Robust fitting

In [Foi et al.,2008], the authors model each local mean-standard deviation pair as a bi-variate Gaussian random variable centered on the standard deviation curve. Considering each estimate independent, they maximize a likelihood function in which the joint probability is simply given by the product of each distribution. However, in challenging cases, e.g. highly textured images, a large number of outliers might be present among the mean-standard deviation pairs. We cope with this possibility by modifying the distribution model that represent each local estimate. In particular, we model the standard deviation estimates as realizations of a mixture of zero-mean Gaussian and Cauchy distributions, where the mixture coefficient adapt to the particular image under interest [Publication I]. Since there are no outliers among the estimates of the local means, we model the mean estimates as realizations of Gaussian random variables.

The main advantage of using a Gaussian-Cauchy mixture is that the Cauchy distribution better adapts to the possibility to have samples far from the mean value. In particular, since the Cauchy distribution has heavier tails compared to the Gaussian distribution, it better models outliers that lie far from the curve that we want to fit. In normal conditions, i.e. using only a Gaussian distribution, the outliers would heavily influence the final estimate; the adoption of Cauchy distributions, instead, includes them as

3.2. Robust noise estimation for any parametric variance function

Figure 3.5: Underexposed raw image and the relative scatterplot of mean-standard deviation pairs, severely corrupted by outliers.

part of the model, and models them as improbable realizations far from the distribution mean.

We also include the mixture coefficient in the optimization process, i.e.

the likelihood is optimized considering as variables the noise parameters and the mixture coefficient. In this way we adapt our model to the amount of textured areas in the observed image. In particular, when the final estimate of the mixture coefficient gives a distribution that is mostly Gaussian, we can assume that the amount of texture (or/and edges) in the observed image is relatively low. On the other hand, if the final mixture coefficient gives more weight to the Cauchy distribution, it is reasonable to assume that the image contains large amount of texture and edges.

3.2.4 Results

We now show the advantage of using a robust estimator compared to the non-robust version from [Foi et al., 2008]. Figure 3.5, reproduced from Publication I, shows a typical challenging case, in which the estimation is severely affected by the amount of texture and by the lack of scatter-points.

On the left side of the figure we show the image on which we perform noise estimation, and on the right we show the results of the proposed

method compared to the original one. In the scatterplot we report the mean-standard deviation pairs (red dots), the ground-truthσ (y), the curves estimated with the proposed algorithmσˆ_new(y) and ˆσ˜_new(y), and with the reference one σˆ_old(y) and ˆσ˜_old(y). We denote with the symbol ˆ˜ on top of a variable the curve estimated from the clipped data, and with the symbol ˆ the final estimate.

Since the observed picture is underexposed, the algorithm segments the image in a limited amount of bins. As a consequence, we have only few local estimate, each corresponding to a bin. Furthermore, several of the local estimates are outliers, that has been miscalculated due to the lack of samples per bin, or mostly due to the heavy presence of texture and edges in the image. Note how, in fact, the reference algorithm is influenced by these outliers, and fails to fit a correct curve over the scatter-points. On the other hand, the presented robust estimator, although partially misled by outliers, well estimates most of the standard deviation curve.

Chapter 4

Contribution to denoising

In the first part of this chapter we introduce a spatially adaptive 1-D group transform for BM3D based on the spatial coordinates of similar blocks.

The adaptive transform increases the sparsity of the transform coefficients, and consequently improves the filtering results from shrinkage. We also use the same spatial information to design an algorithm based on alpha-rooting [Dabov et al.,2007c] that, adaptively, enhances local image features depending on their orientation with respect to the coordinates of similar features at other locations.

In the second part of this chapter we propose a model for iterative denoising of Poisson images. The proposed model is based on the VST denoising framework, and exploits convex combinations of noisy signal and its previous estimate to improve the signal-to-noise-ratio (SNR) of the signal to be stabilized. We prove that, especially for low signal intensities, the performance of the stabilization, and in general of the denoising filter, are vastly improved, outperforming the state of the art.

4.1 Collaborative filtering based on coordinates of similar features

As mentioned in Section 2.5, the BM3D algorithm performs a 3-D transform of groups of similar blocks via separable D+1-D transform, in which a 2-D 2-DCT (or wavelet) applied separately to each block is followed by a 1-2-D Haar transform in the orthogonal (nonlocal) direction. It is clear that in this scheme the spatial information (i.e. coordinates) of similar blocks is never used. However, the spatial coordinates of the grouped blocks is a feature that could be exploited to enhance the sparsity of the 3-D group spectrum.

Let us consider, for example, the case in which similar blocks lie on a surface that smoothly changes its average intensity. Since, before applying the 3-D transform, the blocks are reordered according to their similarity with respect to the reference block, the Haar transform may not be effective at sparsifying the spectra of the blocks. By using, instead, polynomials that approximate the smooth variation of the surface from which the blocks are extracted, we can improve the sparsity of the 3-D spectrum coefficients. From this idea, we propose a novel 1-D transform based on the spatial coordinates of the similar blocks, that adapts to the particular group that we are processing [Publication III]. We then use the principal directions of the coordinates of similar blocks to derive an enhancement algorithm that adaptively chooses whether to soften or sharpen selective coefficients of the group spectrum.

4.1.1 Orthogonal polynomials transform

We now describe how to generate the set of orthogonal polynomial (OP) functions that we use as 1-D transform basis in the presented extension of BM3D. Denoting with n_B the number of similar blocks in a group, we consider a collection ofnB bi-variate polynomial functions defined over the image spatial domain, and linearly independent over the coordinates of sim-ilar blocks. We then sample the functions at the spatial coordinates of the blocks, and we rearrange them as columns of a matrix P . We compute the QR decomposition of P (Gram-Schmidt factorization) to generate an orthogonal basisQ whose elements (matrix columns) are bi-variate

polyno-4.1. Collaborative filtering based on coordinates of similar features

Figure 4.1: Example of adaptive basis functions generated by orthogonal polynomials for two different groups of similar blocks (purple areas). At each row we show the first 6 basis functions of the corresponding group.

Observe how the basis functions adapt to the group coordinates.

mial functions sampled at the spatial coordinates of the similar blocks. We use the matrix Q as 1-D transform in BM3D.

In Figure 4.1, reproduced from Publication III, we show two examples of basis functions (visualizing the first 6 basis functions for each basis) obtained from the block coordinates of two different groups. On the top row we show the basis functions of a group aligned along an edge, while on the bottom row we plot the basis functions of a group from a uniform region. Note how the basis functions strongly adapt to the particular blocks spatial coordinates, and how the proposed 1-D transform introduces a new level of adaptability to BM3D.

The scenario in which one can distinctly appreciate the advantages of the proposed adaptive transform compared to the standard Haar transform is when the search window is on an area that smoothly varies, like for example the shoulder of Lena (shown in Figure 4.2). In this scenario, the polynomial functions better represent such smooth changes compared to the Haar transform.