Block-matching and 3D filtering

Block-matching and 3D filtering, known as BM3D, is a block-matching algorithm used mainly to reduce noise in images, and is one of the expansions of the non-local means approach. It works in two cascades, a hard-thresholding stage and a Wiener filter stage, each made of grouping, collaborative filtering and aggregation. In grouping, image fragments of equal size are collected when their dissimilarity from a reference fragment falls below a threshold, and, unlike clusters from k-means, the groups need not be disjoint. The fragments of a group are stacked into a three-dimensional shape, a linear transform is applied along with transform-domain shrinkage such as Wiener filtering, and the transform is inverted. The overlapping filtered fragments are then weight-averaged back into a two-dimensional image. Colour images are handled by grouping on the luminance channel, which saves roughly a third of the computing time. Extensions include IDD-BM3D, which performs decoupled deblurring and denoising, and a variant that integrates a convolutional neural network.