by maximizing a likelihood function that takes into consideration the possi-ble overestimation of the scatter-points variances. Similarly,Foi et al.[2008]
filter the observed image, segment it, and then maximize a likelihood func-tion to estimate the noise parameters. A major novelty introduced by this work is that it takes under consideration the clipping of the data, i.e. values of the image that exceed certain boundaries are replaced by the values of the boundaries.
2.4.3 Alternative approaches
In our work [Publication II] we show that the restriction on the use of ho-mogeneous samples is not necessary to correctly estimate the affine noise variance in (2.20), and like the methods [Mäkitalo and Foi,2014; Pyatykh et al., 2013] we estimate the noise parameters without any segmentation of the image. The algorithm in [Mäkitalo and Foi, 2014] estimates the noise by exploiting the stabilizing capability of the generalized Anscombe transformation. In particular, the authors try to find the parameters for the transformation that best stabilizes the standard deviation of the trans-formed image, and assume those as the final estimates. The algorithm in [Pyatykh et al.,2013], instead, estimates the noise parameters by analyzing the last eigenvalues of the singular value decomposition (SVD) of homoge-neous patches from a noisy image. These eigenvalues depends only on the noise, and therefore are used to estimate the noise parameters.
2.5 Block matching and 3-D filtering (BM3D)
Since the denoising methods fromPublication Iand Publication IIIinvolve the use of the BM3D algorithm, in this section we discuss the key features of its original implementation [Dabov et al., 2007b], also useful for under-standing Chapter 4. Although introduced a decade ago, BM3D still gives state-of-the-art results. It can be considered a hybrid algorithm that per-forms collaborative filtering by exploiting the sparsity of groups of similar patches extracted from different parts of the image (i.e. nonlocality).
2.5.1 Grouping and transform
The first step of BM3D is the grouping of similar blocks. In this step the ob-served image is divided into overlapping reference blocks; for each reference block the algorithm looks for similar blocks in its surroundings (search win-dow). The similarity criterion is the Euclidean patch difference, intended as the sum of the squared pointwise differences between corresponding pix-els in the two blocks (i.e. squared`2-norm). Each group of similar blocks is then collected and stacked together in a 3-D volume, that is eventually transformed to decorrelate its pixels, and to collect most of the group energy into few transform coefficients. In BM3D, a separable 3-D transform is ap-plied to the group: first, a 2-D transform (e.g. 2-D DCT or 2-D wavelet) is applied to each block individually, then, a 1-D transform (Haar) is applied to the obtained 2-D coefficients along the remaining orthogonal dimension.
Since multiple reference blocks overlap, and 3-D volumes usually share common pixels, eventually the algorithm produces multiple estimates of the same pixel. The overlapping will be taken into account when producing the final estimated image.
2.5.2 Collaborative filtering
Once the 3-D spectrum is computed, we can proceed to filter the group.
Since the amount of redundant information of a group of similar patches is abundant, the 3-D transform should promote sparsity of the spectrum coefficients. A very sparse 3-D spectrum may result from spatial local regu-larity of the patch content, nonlocal reguregu-larity (i.e. repeated spatial content across the different patches in the group), or a combination of both. In or-der to perform denoising we apply a hard-thresholding operator to the 3-D spectrum to zero the small coefficients that only depend on the noise power (variance). The threshold used for shrinkage is set proportional to the stan-dard deviation of the noise affecting the signal.
2.5. Block matching and 3-D filtering (BM3D)
2.5.3 Aggregation
Once all the groups are filtered, we compute the inverse transform and we aggregate the estimates into the final image. In BM3D the aggregation is done via convex combination of all the overlapping estimates, where the weights are inversely proportional to the residual variance of the correspond-ing group. In this way, we give larger weights to pixels belongcorrespond-ing to groups enjoying a sparser 3-D spectrum.
2.5.4 Second iteration with Wiener filter
A second iteration of the algorithm is usually performed to improve the denoising results. Since we have already an approximation of the noise-free data, in this second stage the block matching is performed on the previously denoised image. Then, the similar blocks are extracted from the noisy image at the corresponding coordinates. The group is transformed like in the previous stage, and shrunk by Wiener filtering. In particular, we extract also the corresponding group from the previously denoised image, and we use its spectrum as pilot signal for the Wiener filter. Note that the Wiener filter is applied only in the second stage because it requires a pilot signal for denoising, and at the first iteration we do not have at our disposal such signal.
As a side note, it has been tested that iterating BM3D more than once does not provide any benefit, and does not improve the estimation. On the contrary, in most cases the PSNR of the denoised image in successive iterations decreases compared to the output of the first Wiener stage.
2.5.5 BM3D extensions
Several BM3D extensions have been developed in the last ten years [Bo-racchi and Foi,2008;Dabov et al.,2007a;Maggioni et al.,2014]. A direct improvement, for example, is presented in [Dabov et al., 2009], where the 2-D DCT transform is replaced by an adaptive PCA transform. This al-gorithm gives state-of-the-art performance due its outstanding capability of both, sparse representing piece-wise smooth areas from the image, while
preserving fine details due to the PCA adaptability. However, perform-ing PCA for each group of similar patches is more time consumperform-ing than adopting a fixed 2-D transform.