Unwrapping magnetic resonance phase maps with Chebyshev polynomials

Abstract 

A phase-unwrapping algorithm, based on the method of moments, is introduced in this work. The proposed algorithm expands the phase map in terms of a two-dimensional Chebyshev series. The expansion coefficients are calculated by exploiting the orthogonality of Chebyshev polynomials of the first kind. The performance of the proposed phase-unwrapping algorithm is tested on a synthetic phase map and experimental phase maps of a uniform phantom, a human brain and a mouse torso, all acquired from 3-T magnetic resonance (MR) scanners. To impose additional burdens on the algorithm, we introduced magnetic field inhomogeneities to the phantom and human brain data by an external gradient coil. The proposed phase-unwrapping algorithm is found to perform well on the phantom data set in a low signal-to-noise ratio (SNR) environment and on the synthetic data set. The proposed algorithm is also found to perform well in in vivo data sets of the human brain and mouse torso. Results obtained from the in vivo MR data sets show that the proposed algorithm produced unwrapped phase maps that are nearly identical to those produced by a widely used phase-unwrapping algorithm, PRELUDE 2D in the fMRI Software Library.

Keywords: Magnetic resonance imaging, Phase unwrapping, Chebyshev polynomial, Method of moments