sev-eral consecutive frames in order to simultaneously exploit spatial and temporal redundancy. In particular, since an exhaustive spatiotem-poral search would be computationally not feasible, V-BM3D uses a technique based on a data-adaptive predictive-search block-matching procedure which progressively refines the position and size of the search neighborhoods using the information of the blocks matched in the previous frames.
2.5 Assessing Image Quality
In this thesis we mainly restrict to the peak signal-to-noise ratio (PSNR), being widely-used in the field of image restoration and thus allowing for an easy comparison with respect to methods proposed in the literature. The PSNR is formally defined in logarithmic scale as
P SN R= 10 log10 Imax2
M SE, (2.9)
beingImax the maximum intensity value of the signal, hence express-ing the ratio between the maximum possible power of the signal versus the power of the corrupting noise as measured by the mean squared error (MSE)
M SE = 1
|X| X
x2X
(y(x) y(x))ˆ 2,
which corresponds to the dissimilarity magnitude between the origi-nal sigorigi-naly and the estimated one ˆy averaged over all image domain X, being |X| the cardinality of X.
However, a high PSNR (or low MSE) value does not always cor-respond to a signal with perceptually high quality. Image quality assessment (IQA) aims at measuring the quality of a given image using objective metrics designed to agree with human visual judg-ment. This is by itself a difficult problem and still an open research
topic, and thus many IQA algorithms have been proposed by many researchers [18], with the final goal to define a procedural metric able to objectively measure the quality of di↵erent image estimates while also providing a quality assessment that correlates to human perception. In the remainder of this thesis, we make also use of ob-jective metrics that are expected to be more consistent with human judgment, i.e. the structural similarity SSIM index [124] and the motion-based video integrity evaluation MOVIE index [114].
Chapter 3
Volumetric Filtering
In this chapter we introduce a denoising filter for volumetric data based on the BM3D filtering paradigm [25]. In the proposed algo-rithm, denoted BM4D, we naturally utilize cubes of voxels as basic filtering elements, and hence we form 4-D groups by stacking together mutually similar cubes. The fourth dimension, along which the cubes are stacked, embodies the nonlocal correlation across the data. The groups are collaboratively filtered by simultaneously exploiting the local correlation present among voxels in each cube as well as the nonlocal correlation between the corresponding voxels of di↵erent cubes. Thus, the spectrum of the group is highly sparse, leading to a very e↵ective separation of signal and noise by coefficient shrink-age. After inverse transformation, we obtain the estimates of each grouped cube, which are then aggregated at their original locations using adaptive weights.
We apply the BM4D algorithm for noisy data corrupted by Gaus-sian as well as Rician noise, leveraging the VST approach proposed in [47]. Adaptive noise variance estimation is also implemented by exploiting the sparsity of the representation of the group in transform domain, where the local groupwise standard deviation is accurately estimated from the outcome of robust median operations applied to the coefficients of the group spectrum [30].
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Additionally, we apply BM4D as a regularizer operator for the re-construction of incomplete volumetric data. In several inverse imag-ing applications, and particularly in MRI, the observed (acquired) measurements are a severe subsample of a transform-domain repre-sentation of the original unknown signal. The most popular recon-struction techniques are formulated as a convex optimization, usually solved by mathematical programming algorithms, that yields the so-lution most consistent with the available data. The optimization is typically constrained by a penalty term expressed as `0 or`1 norms, which are exploited to enable the sparsity of the assumed image priors [39, 74, 75, 122]. Di↵erently, the proposed procedure addresses the reconstruction of volumetric data having non-zero phase from a set of incomplete and noisy transform-domain measurements, replacing the common parametric modeling of the solution with a nonpara-metric one implemented by the use of a spatially adaptive denoising filter. Our reconstruction procedure works iteratively. In each itera-tion the missing part of the spectrum is excited with random noise;
then, after transforming the excited spectrum to the voxel domain, the BM4D filter attenuates the noise present in both magnitude and phase of the data, thus disclosing even the faintest details from the incomplete and degraded observations. The overall procedure can be interpreted as a progressive approximation in which the denoising filter directs the stochastic search towards the solution.
In Section 3.1 we will first introduce the formalization and imple-mentation of the basic BM4D volumetric filter, and then in Section 3.2 and 3.3 we will present its application in volumetric data denois-ing and reconstruction, respectively.
3.1 Basic Algorithm
The basic BM4D algorithm comprises grouping, collaborative filter-ing and aggregation [25], with an optional additional step for the groupwise noise variance estimation, which is enabled whenever the
3.1. Basic Algorithm 33
R
Figure 3.1. Schematic illustration of the BM4D grouping procedure.
The reference cube “R” is shown in blue.
variance of the noise is unknown. In what follows, we describe the general steps of the algorithm for the filtering of data corrupted by either Gaussian (2.2) or Rician noise (2.5). Note that, the noise vari-ance estimation can be also used to change the filtering strength in the presence of spatially varying noise, as the estimation is performed in a groupwise fashion and thus adapts to the local characteristics of the noise.
3.1.1 Grouping
In the grouping step, any given reference 3-D cube C(xR) of 3-D spatial coordinate xR 2 X ⇢ Z3 are extracted from the noisy data z and then tested for similarity against all cubes within a local 3-D neighborhood around the reference voxelxR. The similarity between two blocks is typically measured using a distance metric, e.g., the`2 -norm of the cubes di↵erence, and two blocks are considered similar if such distance is smaller than or equal to a predefined threshold. A schematic illustration of the grouping is provided in Fig. 3.1.
As a result, for each reference cubeC(xR), a groupG(xR) is build by stacking together mutually similar 3-D cubes along an additional fourth dimension, hence creating a 4-D group.
3.1.2 Adaptive Groupwise Noise Variance Esti-mation
Assuming that the noise variance is slowly varying, and since the grouped cubes have typically nearby coordinates, we can reasonably treat the noise level within each group as a constant. Therefore, only a single noise variance estimate is needed for each group. A precise estimation of the variance is a crucial task, because the amount of filtering operated on the noisy observations is proportional to the strength of the corrupting noise.
After the application of a sparsifyingT4D transform, the energy of the signal and that of the noise are well localized in the low- and high-frequencies portions of the group spectrum, respectively. Thus, in the case of Gaussian noise, an accurate groupwise variance estimation can be directly obtained from the median of absolute deviation [57, 40]
(MAD) of the high-frequencies coefficients of the group spectrum [30]. Di↵erently, if the noise follows a Rician distribution, we first need to estimate the mean-variance pair of the median value of the underlying noise-free group so that we can univocally and directly obtain a robust estimate of the scale parameter of the Rician noise in (2.5) [47].
3.1.3 Collaborative Filtering
Before the collaborative filtering, if the noise is Rician, a VST specif-ically designed for the Rice distribution [47] is applied to the group in order to remove the dependencies between the noise and the un-derlying data [6]. In this way, the stabilized group can be filtered using the constant standard deviation value induced by the VST.
During collaborative filtering, the group is first transformed by a decorrelating separable four-dimensional transform T4D, then the coefficients of the so-obtained spectrum are thresholded through a coefficient shrinkage operator (e.g., hard thresholding or Wiener fil-tering) scaled by the noise standard deviation level. An estimate of
3.2. Volumetric Data Denoising 35