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A Novel Phase-Unwrapping Method Based on PixelClustering and Local Surface Fitting With Applicationto Dixon Water–Fat MRI
Junying Cheng,1,2 Yingjie Mei,2 Biaoshui Liu,2 Jijing Guan,2 Xiaoyun Liu,1 Ed X. Wu,3,4
Qianjin Feng,2 Wufan Chen,1,2* and Yanqiu Feng2*
Purpose: To develop and evaluate a novel 2D phase-
unwrapping method that works robustly in the presence ofsevere noise, rapid phase changes, and disconnected regions.
Theory and Methods: The MR phase map usually varies rapid-ly in regions adjacent to wraps. In contrast, the phasors can varyslowly, especially in regions distant from tissue boundaries.
Based on this observation, this paper develops a phase-unwrapping method by using a pixel clustering and local surface
fitting (CLOSE) approach to exploit different local variation char-acteristics between the phase and phasor data. The CLOSEapproach classifies pixels into easy-to-unwrap blocks and
difficult-to-unwrap residual pixels first, and then sequentiallyperforms intrablock, interblock, and residual-pixel phaseunwrapping by a region-growing surface-fitting method. The
CLOSE method was evaluated on simulation and in vivo water–fat Dixon data, and was compared with phase region expanding
labeler for unwrapping discrete estimates (PRELUDE).Results: In the simulation experiment, the mean error ratio byCLOSE was less than 1.50%, even in areas with signal-to-noise ratio
equal to 0.5, phase changes larger than p, and disconnectedregions. For 350 in vivo knee and ankle images, the water–fat swap
ratio of CLOSE was 4.29%, whereas that of PRELUDE was 25.71%.Conclusions: The CLOSE approach can correctly unwrap phasewith high robustness, and benefit MRI applications that require
phase unwrapping. Magn Reson Med 000:000–000, 2017. VC 2017International Society for Magnetic Resonance in Medicine.
Key words: phase unwrapping; pixel clustering; local polyno-
mial surface fitting; thresholding; water–fat separation
INTRODUCTION
The performance of phase-related MRI techniques, suchas chemical shift mapping (1–3), magnetic susceptibilitymapping (4,5), blood flow imaging (6,7), and brain phaseimaging (8,9), usually depends on the accurate acquisi-tion of phase information. The phase calculated from thecomplex MR data is generally wrapped into the range of(�p, p], and phase unwrapping is usually required torecover the underlying true phase. A large number ofphase-unwrapping algorithms have been proposed(2,10–14). Phase-unwrapping methods are generallybased on the assumption that the underlying true phaseis smooth and the phase difference between adjacentpixels is less than p. If this assumption is satisfied, thenthe true phase map can be easily obtained (10). In thepresence of strong noise and rapid phase changes, thetrue phase difference between adjacent pixels may beabove p (2). In this scenario, accurately recovering theunderlying true phase is a nontrivial task, especially fortwo and higher dimensional data. A further potentialchallenge of phase unwrapping is the presence of dis-connected areas in the region of interest (ROI). Currentphase-unwrapping algorithms tackle these challengesusing various approaches, which can be divided primari-ly into two categories: path-following approaches (15–19)and minimum-norm methods (20–23).
Path-following approaches exploit the phase gradientof adjacent pixels to realize line integration in the entiremap, according to Itoh’s theory (24). If poles exist, thenthe unwrapped result will depend on the integrationpath. Most path-following methods optimize the integra-tion path to resolve the inconsistency caused by poles(2,10). For example, in Goldstein’s branch cut algorithm(15), poles in the wrapped image are identified and con-nected by branch cuts, and the unwrapped phase is cal-culated by the line integral along a path that does notcross any branch cuts. Phase unwrapping based on theregion-growing approach can be considered a kind ofpath-following method. In region-growing methods, apixel in a region with relatively uniform phase is select-ed as the starting seed, and phase information fromalready unwrapped regions is then used to determine thelocation of the growing pixel and to predict the correctphase value of this pixel. The performance of region-growing methods depends on the selection of seeds andgrowing pixel (ie, growing path) (25–28). Xu and Cum-ming (25) proposed to guide the unwrapping path alongthe most reliable directions by checking the consistencyof phase predictions. A quality map represents the
1School of Automation Engineering, University of Electronic Science andTechnology of China, Chengdu, China.2Guangdong Provincial Key Laborary of Medical Image Processing, Schoolof Biomedical Engineering, Southern Medical University, Guangzhou, China.3Laboratory of Biomedical Imaging and Signal Processing, the University ofHong Kong, Hong Kong SAR, China.4Department of Electrical and Electronic Engineering, the University ofHong Kong, Hong Kong SAR, China.
Grant sponsor: National Natural Science Foundation of China; Grant num-bers: 61671228; 61471188; Grant sponsor: National Key Technology R&DProgram of China; Grant number: 2015BAI01B03; Grant sponsor: NationalKey Research and Development Plan; Grant number: 2016YFC0104003.
*Correspondence to: Yanqiu Feng, Ph.D., School of Biomedical Engineer-ing, Southern Medical University, No. 1023 Shatai Nan Rd, Guangzhou,China 510515. Tel: 1 86 20 6164 8294; Fax: 1 86 20 6164 8274;E-mail: [email protected].
Correction added after online publication 03 April 2017. The authorsupdated the model of the MR Scanner from “XGR-OPER” to “XGY-OPER”in the In Vivo Data Acquisition section.
Received 30 August 2016; revised 19 January 2017; accepted 25 January2017
DOI 10.1002/mrm.26647Published online 00 Month 2017 in Wiley Online Library (wileyonlinelibrary.com).
Magnetic Resonance in Medicine 00:00–00 (2017)
VC 2017 International Society for Magnetic Resonance in Medicine 1
quality of phase in each pixel in the given image, andcan be used to guide the region-growing path. Existingquality maps include the local first-order phase deriva-tive variances (27), the local second-order partial deriva-tive variances (26) of the phase, and the magnitudeinformation of each pixel (28). In the path-dependentapproaches, errors occurring at any point along an inte-gral or region-growing path will propagate to theunwrapping of the following pixels, and accumulate toproduce severe errors in results.
Minimum-norm methods achieve phase unwrappingby minimizing the difference between the local deriva-tive of the true phase and that of the wrapped phase(10). The simplest minimum-norm phase-unwrappingapproach is perhaps the least-squares method, whichminimizes the sum of the squared difference betweenthe derivatives of the wrapped and estimated phase (2).The minimum-norm approach can be improved by intro-ducing weights to the cost function or injecting a maskto mask out inconsistent pixels (2,20). The true phase inthe entire map can be modeled as an empirical mathe-matical function, such as a polynomial models (29,30),truncated Taylor series (21), and Markov model (31,32).With these models, phase unwrapping is translated intoa parameter-estimation problem. Model-based minimum-norm methods are less susceptible to phase errors inlocal regions. Even in the presence of regions that areseparated by large void signal areas, model-based meth-ods can still obtain accurate unwrapped results (2). Themodel-based methods can be further improved usinglocal models, such as a local polynomial model, to accu-rately describe the local rapid or complex changes in thephase map (26,29).
An efficient phase-unwrapping approach is the block-based method using a “split and merge” strategy (23,33).This approach tessellates the wrapped phase map intosmall blocks, conducts intrablock unwrapping, and per-forms interblock unwrapping to merge all the unwrappedblocks using a block-wise region-growing method. A typi-cal split and merge phase-unwrapping algorithm is phaseregion expanding labeler for unwrapping discrete esti-mates (PRELUDE) (33). In this method, the interval (�p,p] is partitioned into an even number of equispaced sub-intervals first, and then these subintervals are exploited tosplit the phase image into blocks without intrablockwraps. Thus, phase unwrapping is implemented betweenblocks. The PRELUDE algorithm will fail if a single blockcontains two areas with phase difference above 2p, whichmay occur in high-resolution imaging because of a lowsignal-to-noise ratio (SNR) (26,33).
Considering the different characteristics between thelocal variation of the wrapped phase and the local varia-tion of phasors (34), this paper presents a novel 2Dphase-unwrapping algorithm based on pixel clusteringand local surface fitting (CLOSE). The CLOSE algorithmclusters the pixels into easy-to-unwrap blocks anddifficult-to-unwrap residual pixels first, and thensequentially performs intrablock, interblock, andresidual-pixel phase unwrapping using a region-growinglocal polynomial surface-fitting method. Simulation andin vivo water–fat Dixon separation experiments wereimplemented to evaluate the performance of the pro-posed method as compared with PRELUDE.
THEORY
Problem of Phase Unwrapping
Figure 1a shows the magnitude image of a representative
axial knee image, and Figure 1b shows the wrapped
phase map. Let u denote the wrapped phase, which is
derived from the angle extraction operation on the com-
plex MR data. The true phase / can be estimated as
fðx;yÞ ¼ wðx;yÞ þ 2kðx;yÞp; [1]
where ðx; yÞ denotes the spatial index of a pixel, and k is
an integer. Phase unwrapping aims to find the correct
integer k at each pixel to restore the underlying true
phase from the wrapped phase, based on the assumption
that the true phase is spatially smooth (2,10).
Related Work: Phase Unwrapping by Local PolynomialSurface Fitting
Modeling the true phase using a smooth function is a
key component in various phase-unwrapping methods
(21,29). To approximate the smooth true phase in a local
window, the following polynomial function is used in
the methods based on local polynomial surface fitting
(LPSF):
fðx;yÞ ¼XMm¼0
XNn¼0
Cm;nxmyn þ rðx;yÞ; [2]
where Cm;n denotes a polynomial coefficient, M and N
are the orders of the polynomial function along the x-
and y-directions, and rðx;yÞ represents the approximation
error of the polynomial model.With the LPSF model, the phase-unwrapping problem
is effectively converted to a parameter-estimation prob-
lem. The model coefficients Cm;n can be determined by
fitting the model function in Equation [2] on the already
unwrapped pixels in a local window. Given P
unwrapped pixels in a local window centered at the
growing pixel ðx0; y0Þ, Equation [2] can be expressed in a
matrix formation as
F ¼ Xc þ R; [3]
where F is a column vector containing the model phase
of these P pixels, X is a P � ðM þ 1ÞðN þ 1Þ matrix with
each row consisting of polynomial bases for each
unwrapped pixel (eg, the row vector for the pixel at ðx; yÞ is½1; x; y ;xy ; :::; xM ; yN ;xM yN �), c is a column vector
containing polynomial coefficients, and R is a residual
error column vector. By fitting Equation [3] to already
unwrapped phase values of these P pixels using the
least-squares method, the coefficients c can be estimated.
Then, the estimated true phase value of the growing pix-
el can be calculated as
fðx0 ;y0Þ ¼ Xðx0 ;y0Þc; [4]
where Xðx0;y0Þ ¼ ½1; x0; y0; x0y0; :::; x0M ; y0
N ;x0M y0
N � is the
polynomial basis for the growing pixel ðx0; y0Þ. Finally,
2 Cheng et al
the integer offset for the growing pixel can be calculated
as
kðx0 ;y0Þ ¼ roundfðx0 ;y0Þ � wðx0 ;y0Þ
2p
( ); [5]
where round{z} is a function to calculate the closest inte-
ger to z.
Proposed CLOSE Method
Phasors, calculated as the complex MR data divided by
their magnitude, usually vary slowly inside tissues, but
rapidly near tissue boundaries because of susceptibility
differences between tissues. The local variation of
phasors reflects the local variation of the underlying true
phase (LDTP hereinafter). The LDTP value is calculated
as
LDTP ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXðx;yÞ2Nðx0;y0Þ
j/ðexpðiwðx;yÞÞ � expð�iwðx0 ;y0ÞÞÞj2
s;
[6]
where N(x0, y0) denotes the neighborhood centering at
the pixel (x0, y0), and /(�) the function to acquire the
angle in (�p, p] from a complex number. The LDTP
value is unaffected by wrapping because its calculation
is before the angle operation. The pixels with low LDTP
values locate in regions where the true phase changes
slowly. Figure 1c shows the LDTP map. The high-
intensity pixels in Figure 1c correspond to locations
with rapid variations of true phase, usually at tissue
boundaries (as compared with Fig. 1a).The local difference of the wrapped phase (LDWP hereinaf-
ter) is defined as the root square of phase difference between
pixel (x0, y0) and its surrounding pixels (x, y) as follows:
LDWP ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXðx;yÞ2Nðx0 ;y0Þ
jwðx;yÞ � wðx0;y0ÞÞj2
s: [7]
The LDWP value is affected by the underlying true phase
and existing wraps. The high-intensity pixels in the
LDWP map correspond to locations where phase wraps
or varies rapidly. Figure 1d shows the LDWP map. Apart
from tissue boundaries, the LDWP values are also high
near wrapping locations.For each pixel, the difference between LDWP and
LDTP is caused by wrapping, and the LDTP value is the-
oretically less than or equal to the LDWP value (see
proof in Appendix). The nonzero pixels in Figure 1e
indicate locations where the LDWP values are higher
than the LDTP values. The comparison between Figures
1e and 1b shows that these nonzero pixels agree well
with the locations where phase wrapping occurs.The pixels in an ROI can be clustered into three
groups according to their LDTP and LDWP values. The
first group contains pixels with small LDTP and LDWP
values. This condition means that no rapid phase
changes and no wrappings occur inside this group. The
second group consists of pixels with small LDTP values
but large LDWP values. This condition means that wrap-
ping pixels exist in a smooth region, and such wrappings
can be easily unwrapped. The third group contains
FIG. 1. Representative axial knee image data illustrating the foundation of the proposed CLOSE method: (a) magnitude map, (b) phasemap, (c) LDTP map, (d) LDWP map, and (e) the difference in subtracting LDTP from LDWP. The nonzero pixels in the difference map
are near the locations where wrapping occurs.
Phase Unwrapping Based on Pixel Clustering and Local Surface Fitting 3
pixels with large LDTP and LDWP values. These pixelsusually locate near tissue boundaries, and the phase
unwrapping of these pixels is challenging.The proposed CLOSE method is based on these findings.
Figure 2 shows the flowchart of CLOSE. By thresholding
the LDTP map, CLOSE clusters the pixels into easy-to-unwrap blocks and difficult-to-unwrap residual pixels first.
Thereafter, CLOSE sequentially performs intrablock, inter-block, and residual-pixel phase unwrapping. For intrablock
unwrapping, CLOSE performs the LDWP thresholding firstto segment each block into subblocks without phase wrap-
pings and residual pixels, and then sequentially performsinter-subblock and residual-pixel unwrapping using the
LPSF-based approach. The unwrapping starts with the larg-
est subblock and proceeds with the closest subblock and
residual pixels. Interblock unwrapping is implemented
similarly as the inter-subblock unwrapping. The phase
unwrapping of residual pixels is then performed in a simi-
lar way to the residual-pixel phase unwrapping inside a
block. The main steps of CLOSE are described in detail in
the following subsections.
Pixel Clustering: Blocks and Residual Pixels.
The pixels in the ROI are clustered into blocks with
smooth true phase, and residual pixels with rapid-
FIG. 2. Flowchart of the proposed CLOSE method.
4 Cheng et al
changing true phase by thresholding the LDTP map. The
small blocks with pixels less than a predefined number
are also classified as residual pixels. An index is
assigned to each block for identification. In each block,
the underlying true phase may be wrapped but does not
contain rapid changes. Thus, phase wraps inside each
block can be identified easily and corrected with a high
degree of confidence. The residual pixels locate where
the true phase varies rapidly. The unwrapping on these
residual pixels is challenging but can be solved using
the knowledge of already unwrapped regions in CLOSE.
Intrablock Phase Unwrapping
The phase unwrapping inside one block is performed in
the presence of wraps. By thresholding the LDWP map, a
block is segmented into subblocks with smoothly
wrapped phase and residual pixels near wrapping loca-
tions. No phase wraps are present inside each subblock
(ie, the pixels in a subblock show a common identical kvalue). Thus, the intrablock phase unwrapping can be
achieved by sequentially performing LPSF-based inter-
subblock and residual-pixel phase unwrapping.
Inter-subblock Phase Unwrapping
The inter-subblock phase unwrapping uses a region-
growing strategy by considering each subblock as a basic
element. The largest subblock is selected as the starting
subblock that assumes the phase inside it has been
unwrapped. The next subblock to be unwrapped is con-
sidered a growing subblock. The subblock closest to the
already unwrapped regions is selected as the growing
subblock based on the Euclidean distance between clos-
est pixels. In the presence of identical pixel-based
Euclidean distance, the growing subblock is further
determined by considering the Euclidean distance
between the mass center of the subblock and that of
already unwrapped regions.The phase unwrapping of the growing subblock is
equivalent to finding an optimal k for all of the pixels in
this subblock. The LPSF method is implemented on the
adjacent pixels between the growing subblock and the
already unwrapped regions. The integer offset for
unwrapping the growing subblock can be obtained by
minimizing the fitting errors in a short integer interval.
Intrablock Residual-Pixel Phase Unwrapping
After unwrapping and merging all of the subblocks, the
unwrapping of residual pixels inside the block is imple-
mented using a pixel-wise region-growing LPSF method.
The detailed unwrapping procedure is outlined as
follows:
1. Select those pixels that are contiguous to the
already unwrapped regions;2. Sort the aforementioned selected pixels according to
their LDTP values in ascending order;3. Perform LPSF-based phase unwrapping by pixels
according to the order determined in Step 2; and4. Repeat these steps until all residual pixels are
unwrapped.
The phase unwrapping in Step 3 uses the polynomialsurface fitted from already unwrapped pixels in a localwindow centered at each growing pixel to estimate itsoptimal k according to Equations [3] to [5].
Interblock and Residual-Pixel Phase Unwrapping
After intrablock phase unwrapping, each block can betreated as a single entity and assigned a common offset kin the following operations. Interblock phase unwrap-ping is then performed to eliminate potential wrapsbetween blocks by using a similar method as in theaforementioned inter-subblock phase unwrapping. Aftermerging all of the blocks, the residual pixels in the ROIwith rapid true phase changes are unwrapped in a simi-lar way to the residual-pixel phase unwrapping inside ablock.
METHODS
Simulation
A phase map with a size of 256� 256 was synthesizedusing a Gaussian function with a standard deviation (SD)of 40 pixels. The magnitude of simulation data was cre-ated as a disk with 12 equispaced sectors. The magni-tude of sectors decreased from 120 to 10 with adecrement of 10. To test the performance of CLOSE onthe data under different conditions, three simulated datasets were generated separately. Each data set had differ-ent properties, which are listed as follows:
To test the performance of CLOSE under differentSNRs, zero-mean Gaussian noise with an SD of 20 wasadded to the real and imaginary parts of the synthesizedcomplex data. Thus, the SNR of synthesized datadecreased by sectors from 6 to 0.5 with a decrement of0.5 (35). The phase range was set as 30 rad, as shown inFigure 3;
To test the performance of CLOSE under varying phasechange levels, the complex data were synthesized withphase ranges varying from 50 to 200 rad by an incrementof 50 rad (Fig. 4). Zero-mean Gaussian noise with an SDof 10 was added to the real and imaginary parts of thesynthesized complex data; and
To test the performance of CLOSE in the presence ofdisconnected regions, the complex data were synthesizedwith phase ranges of 30 rad, and the pixels in three pre-defined ring regions were set to zero (Fig. 5). Finally,zero-mean Gaussian noise with an SD of 10 was addedto the real and imaginary parts of the synthesized com-plex data.
In Vivo Data Acquisition
The knee and ankle data of 13 healthy adult human vol-unteers were acquired on a 0.35 Tesla (T) permanentmagnet MR scanner (XGY-OPER, Ningbo Xingaoyi Co,Ningbo, China) using a quadrature knee coil and a quad-rature ankle coil. The three-point spin-echo Dixonsequence (1) was used to collect data for water–fat sepa-ration. The in-phase image was acquired with the follow-ing parameters: repetition time (TR)/echo time(TE)¼580/28 ms, matrix size of 256� 256, field of view(FOV)¼240�240 mm2, 10 slices for sagittal and
Phase Unwrapping Based on Pixel Clustering and Local Surface Fitting 5
transverse knee, 5 slices for sagittal ankle, and thick-
ness¼ 8.0 mm. Two out-phase images were obtained by
shifting the refocusing pulse 4.8 ms from the center
between the excitation pulse and the echo in this in-
phase acquisition.The knee data of one healthy volunteer was acquired
on a 3T MR scanner (Philips, Achieva, the Netherlands)
using an eight-channel knee coil. Two-dimensional dual-
echo gradient echo images were obtained with following
parameters: TR/TE1/TE2¼ 152/2.30/3.43 ms, matrix size
192� 192, FOV¼ 180� 180 mm2, 25 slices for sagittal
knee, thickness¼3.0 mm, and without parallel accelera-
tion. The complex images were reconstructed using the
adaptive combination of phased array data (36).All in vivo data sets were acquired with Cartesian
sampling. Written, informed consent was obtained from
each participant in accordance with the institutional
review board policies.
Implementation
The CLOSE algorithm was implemented in MATLAB
(R2010a; the MathWorks, Natick, MA, USA) on a desk-
top computer (Dell, Intel Core 2 Duo, 3.5 GB RAM). The
PRELUDE algorithm was implemented in the fMRI Soft-
ware Library (FSL; Oxford Center for Functional Magnet-
ic Resonance Imaging of the brain, United Kingdom) (37)
using its default parameters in 2D mode. Masks were
used in both methods to exclude pixels in background
regions. To obtain masks, a computational edge–detec-
tion algorithm (38) was implemented to detect edges in
magnitude images. Then, the detected outer edges were
determined as the outline of the imaging object. Morpho-
logical operations were finally implemented to further
refine the mask (39).
The LDTP and LDWP values were calculated using the8-connected neighborhood, which was more robust tonoise and could reflect phase variation along more direc-tions than the 4-connected neighborhood. Appropriateclustering of pixels into subblocks and residual pixels isvital for the performance of CLOSE and depends on theLDTP and LDWP thresholds. An empirical value of p/4was chosen as the clustering threshold using simulateddata and applied to in vivo data. To eliminate the effectof small subblocks, the regions with less than 50 pixelsafter thresholding were classified into residual pixels.
The parameters in the LPSF for phase unwrappinginvolve the orders of polynomial functions and the localfitting window. The polynomial orders M and N were bothset as 2. The second-order polynomial function was foundto accurately model the phase in the applied local windowin this study, whereas further increasing the order of poly-nomial function would slightly improve the fitting qualitybut at the expense of significantly increased computationtime. In inter-subblock phase unwrapping, 50 pixels in thegrowing subblock that are closest to the unwrapped regionand 50 pixels in the unwrapped regions that are closest tothe growing subblock were selected as fitting points. Thefitting size of 50 pixels was determined experientially tobe equal to the minimum subblock size. In residual-pixelunwrapping, the influence of the fitting window size wasstudied using the first simulation data set under differentSNRs. On the basis of the observation from the simulation,the fitting window with a size of 19 was used in all of thefollowing experiments.
Evaluation
The reference-phase image for evaluating simulationresults was defined as the sum of synthesized phase andnoise-caused phase variation. Such reference phase was
FIG. 3. Phase-unwrapping results of PRELUDE and CLOSE under varying SNRs. Images in the first column are reference phase, phaseand magnitude maps. Images in the second column are the unwrapped results of PRELUDE and CLOSE. Errors maps display pixels
with the absolute difference between the unwrapped and reference phase larger than p/10.
6 Cheng et al
subtracted from the unwrapped phase image, and thepixels with an absolute difference larger than p/10 radwere considered to be incorrectly unwrapped. The mis-classification ratio (MCR) (33) was calculated as thenumber of incorrect unwrapped pixels divided by thetotal number of pixels in an ROI to quantitatively evalu-ate the performance of CLOSE. Each simulation wasrepeated 100 times, and the corresponding means andSDs of MCRs (%) were calculated for different SNR lev-els, varying phase slopes, and disconnected regions.
For evaluating the performance of CLOSE on the in vivodata, the unwrapped results were implemented in Dixontechniques (1,3) to generate water and fat images. If thephase-unwrapping method does not obtain a correct phase
map, then the water–fat separation results will exhibitswaps. The water–fat decomposition results of every slicewere evaluated in each data set by a blinded board-certified radiologist using a four-point scale as follows(40): (i) many swaps (slices), (ii) few swaps (slices), (iii)total swap slices, and (iv) error ratio. The error ratio wasexpressed as a percentage and calculated as the number ofslices with swaps divided by the total number of slices.
RESULTS
Evaluation of Simulation Data
Figure 3 shows the phase-unwrapping results of CLOSEand PRELUDE under varying SNRs. The phase unwrapped
FIG. 4. Comparison of CLOSE with PRELUDE using the data set with slow to rapid phase variations from left to right. The numbers on
the top of figures in the last two rows indicate the means and SDs of MCR from 100 independent repetitions (mean 6 SD, %).
Phase Unwrapping Based on Pixel Clustering and Local Surface Fitting 7
by PRELUDE exhibits obvious wraps in low SNR regions.In contrast, the unwrapping error is substantively reducedin the results of CLOSE. Table 1 presents the means andSDs of MCR of the two methods over 100 repetitions. TheCLOSE and PRELUDE algorithms generated accurateunwrapped results with mean MCR below 0.005% for sec-tors with SNR above 2.5. The CLOSE approach consistent-ly produced a substantially lower mean and SD of MCRthan those of PRELUDE when SNR is below 2. At the SNRof 0.5, the mean MCR of CLOSE is approximately one-ninth that of PRELUDE.
The performance of CLOSE and PRELUDE under vary-ing phase variation levels is compared in Figure 4. ThePRELUDE approach achieves accurate phase unwrappingat the phase heights of 50 and 100 rad. As the phaseheight increased to 150 and 200 rad, PRELUDE-unwrapped results contained obvious wraps. In contrast,CLOSE consistently produced accurate unwrappedresults for varying phase height from 50 to 200 rad. Themean MCR of CLOSE was significantly lower than thatof PRELUDE.
The evaluation of the performance of PRELUDE andCLOSE in the presence of disconnected regions is shownin Figure 5. The PRELUDE algorithm generated resultswith serious unwrapping errors, and the mean and SD ofMCR was 18.76 6 0.31. In contrast, CLOSE produced anaccurate unwrapped phase with few errors, with themean and SD of MCR at 0.04 6 0.01.
Performance on In Vivo Data
Figures 6 and 7 compare the performance of PRELUDEand CLOSE on the 0.35T in vivo water–fat Dixon data.Figure 6 presents the phase-unwrapping and water–fatseparation results of an axial slice and a sagittal slice inthe knee data set. As indicated by the arrows, the PRE-LUDE results contain residual phase wraps. This phe-nomenon leads to serious swaps in the water and fatimages. In contrast, no obvious phase wraps and water–fat swaps are observed in the CLOSE results. Figure 7
FIG. 5. Comparison of PRELUDE and CLOSE using the simulated data sets in the presence of disconnected regions. The numbers onthe top of figures in the second row indicate the mean and SD of MCR from 100 independent repetitions (mean 6 SD, %).
Table 1Means and SDs of MCR (%) Using PRELUDE and CLOSE for 100
Repetitions Under Different SNRs
Sector SNR PRELUDE CLOSE
1 6 0.00 6 0.00 0.00 6 0.00
2 5.5 0.00 6 0.00 0.00 6 0.003 5 0.00 6 0.00 0.00 6 0.004 4.5 0.00 6 0.00 0.00 6 0.00
5 4 0.00 6 0.00 0.00 6 0.006 3.5 0.00 6 0.00 0.00 6 0.00
7 3 0.00 6 0.00 0.00 6 0.008 2.5 0.01 6 0.01 0.01 6 0.019 2 0.10 6 0.06 0.01 6 0.02
10 1.5 0.45 6 0.10 0.09 6 0.0411 1 1.61 6 0.26 0.19 6 0.06
12 0.5 7.16 6 1.57 0.81 6 0.34
8 Cheng et al
shows the comparison results on a representative sagittalankle slice. The data set is challenging because of thesevere field inhomogeneity in a large FOV, and the exis-tence of complex tissue structures, especially in regionswith low SNRs and/or near FOV margins (eg, toe region).
As indicated by the arrows, the PRELUDE results containresidual phase wraps and water–fat swaps near the FOVmargins. Meanwhile, CLOSE successfully eliminateswraps in the phase map and produces accurate water–fatseparation.
FIG. 6. Phase-unwrapping and water–
fat separation results of representativeaxial (a) and sagittal (b) slice in the0.35T knee data set using PRELUDE
and CLOSE. Arrows indicate the loca-tions where the two methods produced
different results.
Phase Unwrapping Based on Pixel Clustering and Local Surface Fitting 9
Figure 8 presents the phase-unwrapping and water–fat
separation results of one representative 3T sagittal knee
slice. As indicated by the arrows, the PRELUDE results
contain residual phase wraps and water–fat swaps. In
contrast, no obvious phase wraps and swaps are
observed in the CLOSE results. The results of all 25 sli-
ces are shown in the Supporting Information Figures S1–
S25. The PRELUDE algorithm generated six times as
many swaps and 15 times fewer swaps for all slices,
whereas CLOSE yielded only two times as many swaps
and one times fewer swaps near FOV margins, and con-
sistently produced correct phase-unwrapping and water–
fat separation results for all other slices.Table 2 presents the statistical results of slices with
swaps in all of the 0.35T water–fat separation data sets
of 13 volunteers and the 3T data set of one volunteer (a
total of 350 slices consisting of 130 axial knees, 155 sag-
ittal knees, and 65 sagittal ankles). In general, the PRE-
LUDE results had 12 times as many swaps and 78 times
fewer swaps out of the 350 outputs. Meanwhile, CLOSE
produced only 2 times as many swaps and 13 times few-
er swaps. The total error ratio of PRELUDE was 25.71%,
and that of CLOSE was only 4.29%.
DISCUSSION
This paper presents a novel 2D phase-unwrapping meth-
od called CLOSE that exploits the difference between the
local variation of phasors and the local variation of the
wrapped phase. The LDTP value is unaffected by the
angle operation and reflects the local variation of the
underlying true phase. The CLOSE method performs the
FIG. 7. Phase-unwrapping and water–at separation results of one representative 0.35T sagittal ankle slice using PRELUDE and CLOSE.
Arrows indicate the locations where the two methods produced different results.
10 Cheng et al
FIG. 8. Phase-unwrapping and water–fat separation results of one representative 3T sagittal knee slice using PRELUDE and CLOSE.
Arrows indicate the locations where the two methods produced different results.
Table 2Statistical Results of Slices With Swaps in the Water–Fat Separation Data Sets of 14 Volunteers
Data MethodMany
Swaps (Slices)Few
Swaps (Slices)Total Error
SlicesError
Ratio (%)
Axial Knee (0.35 T, 130 slices) PRELUDE 2 13 15 11.54CLOSE 0 2 2 1.54
Sagittal Knee (0.35 T, 130 slices) PRELUDE 3 25 28 21.54CLOSE 0 5 5 3.85
Sagittal Ankle (0.35 T, 65 slices) PRELUDE 1 25 26 40.00CLOSE 0 5 5 7.69
Sagittal Knee (3.0 T, 25 slices) PRELUDE 6 15 21 84.00
CLOSE 2 1 3 12.00Total (350 slices) PRELUDE 12 78 90 25.71
CLOSE 2 13 15 4.29
Phase Unwrapping Based on Pixel Clustering and Local Surface Fitting 11
LDTP thresholding first to cluster the pixels into smoothblocks that are easy to unwrap, and residual pixels neartissue boundaries with rapid true phase changes that aredifficult to unwrap. Then, the method sequentially per-forms intrablock, interblock, and residual-pixel phaseunwrapping using a region-growing LPSF method. Thesimulation study demonstrates that CLOSE outperformsPRELUDE in phase unwrapping and achieves accurateresults, even in the presence of severe noise, rapid phasechanges, and disconnected regions. The in vivo experi-ments demonstrate that CLOSE generates the unwrappedphase map with few errors, and results in the water andfat images with substantially reduced swap ratios.
The excellent performance of CLOSE can be attributedto two main reasons. First, CLOSE unwraps smooth,easy-to-process blocks first, and then processes thedifficult-to-unwrap residual pixels using the knowledgeof phases in already unwrapped regions. This strategycan avoid the early appearance of boundary pixels in theregion-growing path, and thus decrease the possibility oferror propagation and accumulation. Note that pixels inregions with pure noise inside the imaging object will beclassified as residual pixels, and therefore have a negligi-ble effect on the unwrapping of the other pixels. Second,CLOSE models the underlying true phase using a localpolynomial surface, which is insensitive to the noise anddisconnected regions; thus, it can achieve accurate phaseunwrapping even when the adjacent phase differenceexceeds p.
The performance of CLOSE depends on the choice ofthe pixel-clustering threshold. An identical thresholdwas used for thresholding LDTP and LDWP. A largethreshold may introduce many pixels near tissuesboundaries into smooth blocks, which can increase thedifficulty in the intrablock phase unwrapping and resultin unwrapping errors. A small clustering threshold cancreate subblocks with a reduced size and increased dis-tances between subblocks, which may undermine theeffectiveness of modeling the inter-subblock phase varia-tion by an LPSF function. Given the diverse content ofthe MR image, a universal threshold is not obtained forpixel clustering in CLOSE. In this paper, a threshold ofp/4 was empirically set but not optimally determined.However, using this threshold, the proposed methodconsistently produced satisfactory unwrapped results inall of the simulated and in vivo data sets in this study.Further study on the optimization of the clusteringthreshold is warranted in future work.
The performance of CLOSE also depends on theparameters in LPSF: the order of polynomial function,and the data to fit. In this study, the second-order poly-nomial function demonstrated to be capable of modelingthe phase in a local window, even in the presence ofrapid phase changes, as shown in the simulation. This isconsistent with the report in (29). A proper choice of fit-ting data in LPSF is important for phase unwrapping.More data may improve the robustness of fitting, butwill reduce the fitting accuracy because the model maynot accurately reflect the phase in a large window. Weadopted two different approaches to select fitting data ininter-subblock unwrapping and residual-pixel unwrap-ping. In LPSF for inter-subblock unwrapping, a number
of close and already unwrapped pixels in two data sets
were fitted. In LPSF for residual-pixel unwrapping, a
number of already unwrapped pixels in a local window
with a fixed radius around the residual pixel were select-
ed as fitting data. These parameters were determined
experientially in this study, and should be carefully
tuned in practice.Block-based phase-unwrapping algorithms using the
“split and merging’’ strategy are inherently more efficient
than pixel-wise region-growing methods. Considering
that CLOSE involves both the block and pixel-based
operations, the efficiency of this method should be
higher than the pixel-wise region-growing methods, but
lower than purely block-based methods, such as PRE-
LUDE. In the current implementation in this study, the
mean running times of CLOSE for the three 0.35T in
vivo data sets were 10.58 6 3.61, 60.39 6 19.10, and
66.66 6 9.07 s, and those of PRELUDE were 1.10 6 0.16,
6.83 6 0.60, and 8.05 6 0.79 s. The CLOSE method is
approximately one order of magnitude slower than PRE-
LUDE. Notably, PRELUDE is programmed in the C lan-
guage and runs on a Linux system, which is by nature
more efficient than MATLAB on Windows, where
CLOSE was implemented. The computational efficiency
of CLOSE can be improved by further code optimization.The CLOSE method combines the advantage of block-
based and per-pixel region-growing phase-unwrapping
methods by unwrapping smooth blocks first and then
residual pixels with a quality-guided per-pixel region-
growing method. Given the validity of a smooth assump-
tion inside a block, accurate intrablock phase unwrap-
ping can be achieved easily using the most available
methods with a high degree of confidence. In this study,
the LPSF method based on the region-growing approach
was used for consistency with interblock and residual
phase unwrapping. Intrablock phase unwrapping by
using purely block-based methods, such as PRELUDE,
may further increase the efficiency. Our future work will
investigate this possibility and extend CLOSE to 3D
phase unwrapping, which is more challenging than the
2D problem.
CONCLUSIONS
This paper presents a novel phase-unwrapping method
that first clusters pixels into smooth blocks that are easy
to unwrap, and residual pixels with rapid phase changes
that are difficult to unwrap. Then, the proposed method
sequentially performs intrablock, interblock, and
residual-pixel unwrapping using a region-growing local
polynomial surface fitting method. The simulation
results demonstrate that the proposed method can
achieve robust and accurate phase unwrapping, even in
the presence of serious noise, rapid phase changes, and
disconnected regions. The application on water–fat Dix-
on MRI data shows that the proposed method can suc-
cessfully separate water and fat with few swaps.
Therefore, the proposed method therefore is beneficial to
phase-related medical imaging applications, such as field
mapping and brain phase imaging.
12 Cheng et al
APPENDIX
In this paper, the LDTP value of a pixel is not larger thanits LDWP value. According to Equation [1], the relation-ship among the true phase fðx0;y0Þ, the wrapped phasewðx0;y0Þ, and the integer offset kðx0;y0Þ can be written as
fðx0 ;y0Þ ¼ wðx0 ;y0Þ þ 2kðx0;y0Þp: [A1]
The value of wðx0 ;y0Þ is in the range of (�p, p]. Assumingthat the pixel at ðx0; y0Þ is adjacent to the pixel at ðx; yÞ,their phasors can be separately defined as
Pðx0 ;y0Þ ¼ exp½1i � ðwðx0;y0Þ þ 2kðx0 ;y0ÞpÞ�; [A2]
Pðx;yÞ ¼ exp½1i � ðwðx;yÞ þ 2kðx;yÞpÞ�: [A3]
Thus, the difference between Pðx0 ;y0Þ and Pðx;yÞ can be cal-culated as
DPðx;yÞ ¼ Pðx;yÞ � ðPðx0;y0ÞÞ�
¼exp½1i
� ðwðx;yÞ þ 2kðx;yÞp� wðx0;y0Þ � 2kðx0 ;y0ÞpÞ�¼exp½1i
� ððwðx;yÞ � wðx0;y0ÞÞ þ 2ðkðx;yÞ � kðx0 ;y0ÞÞpÞ�; [A4]
where ðÞ� represents the complex conjugate operation.The difference between kðx;yÞ and kðx0 ;y0Þ is an integer.
Let Dwðx;yÞ denote the difference between wðx;yÞ andwðx0;y0Þ, then the phase of DPðx;yÞ can be calculated by theangle operation as follows:
WðDwðx;yÞÞ ¼ angleðDPðx;yÞÞ
¼ angle½expð1i � Dwðx;yÞÞ�[A5]
The value of WðDwðx;yÞÞ and the absolute value of WðDwðx;yÞÞ are in the ranges of (�p, p] and [0, p].
The value of Dwðx;yÞ is in the range of (�2p, 2p). If thevalue of Dwðx;yÞ is in the range of (�p, p), then Equation[A5] can be rewritten as
WðDwðx;yÞÞ ¼ angle½expð1i � Dwðx;yÞÞ�
¼ Dwðx;yÞ:[A6]
If the value of Dwðx;yÞ is in the range of (�2p, �p] or [p,2p), then its absolute value is in the range of [p, 2p).Moreover, the absolute value is not less than the abso-lute value of WðDwðx;yÞÞ in the range of [0, p].
Therefore, given an arbitrary Dwðx;yÞ, the absolute val-ue of WðDwðx;yÞÞ is not larger than the absolute value ofDwðx;yÞ:
jWðDwðx;yÞÞj � jDwðx;yÞj [A7]
According to Equations [6] and [7], LDTP and LDWP canbe calculated separately as
LDTP ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXðx;yÞ2Nðx0;y0Þ
jangleðexpðifðx;yÞÞ � expð�ifðx0;y0ÞÞÞj2
s
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXðx;yÞ2Nðx0;y0Þ
jWðDwðx;yÞÞj2;s
[A8]
LDWP ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXðx;yÞ2Nðx0 ;y0Þ
jwðx;yÞ � wðx0;y0Þj2
s
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXðx;yÞ2Nðx0 ;y0Þ
jDwðx;yÞj2:s [A9]
Therefore, according to Equations [A7–A9], the LDTP
value of a pixel is less than or equal to its LDWP value.
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SUPPORTING INFORMATION
Additional supporting information may be found in the online version of thisarticleFigs. S1–S25. Phase-unwrapping and water–fat separation results of the1st through 25th slices in the 3T knee data using PRELUDE and CLOSE.Arrows indicate the locations where the two methods produced differentresults.
14 Cheng et al
