A fast implementation of the minimum spanning tree method for phase unwrapping

Phase unwrapping using deep learning in holographic tomography

Holographic tomography (HT) is a measurement technique that generates phase images, often containing high noise levels and irregularities. Due to the nature of phase retrieval algorithms within the HT data processing, the phase has to be unwrapped before tomographic reconstruction. Conventional algorithms lack noise robustness, reliability, speed, and possible automation. In order to address these problems, this work proposes a convolutional neural network based pipeline consisting of two steps: denoising and unwrapping. Both steps are carried out under the umbrella of a U-Net architecture; however, unwrapping is aided by introducing Attention Gates (AG) and Residual Blocks (RB) to the architecture. Through the experiments, the proposed pipeline makes possible the phase unwrapping of highly irregular, noisy, and complex experimental phase images captured in HT. This work proposes phase unwrapping carried out by segmentation with a U-Net network, that is aided by a pre-processing denoising step. It also discusses the implementation of the AGs and RBs in an ablation study. What is more, this is the first deep learning based solution that is trained solely on real images acquired with HT.

Cyclic Continuous Max-Flow: A Third Paradigm in Generating Local Phase Shift Maps in MRI

Sensitivity to phase deviations in MRI forms the basis of a variety of techniques, including magnetic susceptibility weighted imaging and chemical shift imaging. Current phase processing techniques fall into two families: those which process the complex image data with magnitude and phase coupled, and phase unwrapping-based techniques that first linearize the phase topology across the image. However, issues, such as low signal and the existence of phase poles, can lead both methods to experience error. Cyclic continuous max-flow (CCMF) phase processing uses primal-dual-variational optimization over a cylindrical manifold, which represent the inherent topology of phase images, increasing its robustness to these issues. CCMF represents a third distinct paradigm in phase processing, being the only technique equipped with the inherent topology of phase. CCMF is robust and efficient with at least comparable accuracy as the prior paradigms.

A Robust and Simple Measure for Quality-Guided 2D Phase Unwrapping Algorithms

Quality-based 2D phase unwrapping algorithms provide one of the best tradeoffs between speed and quality of results. Their robustness depends on a quality map, which is used to build a path that visits the most reliable pixels first. Unwrapping then proceeds along this path, delaying unwrapping of noisy and inconsistent areas until the end, so that the unwrapping errors remain local. We propose a novel quality measure that is consistent, technically sound, effective, fast to compute, and immune to the presence of a carrier signal. The new measure combines the benefits of both the quality-guided and the residue-based phase unwrapping approaches. The quality map is justified from the two different theoretical points of view. Exhaustive tests on a variety of artificially generated and real 2D wrapped phase signals illustrate its potential usefulness in the field of fringe projection profilometry.

Quantification of susceptibility change at high-concentrated SPIO-labeled target by characteristic phase gradient recognition

Phase map cross-correlation detection and quantification may produce highlighted signal at superparamagnetic iron oxide nanoparticles, and distinguish them from other hypointensities. The method may quantify susceptibility change by performing least squares analysis between a theoretically generated magnetic field template and an experimentally scanned phase image. Because characteristic phase recognition requires the removal of phase wrap and phase background, additional steps of phase unwrapping and filtering may increase the chance of computing error and enlarge the inconsistence among algorithms. To solve problem, phase gradient cross-correlation and quantification method is developed by recognizing characteristic phase gradient pattern instead of phase image because phase gradient operation inherently includes unwrapping and filtering functions. However, few studies have mentioned the detectable limit of currently used phase gradient calculation algorithms. The limit may lead to an underestimation of large magnetic susceptibility change caused by high-concentrated iron accumulation. In this study, mathematical derivation points out the value of maximum detectable phase gradient calculated by differential chain algorithm in both spatial and Fourier domain. To break through the limit, a modified quantification method is proposed by using unwrapped forward differentiation for phase gradient generation. The method enlarges the detectable range of phase gradient measurement and avoids the underestimation of magnetic susceptibility. Simulation and phantom experiments were used to quantitatively compare different methods. In vivo application performs MRI scanning on nude mice implanted by iron-labeled human cancer cells. Results validate the limit of detectable phase gradient and the consequent susceptibility underestimation. Results also demonstrate the advantage of unwrapped forward differentiation compared with differential chain algorithms for susceptibility quantification at high-concentrated iron accumulation.

Field inhomogeneity correction using partial differential phases in magnetic resonance imaging

Correction of an inhomogeneous magnetic field is proposed using partial differential phases in magnetic resonance imaging. Estimation of the inhomogeneous magnetic field from a measured phase is not an easy task due to phase wrapping and chemical-dependent phase shifts. Using the proposed partial differential phase technique, such problems are resolved. The proposed technique uses most of the 3D pixel data regardless of chemical compounds for the estimation of the inhomogeneous magnetic field. A large number of partial difference data compared to the number of expansion terms for the model of inhomogeneous magnetic field provides a very stable estimation, robust to noise. The technique is applicable to in vivo shimming, water-fat imaging, eddy current compensation, and most phase-related measurements and imaging. The efficacy of the proposed technique is demonstrated with in vivo water-fat imaging.

Robust phase unwrapping for MR temperature imaging using a magnitude-sorted list, multi-clustering algorithm

PURPOSE

Several methods in MRI use the phase information of the complex signal and require phase unwrapping (e.g., B0 field mapping, chemical shift imaging, and velocity measurements). In this work, an algorithm was developed focusing on the needs and requirements of MR temperature imaging applications.

METHODS

The proposed method performs fully automatic unwrapping using a list of all pixels sorted by magnitude in descending order and creates and merges clusters of unwrapped pixels until the entire image is unwrapped. The algorithm was evaluated using simulated phantom data and in vivo clinical temperature imaging data.

RESULTS

The evaluation of the phantom data demonstrated no errors in regions with signal-to-noise ratios of at least 4.5. For the in vivo data, the algorithm did not fail at an average of more than one pixel for signal-to-noise ratios greater than 6.3. Processing times less than 30 ms per image were achieved by unwrapping pixels inside a region of interest (53 × 53 pixels) used for referenceless MR temperature imaging.

CONCLUSIONS

The algorithm has been demonstrated to operate robustly with clinical in vivo data in this study. The processing time for common regions of interest in referenceless MR temperature imaging allows for online updates of temperature maps without noticeable delay.

3D phase unwrapping using global expected phase as a reference: application to MRI global shimming

MRI phase data often suffers from phase wrapping (i.e., phase may be discontinuous by 2π jumps). Numerous MRI phase unwrapping strategies were developed in the past using a criterion based on phase information of local or neighboring voxels. In this study, an alternative and novel three dimensional phase unwrapping strategy is introduced. This method considers the global character of the phase distribution and utilizes continuous trigonometric functions to construct an expected phase map as an unwrapping reference, which is then used to guide the phase correction of every individual voxel. The original phase is estimated by analyzing the derivative of the wrapped phase image. Simulations of various phase wrapped situations were performed and this new method was also used for an in vivo application (i.e., MRI automatic global shimming). Both simulated and experimental results demonstrate that our proposed method is more reliable and robust than traditional algorithms at obtaining correct phase maps, especially in regions of low-signal and air cavities, such as the abdomen and pelvis.

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).

Clustering-driven residue filter for profile measurement system

The profile measurement system is widely used in industrial quality control, and phase unwrapping (PU) is a key technique. An algorithm-driven PU is often used to reduce the impact of noise-induced residues to retrieve the most reliable solution. However, measuring speed is lowered due to the searching of optimal integration paths or correcting of phase gradients. From the viewpoint of the rapidity of the system, this paper characterizes the noise-induced residues, and it proposes a clustering-driven residue filter based on a set of directional windows. The proposed procedure makes the wrapped phases included in the filtering window have more similar values, and it groups the correct and noisy phases into individual clusters along the local fringe direction adaptively. It is effective for the tightly packed fringes, and it converts the algorithm-driven PU to the residue-filtering-driven one. This improves the operating speed of the 3D reconstruction significantly. The tests performed on simulated and real projected fringes confirm the validity of our approach.

Clustering-based robust three-dimensional phase unwrapping algorithm

Relatively recent techniques that produce phase volumes have motivated the study of three-dimensional (3D) unwrapping algorithms that inherently incorporate the third dimension into the process. We propose a novel 3D unwrapping algorithm that can be considered to be a generalization of the minimum spanning tree (MST) approach. The technique combines characteristics of some of the most robust existing methods: it uses a quality map to guide the unwrapping process, a region growing mechanism to progressively unwrap the signal, and also cut surfaces to avoid error propagation. The approach has been evaluated in the context of noncontact measurement of dynamic objects, suggesting a better performance than MST-based approaches.

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.

Linear phase-error correction for improved water and fat separation in dual-echo dixon techniques

Large and spatially-linear phase errors along the frequency-encode direction may be induced by several common and hard-to-avoid system imperfections such as eddy currents. For data acquired in dual-echo Dixon techniques, the linear phase error can be more aggravated when compared to that acquired in a single echo and can pose challenges to a phase-correction algorithm necessary for successful Dixon processing. In this work, we propose a two-step process that first corrects the linear component of the phase errors with a modified Ahn-Cho algorithm (Ahn CB and Cho ZH, IEEE Trans. Med. Imaging 6:32, 1987) and then corrects the residual phase errors with a previously-developed region-growing algorithm (Ma J, Magn. Res. Med. 52:415, 2004). We demonstrate that successive application of the two-step process to data from a dual-echo Dixon technique provides a "1-2 punch" to the overall phase errors and can overcome local water and fat separation failures that are observed when the region-growing-based algorithm is applied alone.

Dixon techniques for water and fat imaging

In 1984, Dixon published a first paper on a simple spectroscopic imaging technique for water and fat separation. The technique acquires two separate images with a modified spin echo pulse sequence. One is a conventional spin echo image with water and fat signals in-phase and the other is acquired with the readout gradient slightly shifted so that the water and fat signals are 180 degrees out-of-phase. Dixon showed that from these two images, a water-only image and a fat-only image can be generated. The water-only image by the Dixon's technique can serve the purpose of fat suppression, an important and widely used imaging option for clinical MRI. Additionally, the availability of both the water-only and fat-only images allows direct image-based water and fat quantitation. These applications, as well as the potential that the technique can be made highly insensitive to magnetic field inhomogeneity, have generated substantial research interests and efforts from many investigators. As a result, significant improvement to the original technique has been made in the last 2 decades. The following article reviews the underlying physical principles and describes some major technical aspects in the development of these Dixon techniques.

Automated phase correction via maximization of the real signal

Due to improved quantification capabilities and enhanced signal-to-noise ratio (SNR), phase-corrected real reconstruction in magnetic resonance imaging is superior to the common magnitude reconstruction, especially at low SNR. This requires the development of an automated phase-correction algorithm. Existing methods are not well suited for multiple unconnected regions of very low SNR. For this situation, a method based on the real-signal maximization is implemented, in which the experimental image phase is approximated by a three-dimensional polynomial of up to third order. The presented implementation was successfully applied to data originating from different samples and pulse sequences.

Adaptive postprocessing techniques for myocardial tissue tracking with displacement-encoded MR imaging

The purpose of this study was to prospectively assess the effects of two adaptive postprocessing techniques on the evaluation of myocardial function with displacement-encoded magnetic resonance (MR) imaging, including sensitivity for abnormal wall motion, with two-dimensional echocardiography as the reference standard. Sixteen patients (11 men, five women; age range, 26-74 years) and 12 volunteers (six men, six women; age range, 29-53 years) underwent breath-hold MR imaging. Institutional review board approval and informed consent were obtained. Adaptive phase-unwrapping and spatial filtering techniques were compared with conventional phase-unwrapping and spatial filtering techniques. Use of the adaptive techniques led to a reduced rate of failure with the phase-unwrapping technique from 18.9% to 0.6% (P < .001), resulted in lower variability of segmental strain measurements among healthy volunteers (P < .001 to P = .02), and increased the sensitivity of quantitative detection of abnormal segments in patients from 82.5% to 87.7% (P = .034). The adaptive techniques improved the semiautomated postprocessing of displacement-encoded cardiac images and increased the sensitivity of detection of abnormal wall motion in patients.

SUPPLEMENTAL MATERIAL

http://radiology.rsnajnls.org/cgi/content/full/246/1/229/DC1.

Unwrapping MR phase maps with Chebyshev moments

Previous polynomial model based phase unwrapping algorithms relied on least squares fitting to determine the polynomial coefficients. This work presents a phase unwrapping algorithm, based on the method of moments, and calculates the polynomial coefficients using integration. The phase unwrapping algorithm is developed using Chebyshev polynomials. The phase unwrapping algorithm implemented in this work is tested on 2-D phase maps obtained from a 3 Tesla magnetic resonance scanner. In this work, the effectiveness of the phase unwrapping algorithm on MR phase maps is investigated.

Unwrapping of MR phase images using a Markov random field model

Phase unwrapping is an important problem in many magnetic resonance imaging applications, such as field mapping and flow imaging. The challenge in two-dimensional phase unwrapping lies in distinguishing jumps due to phase wrapping from those due to noise and/or abrupt variations in the actual function. This paper addresses this problem using a Markov random field to model the true phase function, whose parameters are determined by maximizing the a posteriori probability. To reduce the computational complexity of the optimization procedure, an efficient algorithm is also proposed for parameter estimation using a series of dynamic programming connected by the iterated conditional modes. The proposed method has been tested with both simulated and experimental data, yielding better results than some of the state-of-the-art method (e.g., the popular least-squares method) in handling noisy phase images with rapid phase variations.

Automatic algorithm for correcting motion artifacts in time-resolved two-dimensional magnetic resoance angiography using convex projections

Time-resolved contrast enhanced magnetic resonance angiography (MRA) may suffer from involuntary patient motion. It is noted that while MR signal change associated with motion is large in magnitude and has smooth phase variation in k-phase, signal change associated with vascular enhancement is small in magnitude and has rapid phase variation in k-space. Based upon this observation, a novel projection onto convex sets (POCS) algorithm is developed as an automatic iterative method to remove motion artifacts. The presented POCS algorithm consists of high-pass phase filtering and convex projections in both k-space and image space. Without input of detailed motion knowledge, motion effects are filtered out, while vasculature information is preserved. The proposed method can be effective for a large class of nonrigid motions, including through-plane motion. The algorithm is stable and converges quickly, usually within five iterations. A double-blind evaluation on a set of clinical MRA cases shows that a completely unsupervised version of the algorithm produces significantly better rank scores (P=0.038) when compared to angiograms produced manually by an experienced radiologist.

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.

Robust field map generation using a triple-echo acquisition

PURPOSE

To establish a fast and robust technique for generating magnetic field maps for the correction of geometric distortions in echo-planar magnetic resonance (MR) images.

MATERIALS AND METHODS

Multislice gradient-echo (GE) images were acquired at echo times of 6, 6.5, and 7.5 msec in order to cover a field shift range of +/-666 Hz in the resulting B0 maps. To account for possible phase wrap scenarios, seven phase triples were calculated for each pixel. Linear regression of the phase vs. echo time was performed for each set. The slope of the set with the minimum fitting error was taken as the true magnetic field in the respective pixel.

RESULTS

Based on the fitting error distribution, the technique is shown to be feasible and effective for assessing the field distribution in the brain at 3 T, especially in inferior brain areas (amygdalae, hippocampus). Examples of echo-planar images distortion corrected using the calculated field maps are shown.

CONCLUSION

The approach presented yields robust estimation of magnetic field maps and requires under a minute of additional acquisition time and only seconds of computational time. As such, it is easily possible to apply image distortion correction in routine functional MR imaging (fMRI) studies, enabling improved coregistration of brain activation maps with structures on anatomical images.

Breath-hold water and fat imaging using a dual-echo two-point dixon technique with an efficient and robust phase-correction algorithm

A two-point Dixon technique using a novel phase-correction algorithm and commercially available dual-echo fast gradient-echo pulse sequence is presented. The phase-correction algorithm determines the directional rather than phase distribution of signals due to field inhomogeneities. Specifically, a region-growing scheme uses precalculated spatial gradients of the signal phase to guide the growth sequence, so there is no need to manually select the seeds or use an empirical angular threshold. Further, the determination of the signal direction of a given pixel is based on both the amplitude and phase of the surrounding pixels, the direction of which has already been determined. The advantages of this algorithm include its easy implementation, computational efficiency, and robustness in the presence of pixels with large phase uncertainty. The feasibility and usefulness of the technique are demonstrated in vivo with artifact-free water and fat images of an entire abdomen in a single breath-hold.

Fast, automated, N-dimensional phase-unwrapping algorithm

This work investigates the general problem of phase unwrapping for arbitrary N-dimensional phase maps. A cost function-based approach is outlined that leads to an integer programming problem. To solve this problem, a best-pair-first region merging approach is adopted as the optimization method. The algorithm was implemented and tested with 3D MRI medical data for venogram studies, as well as for fMRI applications in EPI unwarping and rapid, automated shimming.

Understanding phase maps in MRI: a new cutline phase unwrapping method

This paper describes phase maps. A review of the phase unwrapping problem is given. Different structures, in particular fringelines, cutlines, and poles, contained within a phase map are described and their origin and behavior investigated. The problem of phase unwrapping can then be addressed with a better understanding of the source of poles or inconsistencies. This understanding, along with some assumptions about what is being encoded in the phase of a magnetic resonance image, are used to derive a new method for phase unwrapping which relies only on the phase map. The method detects cutlines and distinguishes between noise-induced poles and signal undersampling poles based on the length of the fringelines. The method was shown to be robust to noise and successful in unwrapping challenging clinical cases.