Super-resolution is a challenging technique for recovering lost information according to natural image priori. As an important priori

sparse has been widely studied in the field of image processing

such as in image recovery

inpainting

demosaicing

and denoising. Given the development of compressed sensing theory and the L optimization method

a large number of super resolution methods have been proposed based on sparse representation. An example is the single image super-resolution based on sparse representation

which has been widely studied in recent years. Based on the super-resolution model through sparse representation

where the feature image patch can be represented sparsely

a novel method conducted in the wavelet domain is proposed in this work. Our method is based on the sparse of high-frequency image patches and high-frequency image patches in redundant dictionaries. First

a decomposed coefficient image is obtained after using the discrete wavelet transform in a low-resolution image. Second

in connection with the high-frequency coefficients of the low-resolution image

a double sparse model with super-resolution is established to recover the detail coefficients of a high-resolution image correspondingly. Third

the decomposed coefficients of the low-resolution image and the recovered high-frequency coefficients of the high-resolution image are merged into wavelet coefficients for second floor decomposition. Finally

with the multi-scale property of wavelet and the assumption that a low-resolution image can be used as a high-resolution image of low frequency coefficient approximation

a super-resolution image is reconstructed with two layers of inverse wavelet transformation with low-resolution image wavelet decomposition and estimated high-frequency coefficients of high-resolution images. In the model solving process

we adopt a fast solving method called the constrained splitting Bregman method

which is widely used to solve the L problem. Unlike the method for joint feature spaces

the constrained splitting Bregman method uses two dictionaries with high and low feature spaces. The low-resolution dictionary is learned from a low-resolution feature space using the k-svd method while the resolution dictionary is learned from least square approximation. The sparse model of redundant dictionary is known to recover good texture and denoise at the same time. The double sparse method has the advantage of the famous SR method through sparse representation and obtains good denoising performance. Through several experiments for standard pictures

the double sparse method restores image local texture and edges well and achieves a good effect on a noised image because of the use of multi-scale property and sparse with high-frequency coefficients. Our method involves less computational complexity compared with the popular sparse super-resolution method because of the use of only three quarters of the image patches of the original image. The sparse model is widely used in image recovery. Based on the sparse of detail coefficients in wavelet domain and on the sparse of feature image patches under redundant dictionaries

a novel single image super-resolution method in wavelet domain is proposed in this work. The double sparse method preserves good local texture and edges and obtains better results for noised image with less computational complexity compared with the conventional method via sparse representation.