@article{b81cdb98be364ea9aad6aea5de27525d,
title = "A model-based method for phase unwrapping",
abstract = "This paper presents a model-based phase unwrapping method which represents the unwrapped phase function by a truncated Taylor series and a residual function. An efficient, noniterative computational algorithm is also proposed for calculating the model parameters from the phase derivatives. Sample experimental results are shown to demonstrate the effectiveness of the algorithm for extracting unwrapped phase images from two-dimensional (2-D) magnetic resonance imaging (MR1) data.",
author = "Zhi-Pei Liang",
note = "Funding Information: In many imaging modalities, physical quantities of interest are often transduced to the phase of a complex signal. In magnetic resonance imaging (MRI), for example, phase images can provide useful information about magnetic field inhomogeneities, magnetic susceptibility variations, velocity of flowing spins, etc. [ 11. However, for a measured complex image p(z, y) = Ip(z, y)le”(”{\textquoteright}Y), extracting the phase p(z, y) is nontrivial. The main difficulty lies in the fact that p(z, y) is uniquely defined only in the principal value range of (-T, 7r]. Any value outside this interval will be wrapped around to produce the so-called wrappedphase, @(z,y ), which is related to P(X, Y) by P(X, Y) = +(? Y) * l(z, Y)2T (1) for some integer Z(z, y). The goal of phase unwrapping is to reconstruct the actual phase ~ ( zy,) either from the wrapped phase @(z, y) or from the original complex image data p(z, y). Over the past two decades, a large number of phase unwrapping algorithms have been proposed. The conventional approach is to detect the phase discontinuities in the wrapped phase function based on a predetermined threshold and then remove them by adding or subtracting multiples of 27r [2]. This method works well for noiseless continuous signals. However, for noisy sampled data, detection of discontinuities may be erroneous, resulting in incorrect unwrapped phase. To overcome this problem, Tribolet [3] proposed an adaptive integration method to determine when multiples of 27r should be added or subtracted from the wrapped phase values. This method is rather successful for one-dimensional signals but has trouble handling multidimensional data because integration of the phase derivative becomes path-dependent in higher dimensions. Recently, various methods have been proposed for the multidimensional phase unwrapping problem using path-following schemes [5], [6] or reformulating the problem as a solution of the Poisson equation [4], [7]. In this paper, we propose a model-based method for phase unwrapping. This method represents the unwrapped phase function by a truncated Taylor series and a residual function, making the earlier polynomial models [8]-[ 121 more general and flexible. In addition, Manuscript received March 22, 1996; revised August 26, 1996. This work was supported in part by the National Science Foundation under Grants NSF- BES-95-02121 and NSF-MIP-94-10463. The Associate Editor responsible for coordinating the review of this paper and recommending its publication was M. W. Vannier.",
year = "1996",
doi = "10.1109/42.544507",
language = "English (US)",
volume = "15",
pages = "893--897",
journal = "IEEE transactions on medical imaging",
issn = "0278-0062",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
number = "6",
}