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

Abstract

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.

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“…As shown in Figure 4a, it is the original signal, the square wave represents the structure of the image, and the triangle wave represents texture details. Figure 4b shows the signal with Gaussian noise, expressed with g. It is desired to denoise and maintain the texture details of g. There are three models to solve this problem, as shown in Formula (23)- (25):…”

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