Branch cut surface placement for unwrapping of undersampled three-dimensional phase data: application to magnetic resonance imaging arterial flow mapping

Divergence-Free Constrained Phase Unwrapping and Denoising for 4D Flow MRI Using Weighted Least-Squares

A novel divergence-free constrained phase unwrapping method was proposed and evaluated for 4D flow MRI. The unwrapped phase field was obtained by integrating the phase variations estimated from the wrapped phase data using weighted least-squares. The divergence-free constraint for incompressible blood flow was incorporated to regulate and denoise the resulting phase field. The proposed method was tested on synthetic phase data of left ventricular flow and in vitro 4D flow measurement of Poiseuille flow. The method was additionally applied to in vivo 4D flow measurements in the thoracic aorta from 30 human subjects. The performance of the proposed method was compared to the state-of-the-art 4D single-step Laplacian algorithm. The synthetic phase data were completely unwrapped by the proposed method for all the cases with velocity encoding (venc) as low as 20% of the maximum velocity and signal-to-noise ratio as low as 5. The in vitro Poiseuille flow data were completely unwrapped with a 60% increase in the velocity-to-noise ratio. For the in-vivo aortic datasets with venc ratio less than 0.4, the proposed method significantly improved the success rate by as much as 40% and reduced the velocity error levels by a factor of 10 compared to the state-of-the-art method. The divergence-free constrained method exhibits reliability and robustness on phase unwrapping and shows improved accuracy of velocity and hemodynamic quantities by unwrapping the low-venc 4D flow MRI data.

One-Step Three-Dimensional Phase Unwrapping Approach Based on Small Baseline Subset Interferograms

One of the most critical steps in a multitemporal D-InSAR analysis is the resolution of the phase ambiguities in the context of phase unwrapping. The Extended Minimum Cost Flow approach is one of the potential phase unwrapping algorithms used in the Small Baseline Subset analysis. In a first step, each phase gradient is unwrapped in time using a linear motion model and, in a second step, the spatial phase unwrapping is individually performed for each interferogram. Exploiting the temporal and spatial information is a proven method, but the two-step procedure is not optimal. In this paper, a method is presented which solves both the temporal and spatial phase unwrapping in one single step. This requires some modifications regarding the estimation of the motion model and the choice of the weights. Furthermore, the problem of temporal inconsistency of the data, which occurs with spatially filtered interferograms, must be considered. For this purpose, so called slack variables are inserted. To verify the method, both simulated and real data are used. The test region is the Lower-Rhine-Embayment in the southwest of North Rhine-Westphalia, a very rural region with noisy data. The studies show that the new approach leads to more consistent results, so that the deformation time series of the analyzed pixels can be improved.

Hemodynamic Measurement Using Four-Dimensional Phase-Contrast MRI: Quantification of Hemodynamic Parameters and Clinical Applications

Recent improvements have been made to the use of time-resolved, three-dimensional phase-contrast (PC) magnetic resonance imaging (MRI), which is also named four-dimensional (4D) PC-MRI or 4D flow MRI, in the investigation of spatial and temporal variations in hemodynamic features in cardiovascular blood flow. The present article reviews the principle and analytical procedures of 4D PC-MRI. Various fluid dynamic biomarkers for possible clinical usage are also described, including wall shear stress, turbulent kinetic energy, and relative pressure. Lastly, this article provides an overview of the clinical applications of 4D PC-MRI in various cardiovascular regions.

Intervention-based multidimensional phase unwrapping using recursive orthogonal referring

We present a new intervention-based phase unwrapping algorithm, which solves the inherent integration-path-dependent problem (typically resulting in streaks), by using a 2D recursive orthogonal referring (PUROR) approach. The streaks were removed by three consecutive procedures: intra-image phase unwrapping, inter-image cross-referring a "good-strip," and cross-referring line segments. The application of these procedures results in streak-free 2D phase images. The phase inconsistencies across slices in a 3D image were removed using a hybrid 3D PUROR algorithm: the two step approach involves stacking the individual slices, by using the mean phase values of each slice, then applying the 2D PUROR algorithm to reformatted 2D images that include the slice direction. The described approach was tested with in vivo multislice phase images acquired in the axial, sagittal, and coronal orientation. The results of the unwrapped phase volume recovered using the PUROR algorithm have equivalent quality to that achieved by using established methods, but the PUROR algorithm is about two orders of magnitude faster (between 1 and 5 s per 256×256 slice; independent of slice orientation and echo time).

Period coded phase shifting strategy for real-time 3-D structured light illumination

Phase shifting structured light illumination for range sensing involves projecting a set of grating patterns where accuracy is determined, in part, by the number of stripes. However, high pattern frequencies introduce ambiguities during phase unwrapping. This paper proposes a process for embedding a period cue into the projected pattern set without reducing the signal-to-noise ratio. As a result, each period of the high frequency signal can be identified. The proposed method can unwrap high frequency phase and achieve high measurement precision without increasing the pattern number. Therefore, the proposed method can significantly benefit real-time applications. The method is verified by theoretical and experimental analysis using prototype system built to achieve 120 fps at 640 × 480 resolution.

Recursive approach to the moment-based phase unwrapping method

The moment-based phase unwrapping algorithm approximates the phase map as a product of Gegenbauer polynomials, but the weight function for the Gegenbauer polynomials generates artificial singularities along the edge of the phase map. A method is presented to remove the singularities inherent to the moment-based phase unwrapping algorithm by approximating the phase map as a product of two one-dimensional Legendre polynomials and applying a recursive property of derivatives of Legendre polynomials. The proposed phase unwrapping algorithm is tested on simulated and experimental data sets. The results are then compared to those of PRELUDE 2D, a widely used phase unwrapping algorithm, and a Chebyshev-polynomial-based phase unwrapping algorithm. It was found that the proposed phase unwrapping algorithm provides results that are comparable to those obtained by using PRELUDE 2D and the Chebyshev phase unwrapping algorithm.

Improved SNR in phase contrast velocimetry with five-point balanced flow encoding

Phase contrast velocimetry can be utilized to measure complex flow for both quantitative and qualitative assessment of vascular hemodynamics. However, phase contrast requires that a maximum measurable velocity be set that balances noise and phase aliasing. To efficiently reduce noise in phase contrast images, several investigators have proposed extended velocity encoding schemes that use extra encodings to unwrap phase aliasing; however, existing techniques can lead to significant increases in echo and scan time, limiting their clinical benefits. In this work, we have developed a novel five-point velocity encoding scheme that efficiently reduces noise with minimal increases in scan and echo time. Investigations were performed in phantoms, demonstrating a 63% increase in velocity-to-noise ratio compared to standard four-point encoding schemes. Aortic velocity measurements were performed in healthy volunteers, showing similar velocity-to-noise ratio improvements. In those volunteers, it was also demonstrated that, without sacrificing accuracy, low-resolution images can be used for the fifth encoding point, reducing the scan time penalty from 25% down to less than 1%.

Hybrid robust and fast algorithm for three-dimensional phase unwrapping

We present a hybrid three-dimensional (3D) unwrapping algorithm that combines the strengths of two other fast and robust existing techniques. In particular, a branch-cut surface algorithm and a path-following method have been integrated in a symbiotic way, still keeping execution times within a range that permits their use in real-time applications that need a relatively fast solution to the problem. First, branch-cut surfaces are calculated, disregarding partial residue loops that end at the boundary of the 3D phase volume. These partial loops are then used to define a quality for each image voxel. Finally, unwrapping proceeds along a path determined by a minimum spanning tree (MST). The MST is built according to the quality of the voxels and avoids crossing the branch-cut surfaces determined at the first step. The resulting technique shows a higher robustness than any of the two methods used in isolation. On the one hand, the 3D MST algorithm benefits from the branch-cut surfaces, which endows it with a higher robustness to noise and open-ended wraps. On the other hand, incorrectly placed surfaces due to open loops at the boundaries in the branch-cut surface approach disappear.

Robust three-dimensional best-path phase-unwrapping algorithm that avoids singularity loops

In this paper we propose a novel hybrid three-dimensional phase-unwrapping algorithm, which we refer to here as the three-dimensional best-path avoiding singularity loops (3DBPASL) algorithm. This algorithm combines the advantages and avoids the drawbacks of two well-known 3D phase-unwrapping algorithms, namely, the 3D phase-unwrapping noise-immune technique and the 3D phase-unwrapping best-path technique. The hybrid technique presented here is more robust than its predecessors since it not only follows a discrete unwrapping path depending on a 3D quality map, but it also avoids any singularity loops that may occur in the unwrapping path. Simulation and experimental results have shown that the proposed algorithm outperforms its parent techniques in terms of reliability and robustness.

Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction

A multilevel quality-guided phase unwrapping algorithm for real-time 3D shape measurement is presented. The quality map is generated from the gradient of the phase map. Multilevel thresholds are used to unwrap the phase level by level. Within the data points in each level, a fast scan-line algorithm is employed. The processing time of this algorithm is approximately 18.3 ms for an image size of 640x480 pixels in an ordinary computer. We demonstrate that this algorithm can be implemented into our real-time 3D shape measurement system for real-time 3D reconstruction. Experiments show that this algorithm improves the previous scan-line phase unwrapping algorithm significantly although it reduces its processing speed slightly.