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A Fractional-Order Variational Framework for Retinex: Fractional-Order Partial Differential Equation

A Fractional-Order Variational Framework for Retinex: Fractional-Order Partial Differential Equation

Abstract:

The present work 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, 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 work, 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 optimisation algorithm and the fractional-order Courant-Friedrichs-Lewy condition. Third, we perform experiments to analyse 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.

 

Existing System:

Fractional calculus has evolved as an important, contemporary branch of mathematical analyses, which can be potentially employed in a variety of engineering and other problems. Fractional calculus is as old as the integer-order calculus, although until recently, its applications were exclusively in the domain of mathematics and now seems to be gaining its acceptance as a novel promising mathematical method among the physical scientists and engineering technicians. Several scientific studies have demonstrated that a fractional-order or a fractional dimensional approach is now the best way that many natural phenomena can be described. This approach has obtained promising results and ideas demonstrating that fractional calculus can be an interesting and useful tool in many scientific fields such as diffusion processes, viscoelasticity theory, fractal dynamics, fractional control, signal processing, and image processing.

Proposed System:

The fractional differential-based approach utilizes directly fractional differential masks, fractional differential operators, for the purpose of processing the grey values of an image, while the fractional-order variational framework for retinex proposed by this work uses the FPDE to implement a restrictive fractional-order optimization algorithm. The numerical implementation of the FPDE makes use of the fractional differential masks proposed. The capability of preserving the edges and textural details of the FPDE is an important advantage that leads to the superiority of the proposed approach compared to the traditional integer-order computation-based contrast enhancement algorithms, especially for images rich in textural details.

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