Magnetic Particle Imaging (MPI) is a preclinical imaging technique capable of visualizing the spatio-temporal distribution of magnetic nanoparticles. The image reconstruction of this fast and dynamic process relies on efficiently solving an ill-posed inverse problem. Current approaches to reconstruct the tracer concentration from its measurements are either adapted to image characteristics of MPI but suffer from higher computational complexity and slower convergence or are fast but lack in the image quality of the reconstructed images. In this work we propose a novel MPI reconstruction method to combine the advantages of both approaches into a single algorithm. The underlying sparsity prior is based on an undecimated wavelet transform and is integrated into a fast row-action framework to solve the corresponding MPI minimization problem. Its performance is numerically evaluated against a classical FISTA approach on simulated and real MPI data. We also compare the results to the state-of-the-art MPI reconstruction methods. In all cases, our approach shows better reconstruction results and at the same time accelerates the convergence rate of the underlying row-action algorithm.
Universität Heidelberg（海德堡大学）；Hochschule Darmstadt（达姆施塔特应用技术大学）；Helmholtz Zentrum München（慕尼黑亥姆霍兹中心）
In this paper, we consider the sparse regularization of manifold-valued data with respect to an interpolatory wavelet/multiscale transform. We propose and study variational models for this task and provide results on their well-posedness. We present algorithms for a numerical realization of these models in the manifold setup. Further, we provide experimental results to show the potential of the proposed schemes for applications.
The \emph{deterministic} sparse grid method, also known as Smolyak's method, is a well-established and widely used tool to tackle multivariate approximation problems, and there is a vast literature on it. Much less is known about \emph{randomized} versions of the sparse grid method. In this paper we analyze randomized sparse grid algorithms, namely randomized sparse grid quadratures for multivariate integration on the $D$-dimensional unit cube $[0,1)^D$. Let $d,s \in \mathbb{N}$ be such that $D=d\cdot s$. The $s$-dimensional building blocks of the sparse grid quadratures are based on stratifie…
In this paper, we present modifications of the iterative hard thresholding (IHT) method for recovery of jointly row-sparse and low-rank matrices. In particular a Riemannian version of IHT is considered which significantly reduces computational cost of the gradient projection in the case of rank-one measurement operators, which have concrete applications in blind deconvolution. Experimental results are reported that show near-optimal recovery for Gaussian and rank-one measurements, and that adaptive stepsizes give crucial improvement. A Riemannian proximal gradient method is derived for the spe…
Wavelet theory has been well studied in recent decades. Due to their appealing features such as sparse multiscale representation and fast algorithms, wavelets have enjoyed many tremendous successes in the areas of signal/image processing and computational mathematics. This paper primarily intends to shed some light on the advantages of using wavelets in the context of numerical differential equations. We shall identify a few prominent problems in this field and recapitulate some important results along these directions. Wavelet-based methods for numerical differential equations offer the advan…
The purpose of sparse modelling of ECG signals is to represent an ECG record, given by sample points, as a linear combination of as few elementary components as possible. This can be achieved by creating a redundant set, called a dictionary, from where the elementary components are selected. The success in sparsely representing an ECG record depends on the nature of the dictionary being considered. In this paper we focus on the construction of different families of wavelet dictionaries, which are appropriate for the purpose of reducing dimensionality of ECG signals through sparse representatio…
Sparse Wavelet Representations of Spatially Varying Blurring Operators
Institut Supérieur de l’Aéronautique et de l’Espace (ISAE)（国立高等航空航天学院）；Institut des Technologies Avancées en Sciences du Vivant, ITAV-USR3505 and Institut de Mathématiques de Toulouse, IMT-UMR5219, CNRS and université de Toulouse（生命科学先进技术研究所及图卢兹数学研究所）
In this paper, we present, evaluate and analyse the performance of parallel synchronous Jacobi algorithms by different partitioned procedures including band-row splitting, band-row sparsity pattern splitting and substructuring splitting, when solving sparse large linear systems. Numerical experiments performed on a set of academic 3D Laplace equation and on a real gravity matrices arising from the Chicxulub crater are exhibited, and show the impact of splitting on parallel synchronous iterations when solving sparse large linear systems. The numerical results clearly show the interest of substr…
In this effort, we propose a convex optimization approach based on weighted $\ell_1$-regularization for reconstructing objects of interest, such as signals or images, that are sparse or compressible in a wavelet basis. We recover the wavelet coefficients associated to the functional representation of the object of interest by solving our proposed optimization problem. We give a specific choice of weights and show numerically that the chosen weights admit efficient recovery of objects of interest from either a set of sub-samples or a noisy version. Our method not only exploits sparsity but also…
Multiscale and multiphysics problems need novel numerical methods in order for them to be solved correctly and predictively. To that end, we develop a wavelet based technique to solve a coupled system of nonlinear partial differential equations (PDEs) while resolving features on a wide range of spatial and temporal scales. The algorithm exploits the multiresolution nature of wavelet basis functions to solve initial-boundary value problems on finite domains with a sparse multiresolution spatial discretization. By leveraging wavelet theory and embedding a predictor-corrector procedure within the…
