Image Restoration Models Based on Dyadic Hardy Space and Dyadic Bounded Mean Oscillation Space

Abstract

Texture is widely existed in various images and plays an important role in many area such as medical image diagnosis, remote sensing, etc. However, the image in texture regions is tend to be deteriorated during restoration process. In this paper, we apply the dyadic Hardy space $H_{d}^{1}$ and dyadic Bounded Mean Oscillation (BMO) space in the texture preserving image restoration model. We propose a $H_{d}^{1}$ regularized minimization model to extract texture from noisy data. In this model, $H^{1}_{d}$ norm is taken as regularizer to enforce the prior that the local variance of the noise is below certain level depending on the regularization parameter. We also analyze the mathematical properties of this model which indicate the mechanism of $H^{1}_{d}$ regularizer to control the local variance. For the numerical solution of the model, we transform it into wavelet domain based on the wavelet characterization of dyadic Hardy space and dyadic BMO space, and solve it by the fixed iteration algorithm. Combing the total variation (TV) regularization method and frame based regularization method, a two-layers regularization model is proposed for edge and texture preserving, and then analyzed and solved in the frame of split Bregman method. Finally, we present various numerical results on images to demonstrate the potential of our methods.