# Learning Weighted Forest and Similar Structure for Image Super Resolution
## 1 Introduction  
Image SR technique aims at reconstructing a HR image from one or multiple LR images. The SR technique is closely related to computer vision problems with extensive applications ranging from astronomical and medical imaging to video surveillance. It has received widespread attention since put forward by [PERSON] and [PERSON] [1]. The study focuses on single image SR approaches. Mainstream approaches for SR reconstruction can be subdivided into three categories: Interpolation-based approaches [2; 3; 4; 5; 6], reconstruction-based approaches [7; 8; 9; 10; 11], and example learning-based approaches [12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29]. The SR approaches based on interpolation generally exploit the kernel function [2; 3] to estimate a large number of unknown pixels in HR meshes. Although they offer simplicity and rapidity, the restored HR image usually has obvious blurring and jagged artifacts. Therefore, the performance of such approaches is unattractive in reality. The reconstruction-based approaches generally assume that the observation LR image results from multiple degeneration factors like warping, blurring, down-sampling, and noise [9]. Since aLR pixel can correspond to multiple HR pixels, the SR reconstruction is an ambiguous and ill-posed problem. For the purpose of obtaining a credible and unique HR image, certain prior knowledge such as edge-directed prior [11] needs to be added to the reconstructed image. However, these approaches cannot impose adequate novel details on the target HR image if up-sampling factor is greater than 2, and result in quality decline of the resultant image. Example learning-based approaches usually exploit a variety of machine-learning algorithms to obtain a mapping relationship from LR to HR image by using training dataset including millions of LR-HR exemplar patch pairs [12]. Using the co-occurrence LR-HR patches as priors, more high frequency details can be reconstructed and imposed on the LR test image to promote the quality of the reconstructed image. However, if external training dataset lacks a correlation with the LR input image, example learning-based methods tend to bring unpleasing artifacts to the restored image. For single image SR reconstruction, only one LR input is available. The degradation process from the original HR image to the LR image is formulated as:  
\[X=DBY+n \tag{1}\]  
where \(X\) is observation LR image, \(Y\) expresses objective HR image, \(B\) represents blurring matrix, \(D\) stands for the down-sampling matrix, and n is noise. Because of factors such as down-sampling, blurring and noise, an input LR image is able to match multiple varying HR images, so the issue of SR reconstruction is seriously underdetermined. Thus, to accurately realize estimation for HR images, some priors or regularization constraint items need to be introduced into the reconstruction process. The reconstruction prediction for SR based on regularization constraint is formulated as:  
\[\hat{Y}=\underset{Y}{\text{argmin}}\gamma\|X-DBY\|_{F}^{2}+R(Y) \tag{2}\]  
where the first term \(\|X-DBY\|_{F}^{2}\) denotes the reconstruction error, and \(\|\cdot\|_{F}^{2}\) stands for Frobenius norm. The second term \(R(Y)\) denotes the regularization constraint item. Here \(\gamma\) is the balance parameter, which is used to adjust weight between the first and second items. Example learning-based SR can successfully obtain high-frequency details by depending on the external image training dataset. Simultaneously, the reconstruction-based SR algorithm is able to successfully and effectively utilize reasonable prior knowledge. To obtain better reconstructed quality, we propose a learning weighted forest from an external training dataset and use a similarity structure inside the initial HR image patches as a reasonable regularization prior.