Electronic Speckle Pattern Interferometry Map with Least Number of Residues
M.J. Huang* and B.S. Yun, Professor* and his Graduate Student, Mechanical Engineering,
National Chung Hsing University,
250, Kuo-Kuang Road, Taichung, Taiwan 40227, R.O.C.
ABSTRACT
There are three types of phase change calculation methods in Electronic speckle pattern interferometry (ESPI) methodology.
One is the so-called “difference of phase” phase extracting method. The second method is the so-called “phase of difference”
phase extracting method. The third method is direct phase change calculation method. In this paper, different from most of the
Electronic speckle pattern interferometry (ESPI) approaches, which involve the correlation fringe formulation followed by
speckle noise elimination (or filtering), to develop a wrapped phase map, the “phase of difference” method is adopted here to
extract phase change. However, this approach is so critical of its applying interferograms, the influence of which is dependent
on the robustness of the applied phase-shifting algorithm and the accuracy of the phase shifter. The self-marking technique
proposed by Huang and Chou in 2000 is adopted to help overcome any unfavorable conditioning, including hysteresis,
nonlinearity or plane tilting, of the pieozo-electrical transducer (PZT), to enable the successful implementation of the “phase of
difference” method. Thus, the required (i.e., five herein) phase stepping frames can be accordingly decided and their further
calculation will yield a phase map with least number of residues.
1. Introduction
Phase shifting techniques [1-4] are important in interference fringe analysis, to convert interferograms into phase map. Further
phase unwrapping can eliminate any 2π phase ambiguity of the map and thereby enable the phase to be calculated to within an
accuracy of better than λ/1000. Phase shifting interferometry (PSI) has represented a major step in the development of
interferometry. Of all the phase-shifting mechanisms, the pieozo-electrical transducer (PZT) is the easiest and most effective. It
is usually arranged in the reference arm of the interferometer to ensure precise phase stepping, such that the precision of its
movement is very important.
However, two fundamental problems are associated with the motion of PZT – its unknown sensitivity and nonlinearity. Both
problems yield a sinusoidal phase error. Several workers [5-9] have presented various algorithms to compensate for errors.
However, many other sources of error exist, such as the variation of the gradient of the environmental conditions and the
angular accuracy of the optical set-up. To build up as precise the angles of the illuminating light and the viewing CCD of a tested
sample as those that are fed into the sensitivity vector calculation is difficult practically. Thus, in 2000, Huang (one of the
authors of this article) and Chou [10] presented an automatic self-marking phase shifting system, which effectively overcame
the aforementioned problems and had been successfully verified to be useable in the measurement of the reflected wavefront
of an optical mirror. Huang et al. [11] in 2002 further applied the self-marking technique [10] into ESPI, in combination with the
“phase of difference” method [12], to mark accurately the 2π phase of the PZT phase shifter and to achieve the required
number of correlation frames. However, since each of the obtained phase-shifted frames was a subtraction between
interferograms after and before deformation, a FFT filtering rule was applied to extract the phase signal from the correlation
fringe pattern. Unavoidably, the filtered phase-shifted frames were distorted and their further calculated map was distorted too.
Therefore, in this study, the “difference of phase” method (also according to Ref. 12) is adopted to overcome the weakness of
the “phase of difference” method. In this method, a phase map is independently calculated for both the un-deformed and
deformed object. The phase change due to deformation is then calculated by subtracting the two phase maps. Its success thus
strongly depends on the stepping accuracy of the PZT phase shifter. As a simple and effective phase-stepping calibrator, the
self-marking technique can help engineers acquire the accurate phase-stepping frames for map calculations. Each incremental
phase shifted speckle pattern is captured using a CCD camera and input to a host computer to determine the root mean square
(RMS) of the grabbed interferogram. The self-marking plot specifies the complete phase shifting procedure and thus, the whole
phase shifting procedure is fully monitored and clearly presented. Accordingly, only interferograms corresponding to good
phase-stepping interval are chosen for phase map calculation. Any badly-distributed period (i.e., deviated from sinusoidal curve)
is rejected and not considered for further wrapped calculation. Hence, any ill control of the phase-stepping factors such as
pieozo-electrical transducer, environment, or electronic noise can be limited to their least extends. In this paper, a practical
ESPI work is investigated and the number of residues of the calculated map by five-step methods with and without self-marking
is 9670 and 9984, respectively. Thus, with self-marking, the phase inconsistency of the calculated wrapped phase map is as low
as possible. Consequently, the calculated phase map exhibits less phase distortion and the subsequent unwrapping jobs are
facilitated.
2. Phase-stepping ESPI
The ESPI technique [13, 14] was developed in the early 70s to produce interferometric data without applying traditional
holographic recording techniques. The premise was to use a video camera rather than a holographic plate to record a fringe
pattern with a low spatial frequency. With ESPI, measured phases were recorded electronically in form of speckle pattern.
Various methods have been developed over the last three decades for using speckle to determine the phase of a deformation.
The methods can be roughly classified into two main groups - correlation-fringe-extraction methods and
direct-phase-calculation methods. (Some divided them into three types [12].)
2.1 Correlation-fringe-extraction method
In ESPI, light from a distorted object reaches a recording plane and interferes with the reference beam to produce an
interferometric pattern. The intensity distribution is then subtracted from the original pattern to yield the ESPI correlation fringe
pattern. Mathematically, the light distribution Ibf of the original pattern is,
I bf = A1 + A2 + 2 A1 A2 cosθ s ,
2
2
(1)
where A1 and A2 are the amplitudes of the two interfering beams, and θs is the random interferometric phase of the original
speckle field, representing highly modulated speckle noise. The intensity field is photographed using a CCD camera and stored
in the frame grabber of the host computer.
After deformation, the distorted light distribution Iaf is expressed as,
I af = A1 + A2 + 2 A1 A2 cos(θ s + ∆φ ) ,
2
2
(2)
where ∆ψ is the phase change due to the distorted light.
In ESPI, the image of the undeformed test surface, stored in the frame grabber, is subtracted from the real time image of the
deformed surface at a refreshing rate of 30 frames/s. The subtracted intensity is given by,
| I af − I bf |= 4 A1 A2 | sin 12 ∆φ sin 12 ( 2θ s + ∆φ ) | ,
(3)
where sin[1/2(2θs+∆ψ)] is the random speckle phase term; sin(1/2∆ψ) is the phase change term of the distorted object, and the
absolute value operation prevents the subtraction from yielding any negative result. Equation (3) clearly shows that only the
slowly varying signal term sin(1/2∆ψ) is of interest, which is modulated by a highly varying speckle noise term sin[1/2(2θs+∆ψ)].
Various filtering rules [15-18] are therefore applied to the correlation fringes to extract the signals of interest. However,
depending on the filtering rules, more or less distortions are introduced into the calculated phase data, causing an error in the
results.
Huang et al. [11] applied the method in 2002 to study the sagging problem of CRT (cathode ray tube) panel. Fig. 1(a) illustrates
one of the grabbed phase-stepping frames in the experiment. The finally conducted phase map is also presented in Fig. 1(b).
Obviously, the calculated phase map (of module-π, see Fig. 1(b)) is distorted from its original correlation fringe pattern (see the
correlation fringes of Fig. 1(a)). The derivation is primarily owing to the filtering work, where a FFT followed by a low pass filter
and an inverse FFT are applied. Different filtering algorithms had been proposed to circumvent the aforementioned problems,
such as the correlation filter proposed by Kao et al [17] in 2002. But, no matter what filtering algorithm is applied, a certain
degree of distortion is unavoidable.
(a)
(b)
(c)
Fig.1. ESPI fringe pattern of a tested CRT specimen under loading of shrink band. (a) Subtraction-correlated fringe, (b) wrapped
map derived from filtered interferograms in frequency domain, and (c) wrapped map derived from the direct calculation. It is
shown that (b) is somehow distorted and (c) is well similar to the correlation fringes of (a).
2.2 Direct-phase-calculation method
Using a method other than the method of correlation extraction described above, K. Creath [19] took four frames of intensity
data, before and after deformation, while shifting the phase of the referenced beam in the interferometer, to find the phase of
deformation using speckle interferograms. Using these data, the phase of the deformation can be calculated from,
∆φ = [θ s + ∆φ ] − θ s ,
(4)
where [θs+∆ψ] and θs are the phases of the speckle pattern after and before deformation, respectively. Since the phases
( [θs+∆ψ] and θs ) on the right hand part of Eq. (4) are conducted independently from the grabbed phase-stepping frames after
and before deformation, respectively, each of them should be a vector sum of the scattered light field of the specimen under
corresponding state. In addition, since the tested object of ESPI generally is of scattering nature, therefore the calculated
phases ( either [θs+∆ψ] or θs ) behave like speckle and are wrap-formatted. Further subtraction yields the wrapped phase map.
The true phases of interest can then be restored if a proper phase unwrapping algorithm is applied. Therefore, neither
correlation subtraction nor filtering operation of the speckle interferograms between deformations would be needed. A direct
and clear result can be reached.
Within maps calculation, if the phase maps before and after deformation are calculated respectively, they will be wrapped into a
respective interval of [-π, π) and the subtracted phase map of the deformation will range in an interval of (-2π, 2π), which is
termed as map with random wrapping [20]. If these values are further wrapped into the range of [-π, π) by the addition or
subtraction of 2π to each value that lies below –π or above +π, respectively, the effect of random wrapping are eliminated and
the result is exactly equivalent to a direct calculation [ref12, 21]. In most applications this approach is better since it involves
less computation. With this method, eight speckle interferograms, four un-deformed and four deformed, of a tested sample with
π/2 phase steps are used. The deformation-equivalent phase change ∆ψ is formulated as
∆φ = tan −1
u
I 42d I 13u − I 13d I 24
u
I 13d I 13u + I 42d I 42
,
(5)
where Imn = Im – In; I1 ~ I4 represents the corresponding four phase-shifting buckets; and the superscript “d” and “u” represent the
deformed and the un-deformed states, respectively. The CRT-panel sagging study of Ref. 11 is also carried out by this method
and its wrapped map is shown in Fig. 1(c). Compared with the result (See Fig. 1(b)) from the correlation-fringe-extraction
method, it is shown that the discontinuities distribution of Fig. 1(c) is more similar to the correlation fringe distribution of Fig. 1(a)
than that of Fig. 1(b) is. Even that the similarity between Figs. 1(a) and 1(c) is good, but the wrapped map of Fig. 1(c) is difficult
(or even unable) to be restored. Therefore, ESPI workers or researchers in the past preferred not to follow this track to avoid
difficulties of retrieval. However, in the year 2002, Huang and his students proposed two powerful unwrapping algorithms [22,
23], which were based on a simple and innovative region-referenced criterion and had been proved to be effective (, which
means no further processing would have been needed to restore it) for wrapped maps such as the one (Fig. 1(c)) to be restored
here. Hence, provided that the calculated phase map can be successfully unwrapped, the direct-phase-calculation method is
preferred rather than the correlation-fringe-extraction method is for phase map calculations.
Nevertheless, to the best of our knowledge, to accurately shift phases of the referenced beam is a hard job for a noisy speckle
pattern here. The difficulties may result from the nonlinearity and hysterisis of the pieozo-electrical transducer (PZT), the tilting
of PZT rod during travel or the unwanted variation of environmental conditions. Any of them can cause a deviation from the ideal.
Though some algorithms, such as Carre’s algorithm [25] and the method of Ref. 26, do not require e.g., a π/2 phase step
between frames. However, they still require the phase-steps between frames to be constant. Such requirement can also conflict
in practical ESPI experiments. Hence, the results will always include errors.
3 Self-marking Phase-stepping ESPI
In 2000, Huang and Chou [10] developed an automated self-marking phase-shifting system for measuring the full-field phase
distribution of interference fringe patterns. The voltage required by the PZT to yield a 2π phase shift toward the interference
system is obtained by determining the point of the minimum RMS value of intensity difference between phase-shifted and
non-shifted interferograms vs. voltage applied. Each of the shifted interference fringe patterns associated with various amounts
of the PZT shifter is compared with the original non-shifted fringe pattern by designation the non-shifted interference fringe
pattern as the referenced fringe pattern. The intensity of the pixels that correspond to the shifted interference fringe patterns is
subtracted from that of those of the original pattern, and then the root mean square of the shifted interferogram with respect to
the original interferogram is taken to make the comparison. The (RMS, root mean square) operation can be mathematically
expressed as,
RMS = { N1
∑[I
( m , n )∈R
0
( m, n) − I k ( m, n)] 2 }1 / 2 ,
(6)
where R represents the covered region of the CCD; N is the total number of pixels of the CCD; I0 is the intensity distribution of
the original interference fringe pattern; Ik is the kth interference fringe pattern under PZT driven by k units of voltage, and m and
n are the horizontal and vertical indices of the interference fringe pattern, respectively. The proposed technique has been
successfully verified to be useable in the measurement of the reflected wavefront of an optical mirror (see Ref. 10).
4 Experiments and Results
Fig. 2 schematically depicts the experimental setup. The adopted system is an electronic speckle pattern shearing
interferometry (ESPSI) system. The shearing mechanism is the Michelson interferometer. In this system, the two interference
arms travel apart only during the period between the splitting and the combining of the beam. Accordingly, any (spatial)
environmental gradient influences the phase detection only to a very limited extent.
2
The tested sample was an aluminum plate of area 10 x 10 cm . It is fixed along its two vertical edges and loaded with a
concentrated load at the center of the plate. The illuminating light source is a He-Ne cw laser with a wavelength of 632.8 nm.
The images reflected from mirror 2 and 3 are recombined and photographed using a CCD camera (Pulnix 7X-7CN) built after
the beam combining stage. In the setup herein, the two reflected images of the tested object from mirror 2 and 3 are horizontally
shifted by an amount of 3.1 mm over the total of 10 cm width of the tested plate. The tested sample has an inherently scattering
surface so the photographs are coded with speckle noise randomly. The two sheared images of the tested sample are captured
by the CCD camera to yield the sheared interferogram. Fig. 3 presents the resulting image, which is digitized and sent through
the frame grabber card (National Instruments IMAQ PCI-1408) to a host computer for further calculation and monitoring.
Basically, as revealed by the interferogram, the noisy speckle dominates the entire image. However, regardless of the noise,
engineers in this field are to record the noisy pattern rather than the intensity themselves. Further correlating two correlated
speckle patterns of a deformation can yield the deformed message of interest.
Fig. 2. Schematic plot of the experiment (an electronic speckle pattern shearing interferometry work).
In this work, mirror 3 is attached to a computer-controlled piezoelectric transducer (PZT), Physik Instrumente model P841.60, to
perform phase stepping. The movement of the PZT is precisely controlled by the computer-driven-signal. This signal is
converted using a 16bit D/A card, National Instruments model NI PCI-MIO16XE-10, into an analog signal and then fed to the
PZT to perform precise phase stepping. A discretely increasing voltage with increments of 1 mV was fed to the PZT through the
D/A card. The shifted interferogram was grabbed by the frame grabber and sent to the host computer. The square of the RMS
value of each shifted interferogram was determined whenever a phase stepping is performed. The result was plotted, with the
square of RMS value of the shifted interferogram on the vertical axis and the voltage applied on the horizontal axis. The total
experimental interval was about two to three periods. More than one period are taken can provide intercomparison between
them. It won’t differ much in applying which period for self-marking. From our experience, the second period is easier for
identifying the required frames than others are. Accordingly, the data that corresponds to the second period in this experiment
were used to calculate the phase map.
(b)
(b)
(c)
Fig.3. Calculated maps before (a) and after (b) deformation. (c) shows their deformed phase map without random wrapping.
5 Discussion and Conclusion
Phase shifting interferometry, Fourier analysis of interferogram, and automated interferogram analysis involve the phase
retrieval of wrapped map. However, phase unwrapping amounts to integrating gradients in some fashion. The integration of
wrapped phase difference over a path encompassing residues will yield inconsistent values. Accordingly, number of residues
can be treated as a signal for distortion of the calculated map and easiness of the unwrapping work.
This study has integrated the self-marking technique, the 5-frame algorithm, and the direct-phase-calculation method to
minimize the residues of the ESPI phase map. With these hybrid techniques, the most accurate frames can be definitely
obtained and suitably fed into the five-frame algorithm to manipulate phase map. Thus, neither correlation subtraction nor
speckle noise filtering are required. The achieved phase image is therefore less distorted than others, and simplifying the
unwrapping of the phase, and yielding a better final true phase field.
The self-marking technique is a powerful and successful monitor for phase-stepping work. With its help, the whole phase
shifting procedure can be fully monitored and clearly presented. Any badly-distributed period (i.e., deviated from sinusoidal
curve) is rejected and not considered. Only those interferograms that associated with good phase-shifting periods are chosen
for further wrapped calculation. The required phase-shifted frames that correspond to 0, π/2, π, 3π/2 and 2π, respectively, can
be determined easily and accurately. Hence, any ill control of the phase-stepping factors such as pieozo-electrical transducer,
environment, or electronic noise can be limited to their least extends. The experiment shows that the number of residues of the
calculated map by five-step methods with and without self-marking is 9670 and 9984, respectively. Thus, with self-marking, the
phase inconsistency of the calculated wrapped phase map is as low as possible. Consequently, the calculated phase map
exhibits less phase distortion and the subsequent unwrapping jobs are facilitated.
6 Acknowledgments
The authors would like to thank the National Science Council of the Republic of China for financially supporting this research
under contract Numbers NSC93-2212-E-005-013.
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