Quality-guided phase unwrapping
technique: comparison of quality
maps and guiding strategies
Ming Zhao,1Lei Huang,2Qican Zhang,3Xianyu Su,3Anand Asundi,2and Qian Kemao1,*
1School of Computer Engineering, Nanyang Technological University, 639798, Singapore
2School of Mechanical and Aerospace Engineering, Nanyang Technological University, 639798, Singapore
3Opto-Electronics Department, Sichuan University, Chengdu 610064, China
*Corresponding author: email@example.com
Received 31 May 2011; revised 19 August 2011; accepted 20 August 2011;
posted 8 September 2011 (Doc. ID 148376); published 16 November 2011
Quality-guided phase unwrapping is a widely used technique with different quality definitions and guid-
ing strategies reported. It is thus necessary to do a detailed comparison of these approaches to choose the
optimal quality map and guiding strategy. For quality maps, in the presence of noise, transform-based
methods are found to be the best choice. However in the presence of discontinuities, phase unwrapping is
itself unresolved and hence quality-guided phase unwrapping is not sufficient. For guiding strategies,
classical, two-section, and stack-chain guiding strategies are chosen for comparison. If accuracy is the
foremost criterion then the classical guiding strategy with a data structure of indexed interwoven linked
list is best. If speed is of essence then the stack-chain guiding strategy is the one to use.
Society of America
© 2011 Optical
In many phase-measuring techniques, such as inter-
ferometric synthetic aperture radar (InSAR) ,
fringe projection profilometry (FPP) [2,3], and optical
interferometric methods [4,5], the retrieved phase is
wrapped within a range from −π to π, i.e., modulo 2π.
The wrapped phase needs to be unwrapped in order
to construct a continuous phase distribution. This
process is called phase unwrapping.
Many phase unwrapping algorithms have been
proposed, including temporal phase unwrapping
methods such as dynamic unwrapping method ,
multifrequency method [7–9], and heterodyne meth-
od  and spatial phase unwrapping methods such
as Goldstein’s method , quality-guided method
, Flynn’s method , and minimum Lp-norm
method . Temporal phase unwrapping methods
are effective and robust, but require multiframes
of wrapped phase along the time axis, or multifre-
quency fringe patterns. On the other hand, spatial
phase unwrapping methods have fewer restrictions,
but disjoint regions and phase discontinuities are
difficult to handle.
The quality-guided phase unwrapping (QGPU)
technique has been well studied in the past two dec-
ades [15–26] as an efficient, fast, and automatic spa-
tial phase unwrapping method. Figure 1 shows the
profile of a real box measured by fringe projection
profilometry to exemplify the concept of QGPU.
The wrapped phase in Fig. 1(a) is unwrapped into
Fig. 1(d) by following a certain unwrapping path in
Fig. 1(c) according to the quality map in Fig. 1(b).
The complicated quality map is due to the texture
of the box which is intentionally selected to demon-
strate the capability of QGPU. For the unwrapping
path, the pixels in light yellow are unwrapped first
and those in dark red are unwrapped last. In QGPU,
definition of quality and strategies for guiding the
© 2011 Optical Society of America
6214APPLIED OPTICS / Vol. 50, No. 33 / 20 November 2011
unwrapping are two major aspects affecting the
accuracy and speed of the result. Many quality maps
have been proposed, including the correlation coef-
ficient map in InSAR , modulation map or relia-
bility map from optical metrology [16,18], pseudo
correlation coefficient map , phase derivative var-
iance map , maximum phase gradient map ,
first phase difference map , second phase differ-
ence map [19,30,31], and amplitude map or ridge
map from transform methods [16,21,22]. For the guid-
ing strategy, the most widely used approach is the
classical quality guiding strategy , which always
guiding strategies include two-section guiding strat-
egy and stack-chain guiding strategy .
How to select a proper quality map and guiding
strategy is thus important and relevant in fringe pro-
cessing. This paper provides a detailed comparison of
the different quality maps and guiding strategies.
The principle and problems of spatial phase unwrap-
ping and the process of QGPU are introduced in Sec-
tion 2. The different quality maps are compared in
Section 3. The guiding strategies for flood fill and re-
lated programming using MATLAB and C++ are dis-
cussed in Section 4. The discussion and conclusions
on the choice of quality maps and guiding strategies
are highlighted in Section 5.
Unwrapping and QGPU
Principle and Problems of Spatial Phase
In spatial phase unwrapping, a pixel ‘a’ that neigh-
bors an unwrapped pixel ‘b’, can be unwrapped as
Basic Principle of Spatial Phase Unwrapping
aþ 2π × round
where the function of roundð·Þ converts the input to
its nearest integer; φu
el b; φw
phase of pixel a, respectively. The basic requirement
of Eq. (1) is that the difference of the unwrapped
phase between neighboring pixels is within (−π, π].
In Fig. 2, the unwrapped dark green pixels are used
along with Eq. (1) to unwrap the light pink pixels. If
noise reduction techniques are adopted, the unwrap-
ping results of original noisy wrapped phase maps
bis the unwrapped phase of pix-
arepresent wrapped and unwrapped
a, and φu
can also be obtained from the denoised unwrap-
ped phase maps through a phase congruence
Although the phase unwrapping process described in
Eq. (1) seems straightforward, problems arise if:
i.Undersampling causes the absolute phase dif-
ference between two neighboring pixels to exceed the
range of (−π, π] and thus violates the requirement of
Eq. (1). This situation is not considered in this paper.
ii.Noise can give wrong unwrapping values
in Eq. (1) and result in the failure of phase unwrap-
ping. Suppressing noise before phase unwrapping is
iii.Invalid areas do not reflect the actual physical
quantity. Decorrelation in InSAR images and the
shadow problems in FPP are examples where invalid
areas occur. The unwrapping results within the inva-
lid areas are not of interest, but they may propagate
errors into other pixels, which is a major concern.
iv.Discontinuities in true phase cannot be recog-
nized by the phase unwrapping algorithm and are
treated as continuous by Eq. (1), resulting in incor-
v.If a pixel is incorrectly unwrapped due to rea-
sons highlighted above, the subsequent unwrapping
based on this pixel will also be wrong.
All light pink pixels in Fig. 2 can be simultaneously
unwrapped based on their unwrapped (dark green)
neighbors using Eq. (1). Rather than unwrapping
all light pink pixels with same priority, they are
placed in a queue called the adjoin list and only the
pixel with the highest quality is chosen for un-
wrapping. Noisy pixels and invalid areas that are
Basic Process of QGPU
unwrapped first and those in dark red are unwrapped last), and (d) unwrapped phase map.
(Color online) Concept of of QGPU. (a) Wrapped phase map, (b) quality map, (c) unwrapping path map (pixels in light yellow are
Fig. 2.(Color online) Diagram of spatial unwrapping process.
20 November 2011 / Vol. 50, No. 33 / APPLIED OPTICS6215
assigned low-quality values are moved to the back of
the queue in a quality-guided phase unwrapping pro-
cess. Problems of discontinuities can also be avoided
by QGPU in some cases. The process can be described
Step 1: Find the seed pixel with the highest quality
from the quality map and push it into the adjoin list.
Step 2: Find the pixel with the highest quality from
the adjoin list, unwrap it, and remove it from the
adjoin list. Push the unprocessed neighbors into
the adjoin list.
Step 3: Repeat step 2 until the adjoin list is empty.
3.Comparison of Quality Maps
There are two possible approaches for obtaining the
quality map from measurement data.
Case A: a quality map is calculated from the raw
measurement data. For example in phase-shifting
methods a complex-valued fringe pattern fccan be
obtained as 
Resources for Quality Calculation
fc¼ M · expðj · φwÞ;
where j is the imaginary unit, φwis the wrapped
phase, and M is the intensity modulation. From fc
both φwand M can be delineated.
Case B: The quality map can be calculated from
the wrapped phase or the normalized complex-
valued fringe pattern fnas:
fn¼ 1 · expðj · φwÞ ¼fc
Quality maps are classified according to resources
used to generate them:
Classification of Quality Maps
i.Quality maps only from case A, including the
correlation coefficient map in InSAR ; the modu-
lation map QMOD, and the reliability map QREL
[16,18] from least squares fitting in phase-shifting
ii. Quality maps only from case B, including the
pseudocorrelation coefficient map QPCC , the
phase derivative variance map QPDV, the first
phase difference map QFPD, the second phase
difference map QSPD, and their combinations
iii.Quality maps from both case A and case B,
which are based on transform methods, such as the
Fourier transform (FT) method [2,34,35], the wind-
owed Fourier filtering (WFF) method [22,32,36,37],
[22,32,36,37], and the wavelet transform (WT) meth-
od [21,38]. The quality maps include the amplitude
map from Fourier transform method QFT, ampli-
tude map from windowed Fourier filtering method
QWFF, ridge map from windowed Fourier ridges
form method QWT.
In summary, quality maps from case A include
QMOD, QREL, QFT, QWFF, QWFR, and QWT, while qual-
ity maps from case B include QPCC, QPDV, QPFG, QSPD,
QFT, QWFF, QWFR, and QWT. These two cases will be
compared separately, where noise and discontinu-
ities are two major concerns.
Comparison of Different Quality Maps for Noisy
A four-step phase-shifted fringe pattern (256 × 256
pixels) with both speckle noise and normally distrib-
uted random noise are simulated as shown in
Fig. 3(a). The true wrapped phase map shown in
Fig. 3(b) is simulated as
Generation of Noisy Phase Maps
8ðx − 128Þ þ 3 × peaksðx;yÞ;
where φ denote the true phase map and the function,
peaks is a MATLAB built-in function. The true mod-
ulation map M in Fig. 3(c) is simulated as
where the function normalizeð·Þ linearly scales
the input so that it is between 0 and 1. The wrapped
phase is then retrieved using the least squares
phase-shifting algorithm and is shown in
phase, (c) true modulation, (d) retrieved noisy wrapped phase, and (e) retrieved unwrapped phase (with phase noise).
Simulated of noisy phase maps. (a) Four-step phase-shifted fringe patterns with speckle and random noise, (b) true wrapped
6216APPLIED OPTICS / Vol. 50, No. 33 / 20 November 2011
Fig. 3(d). The wrapped phase, unwrapped using a
congruence operation based on the simulated true
phase is shown in Fig. 3(e). It has to be noted that
the signal-to-noise ratio is almost zero in regions
of low modulation, and the wrapped phase and un-
wrapped phase results in these regions are not reli-
able. However, this is intentionally simulated to
investigate the performance of the quality maps.
Six quality maps, QMOD, QREL, QFT, QWFF, QWFR, and
QWT, are calculated from raw fringe patterns and
then their performances are compared. The results
using these quality maps are shown in Fig. 4. The
results show that in the presence of noise, transform-
based quality maps, QFT, QWFF, QWFR, and QWT, per-
form better than QMODand QREL.
Comparison of Quality Maps in Case A
Eight quality maps, QPCC, QPDV, QFPD, QSPD, QFT,
QWFF, QWFR, and QWT, are calculated from the
wrapped phase map and then their performances
are compared. The results are given in Fig. 5, which
show that, once again, transform-based quality maps
perform better than others.
Comparison of Quality Maps in Case B
To further compare the performances within trans-
form-based methods, a more complicated example
is simulated, where the fringe patterns, true phase,
true modulation, and retrieved wrapped phase are
Further Comparison of Transform-Based
shown in Figs. 6(a)–6(d), respectively. The phase is
designed to have a wider spectrum with no carrier
Unwrapping results with transform-based meth-
ods in case A are shown in Fig. 7. Figure 7(a) shows
that FTas a global method is not able to reduce noise
in high frequency regions. Figures 7(b) and 7(c) de-
monstrate good results by using two local methods,
WFFand WFR, because of their good noise reduction
abilities. Figure 7(d) shows the unwrapping result by
using the WT method. The WT method is not suitable
in low-frequency regions that require a larger win-
dow. As shown in Fig. 7(d), the phase unwrapping er-
ror is mainly in the low-frequency regions. Similar
results are observed in case B.
To summarize, under noisy condition, transform-
based methods show better performances because
they provide both denoised wrapped phases and rea-
sonable quality maps.
Discontinuous Phase Maps
Comparison of Different Quality Maps for
influence of discontinuity, eight frames of phase-
show the retrieved wrapped phase and modulation
maps, respectively. The unwrapped phase map with
temporal phase unwrapping technique is shown in
Fig. 8(d) for benchmarking.
Generation of Discontinuous Phase Maps
(d) QWFF, (e) QWFR, and (f) QWT, respectively. From left to right, each group shows quality maps, path maps, and unwrapping error maps.
(In unwrappingerror maps, correctly unwrapped pixels are shown in gray scale and incorrectly unwrapped pixels are shown in red or blue.
Red andblueindicatethat theunwrapped phasevalueis larger andsmaller thanitstrue value, respectively. Darkershadesindicatelarger
(Color online) Comparison of phase unwrapping from noisy phase maps. The quality maps used are (a) QMOD, (b) QREL, (c) QFT,
20 November 2011 / Vol. 50, No. 33 / APPLIED OPTICS6217
Six quality maps are compared for discontinuous
phase in case A. The results are shown in Fig. 9,
which indicate that the phase unwrapping procedure
for this discontinuous example is successful with as-
sistance of quality maps of QMOD, QREL, QFT, and
QWFF. The ridges based quality maps of QWFRand
QWTfail in leading the unwrapping process with di-
rectly passing through discontinuity.
Comparison of Quality Maps in Case A
Eight quality maps are compared for discontinuous
phase in case B. The results are shown in Fig. 10,
Comparison of Quality Maps in Case B
which indicate that quality maps of QPDV, QFPD,
QSPD, QFT, and QWFFand QWFRsuccessfully guide
phase unwrapping in this experiment. QPCC and
QWTfail in leading the unwrapping process with di-
rectly passing through discontinuity.
In presence of discontinuity, it is hard to draw a con-
clusion as to which quality map can best guide the
phase unwrapping process. Figure 11 shows an ex-
ample that all quality maps fail.
Phase unwrapping under discontinuity condition
is thus still a challenging problem and unresolved
using QGPU alone.
Failure of All Quality Maps
(d) QSPD, (e) QFT, (f) QWFF, (g) QWFR, and (h) QWT, respectively. From left to right, each group shows quality maps, path maps, and un-
wrapping error maps. The meanings of colors in the unwrapping error maps are the same as in Fig. 4.
(Color online) Comparison of phase unwrapping from noisy phase maps. The quality maps used are (a) QPCC, (b) QPDV, (c) QFPD,
without noise, (c) modulation distribution, and (d) retrieved wrapped phase with phase noise.
Phase unwrapping simulation with wide spectrum and no carrier fringes. (a) Phase-shifted fringe patterns, (b) wrapped phase
6218 APPLIED OPTICS / Vol. 50, No. 33 / 20 November 2011
(b) QWFF, (c) QWFR, and (d) QWT, respectively. From left to right, each group shows quality maps, path maps, and unwrapping error maps.
(Color online) Comparison of unwrapping results from transform-based methods in case A. The quality maps used are (a) QFT,
wrapped phase, (c) retrieved modulation map, (d) unwrapped phase by using temporal phase unwrapping technique.
Influence of fringe discontinuity on phase unwrapping. (a) Eight frames of phase-shifted fringe pattern using FPP, (b) retrieved
(b) QREL, (c) QFT, (d) QWFF, (e) QWFR, and (f) QWT, respectively. From left to right, each group shows quality maps, path maps, and un-
wrapping error maps.
(Color online) Comparison of phase unwrapping under discontinuity condition in case A. The quality maps used are (a) QMOD,
20 November 2011 / Vol. 50, No. 33 / APPLIED OPTICS6219
A good quality map helps to achieve correct phase
unwrapping, while a good guiding strategy helps
to reduce the phase unwrapping time. In this section,
different guiding strategies are introduced, followed
by a comparison of their performances.
A.Introduction to Guiding Strategies
The simplest and classical guiding strategy was de-
scribed in Section 2.C. However, it is time-consuming
to process the pixels in the adjoin list. The pixels in
the adjoin list can be unordered or ordered according
to their quality values. If the pixels are unordered, to
maintain the adjoin list needs three operations:
inserting a new pixel, searching the pixel with the
highest quality value to unwrap, and removing the
unwrapped pixel. On the other hand, if the pixels
are ordered, there are also three operations: locating
the inserting position for a new pixel, inserting the
new pixel, and removing the pixel with the highest
quality value. The ordered list only needs to visit a
part of the adjoin list to find the inserting position,
while the unordered one needs to traverse the
whole list to find the highest quality value. Thus
Classical Quality Guiding Strategy
the ordered adjoin list is preferred and used in this
paper by default.
The adjoin list can be implemented by several data
structures. Figure 12(a) shows the data structure of
an array . The array is stored in a continuous
memory area and it is very inefficient to insert or re-
move pixels due to memory reallocation. Figure 12(b)
shows the data structure of a linked list (LL) . A
node of a linked list contains pixel properties and a
pointer to the next node according to the order of
quality values. Unlike an array, an LL is very effi-
cient to insert or remove pixels by modifying the
pointers. However, traversing the whole LL is still
not efficient because the LL is usually very long.
One possible solution to improve the efficiency of
LL is to cut the LL into several sections based on
the quality values. Each section can be viewed as
a “mini-LL”, while the whole structure forms array
of pointers and each pointer points to a mini-LL. This
data structure is called an indexed linked list (ILL)
and is shown in Fig. 12(c), where quality values de-
crease from an upper row to a lower row and also de-
crease from left to right in each mini-LL. The ILL is
efficient in inserting and removing pixels, similar to
an LL, but its traversal is much faster as the length
of each section is much shorter than that of an LL.
For each mini-LL in an ILL, rather than dynami-
cally allocating and releasing memory for each pixel,
(b) QPDV, (c) QFPD, (d) QSPD, (e) QFT, (f) QWFF, (g) QWFR, and (h) QWT, respectively. From left to right, each group shows quality maps, path
maps, and unwrapping error maps.
(Color online) Comparison of phase unwrapping under discontinuity condition in case B. The quality maps used are (a) QPCC,
6220 APPLIED OPTICS / Vol. 50, No. 33 / 20 November 2011
sufficient memory can be preallocated for all pixels to
reduce time consumption. However, the size of a
mini-LL cannot be predetermined and in the worst
cases its size would be the whole wrapped phase
map. Fortunately, these mini-LLs can be interwoven
in the preallocated array as illustrated in Fig. 12(d).
Pointers can distinguish these mini-LLs without am-
biguity. The array is one-dimensional (1D). The 1D
index of an element of the array can be converted
to the 2D location of a pixel in the wrapped phase
map. The data structure is thus called interwoven in-
dexed linked list (I2L2) and is shown in Fig. 12(d).
From our practical experiences, section number in
ILL and I2L2 is recommended to be the square root
of pixel number of a phase map. To the best of our
knowledge,the data structures ILL andI2L2 are pro-
posed by the authors of this paper, with the hint from
the data structure for stack-chain guiding strategy
which is introduced in Section 4.A.3.
In addition, an algorithm called TRIM , which
splits the original list into two sections if it is too
long, is discussed and implemented. One section is
still called the adjoin list and the other one is called
the postponed list. Once the adjoin list is empty,
result under case A (b)–(g) and case B (h)–(o). The quality maps used are (b) QMOD, (c) QREL, (d) QFT, (e) QWFF, (f) QWFR, (g) QWT, (h) QPCC,
(i) QPDV, (j) QFPD, (k) QSPD, (l) QFT, (m) QWFF, (n) QWFR, and (o) QWT, respectively. From left to right, each group shows quality maps, path
maps, and unwrapping error maps.
(Color online) Failure of QGPU. (a) Wrapped phase and unwrapped phase with temporal phase unwrapping and unwrapping
Data structures for sorting in quality guiding. (a) Array, (b) linked list, (c) indexed linked list, and (d) indexed interwoven
20 November 2011 / Vol. 50, No. 33 / APPLIED OPTICS6221
higher quality pixels from the postponed list will be
inserted into it.
A comparison of the speed for different data struc-
tures is carried out with a result shown in Table 1.
The performances from the worst to the best are gi-
ven by an array, an LL, an ILL, and an I2L2, which is
consistent with our previous analysis. TRIM per-
forms better than an array and an LL, but is worse
than the ILL and the I2L2.
In the classical quality guiding strategy, each pixel’s
quality value is compared with others before it is un-
wrapped. However, pixels with high-quality values
usually give correct results even without quality
comparison. A two-section guiding strategy is thus
proposed: the high-quality pixels form a high-quality
section without any comparison; the remaining pix-
els still need the quality comparison and form a low-
quality section which can adopt the data structure of
I2L2. Since only some of the pixels need to be or-
dered, higher processing speed is expected. However
a threshold level has to be set to separate the two
sections, which can be tricky. The classical quality
guiding strategy introduced in Section 4.A.1 can be
seen as a special case of the two-section guiding
strategy when the threshold is high enough so that
the high-quality section is empty. Experimental re-
sults with different thresholds are shown in Table 2.
The improvement is not significant: to guarantee a
zero-error rate, an improvement of only 0:1s can
be obtained. If a 1% error rate can be tolerated then
the speed can be doubled.
Two-Section Guiding Strategy
Since the data structure in Fig. 12(c) has many sec-
tions and each section consists of pixels with similar
Stack-Chain Guiding Strategy
quality values, the ordering of pixels in each section
can be avoided to reduce the processing time. This
idea was proposed in [15,40] and also formed through
personal communication . In each section, since
no ordering is needed, the concept of a stack with
simple “first in, last out” data management can be
adopted. Similar to Fig. 12(d), many stacks can be
interwoven and the technique is called stack-chain
guiding strategy . Empirically the section num-
ber is set as the square root of pixel number of a
phase map. For the Box example in Table 2, which
has 960 rows and 1,280 columns, the section number
is ð960 × 1280Þ1=2≈ 1109.
B.Comparison and Discussion
To evaluate the performance of all the strategies dis-
cussed above, three wrapped phase maps, Box in
Fig. 1, Noisy Peaks in Fig. 3, and Coffee Cup Cover
in Fig. 8, are used. Four methods, TRIM, I2L2, two-
section, and stack-chain, are implemented in two
programming languages, MATLAB and C++. These
two languages are selected due to their popularity.
MATLAB is preferred by many researchers for algo-
rithm design and C++ is widely used in industry.
MATLAB is an interpreted language and C++ is a
compiled language. The former is usually slower
than the latter. For the two-section strategy, three
thresholds, 25%, 50%, and 75% of the largest quality
value, are used to separate two sections. The results
are shown in Tables 3–5.
From these results, it can be summarized that,
i.I2L2 provides the most reasonable and accep-
table result as all the pixels are ordered. The compu-
tation time is also short due to the effective data
structure. TRIM gives the same unwrapping results
but it is slower.
ii. Stack-chain strategy is fastest among all stra-
tegies. However, it is not error-free, although the er-
ror is often small. This algorithm is most useful when
the speed is an essential criterion.
Table 1. Speed Comparison for Different Data Structures in Classical Quality Guiding (C++)
Data StructuresArray LL ILLI2L2 TRIM
Time (s)Box (960 × 1280) in Fig. 1
Noisy Peaks (256 × 256) in Fig. 3
Coffee Cup Cover (400 × 400) in Fig. 8
Table 2.Threshold Level in Two-Section Guiding of Box
(960 × 1280) in Fig. 1 (C++)
Threshold (Percentage of
the highest quality)Time (s)Error Rate (%)
Table 3.Comparison Results of Box (960 × 1280) in Fig. 1
MATLABC++Error Rate (%)
6222 APPLIED OPTICS / Vol. 50, No. 33 / 20 November 2011
iii. The two-section strategy is less attractive
with regard to both error rate and speed. However,
if a batch of phase maps with similar properties is
to be unwrapped, a proper threshold can usually
be predetermined. In this situation, there is a possi-
bility that the two-section strategy is able to unwrap
the phase maps with zero-error rate but faster than
I2L2. Another situation is when the majority of pix-
els in a wrapped phase map have high quality, and
the two-section approach can also quickly unwrap
the phase without error. However, it can be noticed
that the gain of speed is not much.
The QGPU method is discussed with regard to two
aspects—the quality map and the guiding strategy.
A good quality map is important for successful phase
unwrapping. For noisy phase maps, transform-based
methods perform better but with longer processing
time. The parameter selection in transform-based
methods reduces automation of QGPU. When discon-
tinuities are present, QGPU alone is not always suc-
cessful. A guiding strategy is the other important
aspect in the phase unwrapping process. Three stra-
tegies, classical, two-section and stack-chain, are
discussed. The classical quality guiding strategy ap-
pears to be best and can be accelerated by a proper
data structure such as the interwoven index linked
list. The two-section method runs faster, but its suc-
cess depends on a threshold. The stack-chain method
is fastest but is not error-free.
We would like to acknowledge Prof. Yong Li in Zhe-
jiang Normal University, China, and Dr. Zhiling Hou
in Sichuan University, China, for their contributions.
This work was partially supported by the Singapore
Academic Research Fund Tier 1 (RG11/10). We
thank the Optics and Photonics Society of Singapore
and the Nanyang Technological University for their
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