NOVEL PHASE UNWRAPPING ALGORITHM FOR RETRIEVING ESPI MAP
WITH REAL PHYSICAL DISCONTINUITY
M.J. Huang* and J.K. Liou
Professor/Chair* and his graduate student
Department of Mechanical Engineering
National Chung Hsing University
250, Kuo-Kuang Road, Taichung 40227, Taiwan
[email protected]
ABSTRACT
This paper aims to provide an effective and distortion-free approach for the phase unwrapping of digital speckle pattern
interferometry (ESPI) map with real physical discontinuities. ESPI is a powerful measurement tool for industry and research
work. However, due to the speckle noise, its unwrapping job is quite difficult, especially, when treating ESPI map obtained
from the deformation field of an object with certain degree of tear. The minimum Lp norm method propose by Ghiglia and
Romero [1] can treat the aforementioned problem with acceptable accuracy. However, the time needed is long. Therefore, a
novel method with a hybrid of regional algorithm proposed by Gierloff [2], branch cut method proposed by Goldstein et al. [3]
and parallel unwrapping method with region-referenced algorithm proposed by Huang and He [4] is presented herein. With this
newly developed algorithm, quite noisy map coupled with real physical discontinuities can also be retrieved accurately and
efficiently.
Introduction
As it has been stated above, noisy speckle noise and real physical discontinuity should be simultaneously circumvented under
treating ESPI map with real discontinuity. Quite common the case will be, while applying ESPI experimentally on samples with
certain degree of tear. To date, the most effective method for solving this kind of problem is the minimum Lp norm method
p
proposed by Ghiglia and Romero [1] in 1996. They develop an algorithm for the minimum L norm solution to the twodimensional phase unwrapping problem and show that the governing equations are equivalent to those that describe weighted
least squares phase unwrapping. They also show that the minimum Lp norm solution is obtained by embedding the transformbased methods for unweighted and weighted least squares within a simple iterative structure. The data-dependent weights are
generated within the algorithm and need not be supplied explicitly by the user. However, since it is an iterative approach, quite
much time will be needed to reach a definite accuracy of convergence.
Therefore, in this paper, an innovative algorithm for unwrapping noisy ESPI phase map with real physical discontinuity is
proposed as another solution. The proposed algorithm herein is worked on a hybrid of three main modules, including contour
allocation module, physical discontinuity boundary generating module, and regional phase unwrapping module. Combining
them leads to an efficient and successful retrieval of noisy phase map with real physical discontinuity.
A powerful simple noise-immune algorithm with parallel pixels checking [4], which is designed to bypass the speckle noise
induced inconsistencies, (basically exist in forms of either fringe interruptions or extra fringe pieces) is used to generate fringes
shifting first. Usually, by several times of fringe shifting, the 2π ambiguities of the most isolated jumps can be successfully
eliminated and transferred into a wrapped map with much clearer fringes. More details regarding fringe shifting phase
unwrapping can be found in the papers. [4,5]. Since we are going to unwrap noisy map with real physical discontinuity, thus
the physical discontinuity boundary should be further allocated and set as barriers to avoid any integration across them. After
then, regional phase unwrapping [2] can be easily applied further on and the phase retrieval job can be done with ease and
correctness. On summary, with the aid of the noise-immune algorithm [4] and image processing techniques, the breaking
points of each fringe contour can be found. Further checking each fringe’s shifting direction helps the successful physical
discontinuity barrier generation and the final regional 2π ambiguities elimination.
The Algorithm
Firstly, a noise-immune algorithm [4] is used to phase shift the contour fringes. Its mathematical descriptions are listed as
follows.
[Φ (i, j )]k = [Φ (i, j )]k −1 + 2π [h(i, j )]k ,
where
[h(i, j )]k
⎧⎪ 1,
=⎨
⎪⎩ 0,
and
[F (i, j; r , s )]k
∑ ∑ [F (i, j; r , s)]
k
r
≥ T1
s
(1a)
,
(1b)
otherwise
⎧ 1, if [Φ ( r , s )]k −1 − [Φ (i, j )]k −1 ≥ T2 .
=⎨
otherwise
⎩ 0,
(1c)
where k is the iterative number, (i,j) are the indexes of the pixel of interest, and its phase value is denoted by φ(i,j). For a
rectangular mask of size m x n and pixel (i,j) as its central pixel, its pixel’s indexes r and s run from i - (m-1)/2 to i + (m-1)/2 and
j – (n-1)/2 to j + (n-1)/2, respectively. Whether 2π phase bias is to be added to phase value of the processed pixel (i, j) or not is
decided by the total number of “flags” of the referenced window. Wheneverφ(r,s) is greater thanφ(i,j) by a threshold value of
T2, a flag is set on pixel (r,s) of the associated window of pixel (I,j). In Eq. (1b), F(i,j;r,s) stands for the flag value between pixels
(r,s) and (i,j). This (flag) value is set as one or null according to the phase differences between them (see Eq. (1c)). T1 is a
threshold for number of pixels, which should be less than the value of m by n (the size of referenced window, respectively). T2
is the adaptive threshold for phase differences between two pixels and generally, set as a fraction value between π ~ 2π. In this
study, these two parameters are set fixed for simplicity and the setting values of T1 and T2 are 8 and π, respectively, for a
square referenced window of size 5 x 5.
As it has been depicted in Eq. 1, the unwrapping algorithm will convert the transition zone [6] into a clear phase jump in the
beginning numbers of iterations, e.g., ten times. No matter they exist in the form of either transition zone or clear phase jump,
they are with the same physical meanings, i.e., the contour plots of the unwrapped phase of interest. This concept is adopted
to generate contour plots of the map. Accordingly, for the next (i.e., eleventh) iteration of the unwrapping, all unwrapping pixels
in this iteration are marked and leveled. Similar operations are implemented on the twelfth through fifteenth iteration, and
combining all the unwrapping pixels from the eleventh to fifteenth iteration into one single presentation. According to this result,
the contour distributions can thus be obtained. Moreover, both the physical discontinuity barrier generation and the final
regional phase unwrapping are also based on this result, too.
Any break point of the contour fringes is found. Connecting these break points using branch cut method will generate the
physical discontinuity barrier of the wrapped map. Provided the physical discontinuity barrier is successfully found, regional
phase unwrapping can be easily applied for the phase retrieval of the unwrapped phase distribution.
The Unwrapping details
An experimental wrapped map with real physical discontinuity obtained from electronic speckle pattern interferometry (ESPI) is
used for the demonstrations of the unwrapping process. By combining several phase stepping frames of ESPI interferograms,
the experimental phase map is obtained and is shown as Fig. 1. The wrapped phase level goes from 0 ~ 2π.
Figure 1. Experimental ESPI map with real physical discontinuity
It is very clear from Fig. 1, there exists a real phase discontinuity from the center point to the right middle point of the map.
Quite noisy the map is because of the laser speckle. Besides, all the contour fringes are distributed within a certain broad band
instead of a narrow line. In this study, the fringe shifting algorithm (formulated as Eq. 1) is used to eliminate any local
oscillation fluctuation of the wrapped phenomenon. After ten times of iteration, a wrapped map with clear phase contour “lines”
is achieved and the result is shown in Fig. 2.
Figure 2. Wrapped map after 10 times of iteration
Then, combining all the unwrapping pixels from the eleventh to fifteenth iteration into one single presentation yields the result
of Fig. 3. Each fringe consists of five iterative (i.e., the 11th ~15th, as shown in the zoom in area) results.
Figure 3. Contour fringes obtained from the eleventh to fifteenth iteration
Fringe thinning technique is further applied on each contour fringe of Fig. 3 to yield the map of Fig. 4.
Figure 4. Contour fringe thinning result
Then, any break point of contour line is further located. In this study, two break points (see Fig. 4) are found. In order to avoid
any bad integration of the unwrapping, these two points are combined into a physical discontinuity barrier, which is shown in
Fig. 5.
Figure 5. Physical discontinuity barrier generation
Fig. 4 and Fig. 5 are further combined into one single map (shown as Fig. 6) for the following regional phase unwrapping.
Figure 6. Contour lines and physical discontinuity barrier
Using the contour lines and the physical discontinuity barrier of Fig. 6 as regional boundaries divides the wrapped map into six
regions (see Fig. 7).
Figure 7. Contour lines and physical discontinuity barrier
Finally, according to Fig. 7, eliminating 2π ambiguity of the wrapped map yields the continuous phase distribution with real
physical discontinuity. Fig. 8 show the result from two different perspective views.
Figure 8. 3D plot of the unwrapped result
Experimental results
The proposed algorithm is practically applied to a wrapped map obtained from ESPI experiment for out-of-plane deformation
detection of a centrally loaded plate clamped along its left boundaries. The tested sample plate is with a break-through
horizontal tear from its center to the middle point of the right edge. Therefore, the out-of-plane deformation detection of the torn
plate is in a rotational phase distribution. The wrapped map is shown in Fig. 1 and the unwrapping process has been
demonstrated in details previously. The final result is shown in Fig. 8.
Conclusions
A novel phase unwrapping algorithm is proposed. The proposed algorithm is practically applied to a wrapped map obtained
from ESPI experiment for out-of-plane deformation detection of a centrally loaded plate clamped along its left boundary. Since
the out of plane deformation of the torn plate is in a rotational phase distribution, most phase unwrapping algorithms fail to
retrieve its phase. The experimental phase map has been proven to be able to be unwrapped by the proposed algorithm. The
hybrid approach of this study combines contour fringes marking module, physical discontinuity barrier generation module, and
regional phase unwrapping module. The unwrapping speed is fast and the unwrapped result is accurate.
Acknowledgments
This research was sponsored by the National Science Council of the Republic of China under contract Numbers NSC95-2221E-005-093.
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