Sparse Gradient Image Reconstruction from Incomplete Fourier Measurements and Prior Edge Information

In many imaging applications, such as functional Magnetic Resonance Imaging (fMRI), full, uniformlysampled Cartesian Fourier (frequency space) measurements are acquired to reconstruct an image. In order to reduce scan time and increase temporal resolution for fMRI studies, one would like to accurately reconstruct these images from the smallest possible set of Fourier measurements. The emergence of Compressed Sensing (CS) has given rise to techniques that can provide exact and stable recovery of sparse images from a relatively small set of Fourier measurements. In particular, if the images are sparse with respect to their gradient, e.g., piece-wise constant, total-variation minimization techniques can be used to recover those images from a highly incomplete set of Fourier measurements. In this paper, we propose a new algorithm to further reduce the number of Fourier measurements required for exact or stable recovery by utilizing prior edge information from a high resolution reference image. This reference image, or more precisely, the fully sampled Fourier measurements of this reference image, is obtained prior to an fMRI study in order to provide approximate edge information for the region of interest. By combining this edge information with CS techniques for sparse gradient images, numerical experiments show that we can further reduce the number of Fourier measurements required for exact or stable recovery by an additional factor of 1.6− 3 , compared with CS techniques alone, without edge information. Index Terms Image reconstruction, Fourier transform, compressed sensing, edge detection, sparse recovery, total variation, L1-minimization. K. Chowdhary and J. S. Hesthaven are with the Division of Applied Mathematics, Brown University, Providence, RI, 02912 USA email: [email protected]. and jan [email protected] E. G. Walsh is with Department of Neuroscience, Brown University, Providence, RI, 02912 USA email: Edward [email protected]