Low-light enhancement methods based on image Retinex yield enhanced images through the estimation of illumination and reflectance components. In image Retinex theory, since the reflectance component represents the reflectivity of objects' surface in the scene, it should be constrained between 0 and 1 to reflect its physical meaning. This constraint directly leads to an inequality constraint on the illumination component. Given the challenges of solving optimization problems with two inequality constraints, previous variational Retinex models initially treated these problems as unconstrained ones. To make up for the absence of constraints, hard projections are typically employed in the implementation algorithms. However, such a practice makes the models theoretically deviate from the physical essence of image Retinex and in practice may result in inaccurate estimates of two components, as shown in the analysis of this paper. For this problem, we propose a solution from the perspective of diffusion equations instead of the variational principle. Firstly, a general model for image Retinex is introduced in the framework of diffusion equations, where two inequality constraints are seamlessly incorporated into diffusion equations as source terms. To demonstrate the practicability of this general model, a Fractional-Integer order Diffusion system based on image Retinex (FIDR) is presented for low-light enhancement, which takes advantage of both integer-order diffusion preserving structure and fractional-order diffusion favoring texture. The FIDR is solved numerically by using an alternate iterative scheme based on explicit finite difference and 2D discrete Fourier transform. Subjective and objective evaluations on four low-light image datasets show that the FIDR model achieves higher performance in image decomposition and low-light enhancement, compared to several state-of-the-art methods.
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