Compressive Sensing Ensemble Average Propagator Estimation Via L1 Spherical Polar Fourier Imaging

Jian Cheng^1,2, Sylvain Merlet², Aurobrata Ghosh², Emmanuel Caruyer², Tianzi Jiang¹, Rachid Deriche²

Since Diffusion Tensor Imaging (DTI) cannot detect the fiber crossing, many new works beyond DTI has been proposed to explore the q-space. Most works, known as single shell High Angular Resolution Imaging (sHARDI), focus on single shell sampling and reconstruct the Orientation Distribution Function (ODF). The ODF, which has no radial information at all, is just one of features of Ensemble Average Propagator (EAP). Diffusion Spectrum Imaging (DSI) is a standard method to estimate EAP via numerical Fourier Transform (FT), which needs lots of samples and is impractical for clinical study. Spherical Polar Fourier Imaging (SPFI) [1,2] was proposed to represent the signal using SPF basis, then the EAP and the ODF have analytical closed forms. So the estimation of the coefficients is very important. In [1,2], the coefficients are estimated based on a standard Least Square (LS) with L2 norm regularization (L2-L2). In this paper, we propose to estimate A using LS with L1 norm regularization (L2-L1), also named as Least Absolute Selection and Shrinkage Operator (LASSO). And we prove that the L2-L1 estimation of the coefficients is actually the well known Compressive Sensing (CS) method to estimate EAP, which brings lots of Mathematical tools and possibility to improve the sampling scheme in q-space.