Recovery of piece-wise constant signal

Application to cardiac Cine MRI

We model the dynamic MR images as a piece-wise constant signal in 3-D. We assume that the edges/discontinuities of the signal coincide with the zero level sets of a 3-D bandlimited function.

Construction of the Toeplitz matrix

This enables us to derive an annihilation relation between the 3-D DFT coefficients of the dynamic MR images and a 3-D FIR filter. We show that this relation further translates into a low rank property on a large multi-fold Toeplitz matrix, formed from the Fourier samples. The figure to the left depicts the construction of the Toeplitz matrix. We exploit the low rank property of this matrix to recover the dynamic MR images from heavily undersampled Fourier measurements.

We validate the algorithm on a Breath-held CINE data and show improvements over spatio-temporal TV and temporal Fourier sparsity regularized reconstruction schemes. Comparison of different methods on the recovery of breath held Cine data from 12 golden angle lines is shown below.

References:

Accelerated Dynamic MRI using structured low rank matrix completion.

A. Balachandrasekaran and M. Jacob. ICIP 2016.

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