Vol. 3 No. 4 - Dec 2017

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The Empirical Effect of Gaussian Noise in Undersampled MRI Reconstruction Patrick Virtue and Michael Lustig Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA Corresponding Author: Patrick Virtue, BS Electrical Engineering and Computer Sciences, University of California, Berkeley, 253 Cory Hall, Berkeley, CA 94720; E-mail: Key Words: image reconstruction, noise analysis, MRI, undersampling, compressed sensing Abbreviations: Signal-to-noise ratio (SNR), magnetic resonance imaging (MRI), computed tomography (CT), positron emission tomography (PET), maximum a posteriori (MAP), weighted least squares (WLS), mean squared error (MSE), repetition time (TR), flip angle (FA), echo time (TE), field of view (FOV), 1-dimensional (1D), 2-dimensional (2D), 3-dimensional (3D) In Fourier-based medical imaging, sampling below the Nyquist rate results in an underdetermined system, in which a linear reconstruction will exhibit artifacts. Another consequence is lower signal-to-noise ratio (SNR) because of fewer acquired measurements. Even if one could obtain information to perfectly disambiguate the underdetermined system, the reconstructed image could still have lower image quality than a corresponding fully sampled acquisition because of reduced measurement time. The coupled effects of low SNR and under- determined system during reconstruction makes it difficult to isolate the impact of low SNR on image quality. To this end, we present an image quality prediction process that reconstructs fully sampled, fully determined data with noise added to simulate the SNR loss induced by a given undersampling pattern. The resulting pre- diction image empirically shows the effects of noise in undersampled image reconstruction without any effect from an underdetermined system. We discuss how our image quality prediction process simulates the distri- bution of noise for a given undersampling pattern, including variable density sampling that produces colored noise in the measurement data. An interesting consequence of our prediction model is that recovery from an underdetermined nonuniform sampling is equivalent to a weighted least squares optimization that accounts for heterogeneous noise levels across measurements. Through experiments with synthetic and in vivo data- sets, we demonstrate the efficacy of the image quality prediction process and show that it provides a better estimation of reconstruction image quality than the corresponding fully sampled reference image. INTRODUCTION Undersampling in Fourier-based medical imaging provides a variety of clinical benefits including shorter exam times, re- duced motion artifacts, and the ability to capture fast-moving dynamics, such as cardiac motion. Undersampling reduces ac- quisition time by collecting fewer measurements in the fre- quency domain than required by the Nyquist rate. However, undersampling causes two specific challenges for the recon- struction system, namely, an underdetermined system 1 of lin- ear equations and lower SNR (signal-to-noise ratio) because of reduced measurement time. When reconstruction algorithms are able to overcome these challenges, undersampling can benefit a variety of Fourier-based imaging modalities, including mag- netic resonance imaging (MRI) with parallel imaging or com- pressed sensing (1, 2), computed tomography (CT) with reduced or gated acquisition views (3, 4), and positron emission tomog- raphy (PET) with multiplexed or missing detectors (5, 6). Under- sampling for acceleration is becoming the mainstream approach for fast imaging. In fact, this year, two of the major MRI man- ufacturers have announced products which leverage undersam- pling and a compressed sensing reconstruction that have been approved by the Food and Drug Administration (FDA). While the tools and analysis discussed in this paper apply generally to Fourier-based medical imaging with Gaussian noise, we will direct our numerical modeling, examples, and experiments to the application of compressed sensing MRI. When designing an undersampled reconstruction system, the primary concern is often focused on compensating for the underdetermined system caused by sub-Nyquist sampling, for example choosing a sparse representation for compressed sens- ing. However, we should not overlook the fact that collecting fewer measurements in practice leads to overall lower SNR in the acquired data. If the measurements are too noisy, the low SNR will lead to poor reconstructed image quality even if the recon- struction system were fully determined. On the other hand, with high SNR measurements, the resulting image quality will be 1 In the context of this paper, we specify fully determined and underdeter- mined as follows: for a fixed Cartesian k-space (frequency space) grid with predefined field of view and spatial resolution parameters, fully determined means having at least one measured sample for each k-space grid location, and underdetermined means at least one k-space location has zero samples, in which case we have more unknowns (image pixels) than equations (one per acquired k-space location). RESEARCH ARTICLE ABSTRACT © 2017 The Authors. Published by Grapho Publications, LLC This is an open access article under the CC BY-NC-ND license ( ISSN 2379-1381 TOMOGRAPHY.ORG | VOLUME 3 NUMBER 4 | DECEMBER 2017 211

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