# Retinex Based Image Enhancement via General Dictionary Convolutional Sparse Coding
## 1 Introduction
In such a case, the decomposition of reflectance is not properly performed, and an error in the reflectance function affects the illumination function, which results in the improper decomposition of reflectance and illumination. In particular, in the case of a Retinex model [17] using sparse coding, an advantage is that the reflectance component can be expressed by the dictionary greater detail. The sparse coding consists of a dictionary and a sparse coefficient in a linear combination, constituting a local patch dictionary. A drawback of the dictionary-based sparse coding approaches is that important spatial structures of the signal of interest may be lost because of its subdivision into mutually-independent patches. Further, patches (atoms) of the dictionaries learned using this approach are often redundant and contain shifted versions of the same features. Therefore, it is necessary to construct a reflectance dictionary for each image, which cannot be used as a robust dictionary. Therefore, a new method that accurately generates the components of the reflectance function is needed to overcome these limitations. In this paper, we propose an image enhancement method based on Retinex theory using convolutional sparse coding (CSC). Our approach builds on recent advances in CSC and reconstruction techniques. We show that the CSC reconstruction technique provides a higher quality of high contrast and complex image than the existing patch-based sparse reconstruction techniques. In addition, we observe that CSC, is a particularly well-suited general dictionary for the different types of high-contrast and complex signals present in the reflectance function. The advantage of a general dictionary is that when an arbitrary image is input, the reconstruction can be performed immediately with the learned dictionary without learning a new one. We use singular value decomposition (SVD) in CSC to construct a more compact dictionary in limited memory. In addition, since it is a form of reflectance basis of a general image, it only has additional information that fits the basis. Based on this fact, we can improve the low-light image through our proposed method. Additionally, a halo artifact can occur in a common Reinex-based reconstruction problem, which can be reduced by our proposed method. These are detailed in Section 4. Therefore, we pose the Retinex image enhancement problem as a CSC problem and derive necessary formulations to solve it. We make the following contributions:* We show that the reflectance function of the Retinex model can be learned through a CSC dictionary, leading to efficient image enhancement through the expression of various nonlinear image shapes. * We propose that the reflectance function can be reconstructed using a trained general dictionary using CSC from a large dataset. * We use SVD in CSC to construct a more compact dictionary in limited memory. The remainder of this paper is organized as follows. Section 2 briefly introduces traditional Retinex model, sparse coding for Retinex model, and basic CSC. Section 3 discusses our CSC Retinex algorithm, focusing on the proposed objective function and the general reflectance function dictionary learned with CSC. Obtained experimental results are presented and analyzed in Section 4. Finally, Section 5 summarizes this paper.