# Recovering Texture with a Denoising-Process-Aware LMMSE Filter
## 1 Introduction  
Image denoising methods that can estimate a noiseless, clean natural image (original image) from a noisy observation are actively being studied [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21]. Many previous works have assumed that the noise in such a noisy observation is an additive white Gaussian noise (AWGN). One popular approach for image denoising is to use nonlocal self-similarity (NLSS), in which it is assumed that a local segment of an image (a patch) is similar to other local patches [10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21]. Weighted nuclear norm minimization for image denoising (WNNM) [16] is an optimization-based method based on an NLSS-based objective function. WNNM assumes that a matrix whose columns consist of similar patches extracted from a clean image is low rank and achieves state-of-the-art denoising performance among non-learning-based methods. Image denoising methods such as WNNM can estimate the original image well in terms of the mean squared error (MSE) or the peak-signal-to-noise ratio (PSNR). However, as shown in Figure 1, texture is often lost in the estimated image. Because the texture carries important information about the aesthetic properties of the materials depicted in certain parts of an image (such as the feel of sand, fur, or tree bark), the texture losses severely degrade the subjective image quality. On the other hand, it is difficult to obtain a good description of texture (we refer to such a description as a texture model) and to estimate its parameters because of its stochastic nature, and the performance of texture-aware image denoising methods [8; 19; 20; 21] depends on these models and parameters. Thus, we can classify texture-aware image denoising methods based on the texture models they utilize. The gradient histogram preservation (GHP) method [19] is a prominent texture model for texture-aware image denoising. GHP describes texture features in the form of a histogram of the spatial gradients of the pixel values (gradient histogram). GHP is based on an image denoising method called nonlocally centralized sparse representation [15], and it can recover texture by imposing the condition that the gradient histogram of the output image must be close to that of the original image. The parameters of GHP can be estimated from the observed image by solving an inverse problem. However, GHP does not utilize the relationships between distant pixels. Nevertheless, these relationships can carry important texture information because textures are often repeated over long distances. [PERSON] et al. addressed this problem by proposing a texture-preserving denoising method [20] that groups similar texture patches through adaptive clustering and then applies principal component analysis (PCA) and a suboptimal Wiener filter to each group. The Wiener filter is a special case of a linear minimum mean squared error estimator (LMMSE filter), and it requires an estimation of the covariance matrices of the original signals. In this texture model, the relationships between distant pixels can be expressed by the covariance matrices. The covariance matrices that are used in the suboptimal Wiener filter are calculated from sample observation patches, which are chosen using a nonfixed search window. As another method of managing the distant relationships characterizing texture, a denoising method using the total generalized variation (TGV) and low-rank matrix approximation via nuclear norm minimization has been proposed [21]. In this method, the TGV is used to avoid the oversmoothing that is typically caused by low-rank matrix approximation methods [10,16]. The TGV and nuclear norm minimization are applied to capture the relationships between nearby pixels and distant pixels, respectively. [PERSON] et al. claim that the latter are related to textures with regular patterns. The parameter of their texture model is the weight parameter of the TGV term. An iterative algorithm is required to estimate this parameter.