A Nonlocal Transform-Domain Filter for Volumetric Data Denoising and Reconstruction

We present an extension of the BM3D filter to volumetric data. The proposed algorithm, denominated BM4D, implements the grouping and collaborative filtering paradigm, where mutually similar d-dimensional patches are stacked together in a (d + 1)-dimensional array and jointly filtered in transform domain. While in BM3D the basic data patches are blocks of pixels, in BM4D we utilize cubes of voxels, which are stacked into a four-dimensional “group”. The fourdimensional transform applied on the group simultaneously exploits the local correlation present among voxels in each cube and the nonlocal correlation between the corresponding voxels of different cubes. Thus, the spectrum of the group is highly sparse, leading to very effective separation of signal and noise through coefficients shrinkage. After inverse transformation, we obtain estimates of each grouped cube, which are then adaptively aggregated at their original locations. We evaluate the algorithm on denoising of volumetric data corrupted by Gaussian and Rician noise, as well as on reconstruction of phantom data from sparse Fourier measurements. Experimental results demonstrate the state-ofthe-art denoising performance of BM4D, and its effectiveness when exploited as a regularizer in volumetric data reconstruction.