Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms

Abstract

Phase contrast magnetic resonance velocity imaging is a powerful technique for quantitative in vivo blood flow measurement. Current practice normally involves restricting the sensitivity of the technique so as to avoid the problem of the measured phase being 'wrapped' onto the range -pi to +pi. However, as a result, dynamic range and signal-to-noise ratio are sacrificed. Alternatively, the true phase values can be estimated by a phase unwrapping process which consists of adding integral multiples of 2pi to the measured wrapped phase values. In the presence of noise and data undersampling, the phase unwrapping problem becomes non-trivial. In this paper, we investigate the performance of three different phase unwrapping algorithms when applied to three-dimensional (two spatial axes and one time axis) phase contrast datasets. A simple one-dimensional temporal unwrapping algorithm, a more complex and robust three-dimensional unwrapping algorithm and a novel velocity encoding unwrapping algorithm which involves unwrapping along a fourth dimension (the 'velocity encoding' direction) are discussed, and results from the three are presented and compared. It is shown that compared to the traditional approach, both dynamic range and signal-to-noise ratio can be increased by a factor of up to five times, which demonstrates considerable promise for a possible eventual clinical implementation. The results are also of direct relevance to users of any other technique delivering time-varying two-dimensional phase images, such as dynamic speckle interferometry and synthetic aperture radar.

Phase unwrapping in three dimensions with application to InSAR time series

“…As the edge of any surface is closed loop, the residues along a discontinuity surface edge also form a closed loop. Furthermore, it can be shown that all residues, including those within discontinuity surfaces, form loops in 3D space [12,16]. These loops are referred to in previous publications as "phase singularity loops" [12,16], but we use the term "residue loop" to be consistent with our use of the term "residue" to describe phase singularities.…”

“…Although we expect these surfaces to be rare in well-sampled data sets, fitting a minimal surface to every loop does not allow for this type of surface, hence we refer to it as a quasi-L ϱ approach. Existing branch-cut algorithms [12,[14][15][16]] also fall in this class, although the ways in which truncated residue loops are handled by these algorithms do not minimize the L ϱ -norm for the general case. In the following section we describe an algorithm to implement the quasi-L ϱ approach that applies for all residue loops, including those that are truncated.…”

“…Existing 3D branch-cut algorithms [12,[14][15][16]] make the assumption that discontinuity surfaces are (a) bounded by only one residue loop and (b) have no internal residue loops. Only when this is the case does the ambiguity of pairing positive and negative residues, which exists in two dimensions, disappear.…”

“…Several algorithms have recently been developed that take advantage of the third dimension including branchcut algorithms [12][13][14][15][16] and an integer least-squares algorithm [17]. None of these algorithms are directly applicable to InSAR time series, however.…”

A patch-based spatiotemporal phase unwrapping method for phase contrast MRI using graph cuts

“…The existing PU algorithms can be grouped into three major categories: 1) path-following algorithms [3] [5] [6]; 2) one-dimensional PU [7] [8]; and 3) minimum norm algorithms [2] [9] [10]. The algorithm proposed in this paper falls in the third category.…”