Simultaneously Sparse Solutions to Linear Inverse Problems with Multiple System Matrices and a Single Observation Vector

A linear inverse problem is proposed that requires the determination of multiple unknown signal vectors. Each unknown vector passes through a different system matrix and the results are added to yield a single observation vector. Given the matrices and lone observation, the objective is to find a simultaneously sparse set of unknown vectors that solves the system. We will refer to this as the multiple-system single-output (MSSO) simultaneous sparsity problem. This manuscript contrasts the MSSO problem with other simultaneous sparsity problems and conducts a thorough initial exploration of algorithms with which to solve it. Seven algorithms are formulated that approximately solve this NP-Hard problem. Three greedy techniques are developed (matching pursuit, orthogonal matching pursuit, and least squares matching pursuit) along with four methods based on a convex relaxation (iteratively reweighted least squares, two forms of iterative shrinkage, and formulation as a second-order cone program). The algorithms are evaluated across three experiments: the first and second involve sparsity profile recovery in noiseless and noisy scenarios, respectively, while the third deals with magnetic resonance imaging radio-frequency excitation pulse design.Comment: 36 pages; manuscript unchanged from July 21, 2008, except for updated references; content appears in September 2008 PhD thesi

Sparsity-Promoting Calibration for GRAPPA Accelerated Parallel MRI Reconstruction

The amount of calibration data needed to produce images of adequate quality can prevent auto-calibrating parallel imaging reconstruction methods like generalized autocalibrating partially parallel acquisitions (GRAPPA) from achieving a high total acceleration factor. To improve the quality of calibration when the number of auto-calibration signal (ACS) lines is restricted, we propose a sparsity-promoting regularized calibration method that finds a GRAPPA kernel consistent with the ACS fit equations that yields jointly sparse reconstructed coil channel images. Several experiments evaluate the performance of the proposed method relative to unregularized and existing regularized calibration methods for both low-quality and underdetermined fits from the ACS lines. These experiments demonstrate that the proposed method, like other regularization methods, is capable of mitigating noise amplification, and in addition, the proposed method is particularly effective at minimizing coherent aliasing artifacts caused by poor kernel calibration in real data. Using the proposed method, we can increase the total achievable acceleration while reducing degradation of the reconstructed image better than existing regularized calibration methods.National Science Foundation (U.S.) (CAREER Grant 0643836)National Institutes of Health (U.S.) (Grant NIH R01 EB007942)National Institutes of Health (U.S.) (Grant NIH R01 EB006847)National Institutes of Health (U.S.) (Grant NIH P41 RR014075)National Institutes of Health (U.S.) (Grant NIH K01 EB011498)National Institutes of Health (U.S.) (Grant NIH F32 EB015914)National Science Foundation (U.S.). Graduate Research Fellowship Progra

Accelerated parallel magnetic resonance imaging reconstruction using joint estimation with a sparse signal model

Accelerating magnetic resonance imaging (MRI) by reducing the number of acquired k-space scan lines benefits conventional MRI significantly by decreasing the time subjects remain in the magnet. In this paper, we formulate a novel method for Joint estimation from Undersampled LinEs in Parallel MRI (JULEP) that simultaneously calibrates the GeneRalized Autocalibrating Partially Parallel Acquisitions (GRAPPA) reconstruction kernel and reconstructs the full multi-channel k-space. We employ a joint sparsity signal model for the channel images in conjunction with observation models for both the acquired data and GRAPPA reconstructed k-space. We demonstrate using real MRI data that JULEP outperforms conventional GRAPPA reconstruction at high levels of undersampling, increasing the peak-signal-to-noise ratio by up to 10 dB.National Science Foundation (U.S.) (CAREER Grant 0643836)National Center for Research Resources (U.S.) (P41 RR014075)National Institutes of Health (U.S.) (NIH R01 EB007942)National Institutes of Health (U.S.) (NIH R01 EB006847)Siemens CorporationNational Science Foundation (U.S.). Graduate Research Fellowship Progra

Designing lightweight 3D-printable bioinspired structures for enhanced compression and energy absorption properties

Recent progress in additive manufacturing, also known as 3D printing, has offered several benefits, including high geometrical freedom and the ability to create bioinspired structures with intricate details. Mantis shrimp can scrape the shells of prey molluscs with its hammer-shaped stick, while beetles have highly adapted forewings that are lightweight, tough, and strong. This paper introduces a design approach for bioinspired lattice structures by mimicking the internal microstructures of a beetle’s forewing, a mantis shrimp’s shell, and a mantis shrimp’s dactyl club, with improved mechanical properties. Finite element analysis (FEA) and experimental characterisation of 3D printed polylactic acid (PLA) samples with bioinspired structures were performed to determine their compression and impact properties. The results showed that designing a bioinspired lattice with unit cells parallel to the load direction improved quasi-static compressive performance, among other lattice structures. The gyroid honeycomb lattice design of the insect forewings and mantis shrimp dactyl clubs outperformed the gyroid honeycomb design of the mantis shrimp shell, with improvements in ultimate mechanical strength, Young’s modulus, and drop weight impact. On the other hand, hybrid designs created by merging two different designs reduced bending deformation to control collapse during drop weight impact. This work holds promise for the development of bioinspired lattices employing designs with improved properties, which can have potential implications for lightweight high-performance applications