A Fractional-Order Variational Framework for Retinex: Fractional-Order Partial Differential Equation-Based Formulation for Multi-Scale Nonlocal Contrast Enhancement with Texture Preserving

This paper discusses a novel conceptual formulation of the fractional-order variational framework for retinex, which is a fractional-order partial differential equation (FPDE) formulation of retinex for the multi-scale nonlocal contrast enhancement with texture preserving. The well-known shortcomings of traditional integer-order computation-based contrast-enhancement algorithms, such as ringing artefacts and staircase effects, are still in great need of special research attention. Fractional calculus has potentially received prominence in applications in the domain of signal processing and image processing mainly because of its strengths like long-term memory, nonlocality, and weak singularity, and because of the ability of a fractional differential to enhance the complex textural details of an image in a nonlinear manner. Therefore, in an attempt to address the aforementioned problems associated with traditional integer-order computation-based contrast-enhancement algorithms, we have studied here, as an interesting theoretical problem, whether it will be possible to hybridize the capabilities of preserving the edges and the textural details of fractional calculus with texture image multi-scale nonlocal contrast enhancement. Motivated by this need, in this paper, we introduce a novel conceptual formulation of the fractional-order variational framework for retinex. First, we implement the FPDE by means of the fractional-order steepest descent method. Second, we discuss the implementation of the restrictive fractional-order optimization algorithm and the fractional-order Courant–Friedrichs–Lewy condition. Third, we perform experiments to analyze the capability of the FPDE to preserve edges and textural details, while enhancing the contrast. The capability of the FPDE to preserve edges and textural details is a fundamental important advantage, which makes our proposed algorithm superior to the traditional integer-order computation-based contrast enhancement algorithms, especially for images rich in textural details.

[1]  Alessandro Rizzi,et al.  A computational approach to color adaptation effects , 2000, Image Vis. Comput..

[2]  Michael Elad,et al.  A Variational Framework for Retinex , 2002, IS&T/SPIE Electronic Imaging.

[3]  Yi-Fei Pu,et al.  A texture image denoising approach based on fractional developmental mathematics , 2015, Pattern Analysis and Applications.

[4]  Ahmed S Elwakil,et al.  Fractional-order circuits and systems: An emerging interdisciplinary research area , 2010, IEEE Circuits and Systems Magazine.

[5]  Yi-Fei Pu,et al.  Fractional-Order Euler-Lagrange Equation for Fractional-Order Variational Method: A Necessary Condition for Fractional-Order Fixed Boundary Optimization Problems in Signal Processing and Image Processing , 2016, IEEE Access.

[6]  Robert A. Schowengerdt,et al.  Spatially Variant Contrast Enhancement Using Local Range Modification , 1983 .

[7]  Michael K. Ng,et al.  A Total Variation Model for Retinex , 2011, SIAM J. Imaging Sci..

[8]  M I Sezn,et al.  Automatic anatomically selective image enhancement in digital chest radiography. , 1989, IEEE transactions on medical imaging.

[9]  Yi-Fei Pu,et al.  Measurement Units and Physical Dimensions of Fractance-Part I: Position of Purely Ideal Fractor in Chua’s Axiomatic Circuit Element System and Fractional-Order Reactance of Fractor in Its Natural Implementation , 2016, IEEE Access.

[10]  Zhang Yi,et al.  Fractional Hopfield Neural Networks: Fractional Dynamic Associative Recurrent Neural Networks , 2017, IEEE Transactions on Neural Networks and Learning Systems.

[11]  Yi-Fei Pu,et al.  Fracmemristor: Fractional-Order Memristor , 2016, IEEE Access.

[12]  Zia-ur Rahman,et al.  Multi-scale retinex for color image enhancement , 1996, Proceedings of 3rd IEEE International Conference on Image Processing.

[13]  S. Manabe.  A Suggestion of Fractional-Order Controller for Flexible Spacecraft Attitude Control , 2002 .

[14]  Zhou Ji-liu,et al.  Research on Application of Fractional Calculus , 2011 .

[15]  Necati Özdemir,et al.  Fractional diffusion-wave problem in cylindrical coordinates , 2008 .

[16]  G. Johnson,et al.  Regionally Adaptive Histogram Equalization of the Chest , 1987, IEEE Transactions on Medical Imaging.

[17]  Zia-ur Rahman,et al.  Properties and performance of a center/surround retinex , 1997, IEEE Trans. Image Process..

[18]  P. G. Tahoces,et al.  Enhancement of chest and breast radiographs by automatic spatial filtering. , 1991, IEEE transactions on medical imaging.

[19]  Edoardo Provenzi,et al.  Issues About Retinex Theory and Contrast Enhancement , 2009, International Journal of Computer Vision.

[20]  John J. McCann,et al.  Retinex in Matlab , 2000, CIC.

[21]  Zia-ur Rahman,et al.  A multiscale retinex for bridging the gap between color images and the human observation of scenes , 1997, IEEE Trans. Image Process..

[22]  I. Podlubny,et al.  Analogue Realizations of Fractional-Order Controllers , 2002 .

[23]  John D. Austin,et al.  Adaptive histogram equalization and its variations , 1987 .

[24]  M. Shitikova,et al.  Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids , 1997 .

[25]  Demetri Terzopoulos,et al.  Image Analysis Using Multigrid Relaxation Methods , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[26]  Alessandro Rizzi,et al.  Random Spray Retinex: A New Retinex Implementation to Investigate the Local Properties of the Model , 2007, IEEE Transactions on Image Processing.

[27]  Jinzhu Yang,et al.  An image-enhancement method based on variable-order fractional differential operators. , 2015, Bio-medical materials and engineering.

[28]  Yi-Fei Pu,et al.  Measurement Units and Physical Dimensions of Fractance-Part II: Fractional-Order Measurement Units and Physical Dimensions of Fractance and Rules for Fractors in Series and Parallel , 2016, IEEE Access.

[29]  Robert A. Hummel,et al.  Image Enhancement by Histogram transformation , 1975 .

[30]  Yi-Fei Pu,et al.  Analog Circuit Realization of Arbitrary-Order Fractional Hopfield Neural Networks: A Novel Application of Fractor to Defense Against Chip Cloning Attacks , 2016, IEEE Access.

[31]  Todd J. Freeborn,et al.  A Survey of Fractional-Order Circuit Models for Biology and Biomedicine , 2013, IEEE Journal on Emerging and Selected Topics in Circuits and Systems.

[32]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[33]  Yiguang Liu,et al.  Fractional partial differential equation denoising models for texture image , 2014, Science China Information Sciences.

[34]  S. Pizer,et al.  Adaptive grey level assignment in CT scan display. , 1984, Journal of computer assisted tomography.

[35]  Yuriy Povstenko,et al.  Solutions to the fractional diffusion-wave equation in a wedge , 2014 .

[36]  Yi-Fei Pu,et al.  Defense Against Chip Cloning Attacks Based on Fractional Hopfield Neural Networks , 2017, Int. J. Neural Syst..

[37]  Alan C. Bovik,et al.  Blind Image Quality Assessment: From Natural Scene Statistics to Perceptual Quality , 2011, IEEE Transactions on Image Processing.

[38]  J. W. Thomas.  Numerical Partial Differential Equations: Finite Difference Methods , 1995 .

[39]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[40]  Yi-Fei Pu,et al.  Fractional Differential Mask: A Fractional Differential-Based Approach for Multiscale Texture Enhancement , 2010, IEEE Transactions on Image Processing.

[41]  Om P. Agrawal,et al.  Analysis of an axis-symmetric fractional diffusion-wave problem , 2009 .

[42]  R. Koeller.  Applications of Fractional Calculus to the Theory of Viscoelasticity , 1984 .

[43]  Jean-Michel Morel,et al.  A PDE Formalization of Retinex Theory , 2010, IEEE Transactions on Image Processing.

[44]  Brian V. Funt,et al.  Investigations into Multi-Scale Retinex , 1998 .

[45]  E. Land,et al.  Lightness and retinex theory. , 1971, Journal of the Optical Society of America.

[46]  Sangram R. Jadhav,et al.  Design of Fractional Order Differentiator & Integrator Circuit Using RC Cross Ladder Network , 2014 .

[47]  Tamar Peli,et al.  Adaptive Filtering For Image Enhancement , 1982 .

[48]  Berthold K. P. Horn,et al.  Determining lightness from an image , 1974, Comput. Graph. Image Process..

[49]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[50]  Yi-Fei Pu,et al.  A fractional partial differential equation based multiscale denoising model for texture image , 2014 .

[51]  Yi-Fei Pu,et al.  Fractional Extreme Value Adaptive Training Method: Fractional Steepest Descent Approach , 2015, IEEE Transactions on Neural Networks and Learning Systems.

[52]  E H Land,et al.  An alternative technique for the computation of the designator in the retinex theory of color vision. , 1986, Proceedings of the National Academy of Sciences of the United States of America.

[53]  Yangquan Chen,et al.  A new IIR-type digital fractional order differentiator , 2003, Signal Process..

[54]  西本 勝之,et al.  Fractional calculus : integrations and differentiations of arbitrary order , 1984 .

[55]  Andrew Blake,et al.  Boundary conditions for lightness computation in Mondrian World , 1985, Comput. Vis. Graph. Image Process..

[56]  Malur K. Sundareshan,et al.  Adaptive image contrast enhancement based on human visual properties , 1994, IEEE Trans. Medical Imaging.

[57]  Weixing Wang,et al.  Fractional differential approach to detecting textural features of digital image and its fractional differential filter implementation , 2008, Science in China Series F: Information Sciences.

[58]  E H Land,et al.  Recent advances in retinex theory and some implications for cortical computations: color vision and the natural image. , 1983, Proceedings of the National Academy of Sciences of the United States of America.

[59]  Dingyu Xue,et al.  Non-Local Fractional Differential-Based Approach for Image Enhancement , 2013 .

[60]  Leo Levi,et al.  Unsharp masking and related image enhancement techniques , 1974, Comput. Graph. Image Process..

[61]  Jean-Michel Morel,et al.  Multiscale Retinex , 2014, Image Process. Line.

[62]  G. Leitmann.  The Calculus of Variations and Optimal Control: An Introduction , 2013 .

[63]  김정연,et al.  서브블록 히스토그램 등화기법을 이용한 개선된 콘트래스트 강화 알고리즘 ( An Advanced Contrast Enhancement Using Partially Overlapped Sub-Block Histogram Equalization ) , 1999 .