# A Coupled Variational System for Image Decomposition along with Edges Detection
## 1 Introduction  
Image decomposition is an important research topic in the field of image processing. The purpose is to decompose a given image into the different parts. For example, for a noise-free image, we can split it into cartoon and texture; for a noisy smooth image, we can split it into cartoon and noise. The cartoon part (also called structure component) is ideally piecewise smooth parts corresponding to the main objects; the texture part usually refers to the fine scale details consisting of repetitive patterns presented in the original image. Variational methods for image decomposition have become very popular in recent years. Their aims are to find the appropriate priors to describe the cartoon and texture components so that the two components have respective small norms in their corresponding functional spaces [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15]. For example, a classical variational model was proposed by [PERSON] et al. [1]. The cartoon component belongs to the bounded variation space and the noise or small scale texture component is in \(L^{2}\). The model denoises well for the piecewise-constant images while preserving sharp edges. However, some details can be destroyed. Consequently, [PERSON] [2] introduced the dual spaces of bounded variation space to model the oscillating patterns and pointed out that the oscillating patterns should have small norm in these dual spaces. However, [PERSON]'s model can not directly give its numerical solutions. To solve the difficulty, the authors in [3; 4; 5; 6] designed different approximation models. Other methods based on total variation (TV) can be seen in [7; 8; 9; 10; 11; 12; 13; 14; 15]. As is well known, TV regularization easily tends to make the staircase effect. To reduce the staircase effect, some higher-order and nonlocal variational models were discussed in [16; 17; 18; 19; 20; 21] for image decomposition. Different from the above approaches, [PERSON] et al. [22] proposed a relative TV regularization model with the windowed TV norm. Based on sparse approximation theory, [PERSON] et al. [23] and [PERSON] et al. [24] designed two different dictionaries and tight frames for the cartoon and texture respectively. Based on the discriminative patch recurrence prior, [PERSON] al. [25] constructed an effective approach for cartoon-texture decomposition. Through the relative variation of gradient, [PERSON] et al. [26] designed a fast filter method for cartoon-texture decomposition. By increasing the size of the parameter, multiscale methods for image decomposition were studied in [27; 28; 29; 30]. Using the texture similarity, some low rank techniques for image decomposition were proposed in [31; 32; 33; 34; 35]. With the development of deep learning, some deep unfolding learning methods for image decomposition were studied in [36; 37]. However, on the one hand, it is difficult to find the original cartoon and texture data for training, on the other hand, deep unfolding networks have to adopt the input and the output of each stage as an image, which inevitably results in feature-to-image information distortion [38]. So it is difficult for us to get good decomposition results based on deep learning method. In addition, [36; 37] are based on \(\mathrm{TV}-L^{1}\) model which is not suitable for image decomposition [2]. Recently, [PERSON] et al. [39] introduced a coupled system for image restoration by combining a constrained partial differential equation (PDE) with weighted TV. Substituting the first equation of the coupled system in [39] by the weighted \(\mathrm{TV}-L^{1}\) regularization model, [PERSON] et al. [40] proposed a new coupled system and gained some better image denoising and decomposition results. Because \(\mathrm{TV}-L^{1}\) model is not suitable for additive Gaussian noise, a higher order variational system was proposed by [PERSON] et al. [11]. Inspired by [11; 39; 40], in this paper, we give a new different coupled system which contains two variational minimal models.