Fractional Calculus in Image Processing: A Review

Yang, Qi; Chen, Dali; Zhao, Tiebiao; Chen, YangQuan

doi:10.1515/fca-2016-0063

Cited by 248 publications

(115 citation statements)

References 101 publications

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“…Definition 4.1: When three boundary conditions defined in Eqs. (17), (18), (21) and (22) are drawn in (a, c)−parameter plane together, the plane separates to many regions. The most basic property of these regions is that all points in any region have the same number of stable or unstable roots [39].…”

Section: Parametric Stability Analysis Of Fractional Differential Equmentioning

confidence: 99%

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Stability analysis of fractional differential equations with unknown parameters

Köksal

2019

NAMC

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In this paper, the stability of fractional differential equations (FDEs) with unknown parameters is studied. FDEs bring many advantages to model the physical systems in nature or man-made systems in industry. In fact, real objects are generally fractional and fractional calculus has gained popularity in modelling physical and engineering systems in the last few decades in parallel to advancement of high speed computers. Using the graphical based D-decomposition method, we investigate the parametric stability analysis of FDEs without complicated mathematical analysis. To achieve this, stability boundaries are obtained firstly, and then the stability region set depending on the unknown parameters is found. The applicability of the presented method is shown considering some benchmark equations which are often used to verify the results of a new method. Simulation examples shown that the method is simple and give reliable stability results.

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Section: Parametric Stability Analysis Of Fractional Differential Equmentioning

confidence: 99%

“…(29) whose real and infinite root boundaries are given by Eqs. (17) and (18) are obtained for α 2 = 2α and α 1 = α in Eqs. (21) and (22) as follows:…”

Section: Examplementioning

confidence: 99%

Stability analysis of fractional differential equations with unknown parameters

Köksal

2019

NAMC

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“…Three popular definitions for fractional calculus were given by Grünwald [27]. For numerical calculation of fractional-order derivatives we can use the relation derived from the G-L definition.…”

Section: Linear Spatial Pyramid and Frameworkmentioning

confidence: 99%

Single image super-resolution using self-optimizing mask via fractional-order gradient interpolation and reconstruction

et al. 2018

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Image super-resolution using self-optimizing mask via fractional-order gradient interpolation and reconstruction aims to recover detailed information from low-resolution images and reconstruct them into high-resolution images. Due to the limited amount of data and information retrieved from low-resolution images, it is difficult to restore clear, artifact-free images, while still preserving enough structure of the image such as the texture. This paper presents a new single image super-resolution method which is based on adaptive fractional-order gradient interpolation and reconstruction. The interpolated image gradient via optimal fractional-order gradient is first constructed according to the image similarity and afterwards the minimum energy function is employed to reconstruct the final high-resolution image. Fractional-order gradient based interpolation methods provide an additional degree of freedom which helps optimize the implementation quality due to the fact that an extra free parameter α-order is being used. The proposed method is able to produce a rich texture detail while still being able to maintain structural similarity even under large zoom conditions. Experimental results show that the proposed method performs better than current single image super-resolution techniques.

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“…Because of its ability to preserve the texture details of the smooth region while highlighting the image edge feature and its spatiotemporal memory of the target point neighborhood, a fractional differential is applied to many image processing fields [9], such as image denoising [10], image enhancement [11], and motion estimation [12][13][14][15][16]. In [12,13], the fractionalorder smoothing constraint equation was used in the HS optical flow model to preserve the discontinuity of motion estimation, but it does not consider the correlation of the pixel intensity.…”

Section: Introductionmentioning

confidence: 99%

An Improved Fractional-Order Optical Flow Model for Motion Estimation

Zhu

Tian

et al. 2018

Mathematical Problems in Engineering

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The Horn and Schunck (HS) optical flow model cannot preserve discontinuity of motion estimation and has low accuracy especially for the image sequence, which includes complex texture. To address this problem, an improved fractional-order optical flow model is proposed. In particular, the fractional-order Taylor series expansion is applied in the brightness constraint equation of the HS model. The fractional-order flow field derivative is also used in the smoothing constraint equation. The Euler-Lagrange equation is utilized for the minimization of the energy function of the fractional-order optical flow model. Two-dimensional fractional differential masks are proposed and applied to the calculation of the model simplification. Considering the spatiotemporal memory property of fractional-order, the algorithm preserves the edge discontinuity of the optical flow field while improving the accuracy of the estimation of the dense optical flow field. Experiments on Middlebury datasets demonstrate the predominance of our proposed algorithm.

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Fractional Calculus in Image Processing: A Review

Abstract: Over the last decade, it has been demonstrated that many systems in science and engineering can be modeled more accurately by fractional-order than integer-order derivatives, and many methods are developed to solve the problem of fractional systems. Due to the extra free parameter order

Cited by 248 publications

References 101 publications

Stability analysis of fractional differential equations with unknown parameters

Stability analysis of fractional differential equations with unknown parameters

Single image super-resolution using self-optimizing mask via fractional-order gradient interpolation and reconstruction

An Improved Fractional-Order Optical Flow Model for Motion Estimation

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