Structured low-rank methods for robust 3D multi-shot EPI

Magnetic resonance imaging (MRI) has inherently slow acquisition speed, and Echo-Planar Imaging (EPI), as an efficient acquisition scheme, has been widely used in functional magnetic resonance imaging (fMRI) where an image series with high temporal resolution is needed to measure neuronal activity. Recently, 3D multi-shot EPI which samples data from an entire 3D volume with repeated shots has been drawing growing interest for fMRI with its high isotropic spatial resolution, particularly at ultra-high fields. However, compared to single-shot EPI, multi-shot EPI is sensitive to any inter-shot instabilities, e.g., subject movement and even physiologically induced field fluctuations. These inter-shot inconsistencies can greatly negate the theoretical benefits of 3D multi-shot EPI over conventional 2D multi-slice acquisitions. Structured low-rank image reconstruction which regularises under-sampled image reconstruction by exploiting the linear dependencies in MRI data has been successfully demonstrated in a variety of applications. In this thesis, a structured low-rank reconstruction method is optimised for 3D multi-shot EPI imaging together with a dedicated sampling pattern termed seg-CAIPI, in order to enhance the robustness to physiological fluctuations and improve the temporal stability of 3D multi-shot EPI for fMRI at 7T. Moreover, a motion compensated structured low-rank reconstruction framework is also presented for robust 3D multi-shot EPI which further takes into account inter-shot instabilities due to bulk motion. Lastly, this thesis also investigates into the improvement of structured low-rank reconstruction from an algorithmic perspective and presents the locally structured low-rank reconstruction scheme

Methods for Photoacoustic Image Reconstruction Exploiting Properties of Curvelet Frame

Curvelet frame is of special significance for photoacoustic tomography (PAT) due to its sparsifying and microlocalisation properties. In this PhD project, we explore the methods for image reconstruction in PAT with flat sensor geometry using Curvelet properties. This thesis makes five distinct contributions: (i) We investigate formulation of the forward, adjoint and inverse operators for PAT in Fourier domain. We derive a one-to-one map between wavefront directions in image and data spaces in PAT. Combining the Fourier operators with the wavefront map allows us to create the appropriate PAT operators for solving limited-view problems due to limited angular sensor sensitivity. (ii) We devise a concept of wedge restricted Curvelet transform, a modification of standard Curvelet transform, which allows us to formulate a tight frame of wedge restricted Curvelets on the range of the PAT forward operator for PAT data representation. We consider details specific to PAT data such as symmetries, time oversampling and their consequences. We further adapt the wedge restricted Curvelet to decompose the wavefronts into visible and invisible parts in the data domain as well as in the image domain. (iii) We formulate a two step approach based on the recovery of the complete volume of the photoacoustic data from the sub-sampled data followed by the acoustic inversion, and a one step approach where the photoacoustic image is directly recovered from the subsampled data. The wedge restricted Curvelet is used as the sparse representation of the photoacoustic data in the two step approach. (iv) We discuss a joint variational approach that incorporates Curvelet sparsity in photoacoustic image domain and spatio-temporal regularization via optical flow constraint to achieve improved results for dynamic PAT reconstruction. (v) We consider the limited-view problem due to limited angular sensitivity of the sensor (see (i) for the formulation of the corresponding fast operators in Fourier domain). We propose complementary information learning approach based on splitting the problem into visible and invisible singularities. We perform a sparse reconstruction of the visible Curvelet coefficients using compressed sensing techniques and propose a tailored deep neural network architecture to recover the invisible coefficients

Automatic Music Transcription using Structure and Sparsity

PhdAutomatic Music Transcription seeks a machine understanding of a musical signal in terms of pitch-time activations. One popular approach to this problem is the use of spectrogram decompositions, whereby a signal matrix is decomposed over a dictionary of spectral templates, each representing a note. Typically the decomposition is performed using gradient descent based methods, performed using multiplicative updates based on Non-negative Matrix Factorisation (NMF). The final representation may be expected to be sparse, as the musical signal itself is considered to consist of few active notes. In this thesis some concepts that are familiar in the sparse representations literature are introduced to the AMT problem. Structured sparsity assumes that certain atoms tend to be active together. In the context of AMT this affords the use of subspace modelling of notes, and non-negative group sparse algorithms are proposed in order to exploit the greater modelling capability introduced. Stepwise methods are often used for decomposing sparse signals and their use for AMT has previously been limited. Some new approaches to AMT are proposed by incorporation of stepwise optimal approaches with promising results seen. Dictionary coherence is used to provide recovery conditions for sparse algorithms. While such guarantees are not possible in the context of AMT, it is found that coherence is a useful parameter to consider, affording improved performance in spectrogram decompositions

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum