Wavelet inpainting with the ℓ₀ sparse regularization

We propose a constrained inpainting model to recover an image from its incomplete and/or inaccurate wavelet coefficients. The objective functional of the proposed model uses the ℓ₀ norm to promote the sparsity of the resulting image in a tight framelet system. To overcome the algorithmic difficulty caused by the use of the ℓ₀ norm, we approximate the ℓ₀ norm by its Moreau envelope. A fixed-point proximity algorithm is developed to solve the new approximation optimization model and the convergence analysis of the algorithm is provided. The proposed algorithm can be accelerated by the FISTA technique and we also develop an adaptive method to determine the approximation parameter to further speed up the algorithm. We demonstrate that the rows of the discrete cosine transform matrix can generate a redundant tight framelet system with symmetric boundary condition, which has good ability to extract information from incomplete wavelet coefficients. Using the tight framelet system, our numerical experiments show that the proposed model and the related fixed-point algorithm can recover images with much higher quality in terms of the PSNR values and visual quality of the restored images than the models based on the ℓ₁ norm and the total variation.