Non-Equispaced Fast Fourier Transforms in Turbulence Simulation

Fourier pseudo-spectral method on equispaced grid is one of the approaches in turbulence simulation, to compute derivative of discrete data, using fast Fourier Transform (FFT) and gives low dispersion and dissipation errors. In many turbulent flows the dynamically important scales of motion are concentrated in certain regions which requires a coarser grid for higher accuracy. A coarser grid in other regions minimizes the memory requirement. This requires the use of Non-equispaced Fast Fourier Transform (NFFT) to compute the Fourier transform, by solving a system of linear equations. To achieve similar accuracy, the NFFT needs to return more Fourier coefficients than the number of non-equispaced gridpoints, making it an under-determined system. The minimum L2 norm solution of the system is refined using an iterative reconstruction algorithm, FOCUSS. The NFFT and FOCUSS algorithms yield accurate results with smaller test case of a Direct Numerical Simulation on a grid of 64 gridpoints in each dimension, using Taylor green initial condition. The computational speed for this case was found to be unacceptably slow and few methods to improve the performance have been discussed. The approach of NFFT and FOCUSS was tested on a line extracted from 3-dimensional turbulent flow field. Fourier transform of the extracted line, sampled on 1024 non-equispaced gridpoints, computed for 2048 coefficients and the corresponding numerical derivative are found to be inaccurate. It can be observed that the NFFT and FOCUSS approach works for sparse Fourier transform, but not for turbulent fields having a wideband Fourier transform

Fast image reconstruction with L2-regularization

Purpose We introduce L2-regularized reconstruction algorithms with closed-form solutions that achieve dramatic computational speed-up relative to state of the art L1- and L2-based iterative algorithms while maintaining similar image quality for various applications in MRI reconstruction. Materials and Methods We compare fast L2-based methods to state of the art algorithms employing iterative L1- and L2-regularization in numerical phantom and in vivo data in three applications; (i) Fast Quantitative Susceptibility Mapping (QSM), (ii) Lipid artifact suppression in Magnetic Resonance Spectroscopic Imaging (MRSI), and (iii) Diffusion Spectrum Imaging (DSI). In all cases, proposed L2-based methods are compared with the state of the art algorithms, and two to three orders of magnitude speed up is demonstrated with similar reconstruction quality. Results The closed-form solution developed for regularized QSM allows processing of a three-dimensional volume under 5 s, the proposed lipid suppression algorithm takes under 1 s to reconstruct single-slice MRSI data, while the PCA based DSI algorithm estimates diffusion propagators from undersampled q-space for a single slice under 30 s, all running in Matlab using a standard workstation. Conclusion For the applications considered herein, closed-form L2-regularization can be a faster alternative to its iterative counterpart or L1-based iterative algorithms, without compromising image quality.National Institute for Biomedical Imaging and Bioengineering (U.S.) (Grant NIBIB K99EB012107)National Institutes of Health (U.S.) (Grant NIH R01 EB007942)National Institute for Biomedical Imaging and Bioengineering (U.S.) (Grant NIBIB R01EB006847)Grant K99/R00 EB008129National Center for Research Resources (U.S.) (Grant NCRR P41RR14075)National Institutes of Health (U.S.) (Blueprint for Neuroscience Research U01MH093765)Siemens CorporationSiemens-MIT AllianceMIT-Center for Integration of Medicine and Innovative Technology (Medical Engineering Fellowship

Doctor of Philosophy

dissertationDynamic contrast enhanced magnetic resonance imaging (DCE-MRI) is a powerful tool to detect cardiac diseases and tumors, and both spatial resolution and temporal resolution are important for disease detection. Sampling less in each time frame and applying sophisticated reconstruction methods to overcome image degradations is a common strategy in the literature. In this thesis, temporal TV constrained reconstruction that was successfully applied to DCE myocardial perfusion imaging by our group was extended to three-dimensional (3D) DCE breast and 3D myocardial perfusion imaging, and the extension includes different forms of constraint terms and various sampling patterns. We also explored some other popular reconstruction algorithms from a theoretical level and showed that they can be included in a unified framework. Current 3D Cartesian DCE breast tumor imaging is limited in spatiotemporal resolution as high temporal resolution is desired to track the contrast enhancement curves, and high spatial resolution is desired to discern tumor morphology. Here temporal TV constrained reconstruction was extended and different forms of temporal TV constraints were compared on 3D Cartesian DCE breast tumor data with simulated undersampling. Kinetic parameters analysis was used to validate the methods

Multi-contrast reconstruction with Bayesian compressed sensing

Clinical imaging with structural MRI routinely relies on multiple acquisitions of the same region of interest under several different contrast preparations. This work presents a reconstruction algorithm based on Bayesian compressed sensing to jointly reconstruct a set of images from undersampled k-space data with higher fidelity than when the images are reconstructed either individually or jointly by a previously proposed algorithm, M-FOCUSS. The joint inference problem is formulated in a hierarchical Bayesian setting, wherein solving each of the inverse problems corresponds to finding the parameters (here, image gradient coefficients) associated with each of the images. The variance of image gradients across contrasts for a single volumetric spatial position is a single hyperparameter. All of the images from the same anatomical region, but with different contrast properties, contribute to the estimation of the hyperparameters, and once they are found, the k-space data belonging to each image are used independently to infer the image gradients. Thus, commonality of image spatial structure across contrasts is exploited without the problematic assumption of correlation across contrasts. Examples demonstrate improved reconstruction quality (up to a factor of 4 in root-mean-square error) compared with previous compressed sensing algorithms and show the benefit of joint inversion under a hierarchical Bayesian model

Conjugate Gradient Method Applied to Cortical Imaging in EEG/ERP

International audienc

Low-rank and sparse reconstruction in dynamic magnetic resonance imaging via proximal splitting methods

Dynamic magnetic resonance imaging (MRI) consists of collecting multiple MR images in time, resulting in a spatio-temporal signal. However, MRI intrinsically suffers from long acquisition times due to various constraints. This limits the full potential of dynamic MR imaging, such as obtaining high spatial and temporal resolutions which are crucial to observe dynamic phenomena. This dissertation addresses the problem of the reconstruction of dynamic MR images from a limited amount of samples arising from a nuclear magnetic resonance experiment. The term limited can be explained by the approach taken in this thesis to speed up scan time, which is based on violating the Nyquist criterion by skipping measurements that would be normally acquired in a standard MRI procedure. The resulting problem can be classified in the general framework of linear ill-posed inverse problems. This thesis shows how low-dimensional signal models, specifically lowrank and sparsity, can help in the reconstruction of dynamic images from partial measurements. The use of these models are justified by significant developments in signal recovery techniques from partial data that have emerged in recent years in signal processing. The major contributions of this thesis are the development and characterisation of fast and efficient computational tools using convex low-rank and sparse constraints via proximal gradient methods, the development and characterisation of a novel joint reconstruction–separation method via the low-rank plus sparse matrix decomposition technique, and the development and characterisation of low-rank based recovery methods in the context of dynamic parallel MRI. Finally, an additional contribution of this thesis is to formulate the various MR image reconstruction problems in the context of convex optimisation to develop algorithms based on proximal splitting methods

A swarm based heuristic for sparse image recovery

This paper discusses the Compressive Sampling framework as an application for sparse representation (factorization) and recovery of images over an over-complete basis (dictionary). Compressive Sampling is a novel new area which asserts that one can recover images of interest, with much fewer measurements than were originally thought necessary, by searching for the sparsest representation of an image over an over-complete dictionary. This task is achieved by optimizing an objective function that includes two terms: one that measures the image reconstruction error and another that measures the sparsity level. We present and discuss a new swarm based heuristic for sparse image approximation using the Discrete Fourier Transform to enhance its level of sparsity. Our experimental results on reference images demonstrate the good performance of the proposed heuristic over other standard sparse recovery methods (L1-Magic and FOCUSS packages), in a noiseless environment using much fewer measurements. Finally, we discuss possible extensions of the heuristic in noisy environments and weakly sparse images as a realistic improvement with much higher applicability