Vol. 3 No. 4 - Dec 2017

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limited by how well the reconstruction can constrain the under- determined system. The effects of the underdetermined system and the lower SNR are coupled during the reconstruction pro- cess, making it difficult to analyze one without the other. It is important, however, to analyze how both issues impact the reconstruction system to determine the empirical limits of un- dersampling and gain insight on how to improve undersampled acquisition and reconstruction when targeting specific applica- tions. Compressed sensing theory has provided us with exten- sive analysis on the bounds for the successful signal recovery from undersampled data. Candès (7) describes a bound on the squared error of the recovered signal limited by the under- sampling rate and the sparsity of the data. He also shows that this bound scales linearly with the variance of the noise in the measured data. Candès and Plan (8) provide a more general compressed sensing theory that addresses a combination of practical concerns. For instance, they derive the bounds on squared error of the recovered signal for systems with Fourier encoding matrices, noise measurements, and approximately sparse signals. Unfortunately, while squared error is an impor- tant tool in measuring similarity between signals, it often fails to provide a good measure of perceptual image quality. Wain- wright (9) improves upon the squared error definition of success by studying the undersampling rates and sparsity levels for which there is a high probability of successfully recovering the support of the sparse signal. Although it is important to have a theory showing that reconstruction techniques are mathematically founded, when testing a reconstruction algorithm on a new undersampled clin- ical dataset shows unacceptable image results, it is difficult to leverage the theoretical bounds to understand the cause of the failure. Conversely, when an undersampled reconstruction is successful at a certain undersampling rate, it is natural to then ask, how much further can we push undersampling? In this case, it is difficult to translate theoretic analyses, such as time con- stants for polylogarithmic bounds (8), into practice. Our goal in this paper is to provide the tools to empirically analyze the effects of lower SNR from reduced measurement time using a reconstruction system that is fully determined, rather than un- derdetermined. To this end, we present the image quality pre- diction process (Figure 1). The image quality prediction process takes a Nyquist-sampled (fully determined) reference dataset and adds the proper amount of noise to mimic the lower SNR produced by a given undersampling pattern. By reconstructing this noisy, but still Nyquist-sampled dataset, we have a predic- tion image that has been affected by lower SNR from reduced measurement time but not by artifacts from an underdetermined reconstruction. The image quality prediction process give us the following three benefits: • Comparing the prediction image to the reference recon- struction allows us to see the impact of lower SNR from reduced measurement time on the reconstruction system. • Comparing the prediction image to the underdetermined reconstruction, we are able to assess the added effect of the underdetermined system on the reconstructed image. • The prediction image provides a better estimate of under- sampled image quality than overoptimistically comparing an underdetermined reconstruction to a fully sampled ref- erence reconstruction. As exemplified in Figure 2, for a given clinical application and undersampling pattern, pulse sequence and reconstruction developers can use the image quality prediction process to determine if low SNR, rather than the underdetermined system, is the limiting factor for a successful reconstruction. Specifi- Figure 1. Prediction of image quality: The process to add the proper amount of noise to fully sam- pled reference k-space and recon- struct an image affected by lower SNR because of reduced acquisi- tion time but not affected by an underdetermined system. The ex- pected measurement time at each k-space location, t k , associated with the given sampling pattern is used to calculate the amount of noise (zero mean, complex Gauss- ian with variance s add,k 2 ) to add to each position in the reference k- space. This k-space with added noise is then processed by the re- construction algorithm to produce the prediction image. Empirical Effect of Noise in Undersampled MRI 212 TOMOGRAPHY.ORG | VOLUME 3 NUMBER 4 | DECEMBER 2017

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