Series of first-order phase shifts correct lattice reduction of fractional K-space indices

Abstract

Lattice reduction of K-space acquisition at fractional indices refers to reducing the indices to the smallest nearby integer, thereby generating a Cartesian grid, allowing subsequent inverse Fourier Transformation. For band-limited signals, we show that the error in lattice reduction is equivalent to first order phase shifts, which in the infinite limit approaches W∞=φ(cotφ-i), where φ is a first-order phase shift vector. In general, the inverse corrections can be specified from the binary representation of the fractional part of the K-space indices. For non-uniform sparsity, we show how to incorporate the inverse corrections into compressed sensing reconstructions.