Shearlets: an overview

The aim of this report is a self-contained overview on shearlets, a new multiscale method emerged in the last decade to overcome some of the limitation of traditional multiscale methods, like wavelets. Shearlets are obtained by translating, dilating and shearing a single mother function. Thus, the elements of a shearlet system are distributed not only at various scales and locations – as in classical wavelet theory – but also at various orientations. Thanks to this directional sensitivity property, shearlets are able to capture anisotropic features, like edges, that frequently dominate multidimensional phenomena, and to obtain optimally sparse approximations. Moreover, the simple mathematical structure of shearlets allows for the generalization to higher dimensions and to treat uniformly the continuum and the discrete realms, as well as fast algorithmic implementation. For all these reasons, shearlets are one of the most successful tool for the efficient representation of multidimensional data and they are being employed in several numerical applications

Shearlet-based regularization in statistical inverse learning with an application to X-ray tomography

Statistical inverse learning theory, a field that lies at the intersection of inverse problems and statistical learning, has lately gained more and more attention. In an effort to steer this interplay more towards the variational regularization framework, convergence rates have recently been proved for a class of convex, $p$-homogeneous regularizers with $p \in (1,2]$, in the symmetric Bregman distance. Following this path, we take a further step towards the study of sparsity-promoting regularization and extend the aforementioned convergence rates to work with $\ell^p$-norm regularization, with $p \in (1,2)$, for a special class of non-tight Banach frames, called shearlets, and possibly constrained to some convex set. The $p = 1$ case is approached as the limit case $(1,2) \ni p \rightarrow 1$, by complementing numerical evidence with a (partial) theoretical analysis, based on arguments from $\Gamma$-convergence theory. We numerically demonstrate our theoretical results in the context of X-ray tomography, under random sampling of the imaging angles, using both simulated and measured data

4D Dual-Tree Complex Wavelets for Time-Dependent Data

The dual-tree complex wavelet transform (DT-ℂWT) is extended to the 4D setting. Key properties of 4D DT-ℂWT, such as directional sensitivity and shift-invariance, are discussed and illustrated in a tomographic application. The inverse problem of reconstructing a dynamic three-dimensional target from X-ray projection measurements can be formulated as 4D space-time tomography. The results suggest that 4D DT-ℂWT offers simple implementations combined with useful theoretical properties for tomographic reconstruction.Peer reviewe

Controlled wavelet domain sparsity for x-ray tomography

Tomographic reconstruction is an ill-posed inverse problem that calls for regularization. One possibility is to require sparsity of the unknown in an orthonormal wavelet basis. This, in turn, can be achieved by variational regularization, where the penalty term is the sum of the absolute values of the wavelet coefficients. The primal-dual fixed point algorithm showed that the minimizer of the variational regularization functional can be computed iteratively using a soft-thresholding operation. Choosing the soft-thresholding parameter mu > 0 is analogous to the notoriously difficult problem of picking the optimal regularization parameter in Tikhonov regularization. Here, a novel automatic method is introduced for choosing mu, based on a control algorithm driving the sparsity of the reconstruction to an a priori known ratio of nonzero versus zero wavelet coefficients in the unknown.Peer reviewe

Convex regularization in statistical inverse learning problems

We consider a statistical inverse learning problem, where the task is to estimate a function $f$ based on noisy point evaluations of $Af$, where $A$ is a linear operator. The function $Af$ is evaluated at i.i.d. random design points $u_n$, $n=1,...,N$ generated by an unknown general probability distribution. We consider Tikhonov regularization with general convex and $p-$homogeneous penalty functionals and derive concentration rates of the regularized solution to the ground truth measured in the symmetric Bregman distance induced by the penalty functional. We derive concrete rates for Besov norm penalties and numerically demonstrate the correspondence with the observed rates in the context of X-ray tomography

Shearlet-based regularization in sparse dynamic tomography

Peer reviewe

Deep Neural Networks for Inverse Problems with Pseudodifferential Operators : An Application to Limited-Angle Tomography

We propose a novel convolutional neural network (CNN), called Psi DONet, designed for learning pseudodifferential operators (Psi DOs) in the context of linear inverse problems. Our starting point is the iterative soft thresholding algorithm (ISTA), a well-known algorithm to solve sparsity-promoting minimization problems. We show that, under rather general assumptions on the forward operator, the unfolded iterations of ISTA can be interpreted as the successive layers of a CNN, which in turn provides fairly general network architectures that, for a specific choice of the parameters involved, allow us to reproduce ISTA, or a perturbation of ISTA for which we can bound the coefficients of the filters. Our case study is the limited-angle X-ray transform and its application to limited-angle computed tomography (LA-CT). In particular, we prove that, in the case of LA-CT, the operations of upscaling, downscaling, and convolution, which characterize our Psi DONet and most deep learning schemes, can be exactly determined by combining the convolutional nature of the limited-angle X-ray transform and basic properties defining an orthogonal wavelet system. We test two different implementations of Psi DONet on simulated data from limited-angle geometry, generated from the ellipse data set. Both implementations provide equally good and noteworthy preliminary results, showing the potential of the approach we propose and paving the way to applying the same idea to other convolutional operators which are Psi DOs or Fourier integral operators.Peer reviewe

In-air and in-water performance comparison of Passive Gamma Emission Tomography with activated Co-60 rods

Abstract A first-of-a-kind geological repository for spent nuclear fuel is being built in Finland and will soon start operations. To make sure all nuclear material stays in peaceful use, the fuel is measured with two complementary non-destructive methods to verify the integrity and the fissile content of the fuel prior to disposal. For pin-wise identification of active fuel material, a Passive Gamma Emission Tomography (PGET) device is used. Gamma radiation emitted by the fuel is assayed from 360 angles around the assembly with highly collimated CdZnTe detectors, and a 2D cross-sectional image is reconstructed from the data. At the encapsulation plant in Finland, there will be the possibility to measure in air. Since the performance of the method has only been studied in water, measurements with mock-up fuel were conducted at the Atominstitut in Vienna, Austria. Four different arrangements of activated Co-60 rods, steel rods and empty positions were investigated both in air and in water to confirm the functionality of the method. The measurement medium was not observed to affect the ability of the method to distinguish modified rod positions from filled rod positions. More extended conclusions about the method performance with real spent nuclear fuel cannot be drawn from the mock-up studies, since the gamma energies, activities, material attenuations and assembly dimensions are different, but full-scale measurements with spent nuclear fuel are planned for 2023