Wavelet-based minimal-energy approach to image restoration

The popular mathematical models for digital image restoration, formulated as a minimization problem of certain total energy functionals, give rise to the variational/PDE-based approach to process the input image in the spatial (or physical) domain. In general, the total energy in these models consists of two additive terms, namely: the internal energy for dictating the image quality in terms of image smoothness and image feature preservation, and the external energy for ensuring the output image not to deviate too far from the input image. In formulating the internal energy, a specific energy density function is chosen and applied to the magnitude of the gradient operation for extracting image features to be preserved. In this paper, we replace the gradient operation by some wavelet transform that has better performance in feature extraction and eliminates the need of iterative steps. Thus, the problem of image processing is performed in the wavelet domain instead. By taking advantage of the multi-scale structure of wavelets and the corresponding multi-level singularity detection capability, the proposed approach should facilitate further development of fast and effective algorithms for the variational approach, perhaps even with significant reduction in computational complexity as compared with the traditional approach to digital image restoration.