Rotationally invariant similarity measures for nonlocal image denoising

Abstract

Many natural or texture images contain structures that appear several times in the image. One of the denoising filters that successfully take advantage of such repetitive regions is the nonlocal means filter. It is simple and yields very good denoising results. Unfortunately, the block matching within the standard nonlocal means filter is not able to handle rotation or mirroring. Rotated or mirrored instances are not detected as variations of the corresponding original structures. In this paper, we analyse two natural approaches for a rotationally invariant similarity measure that will be used as an alternative to, respectively a modification of the well-known block matching algorithm in nonlocal means denoising. The first approach is based on similarity distances computed with the help of moment invariants whereas the second one estimates the rotation angle, rotates the block via interpolation and then uses a standard block matching. In contrast to the standard method, the presented algorithms can find similar regions or patches in an image even if they appear in several rotated or mirrored instances. With this modification, the nonlocal means filter is able to find more suitable regions for its weighted average.

Parameter-Free Fast Pixelwise Non-Local Means Denoising

“…whereK(z) = K(−z), * is the discrete convolution product and s t is the square difference image defined in (13). As it is well known, the convolution product may be computed in O(N N c log(N N c )) operations using the 2D discrete Fourier transform (2D-FFT) denoted by F and its inverse F −1 :…”

“…The window Ω x is called the search window at x and, for simplicity and faster computations, in [1] a choice of a square shape of fixed size is made while, according to authors, the search window should cover the entire image plane, hence the non-local nature of the algorithm. However, it has been reported that using for NLM a neighborhood instead of the whole image plane allows to increase the denoising performance [12,13,21,22], see also the discussion in [9] and the specific study in [19] where it is experimentally established that the optimal window size D is very small, when using a variant of the pixelwise NLM. As this present article will establish the best parameters, it will give an answer for the original pixelwise NLM.…”

Foveated self-similarity in nonlocal image filtering

“…10 Several extensions of the patch-similarity measure have been proposed to exploit image redundancy up to a class of geometric transformation: in the basic NL-means, the patch similarity is de…ned up to translations and therefore similar patches can be at di¤erent locations of the image but must share the same scale and orientation. Modi…ed similarity measures that handle rotated patches have been presented in, 11 by exploiting patch clustering, and in, 10 where patch centroids or structure tensors are used to align rotated patches. In 12 patch similarity is extended up to scale and rigid transformations, by enforcing SIFT descriptors 13 to map patches in a canonical form, which does not depend on scale and orientation.…”

Manifold Learning via the Principle Bundle Approach

“…It is well-known that the non-local mean (NLM) [13] algorithm leads to a better edge preservation [14], and this improvement is directly related to the diffusion process on the nonlinear geometric structure [15]. One can further consider the rotational structure of patch spaces to denoise images [16][17][18][19][20][21]. Two patches are viewed the same or called rotationally invariant, if they are the same up to rotation.…”