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Recovering exponential signals is a problem of high significance in many MR imaging applications, including MR parameter mapping, MR spectroscopy, diffusion MRI, and fat/water imaging. The goal is to estimate a spatial map of the exponential parameters, which reflect the underlying tissue microstructure and metabolism, from a time series of images. These parameters typically serve as biomarkers for various pathologies including brain and cardiovascular disorders.
MR parameter mapping
Current approaches include acquiring a series of images by sampling the exponential signal at different time points followed by fitting an exponential signal at every pixel to estimate the parameters. However, the main challenge with these schemes is the long acquisition time, resulting from the need to acquire many high spatial resolution images. Hence one of the ways to speed up acquisition is to collect very few Fourier measurements; the image series is then recovered from these measurements by using appropriate priors which enforce sparsity, smoothness and low-rankness.
In this work, we introduce a novel structured matrix completion algorithm to recover a time series of images from undersampled Fourier measurements. We model the temporal signal at every pixel as a linear combination of a few exponentials, which are characterized by spatially smooth parameters. We exploit the exponential behavior at every pixel, along with the spatial smoothness of the parameters to derive a 3-D annihilation relation in the Fourier domain. We show that this relation translates into a low rank property on a multi-fold Toeplitz matrix formed from the Fourier samples. An illustration of the formation of the Toeplitz matrix is shown in the figure below. We exploit the low rank property of the Toeplitz matrix to recover the series of images from heavily undersampled Fourier measurements.
Construction of the Toeplitz matrix
We demonstrate the algorithm in the MR parameter mapping setting and show improved results over the state-of-the-art methods. Comparison of the different methods on the recovery of multi-channel data at an acceleration factor of 12 is shown below.
References:
References:
A. Balachandrasekaran, V. Magnotta and M. Jacob. IEEE Transactions on Medical Imaging, 2017.
A. Balachandrasekaran and M. Jacob. ISBI 2017.
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