Nonlinear phase correction with an extended statistical algorithm

This paper presents a new magnetic resonance imaging (MRI) phase correction method. The linear phase correction method using autocorrelation proposed by Ahn and Cho (AC method) is extended to handle nonlinear terms, which are often important for polynomial expansion of phase variation in MRI. The polynomial coefficients are statistically determined from a cascade series of n-pixel-shift rotational differential fields (RDFs). The n-pixel-shift RDF represents local vector rotations of a complex field relative to itself after being shifted by n pixels. We have found that increasing the shift enhances the signal significantly and extends the AC method to handle higher order nonlinear phase error terms. The n-pixel-shift RDF can also be applied to improve other methods such as the weighted least squares phase unwrapping method proposed by Liang. The feasibility of the method has been demonstrated with two-dimensional (2-D) in vivo inversion-recovery MRI data.

I. Introduction

In magnetic resonance imaging (MRI), signal consists of real and imaginary components, which can be used to derive a magnitude image and a phase map. Although magnitude images are used much more frequently than phase maps for clinical diagnosis, phase maps should not be undervalued because they often encode important information such as field inhomogeneity, motion, and chemical shift. Because of phase wrapping, interpreting the encoded information requires phase correction and unwrapping in many MRI applications. The phase correction and unwrapping methods developed in the past few years generally fall into two categories: path-following methods and minimum-norm methods [1]. Cut-line methods [2]–[4], disk growing method [5], region growing methods (including maximum [6] and minimum spanning tree methods [7]) and Flynn's minimum discontinuity approach [8] are all path-following methods, which are based on the idea of guiding the integration of the gradient map to avoid any error propagation [2]. Minimum-norm methods are used to minimize the cost function to achieve phase unwrapping [1]. To minimize the cost function, the well-known least squares algorithm [9] or a generalized -norm algorithm [10] with data-dependent weights is generally used, leading to a direct minimum-norm solution to the phase. Alternatively, the phase can be modeled with a function by estimating its coefficients in a weighted least squares sense [1]. Amongst the model-based methods, polynomial fitting is the most commonly used in MRI because phase variation in MRI can often be modeled with a polynomial [11]–[16]. Although polynomials may not necessarily represent all phase maps, especially when poles [2] are present in the phase, polynomial fitting is of great use in the case of disconnected tissues in the field-of-view (FOV) [17], a typical example of which is an image showing two separated legs. A successful example of polynomial fitting is the linear phase correction method using autocorrelation proposed by Ahn and Cho (AC method) [15]. While the AC method is efficient, its straightforward extension to handle higher order nonlinear phase terms fails to yield good results because the signal in the estimated phase terms is substantially contaminated by noise as a result of 1-pixel-shift phase differentials.