Homotopic l0 minimization technique applied to dynamic cardiac MR imaging

Introduction: The l1 minimization technique has been empirically demonstrated to exactly recover an S-sparse signal with about 3S-5S measurements [1]. In order to get exact reconstruction with smaller number of measurements, recently, for static images, Trzasko [2] has proposed homotopic l0 minimization technique. Instead of minimizing the l0 norm which achieves best possible theoretical bound (approximately 2S measurements) but is a NP hard problem or l1 norm which is a convex optimization problem but requires more measurements, the homotopic technique minimizes iteratively the continuous approximations of the l0 norm. In this work, we have extended the use of homotopic l0 method to dynamic MR imaging. For dynamic 2D CINE data, using five different non-convex functional approximations to l0 norm, we have compared the performance of homotopic l0 minimization technique with the standard l1 method. Method: Given a linear system of equations expressed as: Au= b, where u is sparse signal to be recovered, A is a measurement matrix with dimensions MxN (M<<N), b is a set of M measurements; the homotopic l0 minimization problem is solved by iteratively solving a sequence of problems given by b Au s.t. ) σ , (x) u ρ( min arg u t