# Retinex Based Image Enhancement via General Dictionary Convolutional Sparse Coding
## 1 Introduction  
The color of an object determined by a machine visual system (MVS) such as a digital camera is based on the amount of light reflected from it. On the other hand, the human visual system (HVS) determines the color of an object by considering the details of the surrounding environment and changes in overall illumination [1]. The complex HVS automatically recognizes the changes in illumination and easily recognizes the original color of the object. This feature of the HVS is called color constancy and has been studied for a long time [2, 3]. Owing to the inconsistency between the HVS and the MVS under various illumination conditions, a machine cannot obtain the same image as a human. These inconsistencies also cause algorithmic errors in functions such as color separation, pattern recognition and object tracking. Therefore, to improve the performance of MVS, it is important to understand the color constancy of HVS. The Retinex concept by [PERSON] and [PERSON] [4, 5, 6, 7] combines the functions of retina and cortex, and explains how the HVS perceives color. If \(S\) is defined as the value of an image in the spatial domain \(\Omega\), the image value in the Retinex model mainly depends on two factors: the amount of illumination projected onto the object in the image and the amount of illumination reflected by the object in the image. Based on these two components, \(S\) can be constructed with the reflectance function \(R\) and illumination function \(L\)[8] as follows:  
\[S(x)=R(x)\cdot L(x) \tag{1}\]  
where, \(0<R<1\) (reflectivity) and \(0<L<\infty\) (illumination effect), \(\forall x\in\Omega\)The HVS can recognize both the illumination component and the reflectance component, and is able to remove the illumination component. This is called color constancy. Therefore, the HVS can recognize the constant color of an object, while ignoring the changing illumination. Removing the illumination component from the image can cause dark areas in the low-light image enhancement application to represent the original color of the object, which can increase the success rate of other algorithms on the machine. Also, it is possible to improve the edge, texture, etc. so that the reflectance component can exhibit the same effect in reconstruction as a super resolution. Motivated by the color perception characteristics of the HVS, the Retinex theory enhances high-contrast images, allowing more detail and color to be observed in low-light areas [8; 9; 10; 11; 12; 13]. In addition, Retinex theory can be used effectively for shadow removal [14]. A representative method among Retinex theories is the Retinex model using regularization parameters by calculating the total variation in the reflectance function [8; 15]. Recently, a Retinex model [16] has been proposed for effective image enhancement using the sparsity of reflectance in the gradient region and the sparsity of illumination in the frequency domain. The Retinex model in [17] proposed image enhancement using sparse coding. Sparse coding is a method of constructing the basis of an image through a dictionary. However, such conventional Retinex methods have a problem in that when the illumination changes rapidly, details in a complex area within the image are not properly reflected and blurred, or illumination and reflectance cannot be accurately decomposed. In most Retinex models, illumination has smooth conditions and reflectance shows a detailed structure. Therefore, if the reflectance function can be more accurately generated, the detailed description of the complex area can be well preserved when the illumination changes rapidly, and thus, the illumination and reflectance of the image can be accurately resolved. In sparse source separation Retinex model [16] that consider sparsity, it is assumed that the reflectance component is sparse in the frequency domain. An image with a complex region (high rank) has, a number of detailed components.