic Resonance Imaging (MRI) and Computed Tomography (CT) sequences have become essential in healthcare systems and an integral segment of a patient's medical record. Such medical image data needed a huge amount of resources for storage and transmission. Compression of medical images for efficient use of storage space and transmission bit rate has become a necessity. A quasigroup successfully creates an astronomical number of keys which is used to confuse the hackers in determining the original data. However, quasi group encryption is not capable in diffusing the information of the plain text. Hence, in this study the DICOM images are encrypted using a chained Hadamard transforms and Number Theoretic Transforms to introduce diffusion along with the quasigroup transformation. A secure Fast Two Dimensional Discrete Fractional Fourier Transform (DFRFT) and a SPIHT Algorithm with Huffman Encoder is used for compress the image and it provides a secure compression of medical images as compared to the other transform. The experimental results evaluate the performance of the proposed encryption approach based on the PSNR and MSE it shows the proposed approach gives better results.
---
paper_title: Digital image watermarking using fractional fourier transform via image compression

paper_content:

Recently, works are completely based on the internet, so authentication is required to protect the content of the data because owner never wants to degrade the quality of his data. Information hiding via digital watermarks for the multimedia data is the compatible way to provide the protection of data. Watermarks are imperceptible and it is predefined pattern inserting into multimedia data to protect or authenticity. The watermark indicates that data is containing a copyright or not. Many digital watermarking algorithms have been proposed in spatial and transform domain. The performance of the digital watermarking schemes is evaluated as tradeoffs between images embedded on it, robustness against attacks and embedding induced distortions. This work provides Digital image watermarking using Fractional Fourier transform via image compression for better results comparatively previous techniques of information hiding.
---
paper_title: Image Compression by Using Fractional Transforms

paper_content:

This paper deals with the image compression using fractional Fourier transform (FRFT), fractional cosine transform (FRCT), Fourier transform (FT) and cosine transform (CT). A comparison has been made between these techniques and observed that a good fidelity of decompressed image can be achieved at different fractional values of the transforms. The FRFT provides better mean square error (MSE) and peak signal noise ratio (PSNR) for the same compression ratio (CR) as compared to FRCT while processing image quality.
---
paper_title: Blurred defocused image restoration based on FRFT

paper_content:

A technique for restoring the blurred image resulted from defocusing of the lens is proposed in this paper, which is based on fractional Fourier transform (FRFT). The FRFT, as a powerful tool for the analysis of time-varying signals, is closely connected with the optical imaging system. FRFT also can describe optical imaging process just like Fresnel diffractions, so a defocused imaging model based on FRFT is established to explain the blur phenomena of defocusing image. The defocused imaging model is greatly different from the traditional point spread function (PSF) model, and enables to uncover the blur nature of non-focus image. Then, an image restoration method using the novel model is proposed to handle the blurred defocused image. The method adopted a new iterative phase retrieval approach which can approximately estimate phase signals from intensity signals of a single defocused image by means of FRFT. Restoring image may acquire sharp image by implementing inverse FRFT on complex image signal made from the estimated phase signals and intensity signals. Experimental results demonstrate that the method is effective in restoring blurred defocused image.
---
paper_title: A retina-filtering model and its application in image restoration

paper_content:

It is one of the attractive and challenging topics in the field of computer vision to reveal the mechanism of the biological visual system and gain useful insights from it. According to the experimental results of physiology, this paper analyzes the role of the image transmission components in mammalian retina for improving imaging clarity. Then, based on the theory of Bayesian estimation, a kind of retina-filtering model is proposed, of which the parameters were determined by Green's solution of fractional diffusion equation and the optical characteristics of eye. Finally, experiment results in image restoration show the effectiveness and feasibility of the proposed method.
---
paper_title: A fractional order variational model for the robust estimation of optical flow from image sequences

paper_content:

Abstract In this paper, a fractional order variational model for estimating the optical flow is presented. In particular, the proposed model generalizes the integer order variational optical flow models. The fractional order derivative describes discontinuous information about texture and edges, and therefore a more suitable in estimating the optical flow. The proposed variational functional is a combination of a global model of Horn and Schunck and the classical model of Nagel and Enkelmann. This formulation yields a dense flow and preserves discontinuities in the flow field and also provides a significant robustness against outliers. The Grunwald–Letnikov derivative is used for solving complex fractional order partial differential equations. The corresponding linear system of equations is solved by an efficient numerical scheme. A detailed stability and convergence analysis is given in order to show the mathematical applicability of the numerical algorithm. Experimental results on various datasets verify the validity of the proposed model.
---
paper_title: Repeated Filtering in Consecutive Fractional Fourier Domains and Its Application to Signal Restoration

paper_content:

Filtering in a single time domain or in a single frequency domain has been generalized to filtering in a single fractional Fourier domain. We generalize this to repeated filtering in consecutive fractional Fourier domains and discuss its applications to signal restoration through an illustrative example.
---
paper_title: Optimum signal and image recovery by the method of alternating projections in fractional Fourier domains

paper_content:

Abstract This paper presents a signal and image recovery scheme by the method of alternating projections onto convex sets in optimum fractional Fourier domains. It is shown that the fractional Fourier domain order with minimum bandwidth is the optimum fractional Fourier domain for the method employing alternating projections in signal recovery problems. Following the estimation of optimum fractional Fourier transform orders, incomplete signal is projected onto different convex sets consecutively to restore the missing part. Using a priori information in optimum fractional Fourier domains, superior results are obtained compared to the conventional Fourier domain restoration. The algorithm is tested on 1-D linear frequency modulated signals, real biological data and 2-D signals presenting chirp-type characteristics. Better results are obtained in the matched fractional Fourier domain, compared to not only the conventional Fourier domain restoration, but also other fractional Fourier domains.
---
paper_title: An automatic regularization parameter selection algorithm in the total variation model for image deblurring

paper_content:

Image restoration is an inverse problem that has been widely studied in recent years. The total variation based model by Rudin-Osher-Fatemi (1992) is one of the most effective and well known due to its ability to preserve sharp features in restoration. This paper addresses an important and yet outstanding issue for this model in selection of an optimal regularization parameter, for the case of image deblurring. We propose to compute the optimal regularization parameter along with the restored image in the same variational setting, by considering a Karush Kuhn Tucker (KKT) system. Through establishing analytically the monotonicity result, we can compute this parameter by an iterative algorithm for the KKT system. Such an approach corresponds to solving an equation using discrepancy principle, rather than using discrepancy principle only as a stopping criterion. Numerical experiments show that the algorithm is efficient and effective for image deblurring problems and yet is competitive.
---
paper_title: Optimal image restoration with the fractional Fourier transform

paper_content:

The classical Wiener filter, which can be implemented in O(N log N) time, is suited best for space-invariant degradation models and space-invariant signal and noise characteristics. For space-varying degradations and nonstationary processes, however, the optimal linear estimate requires O(N2) time for implementation. Optimal filtering in fractional Fourier domains permits reduction of the error compared with ordinary Fourier domain Wiener filtering for certain types of degradation and noise while requiring only O(N log N) implementation time. The amount of reduction in error depends on the signal and noise statistics as well as on the degradation model. The largest improvements are typically obtained for chirplike degradations and noise, but other types of degradation and noise may also benefit substantially from the method (e.g., nonconstant velocity motion blur and degradation by inhomegeneous atmospheric turbulence). In any event, these reductions are achieved at no additional cost.
---
paper_title: Fractional nonlinear anisotropic diffusion with p-Laplace variation method for image restoration

paper_content:

In this paper, a novel class of fractional-order nonlinear anisotropic diffusion equations based image restoration model is established, which employs the p-Laplace norm of fractional-order gradient of an image intensity function. The role of the fractional-order gradient is to better accommodate the texture details of an image, and the adaptive factor p can be used to diffuse adaptively according to local geometry features, which are fractional-order curvature and fractional-order gradient of an image. Besides removing noise and non-linearly keeping high-frequency edge of images efficiently, our proposed model can enhance the texture details of images and greatly eliminate the staircase effects and also the speckle effects. Fourier transform technique is also proposed to compute the fractional order derivative. Experimental results illustrate that our proposed model can deal with edge preserving and texture enhancing, more efficiently than the other four methods and outperform the other four methods by means of PSNR. Our average PSNR is closed to 1dB higher than the average PSNRs of the other four methods.
---
paper_title: Fractional-order total variation combined with sparsifying transforms for compressive sensing sparse image reconstruction

paper_content:

A FrTV combined with sparsifying transforms for CS sparse image reconstruction model is proposed.A method for estimating the regularization parameter is proposed.A gradient projection algorithm is developed to solve this reconstruction model.The algorithm has a higher PSNR and better detail preservation. The total variation (TV) model has been considered to be one of the most successful and representative model for compressive sensing (CS) sparse image reconstruction due to its advantage of preserving image edges. However, TV regularized term often favors piecewise constant solution and therefore it fails to preserve the details and textures. To overcome this defect and reconstruct the fine details, this paper proposes a two-dimensional CS sparse image reconstruction model by introducing the fractional-order TV (FrTV) regularization constraint into CS optimization problem. Furthermore, in order to achieve sparser representation flexibly, a combination of discrete wavelet transform and curvelet transform ź 1 -norm regularization is also incorporated into the cost function and a method for estimating the regularization parameter that trades off the two terms in the cost function is proposed. By using a smooth approximation of the ź 1 -norm, a gradient projection algorithm is derived to solve the combined FrTV and sparsifying transforms constrained minimization problem effectively. Compared with several state-of-the-art reconstruction algorithms, the proposed algorithm is more efficient and robust, not only yielding higher peak-signal-to-noise ratio, but also reconstructing the fine details and textures more efficiently.
---
paper_title: Image restoration based on the discrete fraction Fourier transform

paper_content:

The fractional Fourier transform is the powerful tool for time-variant signal analysis. For space-variant degradation and non-stationary processes the filtering in fractional Fourier domains permits reduction of the error compared with ordinary Fourier domain filtering. In this paper the concept of filtering in fractional Fourier domains is applied to the problem of estimating degraded images. Efficient digital implementation using discrete Hermite eigenvectors can provide similar results to match the continuous outputs. Expressions for the 2D optimal filter function in fractional domains will be given for transform domains characterized by the two rotation angle parameters of the 2D fractional Fourier transform. The proposed method is used to restore images that have several degradations in the experiments. The results show that the method presented in this paper is valid.
---

Title: Fractional Calculus in Image Processing: A Review

Section 1: Introduction
Description 1: Introduce the historical background of fractional calculus and its growing applications in various fields, including image processing. Also, outline the structure of the paper.

Section 2: Basics Facts on Fractional-order Derivative
Description 2: Provide an overview of fractional-order differentiation and its general principles.

Section 3: Fractional-order Definition
Description 3: Detail the three popular definitions of fractional calculus operators: Grünwald-Letnikov (G-L), Riemann-Liouville (R-L), and Caputo.

Section 4: Fractional-order Discretization Methods and Toolboxes
Description 4: Explain the numerical methods for calculating fractional-order derivatives and discuss available Matlab toolboxes.

Section 5: Grünwald-Letnikov Discretization Method
Description 5: Describe the Grünwald-Letnikov discretization method and its implementation.

Section 6: Matrix Discretization Method
Description 6: Discuss the matrix discretization method for fractional-order differentiation.

Section 7: A Review of Fractional-order Image Processing Methods
Description 7: Summarize and analyze various fractional-order based image processing methods, focusing on different sub-fields.

Section 8: Image Enhancement
Description 8: Examine how fractional-order techniques improve image enhancement, particularly in preserving texture and contour features.

Section 9: Image Denoising
Description 9: Explore fractional-order methods for image denoising, including operator mask-based methods and model-based methodologies.

Section 10: Image Edge Detection
Description 10: Investigate the use of fractional-order differentiation in edge detection and its advantages over integer-order methods.

Section 11: Image Segmentation
Description 11: Discuss the application of fractional-order methods in image segmentation and their impact on accuracy and noise robustness.

Section 12: Image Registration
Description 12: Review the use of fractional-order methods in image registration, including spatial transformations and variational models.

Section 13: Image Recognition
Description 13: Analyze how fractional-order techniques can be used to reduce dimensionality and improve image recognition.

Section 14: Image Fusion
Description 14: Discuss how fractional-order methods enhance image fusion techniques.

Section 15: Image Encryption
Description 15: Explain the role of fractional-order techniques in image encryption and their benefits in security.

Section 16: Image Compression
Description 16: Describe fractional-order methods for image compression and their effectiveness in maintaining image quality.

Section 17: Image Restoration
Description 17: Discuss how fractional-order methods can be applied to restore damaged or degraded images.

Section 18: Conclusion
Description 18: Summarize the key points of the paper and suggest future directions for research in fractional-order image processing.