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Residue determined threshold phase
unwrapping method
Fang, Meiqi, Zhao, Hong, Jia, Pingping
Meiqi Fang, Hong Zhao, Pingping Jia, "Residue determined threshold phase
unwrapping method," Proc. SPIE 11351, Unconventional Optical Imaging II,
113511L (30 March 2020); doi: 10.1117/12.2554971
Event: SPIE Photonics Europe, 2020, Online Only
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Residue determined threshold phase unwrapping method
Meiqi Fang a, Hong Zhao b, # and Pingping Jia c
State Key Laboratory for Manufacturing Systems Engineering, Xi'an Jiaotong University, Xi'an,
Shaanxi, 710049, P. R. China.
a
yqfang@stu.xjtu.edu.cn, bzhaohong@mail.xjtu.edu.cn, cjpp8084@163.com
#
corresponding author
Abstract. In this paper, we propose a robust phase unwrapping algorithm that can be applied to optic
interferometry based on combing the theory of residues and local phase information to mask out the
discontinuous regions in the unwrapping. Unlike previous methods, which require subjective
appraisal to determine the threshold value of the second differences in a locally unwrapped phase
empirically, this technique sets the threshold value in a straightforward way. The technique aims to
minimize the loss of phase information produced by erroneous distinguished pixels, and simplifies
the unwrapping process. Experimental results on complex discontinuous objects are presented to
illustrate the validity of this technique.
Keywords: Phase unwrapping, Phase measurement, Phase difference, Fringe analysis, Fringe
projection profilometry.
1. Introduction
Phase unwrapping is an essential technique command in various techniques that utilize phase
information to obtain desired physical quantities, such as synthetic aperture radar [1], magnetic
resonance imaging [2], digital holography [3] and fringe projection profilometry [4]. The problem is
that, since existed techniques that used to extract phase information from the measured data involve
periodic function calculations, only principal values of the phase lie in (-π, π] are returned [5]. Thus,
these phase values are termed “wrapped”, and a phase unwrapping process is required to remove
these created discontinuities and recover the true original phase.
Phase unwrapping problem is trivial when the measured phase data is obtained in an idealized
situation. However, when work with the data acquired from practical situations in the presence of
noise or genuine discontinuities, phase unwrapping problem becomes difficult and even impossible.
To solve these problems, one major effort based on determining specific paths to guide the
unwrapping process in the wrapped phase image has been investigated [5-7]. By contrast with an
alternative class of methods that estimate the phase data through fitting two-dimensional gradients
[8], this category of path following unwrapping algorithms is so called “path dependent”.
Among the path-following unwrapping methods, branch-cuts method that proposed by Goldstein
et. all [7] is the most representative one. This approach is based on the theory of residues that aims to
restrict the impact caused by noise to local isolated points or regions. Branch-cuts are constructed
between the residues with opposite charges as barriers to prevent any path crossing it during the
unwrapping process. Although this technique is known to be robust, only an approximate unwrapped
phase can be achieved. Different criterion of residue matching will lead to different branch cuts,
which are contradicted with the uniqueness of the desired physical quantity. To solve this
contradiction problem aroused by lack of additional information from the local phase, Bone [9]
proposed a more robust unwrapping technique by investigating the second differences of the locally
unwrapped phase in the wrapped phase map to mask out the regions that corrupted by noise. The
main limitation of this technique stems from the fact that threshold value of the second differences of
the local phase cannot be determined automatically during the computer calculation, it has to be
determined subjectively from the person who implemented this algorithm. More phase information
would be lost if the threshold is set too high, whereas the erroneous points would not be masked out
and may cause error propagation during the unwrapping process when a lower threshold is used.
Unconventional Optical Imaging II, edited by Corinne Fournier, Marc P. Georges, Gabriel Popescu,
Proc. of SPIE Vol. 11351, 113511L · © 2020 SPIE · CCC code: 0277-786X/20/$21 · doi: 10.1117/12.2554971
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In this paper, we proposed an automatic phase unwrapping method based on investigating the
connection between the second differences and the residues within the local wrapped phase map.
Through estimating the second differences of the residues associated points, the threshold value is
determined in automatically. One of the major advantages is that via this approach, the lost phase
information associated with the physical quantity is minimized. Meanwhile, the experimentally
determined threshold value of the local phase proposed by Bone [9] has been displaced, which give
rise to a simplified implementation process of unwrapping. We described the details principle of the
proposed method in section 2 and illustrated a set of experimental results on complex discontinuous
objects to verify this approach in section 3.
2. Algorithm
2.1 The calculation of the second differences threshold
To utilize the relationship between the second differences and the residues with a same wrapped phase
map, we could first calculate each of them simultaneously.
…
(m, n  1)
…
(m  1, n )
(m, n )
(m, n  1)
(m  1, n )
(m  1, n  1)
…
…
Fig. 1. Calculation of the residues and the second differences in an image. The solid black dots represent the detected
residues.
Generally, the phase unwrapping problem can be modeled as
 (m, n)   (m, n)  2 k (m, n),
(1)
where  (m, n) is a two-dimensional discretized phase array of a wrapped phase,  (m, n) is the
original true phase need to be recovered, and k (m, n) is an array of integers chosen so that
- < (m, n)   .
The residue, q (m, n ), can be then found by the relationship [5]
1
 2
q (m, n )  Round 
1
= Round 
 2

4
i   m, n   

i

1
4

i   m , n   



i
1
where the Round operator returns the nearest integer,
 (m, n)   (m, n)  , and i is the difference operator so that
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
(2)
is the wrapping operator satisfies
1   m, n      m, n  1    m, n  ,
 2   m, n      m  1, n  1    m, n  1 ,
3   m, n      m  1, n     m  1, n  1 ,
(3)
 4   m, n      m, n     m  1, n  .
In parallel, the second differences of the local unwrapped phase of  (m, n) could be performed by
the following relationship [9]
xx  m , n      m  1, n     m , n  
=     m , n     m  1, n  
yy  m , n      m , n  1    m , n  
=     m , n     m , n  1  ,
(4)
where  is a simple unwrapping operation to remove any 2π steps between two adjacent points, and
xx , yy are the second differences of the horizontal and vertical directions, respectively. Combining
Eqs. (3) and (4), we can derive
Threshold  Mean2  xx q  m, n +Mean2  yy q  m, n  ,
(5)
where the Mean operator returns the mean value of the summation. Referring to Fig.1, This threshold
only calculated for all the points associated with the detected residues in the field.
2.2 Unwrapping process
Once the threshold value of the second differences is determined, a bit-map mask overlaid on the
phase field can be constructed by selecting the points whose second difference exceed the determined
threshold. Then the mask bit is set to exclude that point from the unwrapping. The mask is initially
assumed to be false in the region to be unwrapped.
Because the basic process conditions have been established, one can then use an algorithm that
able to unwrap regions of arbitrary shape. So we used an algorithm that is essentially the same as that
used by Bone [9].
The algorithm is based on the flood fill algorithm and the counterpart steps required to implement
a simple phase unwrapping operation are as follows:
Step 1. Select a starting pixel from the reset (the bit of the mask that set to false) regions.
Step 2. Check each of the four neighbors of the point. If the bit in the bit map corresponding to a
neighbor that is not set, add 2k to the phase of the point, with k chosen to ensure that the
absolute phase difference from the center point (m, n) is   / 2.
Step 3. Perform Steps 1and 2 repeatedly unless all the isolated regions associated with the reset mark
are unwrapped.
After this last step, each of the identified regions in  (m, n) will be successfully unwrapped. The
remaining area of the masked out regions usually contains no useful phase information or is so
strongly aliased that cannot be recovered.
3. Experimental Results
The results of the application of our algorithm, based on the theory of Section 2, are shown in Fig 2.
Figure 2(a) shows a wrapped phase map obtained by the phase stepping algorithm applied to the
image of a fringe projection pattern projected on the surface of a plaster model of Venus.
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This wrapped phase map has noise and discontinuities because the fringe pattern is affected by
factors such as regions of low luminosity, ambient noise, and the shadows caused by the angle
between the located positions of the projector and the CCD camera.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 2. Experimental results of the proposed algorithm: (a) The wrapped phase map of the Venus plaster model, (b) (c) the
unwrapped phase results in 2D and 3D forms performed by the method [9] proposed by Bone, respectively. (d) residues
determined mask with a threshold second difference of 0.43π, (e) (f) the unwrapped phase results in 2D and 3D forms
performed by the proposed algorithm, respectively.
Figure 2(b) and (c) show the results of the phase unwrapping operation over the phase map of
Fig.2(a) performed by a Bone algorithm [9]. As can be observed, phase errors occur in this unwrapped
phase map, which is not sufficiently recovered from the original wrapped phase.
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Figures 2(d-f) shows the results of the phase unwrapping process applied to the same phase map
of Fig. 2(a), this time using the algorithm proposed in this paper.
Figure 2(d) shows the mask determined by the threshold using the Eq. (5) in Section 2. It can be
observed that they are positioned at locations where the steep gradients occurs, thus verified the
efficiency of our algorithm for detecting and localizing those points that are critical for the phase
unwrapping process. They are the minimum number of points that can cause phase information errors.
The unwrapped results are shown in Figs. 2(e) and (f), respectively. By comparing Figs. 2(e) and (f)
and 2(b) and (c), one can conclude that our algorithm enhanced the quality of the unwrapped results
when compared with a Bone algorithm.
The 3D profile determination of a plaster model represents a practical example of the potential
application of our technique.
4. Conclusion
In conclusion, the proposed algorithm has high immunity to noise and discontinuities with an
automatic determined threshold value of the second differences. This is because of our approach uses
the connection between the residues and the local phase information, always preventing the
unnecessary loss of the phase information caused by an experimentally determined threshold value.
This major advantage simplifies and facilitates the application of the proposed technique in practical
situations when accurate unwrapped phase results are required.
Funding.
The project of National Natural Science Foundation of China (No.61975161).
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