1,720,974 research outputs found
Shearlet-based regularization in statistical inverse learning with an application to x-ray tomography
Full text linkStatistical 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 (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 .," p -norm regularization, with p (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) p → 1, by complementing numerical evidence with a (partial) theoretical analysis, based on arguments from "-convergence theory. We numerically validate our theoretical results in the context of x-ray tomography, under random sampling of the imaging angles, using both simulated and measured data. This application allows to effectively verify the theoretical decay, in addition to providing a motivation for the extension to shearlet-based regularization
Gamma ray emission imaging in the medical and nuclear safeguards fields
No full textGamma rays emitted from within an object can reveal information about that object in a non-destructive way, i.e. without physically opening the object and looking inside. This makes gamma ray emission imaging very useful in widely varying applications. In these notes, we highlight its application to the medical field, where we discuss molecular imaging in nuclear medicine and in vivo dose delivery verification in particle beam radiotherapy, and nuclear safeguards field, where imaging of spent nuclear fuel assemblies is part of monitoring the non-proliferation of nuclear weapons. The purpose and basic principles of gamma ray emission imaging are discussed as the foundation to look in more detail into the essential instrument design considerations and the iterative image reconstruction procedures. These notes are not intended to be a comprehensive review; their purpose is to introduce gamma ray emission imaging to those that are new to this technique. The examples of implementation that are presented were thus chosen in order to introduce the reader to a fairly wide range of applications and practical implementations
4D Dual-Tree Complex Wavelets for time-dependent data
No full textThe 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
Regularization with Optimal Space-Time Priors
No full textWe propose a variational regularization approach based on a multiscale representation called cylindrical shearlets aimed at dynamic imaging problems, especially dynamic tomography. The intuitive idea of our approach is to integrate a sequence of separable static problems in the mismatch term of the cost function, while the regularization term handles the nonstationary target as a spatio-temporal object. This approach is motivated by the fact that cylindrical shearlets provide (nearly) optimally sparse approximations on an idealized class of functions modeling spatio-temportal data and the numerical observation that they provide highly sparse approximations even for more general spatio-temporal image sequences found in dynamic tomography applications. To formulate our regularization model, we introduce cylindrical shearlet smoothness spaces, which are instrumental for defining suitable embeddings in functional spaces. We prove that the proposed regularization strategy is well-defined, and the minimization problem has a unique solution (for p > 1). Furthermore, we provide convergence rates (in terms of the symmetric Bregman distance) under deterministic and random noise conditions, within the context of statistical inverse learning. We numerically validate our theoretical results using both simulated and measured dynamic tomography data, showing that our approach leads to an efficient and robust reconstruction strategy
Controlled Wavelet Domain Sparsity for X-ray Tomography
No full textTomographic 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 μ > 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 μ, 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
CONVEX REGULARIZATION IN STATISTICAL INVERSE LEARNING PROBLEMS
Full text linkWe 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
Trust your source: quantifying source condition elements for variational regularisation methods
Full text linkSource conditions are a key tool in regularisation theory that are needed to derive error estimates and convergence rates for ill-posed inverse problems. In this paper, we provide a recipe to practically compute source condition elements as the solution of convex minimisation problems that can be solved with first-order algorithms. We demonstrate the validity of our approach by testing it on two inverse problem case studies in machine learning and image processing: sparse coefficient estimation of a polynomial via LASSO regression and recovering an image from a subset of the coefficients of its discrete Fourier transform. We further demonstrate that the proposed approach can easily be modified to solve the machine learning task of identifying the optimal sampling pattern in the Fourier domain for a given image and variational regularisation method, which has applications in the context of sparsity promoting reconstruction from magnetic resonance imaging data
Shearlet-based regularization in sparse dynamic tomography
No full textClassical tomographic imaging is soundly understood and widely employed in medicine, nondestructive testing and security applications. However, it still o↵ers many challenges when it comes to dynamic tomography. Indeed, in classical tomography, the target is usually assumed to be stationary during the data acquisition, but this is not a realistic model. Moreover, to ensure a lower X-ray radiation dose, only a sparse collection of measurements per time step is assumed to be available. With such a set up, we deal with a sparse data, dynamic tomography problem, which clearly calls for regularization, due to the loss of information in the data and the ongoing motion. In this paper, we propose a 3D variational formulation based on 3D shearlets, where the third dimension accounts for the motion in time, to reconstruct a moving 2D object. Results are presented for real measured data and compared against a 2D static model, in the case of fan-beam geometry. Results are preliminary but show that better reconstructions can be achieved when motion is taken into account
In-air and in-water performance comparison of Passive Gamma Emission Tomography with activated Co-60 rods
No full textA 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
- …
