Smoothing and Regularization

The recovery of HST images has also been obtained by a method based on the following two steps:

1. The blurred image is partially smoothed using the public domain package WAVE II of S. Mallat;
2. The convolution equation with the data function partially smoothed is then solved by the Tikhonov regularization technique.

By so doing, multiscale edges can be detected and characterized with a wavelet transform, while the location of the edges in the images is provided by the local maxima of the wavelet transform modulus. In this way a first image approximation, from the important edges that are selected by the coding algorithm, is then reconstructed. The edge coding algorithm restores the main image features but removes all the textures as well as fine image details. An error image is then computed by subtracting the coded image from the original one and, in order to recover some of the texture information, the error image is coded within a wavelet orthonormal basis. This orthogonal basis decomposes the error image into details appearing at different resolutions and within different spatial orientations. The error image to be coded is then chosen both from the image fluctuations detected at different resolution scales and the number of pixels of the original image. The smoothed image is finally obtained by adding up the coded error to the coded edge image.

This image is only roughly smoothed. For this reason, in the second step, instead of solving Eq. (1) with this coded image, we solve it in a discrete set of equally spaced points of , by the Tikhonov regularization method of § 2.