Haar wavelets, gradients and approximate total variation regularization

Image denoising by means of total variation (TV) regularization is still a standard procedure. For very large images, especially three-dimensional voxel datasets, however, this can be computationally infeasible. We show how this TV regularization can be approximately performed even in arbitrary dimensions by applying appropriate shrinkage to selected and properly weighted Haar wavelet coefficients, all of which depends even on the dimensionality of the data. Our approach acts entirely on the wavelet coefficients which represent the compressed image, and is therefore suited for the application on large three-dimensional images represented in the Haar wavelet basis, e.g., volumes from computed tomography.