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Computer aided evaluation of fringe patterns
Optics and Lasers in Engineering
Computer
19 (1993) 221-240
Aided Evaluation
Thomas BIAS-Bremer
Institut
of Fringe Patterns
M. Kreis
ftir Angewandte Strahltechnik, Bremen, Germany
Klagenfurter
Str. 2, D 28359,
ABSTRACT The crucial step in computer aided evaluation of interferometric fringe patterns is the determination of the interference phase distribution from the recorded and stored intensity pattern. Methods for determination of the interference phase distributions from fringe patterns are presented and the most important methods, which are skeletonising, temporary heterodyning, phase shifting und Fourier transform evaluation are compared with regard to experimental requirements, achievable resolution and precision, as well as inherent noise suppression and image enhancement. The comparison shows that whenever phase shifting is possible, it is the best choice. If only one interference pattern is offered, Fourier transform evaluation is recommendable. For demodulation of the wrapped phase a path-independent algorithm is presented. Practical examples are given from holographic interferometric measurements.
1 INTRODUCTION
Interferometry is a method for comparing optical wavefronts in precision metrology. Interferometric measurement methods like holographic interferometry, speckle interferometry, moire topography, etc., have found numerous applications. To utilise the full capabilities of interferometry a computer aided evaluation of the resulting interference fringes has to be performed. 221 Optics and Lasers in Engineering 0143-8166/93/$06.00 Ltd, England. Printed in Northern Ireland
0 1993 Elsevier
Science Publishers
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Thomas M. Kreis
The application of computerised methods in interferometry is manifold. Interference patterns can be calculated based on a given measurement arrangement and object form, deformation, or loading. Thus the measurement parameters can be optimised without a lot of expensive experiments. Methods of digital image processing are used to enhance the fringe patterns and suppress disturbances like speckle noise. The measured deformation fields, refractive index fields, or contours can be further processed by, e.g. finite element methods, boundary element methods, fracture mechanics methods, or tomographic methods to produce refined results. Methods employing pattern recognition, neural networks, or knowledge-based systems check for characteristic partial patterns in the fringe patterns to perform an automatic fault detection. But the central part of each quantitative interferometric measurement is the determination of the interference phase distribution from the digitally recorded and stored interference pattern. This phase distribution by the computer is then combined with geometry data describing the optical arrangement to yield the measurement result, which might be optical path differences, displacement vector fields, surface heights or refractive index distributions. Therefore the rest of this paper concentrates on the computer aided methods of deriving the interference phase from the fringe patterns. An economic interference phase determination requires a digital image processing system. The interference pattern is recorded by a TV-camera, digitised into a rectangular array of pixels and quantised into discrete grey values. In practice this intensity is affected by noise and disturbances, so it can be written in the x-y plane, the image plane of the recording optical system by +7 Y, t) = 4x,
Y) + bk
Y > cos [4(x,
Y > + 44x7
Y,
01
(1)
Here a(x, y) contains all the additive disturbances, mainly a varying background illumination and b(x, y) describes the multiplicative degradations like speckle noise and locally varying contrast. The timeconstant interference phase 4(x, y) is related to the quantity to be measured via the sensitivity vectors. A reference phase & may be produced by a frequency difference or by additionally introduced phase shifts. When determining the interference phase 4(x, y) for the intensity Z(x, y), one has to recognise the periodicity and the eveness of the cosine: the sign of the argument of the cosine function as well as additive terms, which are integer multiples of no cannot be determined from one single interference pattern.’ Each path from the left to the
Computer aided evaluation of fringe patterns
223
IZNrl8lXPI (2N+lKttPI I2N+II)XPI [2Nhl2)XP1 IZN+lOlXPI lZN+BlWPI tZN+61)CPI 12N+41XPI IZNr2l+PI IZN+OIWPI
Fig. 1.
Ambiguous
interference
phase distributions.
right through the graph of Fig. 1 represents one phase distribution belonging to the intensity distribution displayed below the graph. Side information like knowledge about the direction of loading, monotonicity, continuity, or smoothness allows us to decide a probable interference phase distribution. There are a number of methods for computer aided interference phase determination. The main methods are based on skeletonising, phase stepping or shifting, Fourier transform evaluation, and temporal heterodyning. Less frequently used are the carrier frequency or spatial heterodyning method and the phase lock method. In the following these methods will be presented and compared with regard to accuracy, resolution, image enhancement capability and inherent noise suppression. Examples are given from holographic interferometry, but the statements hold for all interferometric methods producing twodimensional fringe patterns. 2 INTERFERENCE
PHASE
DETERMINATION
Skeletonising
Fringe skeletonising is based on the former manual fringe counting. Nowadays computer algorithms search for the local brightness maxima and minima in the interference pattern.2,3 At these points the interference phase corresponds to integer multiples of z. The brightness
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Thomas M. Kreis
extrema are localised by segmentation algorithms based on adaptive thresholds, gradient operators, piecewise approximations or thinning procedures, or by fringe tracking, where from a starting point the hills or valleys through the intensity distribution are followed. This results in a skeleton of the interference pattern consisting of lines only one pixel wide. The skeleton has to be improved by removal of artefacts and connection of interrupted lines. A critical step, which in many applications can only be performed interactively, is the numbering of the lines. Side information about interference phase maxima or phase variation directions due, e.g. to known deformation directions are taken into account. The numbering assigns phase values to the bright and dark fringe centres, between these points the interference phase has to be interpolated linearly, by Bezier polynomials, by spline functions, or others. Temporal
heterodyning
In temporal heterodyning the two interfering wavefields have different frequencies.4.s The frequency separation Af is in the kHz-range. This can be accomplished by a two-frequency laser, acoustooptic modulators, rotating radial gratings or rotating birefringent media. The resulting intensity in the real image reconstruction is oscillating with the beat frequency Af, the reference phase in eqn (1) is &(x, y, t) = -47~ Aft. This signal can be resolved by a photodetector. The interesting interference phase is the phase offset of the oscillation. This phase offset one gets by comparison with a reference signal, which is produced by interference of the wavefronts with no interaction of the process to be measured. Another way is the simultaneous recording of the signals at two neighbouring points of the interference pattern. The phase difference between these two signals corresponds to the difference of the interference phase between these points. By mechanical scanning of the whole interference pattern one obtains a field of A subsequent numerical integration yields the phase differences. interference phase distribution. Phase shifting and phase stepping Temporal heterodyning requires sensors with bandwidths higher than the beat frequency A$ This bandwidth is achievable with pointdetectors, but not with the two-dimensional sensors used in digital image processing, like TV-tubes or CCD-arrays. Contrary to the continuous variation of the reference phase in temporal heterodyning,
225
Computer aided evaluation of fringe patterns
the reference phase can be varied in discrete steps and several interference patterns are recorded consecutively after each individual phase variation: rn(% Y>= 4% Y>+ WG y> cm [4(x, Y>+ hxnl, II = 1,. . . ) m, mr3
(2)
At least three interference patterns are necessary to solve this system of equations pointwisely, if the values of &n are known, for there are three unknowns a(x, y), b(x, y) and 4(x, y).“” The variation of the reference phase is performed mostly by shifting a reflecting mirror in one of the two interfering waves; it can be done by a rotating polariser, a moving diffraction grating, or a tilted glass plate. Instead of the discrete phase steps, the reference phase can be shifted linearly during recording of the interference pattern. If it is shifted from #+n - A$/2 to $+_ + A$/2 during one recording, one obtains the integrated intensities
The evaluation
of this integral results in
Z,(x, y) = a(x, y) + sine
(A41b(x, 2
Y> cm
MJ(XJ Y> + 4Rnl
(4)
This expression is equivalent to eqn (2), only the contrast term b(x, y) is modified by a constant factor. Thus phase shifting is equivalent to phase stepping, therefore the notations are used synonymously. The pointwise solution for 4(x, y) of the nonlinear system of equations, eqn (2), becomes particularly simple if the steps between consequctive reference phases are constant 30”, 60”, 90” or 120”. If the value of the phase step is unknown, provided it is constant between all pairs of consecutive recordings, one needs m = 4 interference patterns.”
4(x, y> = 4x9 Y> + m Y> cm [4(x, Y)l 4(x, y) = a@, Y> + w, Y> cos[4(x, Y> + 41 4(x, y> = 4x9 Y> + b(x, Y) cosMk Y> + 2Wl
(5)
&(x, y> = 4% Y>+ be6 Y) cos km Y>+ 3Wl A two-stage evaluation first solves for the phase step A+ A+(x, y) = arccos
Z,(x, y) - 1,(x, y> +
4(x, Y) - I,k
2[&(% Y> - 4,(x, Y>I
Y)
(6)
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Thomas M. Kreis
After discarding the outliers, which occur especially where the denominator is zero or near zero and averaging over the remaining (x, y), the resulting A+ is used for a pointwise determination of the interference phase 4(x, y) by
4(x, y) = arctan
(7)
(ZS- 12)+ (I, - I,) cos A+ + (Z, - Z,) cos 2A4~ (4 - Z3)sin A+ + (Z2- ZJ sin 2A4
or 4(x, y) = arctan
(14- ZJ + (12- ZJ cos A+ + (Z3- ZJ cos 2A4~_ A+ (Z2- Z4)sin A+ + (Z3- ZJ sin 2A4
(g)
This procedure is insensitive to distortions that affect the phase shift linearly, like vibrations or other motion of the measuring arrangement. It allows the detection of systematic errors, and inherently performs an image enhancement making profit from the redundancyP With this method the thermal deformation of a panel consisting of an aluminium sandwich core and a CFRP surface was measured. Figure 2 shows the
Fig. 2.
Four
phase-shifted
holographic interference CFRP panel.
patterns
of a thermally
loaded
Computer aided evaluation of fringe patterns
Fig. 3.
phase-shifted deformation.
Evaluated
interferograms,
Fourier transform
normal
227
deformation.
Fig. 3 displays
the evaluated
normal
evaluation
Since the Fourier transform method deals with a single interference pattern, the reference phase can be set to c#+= 0 without loss of generality. By using Eulers formula, the cosine is expressed as a complex exponential and using the definition
44
Y>
= MA
Y>
exp [j4k
~11
(9)
eqn (1) is reformulated I(x, Y>= 44
Y>+ cc% Y> + c*(x, Y>
(10)
with j denoting the imaginary unit and * indicating complex conjugation.‘“~” The discrete two-dimensional Fourier transform applied to Z(x, y) yields ,a(U, u) = &(U, u) + %(U, u) + %“(U, u) (11) Since Z(x, y) is a real distribution in the spatial domain, $(u, u) is a Hermitean distribution in the spatial frequency domain, which means the real part of 9(u, u) is even and the imaginary part is odd. This shows as an amplitude spectrum that is point-symmetric to the
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Thomas M. Kreis
zero-frequency (u, u) = (0, 0). &I( u, v ) contains the dc-term 9(0,0) and the low frequency variations of the background. The mutually pointsymmetric parts %‘(u, u) and %*(u, V) carry the same information. By bandpass-filtering in the spatial frequency domain, d(u, u) and one of %(u, u) or %*(u, u) are eliminated.‘* The remaining spectrum, %‘*(u, u) or %(u, u) respectively, is no longer Hermitean, so the inverse Fourier-transform applied to %(u, u) gives a complex c(x, y) with non-vanishing real and imaginary parts. By
the interference phase is calculated pointwisely with values between -X and + n, the principal values of the arctangent. The inverse transform of %*(u, u) instead of %?(u, u) would have resulted in -4(x, y). Since it is not known whether %*(u, u) or %(u, u) was eliminated, the correct sign remains unknown. The Fourier transform method can be interpreted as a least squares fit of a linear combination of harmonic functions to the interference pattern, where the set of functions differing in their spatial frequency is defined by the bandpass filter in the spatial frequency domain. The capability of the Fourier transform evaluation is demonstrated in the following example of a plate loaded thermally in a microwave-oven, Fig. 4. The holographic interferogram of the plate is seen in Fig. 4(a). Overlayed is the metallic grid of the window shielding the environment from microwave radiation. The reordered amplitude spectrum,” with the d.c.-term in the centre of the display is shown in Fig. 4(b). Two filters with orthogonally oriented pass-bands are applied in the spatial frequency domain. The resulting interference phase distributions modulo 27~ are given in Figs 4(c) and 4(d). Artificial phases are given to the pixels where no interference was present, these regions are masked out for further processing. An erroneous change in the sign occurs where the direction of the phase should go over from increasing to decreasing or uice uersa, but is not allowed to do so due to the filter definition. This happens only in the horizontal direction, Fig. 4(c), or in the vertical direction, Fig. 4(d), depending on the filter orientation. So regions can be defined where no sign change occurs. These regions have been chosen from both phase distributions to fill the whole image and to provide smooth continuation across the region borders, when taking the absolute values of the phases. Then the sign in each region has to be set in such a way that consistency in the image is reached. A typical division into regions is shown in Fig. 4(e) together with the proposed signs. After this
229
Computer aided evaluation of fringe patterns
(4
(e)
Fig. 4.
Fourier
transform
evaluation
of a plate heated
in a microwave
oven.
230
Thomas M. Kreis
procedure the interference phase distribution is at least consistent in the sign, in most cases it is exact up to a possible global sign change. This sign corrected interference phase distribution, Fig. 4(f), now can be demodulated. The result in perspective display is shown in Fig. 4(g). If only one interferogram of the form of eqn (1) is evaluated, there always remains a sign-ambiguity, at least a global one, which means all phases have the correct or all phases have the wrong sign. An exact determination of the sign-distribution is achieved if an additional phase shifted interferogram is produced with a mutual phase shift &.I3 Theoretically, C& must be in the range 0 < & < n, but in practice values 7c/3 < & < 2n/3 are recommendable. If this condition is fulfilled, the exact value does not need to be known. The phase shifting complicates the experiment, but it is still easier than with the phase shift methods where several constant phase shifts have to be provided. Now there are two intensity distributions. Ux, Y>= a,
Y>+ w, Y>cm Ma Y)l L(x, Y>= 4& y> + a> Y>cm [4(x, Y>+ &I Fourier-transform processing filter parameters yields
of each intensity with the same bandpass
c&, Y>= W,
46
(13)
Y) exp [Mx, ~11 Y>= Nx, Y) exp [i$G Y>+1’4&, ~11
(14)
From eqns (14), the C&(X,y) is calculated pointwisely by &(x, y) = arctan
Re G(X, Y>Im 4.~ y>- Im c&, Y>Re 4x, Y>
Re c&, y>Re 4x, y>+ Im cdx, Y>Im cdx, Y)
(15)
Of this &(x, y), only the sign distribution is of further interest, it is used for determination of the sign-correct interference phase distribution via
Im cdx, Y)
4(x9Y> = sign w&7 Y)l arctan Re
c
(x
1
y)
(16)
2
Figures S(a) and 5(b) sh ow two phase shifted holographic interferograms, the sign of C&(X,y) as calculated by eqn (15) is given in Fig. 5(c). Figure 5(d) displays the phase distribution modulo 27r and Fig. 5(e) shows the sign-corrected phase distribution modulo 27r, calculated by eqn (16). The demodulated interference phase distribution is given in a grey scale display in Fig. 5(f) and in a perspective plot in Fig. 5(g). An improvement of the method is possible by previous recording of the illuminated background, which corresponds to a@, y) in the spatial
Computer aided evaluation of fringe patterns
(4
(e)
(iit) Fig. 5.
Fourier
transform
evaluation
of phase-shifted
interferograms.
231
232
Thomas M. Kreis
domain. For filtering, a normalised version of the transform this background is subtracted from 4i(u, u). I4 Further methods
for interference
&(u, u) of
phase determination
The aforementioned Fourier transform method introduces a sign change in the interference phase distributions, if %‘(u, u) and %‘*(u, V) superpose in the spatial frequency domain. These two components are separated if an additional carrier frequency is introduced, which can be done in a Michelson-interferometer by tilting a mirror.“,” This introduces additional equidistant straight fringes of spatial frequency j, which act as spatial heterodyne carrier. The known carrier fi, is then eliminated by shifting the filtered spectral components by j, in the spatial frequency domain. The main disadvantage of this method is that the condition of equidistant carrier fringes is not fulfilled if the sensitivity vectors of the applied interferometric method vary spatially as they usually do in holographic interferometry. In another version of the spatial carrier method the recorded interference pattern is multiplied pointwisely by cos (2nj)x) and sin (27&x). I7 If the difference between 4(x, y) and 2&x is sufficiently small, one obtains low frequency components M,(x, Y) = ; cos [4(x,
y> -
27&4
and
(17) M,(x, Y > = - 5 sin C#&
which are isolated calculated by
Y>-
27Gl
by a low pass filter. The interference
4(x, y) = 2x$,x + arctan
-M,(x, M,(x,
phase
Y) Y>
then
is
(18)
This version is only feasible if the interference pattern has fringes of nearly equal inclination and density. The phase-lock method uses a sinusoidal phase modulation by, e.g. a piezoelectrically excited axially oscillating mirror.lX The intensity is Z(x, y, t) = a(x, y) + b(x, y) cos [4(x, y) + L sin ot]
(19)
L < h/2 is the amplitude and Y = w/2n the frequency of this oscillation. A bandpass filter, centred to sin wt, determines the amplitude U, = 2b(x, y)J,(L) sin 4(x, y), which is zero at the points (x, y), where 4(x, y) = NZ These points thus can be detected and give a skeleton
Computer aided evaluation of fringe patterns
233
whose lines correspond to interference phase differences of 7rl2. The further processing of these skeletons is as with the fringe tracking method. The electrooptic holography method combines the advantages of phase stepping and electronic speckle pattern interferometry to perform an image enhancement in holographic interferometry for deformation and vibration measurements.‘“*20
3 COMPARISON OF INTERFERENCE DETERMINATION METHODS
PHASE
Given a problem to be solved by computer aided interferometry, one has to choose the appropriate method for automatic phase determination. In the following, the main methods of Section 2 are compared with regard to several criteria when applied to holographic interferometry. The results of this comparison are summarised in Table 1. Phase
Number of interferograms to be recordedResolution (in A) Evaluation between intensity extrema Inherent noise suppression Automatic sign detection Necessary experimental manipulation Experimental requirements Sensitivity to external influences Interaction by the operator Duration of evaluation
TABLE 1 Determination Methods
Fringe tracking
Phase stepping and shifting
Fourier transform evaluation
Temporalheterodyning
1
3 or 4, rarely 25
1 (2)
One per detection point
l-1/10
l/10-l/100
l/10-1/30
l/100-l/2000
No
Yes
Yes
Yes
Partially
Yes
Yes
Partially
No
Yes
No (yes)
Yes
No
Phase
shift
No
(phase
Frequency
shift
shift)
Low
High
Low
Extremely
high
Low
Moderate
Low
Extremely
high
Possible
Not possible
Possible
High
Low
High
Not possible Extremely
high
234
Experimental
Thomas M. Kreis
requirements
The least requirements of the holographic arrangement are demanded when using fringe skeletonising or Fourier transform evaluation, since both methods evaluate a single interference pattern. These methods even allow the evaluation of interferograms given as paper photograph, negative, diapositive, computerfile, etc. But then there remains the sign ambiguity, which has to be solved with the help of side information. This increases the necessary computational effort. The Fourier transform method with one additional phase-shifted pattern to solve for the exact sign requires a phase shift capability in the experimental set-up. This capability is also needed by the phase shift methods, but with the additional requirement of multiple constant phase shifts. The procedures with three patterns offer an easy computation, on the other hand the phase shift must be controlled exactly to a prescribed value. Due to the two reference arms in the interferometer, between which the mutual phase shift is performed, the mechanical stability of the experimental arrangement must be high. Temporal heterodyning, which also needs two reference arms for performing the mutual frequency shift, demands the highest experimental effort. The mechanical scanning is relatively slow, so the long-term stability of the arrangement must be to fractions of the wavelength. Mechanical vibrations, temporal fluctuations and variations of the refractive index of surrounding air have to be excluded. Resolution
and precision
The spatial resolution is defined as the distance between the detection points. For skeletonising this is given by the number of fringes in the interference pattern. Phase shifting and Fourier transform evaluation offer uniform sampling with a spatial resolution only dependent on the detector array, but independent of the fringe pattern. The resolution can be changed with regard to the pattern by varying the magnification of the recording optics. The spatial resolution of the heterodyne method depends on the distance between the detectors and the step width of the mechanical scanning. Changing the size of the real image optically may alter the spatial resolution. The precision of the measured value normally is given in fractions of to an interference phase the wavelength A, which corresponds difference of 2n. The best precision to be reached by skeletonising is A/10, providing a high quality pattern with a reasonable number of fringes and extensive interpolation.
Computer aided evaluation of fringe patterns
235
The Fourier transform evaluation obtains an accuracy and resolution of h/20 at all but the outermost pixel in the marginal five to eight columns and rows of the digitised pattern. For well fitting cutoff filter even better resolutions are frequencies of the bandpass achievable. Phase shifting yields resolutions in the range of A/100. The electronic phase difference measurement of temporal heterodyning enables a resolution of h/500 and better. But these figures are only achievable if all parameters of the process, especially the step width of the detectors and the long-term stability, are kept to the same precision, which is only possible in ideal laboratory environments. Errors and distortions
Skeletonising and Fourier transform evaluation process one single interferogram, and for that reason they are relatively insensitive to external distortions. On the other hand using a single interferogram does not allow detection of whether the interference phase is increasing or decreasing. A change of the direction of the interference phase is recognised only in special cases if closed ring structures exist within the pattern. This change, as well as increasing or decreasing of the phase, always can be detected by phase shifting and temporal heterodyning. These two methods do not exhibit the aforementioned systematic sign error. The sign error is accompanied by a general inaccessability of the additive multiple of 2~. All methods evaluate the interference phase only modulo 27c, which requires a demodulation to correct for the 2x-steps. Thus normally only phase differences between different points are measured, not the absolute interference phase at a single point. Phase shifting is sensitive to non-constant phase shifts and therefore to all distortions which influence the phase shifts. In this regard the procedure with four recordings and an unknown, but constant phase shift is advantageous, because it inherently corrects for additional linear phase shifts. Temporal heterodyning is extremely sensitive to environmental distortions, as was pointed out in Section 2. 4 PATH-DEPENDENT AND PATH-INDEPENDENT DEMODULATION Demodulation in one-dimensional interference phase distributions 4(x) is done by checking differences between adjacent pixels
236
Thomas M. Kreis
4(x + 1) - 4(x). If this difference is less than -rc, 2n is added to 4 from x + 1 on; if the difference is greater than +z, 27~ is subtracted from C$ starting at pixel x + 1. Several of these 2x-terms may accumulate to integer multiples of 27~. If due to noise a wrong difference occurs, the resulting phase error expands up to the outermost pixel (if it is not neutralised by another error in the opposite direction). This one-dimensional demodulation procedure can be transferred to two dimensions along lines and columns of pixels.‘” The result is a path-dependent demodulation. Besides the possible expansion of erroneous phase, path-dependent demodulation may fail with images of complex forms, e.g. containing holes with no interference phase being defined. at all. To circumvent these difficulties, path-independent demodulation procedures are recommended. The following algorithm interpretes the interference phase image modulo 27~ as a graph, where the pixels are the nodes and the arcs are the connections between neighbouring pixels. 4-neighbourhoods or Sneighbourhoods may be used. With each arc a value d,,(p,, p2) is associated, defined by the phase values p, and pz of the two pixels it connects.
&&lyp2) = min{IP,
-A,
Ip, -p2
+ 24,
IpI -p2
-WI
(20)
The values d21rmay be interpreted as a distance modulo 27~ The demodulation now proceeds along paths where these distances are least. Along these paths the probability for an erroneous phase is least. Pixels with wrong phase are surrounded this way, the same is true for regions without interference phase at all. If a pixel possesses an erroneous phase, that cannot be reached correctly along any path, nevertheless the incorrect demodulated pixel remains isolated in the finally resulting interference phase distribution. The algorithm proceeds as follows: 1. 2.
3.
For a starting pixel all emanating arcs are recorded in a list together with their values dxn. The minimal value in the list is searched for. The demodulation term 0, -2n, or +27r for this arc is stored in an extra file. The arc together with its value is discarded from the list and marked to avoid repeated consideration. The final node of the just considered arc acts as a new starting pixel in Step 1. Only these arcs are considered as free, which have not been stored formerly in the extra file in Step 2. If no free arcs emanate from this node, proceed with Step 2 directly.
Computer aided evaluation of fringe patterns
4.
5. 6.
237
If the capacity of the list is exhausted, the list is checked for arcs which have already entered the extra file along another path. These arcs are deleted. Steps 1 to 4 are repeated until all pixels have been an end node of an arc in the extra file. The interference phase distribution is demodulated using the values in the extra file.
A modification of this algorithm, which shortens the computational effort, allows only arcs with values less than a prescribed threshold to be recorded in the list. Thus the number of comparisons for searching the minimum in Step 2 is drastically reduced. This demodulation procedure is demonstrated in the numerical example of Fig. 6. Figure 6(a) shows the phase values of the pixels of an interference phase distribution modulo 27~ in the squares. At the lines between the squares the distances, calculated by dzn, are given. Figure 6(b) shows how the path-independent algorithm detects the best track and circumvents the one bad spot. Figure 6(c) displays the unwrapped phase resulting from demodulation along predetermined horizontal paths and Fig. 6(d) gives the result after path-independent phase demodulation. The result of this demodulation procedure applied to the signcorrected interference phase distribution of Fig. 5(e) is shown in Figs 5(f) and 5(g). 5 CONCLUSIONS At first glance, phase shifting seems to yield less precision than temporal heterodyning. But the precision of phase shifting is valid even for phase differences of points far apart in the interference pattern. This is due to the simulataneous recording of the intensity at all points. In heterodyning small drifts during the evaluation may accumulate and reduce the precision of phase differences between distant points. Since the most important criteria in metrology are accuracy and precision, phase shifting should be chosen if possible. If no phase shifts or multiple interferograms can be recorded, e.g. when using a pulsed laser or using holographic recording in photorefractive crystals which destroy the information during readout, the Fourier transform evaluation is recommended. The conceptual difference between phase shifting and Fourier transform evaluation is the automatic evaluation by phase shifting without interference by the operator, while the Fourier transform method is a tool that allows manifold manipulations with the interference pattern by the user.
238
Thomas M. Kreis
(4
'phase
values
mod
2n
phbse
differences
mod
2n
@I
after
steps
2
after
1 step
after
6
steps
after
7
orter
9
steps
after
10
steps
steps
(c) )I Path-dependent (horizontal
(4
demodulation direction)
3.9
L.l
5.5
6.2
6.1
7.0
7.9
8.8
9.L
3.2
3.8
L.6
5.1
6.1
19.91
7.1
8.1
8.9
2.5
3.1
3.9
L.6
5.6
6.2
6.7
J.5
8.4
Path-independent Fig. 6.
Path-dependent
demodulation and path-independent
demodulation.
Computer aided evaluation of fringe patterns
239
ACKNOWLEDGEMENT The research work reported here was supported partially by the DFG, Deutsche Forschungsgemeinschaft, under grant Kr 953/2-l, which is gratefully acknowledged.
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12.
13.
evaluation of interference patterns. In Industrial Optoelectronic Measurement Systems using Coherent Light, ed. W. F. Fagan, Proc. Sot. Photo-Opt. Instr. Eng., 863 (1987) 68-77. Osten, W., Hofling, R. & Saedler, J., Two computer-aided methods for data reduction from interferograms. In Industrial Optoelectronic Measurement Systems using Coherent Light, ed. W. F. Fagan, Proc. Sot. PhotoOpt. Znstr. Eng., 863 (1987) 105-13. Eichhorn, N. & Osten, W., An algorithm for the fast derivation of line structures from interferograms. J. Mod. Opt., 35(10) (1988) 1717-25. Dandliker, R., Ineichen, B. & Mottier, F. M., High resolution hologram interferometry by electronic phase measurement. Opt. Comm., 9 (1973) 412-16. Dandliker, R. & Thalmann, R., Heterodyne and quasi-heterodyne holographic interferometry. Opt. Eng., 24(5’) (198.5) 824-31. Bruning, J. H., Herriott, D. R., Gallagher, J. E., Rosenfeld, D. P., White, A. D. & Brangaccio, D. J., Digital wavefront measuring interferometer for testing optical surfaces and lenses. Appl. Opt., 13(11) (1974) 2693-2703. Chang, M., Hu, Ch.-P., Lam, P. & Wyant, J. C., High precision deformation measurement by digital phase shifting interferometry. Appl. opt., 24(22) (1985) 3780-3. Creath, K., Phase-measurement interferometry. In Progress in Optics, ed. E. Wolf, 25 (1988) 349-93. Jtiptner, W., Kreis, Th. & Kreitlow, H., Automatic evaluation of holographic interferograms by reference beam phase shifting. In Industrial Applications of Laser Technology, ed. W. F. Fagan, Proc. Sot. Photo-Opt. Instr. Eng., 398 (1983) 22-9. Kreis, Th., Digital holographic interference-phase measurement using the Fourier-transform method. J. Opt. Sot. Amer. A, 3(6) (1986) 847-55. Kreis, Th., Fourier-transform evaluation of holographic interference patterns. In Int. Conf on Photomech. and Speckle Metrol., ed. F.-P. Chiang, Proc. Sot. Photo-Opt. Instr. Eng., 814 (1987) 365-71. Kreis, Th. & Jtiptner, W., Fourier-transform evaluation of interference patterns: the role of filtering in the spatial-frequency domain. In Laser Interferometry: Quantitative Analysis of Interferograms, ed. R. J. Pryputniewicz, Proc. Sot. Photo-Opt. Instr. Eng., 1162 (1989) 116-25. Kreis, Th. & Jtiptner, W., Fourier-transform evaluation of interference patterns: demodulation and sign ambiguity. In Laser Znterferometry IV: Computer-Aided Interferometry, ed. R. J. Pryputniewicz, Proc. Sot. Photo-Opt. Instr. Eng., 1553 (1991) 263-73.
Kreis, Th., Quantitative
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14. Kreis, Th., Automatic evaluation of interference patterns. In Holography Techniques and Applications, ed. W. Jiiptner, Proc. Sot. Photo-Opt. Instr. Eng., 1026 (1988) 80-9. 15. Takeda, M., Ina, H. & Kobayashi, S., Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry. J. Opt., Sot. Amer., 72(l) (1982) 156-60. 16. Takeda, M., Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: An overview. Industr. Metrol., l(2) (1990) 79-99. 17. Womack, K.-H., Interferometric phase measurement using spatial synchronuous detection. Opt. Eng., 23(4) (1984) 391-5. 18. Johnson, G. W., Leiner, D. C. & Moore, D. T., Phase-locked interferometry. Opt. Eng., 18(l) (1979) 46-52. 19. Stetson, K. A. & Brohinsky, W. R., Electra-optic holography system for vibration analysis and nondestructive testing. Opt. Eng., 26(12) (1987) 1234-9. 20. Pryputniewicz, R. J. & Stetson, K. A., Measurement of vibration patterns using electro-optic holography. In Laser Interferometry: Quantitative Analysis of Interferograms, ed. R. J. Pryputniewicz, Proc. Sot. Photo-Opt. Instr. Eng., 1162 (1989) 456-67.
Computer
19 (1993) 221-240
Aided Evaluation
Thomas BIAS-Bremer
Institut
of Fringe Patterns
M. Kreis
ftir Angewandte Strahltechnik, Bremen, Germany
Klagenfurter
Str. 2, D 28359,
ABSTRACT The crucial step in computer aided evaluation of interferometric fringe patterns is the determination of the interference phase distribution from the recorded and stored intensity pattern. Methods for determination of the interference phase distributions from fringe patterns are presented and the most important methods, which are skeletonising, temporary heterodyning, phase shifting und Fourier transform evaluation are compared with regard to experimental requirements, achievable resolution and precision, as well as inherent noise suppression and image enhancement. The comparison shows that whenever phase shifting is possible, it is the best choice. If only one interference pattern is offered, Fourier transform evaluation is recommendable. For demodulation of the wrapped phase a path-independent algorithm is presented. Practical examples are given from holographic interferometric measurements.
1 INTRODUCTION
Interferometry is a method for comparing optical wavefronts in precision metrology. Interferometric measurement methods like holographic interferometry, speckle interferometry, moire topography, etc., have found numerous applications. To utilise the full capabilities of interferometry a computer aided evaluation of the resulting interference fringes has to be performed. 221 Optics and Lasers in Engineering 0143-8166/93/$06.00 Ltd, England. Printed in Northern Ireland
0 1993 Elsevier
Science Publishers
222
Thomas M. Kreis
The application of computerised methods in interferometry is manifold. Interference patterns can be calculated based on a given measurement arrangement and object form, deformation, or loading. Thus the measurement parameters can be optimised without a lot of expensive experiments. Methods of digital image processing are used to enhance the fringe patterns and suppress disturbances like speckle noise. The measured deformation fields, refractive index fields, or contours can be further processed by, e.g. finite element methods, boundary element methods, fracture mechanics methods, or tomographic methods to produce refined results. Methods employing pattern recognition, neural networks, or knowledge-based systems check for characteristic partial patterns in the fringe patterns to perform an automatic fault detection. But the central part of each quantitative interferometric measurement is the determination of the interference phase distribution from the digitally recorded and stored interference pattern. This phase distribution by the computer is then combined with geometry data describing the optical arrangement to yield the measurement result, which might be optical path differences, displacement vector fields, surface heights or refractive index distributions. Therefore the rest of this paper concentrates on the computer aided methods of deriving the interference phase from the fringe patterns. An economic interference phase determination requires a digital image processing system. The interference pattern is recorded by a TV-camera, digitised into a rectangular array of pixels and quantised into discrete grey values. In practice this intensity is affected by noise and disturbances, so it can be written in the x-y plane, the image plane of the recording optical system by +7 Y, t) = 4x,
Y) + bk
Y > cos [4(x,
Y > + 44x7
Y,
01
(1)
Here a(x, y) contains all the additive disturbances, mainly a varying background illumination and b(x, y) describes the multiplicative degradations like speckle noise and locally varying contrast. The timeconstant interference phase 4(x, y) is related to the quantity to be measured via the sensitivity vectors. A reference phase & may be produced by a frequency difference or by additionally introduced phase shifts. When determining the interference phase 4(x, y) for the intensity Z(x, y), one has to recognise the periodicity and the eveness of the cosine: the sign of the argument of the cosine function as well as additive terms, which are integer multiples of no cannot be determined from one single interference pattern.’ Each path from the left to the
Computer aided evaluation of fringe patterns
223
IZNrl8lXPI (2N+lKttPI I2N+II)XPI [2Nhl2)XP1 IZN+lOlXPI lZN+BlWPI tZN+61)CPI 12N+41XPI IZNr2l+PI IZN+OIWPI
Fig. 1.
Ambiguous
interference
phase distributions.
right through the graph of Fig. 1 represents one phase distribution belonging to the intensity distribution displayed below the graph. Side information like knowledge about the direction of loading, monotonicity, continuity, or smoothness allows us to decide a probable interference phase distribution. There are a number of methods for computer aided interference phase determination. The main methods are based on skeletonising, phase stepping or shifting, Fourier transform evaluation, and temporal heterodyning. Less frequently used are the carrier frequency or spatial heterodyning method and the phase lock method. In the following these methods will be presented and compared with regard to accuracy, resolution, image enhancement capability and inherent noise suppression. Examples are given from holographic interferometry, but the statements hold for all interferometric methods producing twodimensional fringe patterns. 2 INTERFERENCE
PHASE
DETERMINATION
Skeletonising
Fringe skeletonising is based on the former manual fringe counting. Nowadays computer algorithms search for the local brightness maxima and minima in the interference pattern.2,3 At these points the interference phase corresponds to integer multiples of z. The brightness
224
Thomas M. Kreis
extrema are localised by segmentation algorithms based on adaptive thresholds, gradient operators, piecewise approximations or thinning procedures, or by fringe tracking, where from a starting point the hills or valleys through the intensity distribution are followed. This results in a skeleton of the interference pattern consisting of lines only one pixel wide. The skeleton has to be improved by removal of artefacts and connection of interrupted lines. A critical step, which in many applications can only be performed interactively, is the numbering of the lines. Side information about interference phase maxima or phase variation directions due, e.g. to known deformation directions are taken into account. The numbering assigns phase values to the bright and dark fringe centres, between these points the interference phase has to be interpolated linearly, by Bezier polynomials, by spline functions, or others. Temporal
heterodyning
In temporal heterodyning the two interfering wavefields have different frequencies.4.s The frequency separation Af is in the kHz-range. This can be accomplished by a two-frequency laser, acoustooptic modulators, rotating radial gratings or rotating birefringent media. The resulting intensity in the real image reconstruction is oscillating with the beat frequency Af, the reference phase in eqn (1) is &(x, y, t) = -47~ Aft. This signal can be resolved by a photodetector. The interesting interference phase is the phase offset of the oscillation. This phase offset one gets by comparison with a reference signal, which is produced by interference of the wavefronts with no interaction of the process to be measured. Another way is the simultaneous recording of the signals at two neighbouring points of the interference pattern. The phase difference between these two signals corresponds to the difference of the interference phase between these points. By mechanical scanning of the whole interference pattern one obtains a field of A subsequent numerical integration yields the phase differences. interference phase distribution. Phase shifting and phase stepping Temporal heterodyning requires sensors with bandwidths higher than the beat frequency A$ This bandwidth is achievable with pointdetectors, but not with the two-dimensional sensors used in digital image processing, like TV-tubes or CCD-arrays. Contrary to the continuous variation of the reference phase in temporal heterodyning,
225
Computer aided evaluation of fringe patterns
the reference phase can be varied in discrete steps and several interference patterns are recorded consecutively after each individual phase variation: rn(% Y>= 4% Y>+ WG y> cm [4(x, Y>+ hxnl, II = 1,. . . ) m, mr3
(2)
At least three interference patterns are necessary to solve this system of equations pointwisely, if the values of &n are known, for there are three unknowns a(x, y), b(x, y) and 4(x, y).“” The variation of the reference phase is performed mostly by shifting a reflecting mirror in one of the two interfering waves; it can be done by a rotating polariser, a moving diffraction grating, or a tilted glass plate. Instead of the discrete phase steps, the reference phase can be shifted linearly during recording of the interference pattern. If it is shifted from #+n - A$/2 to $+_ + A$/2 during one recording, one obtains the integrated intensities
The evaluation
of this integral results in
Z,(x, y) = a(x, y) + sine
(A41b(x, 2
Y> cm
MJ(XJ Y> + 4Rnl
(4)
This expression is equivalent to eqn (2), only the contrast term b(x, y) is modified by a constant factor. Thus phase shifting is equivalent to phase stepping, therefore the notations are used synonymously. The pointwise solution for 4(x, y) of the nonlinear system of equations, eqn (2), becomes particularly simple if the steps between consequctive reference phases are constant 30”, 60”, 90” or 120”. If the value of the phase step is unknown, provided it is constant between all pairs of consecutive recordings, one needs m = 4 interference patterns.”
4(x, y> = 4x9 Y> + m Y> cm [4(x, Y)l 4(x, y) = a@, Y> + w, Y> cos[4(x, Y> + 41 4(x, y> = 4x9 Y> + b(x, Y) cosMk Y> + 2Wl
(5)
&(x, y> = 4% Y>+ be6 Y) cos km Y>+ 3Wl A two-stage evaluation first solves for the phase step A+ A+(x, y) = arccos
Z,(x, y) - 1,(x, y> +
4(x, Y) - I,k
2[&(% Y> - 4,(x, Y>I
Y)
(6)
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Thomas M. Kreis
After discarding the outliers, which occur especially where the denominator is zero or near zero and averaging over the remaining (x, y), the resulting A+ is used for a pointwise determination of the interference phase 4(x, y) by
4(x, y) = arctan
(7)
(ZS- 12)+ (I, - I,) cos A+ + (Z, - Z,) cos 2A4~ (4 - Z3)sin A+ + (Z2- ZJ sin 2A4
or 4(x, y) = arctan
(14- ZJ + (12- ZJ cos A+ + (Z3- ZJ cos 2A4~_ A+ (Z2- Z4)sin A+ + (Z3- ZJ sin 2A4
(g)
This procedure is insensitive to distortions that affect the phase shift linearly, like vibrations or other motion of the measuring arrangement. It allows the detection of systematic errors, and inherently performs an image enhancement making profit from the redundancyP With this method the thermal deformation of a panel consisting of an aluminium sandwich core and a CFRP surface was measured. Figure 2 shows the
Fig. 2.
Four
phase-shifted
holographic interference CFRP panel.
patterns
of a thermally
loaded
Computer aided evaluation of fringe patterns
Fig. 3.
phase-shifted deformation.
Evaluated
interferograms,
Fourier transform
normal
227
deformation.
Fig. 3 displays
the evaluated
normal
evaluation
Since the Fourier transform method deals with a single interference pattern, the reference phase can be set to c#+= 0 without loss of generality. By using Eulers formula, the cosine is expressed as a complex exponential and using the definition
44
Y>
= MA
Y>
exp [j4k
~11
(9)
eqn (1) is reformulated I(x, Y>= 44
Y>+ cc% Y> + c*(x, Y>
(10)
with j denoting the imaginary unit and * indicating complex conjugation.‘“~” The discrete two-dimensional Fourier transform applied to Z(x, y) yields ,a(U, u) = &(U, u) + %(U, u) + %“(U, u) (11) Since Z(x, y) is a real distribution in the spatial domain, $(u, u) is a Hermitean distribution in the spatial frequency domain, which means the real part of 9(u, u) is even and the imaginary part is odd. This shows as an amplitude spectrum that is point-symmetric to the
22x
Thomas M. Kreis
zero-frequency (u, u) = (0, 0). &I( u, v ) contains the dc-term 9(0,0) and the low frequency variations of the background. The mutually pointsymmetric parts %‘(u, u) and %*(u, V) carry the same information. By bandpass-filtering in the spatial frequency domain, d(u, u) and one of %(u, u) or %*(u, u) are eliminated.‘* The remaining spectrum, %‘*(u, u) or %(u, u) respectively, is no longer Hermitean, so the inverse Fourier-transform applied to %(u, u) gives a complex c(x, y) with non-vanishing real and imaginary parts. By
the interference phase is calculated pointwisely with values between -X and + n, the principal values of the arctangent. The inverse transform of %*(u, u) instead of %?(u, u) would have resulted in -4(x, y). Since it is not known whether %*(u, u) or %(u, u) was eliminated, the correct sign remains unknown. The Fourier transform method can be interpreted as a least squares fit of a linear combination of harmonic functions to the interference pattern, where the set of functions differing in their spatial frequency is defined by the bandpass filter in the spatial frequency domain. The capability of the Fourier transform evaluation is demonstrated in the following example of a plate loaded thermally in a microwave-oven, Fig. 4. The holographic interferogram of the plate is seen in Fig. 4(a). Overlayed is the metallic grid of the window shielding the environment from microwave radiation. The reordered amplitude spectrum,” with the d.c.-term in the centre of the display is shown in Fig. 4(b). Two filters with orthogonally oriented pass-bands are applied in the spatial frequency domain. The resulting interference phase distributions modulo 27~ are given in Figs 4(c) and 4(d). Artificial phases are given to the pixels where no interference was present, these regions are masked out for further processing. An erroneous change in the sign occurs where the direction of the phase should go over from increasing to decreasing or uice uersa, but is not allowed to do so due to the filter definition. This happens only in the horizontal direction, Fig. 4(c), or in the vertical direction, Fig. 4(d), depending on the filter orientation. So regions can be defined where no sign change occurs. These regions have been chosen from both phase distributions to fill the whole image and to provide smooth continuation across the region borders, when taking the absolute values of the phases. Then the sign in each region has to be set in such a way that consistency in the image is reached. A typical division into regions is shown in Fig. 4(e) together with the proposed signs. After this
229
Computer aided evaluation of fringe patterns
(4
(e)
Fig. 4.
Fourier
transform
evaluation
of a plate heated
in a microwave
oven.
230
Thomas M. Kreis
procedure the interference phase distribution is at least consistent in the sign, in most cases it is exact up to a possible global sign change. This sign corrected interference phase distribution, Fig. 4(f), now can be demodulated. The result in perspective display is shown in Fig. 4(g). If only one interferogram of the form of eqn (1) is evaluated, there always remains a sign-ambiguity, at least a global one, which means all phases have the correct or all phases have the wrong sign. An exact determination of the sign-distribution is achieved if an additional phase shifted interferogram is produced with a mutual phase shift &.I3 Theoretically, C& must be in the range 0 < & < n, but in practice values 7c/3 < & < 2n/3 are recommendable. If this condition is fulfilled, the exact value does not need to be known. The phase shifting complicates the experiment, but it is still easier than with the phase shift methods where several constant phase shifts have to be provided. Now there are two intensity distributions. Ux, Y>= a,
Y>+ w, Y>cm Ma Y)l L(x, Y>= 4& y> + a> Y>cm [4(x, Y>+ &I Fourier-transform processing filter parameters yields
of each intensity with the same bandpass
c&, Y>= W,
46
(13)
Y) exp [Mx, ~11 Y>= Nx, Y) exp [i$G Y>+1’4&, ~11
(14)
From eqns (14), the C&(X,y) is calculated pointwisely by &(x, y) = arctan
Re G(X, Y>Im 4.~ y>- Im c&, Y>Re 4x, Y>
Re c&, y>Re 4x, y>+ Im cdx, Y>Im cdx, Y)
(15)
Of this &(x, y), only the sign distribution is of further interest, it is used for determination of the sign-correct interference phase distribution via
Im cdx, Y)
4(x9Y> = sign w&7 Y)l arctan Re
c
(x
1
y)
(16)
2
Figures S(a) and 5(b) sh ow two phase shifted holographic interferograms, the sign of C&(X,y) as calculated by eqn (15) is given in Fig. 5(c). Figure 5(d) displays the phase distribution modulo 27r and Fig. 5(e) shows the sign-corrected phase distribution modulo 27r, calculated by eqn (16). The demodulated interference phase distribution is given in a grey scale display in Fig. 5(f) and in a perspective plot in Fig. 5(g). An improvement of the method is possible by previous recording of the illuminated background, which corresponds to a@, y) in the spatial
Computer aided evaluation of fringe patterns
(4
(e)
(iit) Fig. 5.
Fourier
transform
evaluation
of phase-shifted
interferograms.
231
232
Thomas M. Kreis
domain. For filtering, a normalised version of the transform this background is subtracted from 4i(u, u). I4 Further methods
for interference
&(u, u) of
phase determination
The aforementioned Fourier transform method introduces a sign change in the interference phase distributions, if %‘(u, u) and %‘*(u, V) superpose in the spatial frequency domain. These two components are separated if an additional carrier frequency is introduced, which can be done in a Michelson-interferometer by tilting a mirror.“,” This introduces additional equidistant straight fringes of spatial frequency j, which act as spatial heterodyne carrier. The known carrier fi, is then eliminated by shifting the filtered spectral components by j, in the spatial frequency domain. The main disadvantage of this method is that the condition of equidistant carrier fringes is not fulfilled if the sensitivity vectors of the applied interferometric method vary spatially as they usually do in holographic interferometry. In another version of the spatial carrier method the recorded interference pattern is multiplied pointwisely by cos (2nj)x) and sin (27&x). I7 If the difference between 4(x, y) and 2&x is sufficiently small, one obtains low frequency components M,(x, Y) = ; cos [4(x,
y> -
27&4
and
(17) M,(x, Y > = - 5 sin C#&
which are isolated calculated by
Y>-
27Gl
by a low pass filter. The interference
4(x, y) = 2x$,x + arctan
-M,(x, M,(x,
phase
Y) Y>
then
is
(18)
This version is only feasible if the interference pattern has fringes of nearly equal inclination and density. The phase-lock method uses a sinusoidal phase modulation by, e.g. a piezoelectrically excited axially oscillating mirror.lX The intensity is Z(x, y, t) = a(x, y) + b(x, y) cos [4(x, y) + L sin ot]
(19)
L < h/2 is the amplitude and Y = w/2n the frequency of this oscillation. A bandpass filter, centred to sin wt, determines the amplitude U, = 2b(x, y)J,(L) sin 4(x, y), which is zero at the points (x, y), where 4(x, y) = NZ These points thus can be detected and give a skeleton
Computer aided evaluation of fringe patterns
233
whose lines correspond to interference phase differences of 7rl2. The further processing of these skeletons is as with the fringe tracking method. The electrooptic holography method combines the advantages of phase stepping and electronic speckle pattern interferometry to perform an image enhancement in holographic interferometry for deformation and vibration measurements.‘“*20
3 COMPARISON OF INTERFERENCE DETERMINATION METHODS
PHASE
Given a problem to be solved by computer aided interferometry, one has to choose the appropriate method for automatic phase determination. In the following, the main methods of Section 2 are compared with regard to several criteria when applied to holographic interferometry. The results of this comparison are summarised in Table 1. Phase
Number of interferograms to be recordedResolution (in A) Evaluation between intensity extrema Inherent noise suppression Automatic sign detection Necessary experimental manipulation Experimental requirements Sensitivity to external influences Interaction by the operator Duration of evaluation
TABLE 1 Determination Methods
Fringe tracking
Phase stepping and shifting
Fourier transform evaluation
Temporalheterodyning
1
3 or 4, rarely 25
1 (2)
One per detection point
l-1/10
l/10-l/100
l/10-1/30
l/100-l/2000
No
Yes
Yes
Yes
Partially
Yes
Yes
Partially
No
Yes
No (yes)
Yes
No
Phase
shift
No
(phase
Frequency
shift
shift)
Low
High
Low
Extremely
high
Low
Moderate
Low
Extremely
high
Possible
Not possible
Possible
High
Low
High
Not possible Extremely
high
234
Experimental
Thomas M. Kreis
requirements
The least requirements of the holographic arrangement are demanded when using fringe skeletonising or Fourier transform evaluation, since both methods evaluate a single interference pattern. These methods even allow the evaluation of interferograms given as paper photograph, negative, diapositive, computerfile, etc. But then there remains the sign ambiguity, which has to be solved with the help of side information. This increases the necessary computational effort. The Fourier transform method with one additional phase-shifted pattern to solve for the exact sign requires a phase shift capability in the experimental set-up. This capability is also needed by the phase shift methods, but with the additional requirement of multiple constant phase shifts. The procedures with three patterns offer an easy computation, on the other hand the phase shift must be controlled exactly to a prescribed value. Due to the two reference arms in the interferometer, between which the mutual phase shift is performed, the mechanical stability of the experimental arrangement must be high. Temporal heterodyning, which also needs two reference arms for performing the mutual frequency shift, demands the highest experimental effort. The mechanical scanning is relatively slow, so the long-term stability of the arrangement must be to fractions of the wavelength. Mechanical vibrations, temporal fluctuations and variations of the refractive index of surrounding air have to be excluded. Resolution
and precision
The spatial resolution is defined as the distance between the detection points. For skeletonising this is given by the number of fringes in the interference pattern. Phase shifting and Fourier transform evaluation offer uniform sampling with a spatial resolution only dependent on the detector array, but independent of the fringe pattern. The resolution can be changed with regard to the pattern by varying the magnification of the recording optics. The spatial resolution of the heterodyne method depends on the distance between the detectors and the step width of the mechanical scanning. Changing the size of the real image optically may alter the spatial resolution. The precision of the measured value normally is given in fractions of to an interference phase the wavelength A, which corresponds difference of 2n. The best precision to be reached by skeletonising is A/10, providing a high quality pattern with a reasonable number of fringes and extensive interpolation.
Computer aided evaluation of fringe patterns
235
The Fourier transform evaluation obtains an accuracy and resolution of h/20 at all but the outermost pixel in the marginal five to eight columns and rows of the digitised pattern. For well fitting cutoff filter even better resolutions are frequencies of the bandpass achievable. Phase shifting yields resolutions in the range of A/100. The electronic phase difference measurement of temporal heterodyning enables a resolution of h/500 and better. But these figures are only achievable if all parameters of the process, especially the step width of the detectors and the long-term stability, are kept to the same precision, which is only possible in ideal laboratory environments. Errors and distortions
Skeletonising and Fourier transform evaluation process one single interferogram, and for that reason they are relatively insensitive to external distortions. On the other hand using a single interferogram does not allow detection of whether the interference phase is increasing or decreasing. A change of the direction of the interference phase is recognised only in special cases if closed ring structures exist within the pattern. This change, as well as increasing or decreasing of the phase, always can be detected by phase shifting and temporal heterodyning. These two methods do not exhibit the aforementioned systematic sign error. The sign error is accompanied by a general inaccessability of the additive multiple of 2~. All methods evaluate the interference phase only modulo 27c, which requires a demodulation to correct for the 2x-steps. Thus normally only phase differences between different points are measured, not the absolute interference phase at a single point. Phase shifting is sensitive to non-constant phase shifts and therefore to all distortions which influence the phase shifts. In this regard the procedure with four recordings and an unknown, but constant phase shift is advantageous, because it inherently corrects for additional linear phase shifts. Temporal heterodyning is extremely sensitive to environmental distortions, as was pointed out in Section 2. 4 PATH-DEPENDENT AND PATH-INDEPENDENT DEMODULATION Demodulation in one-dimensional interference phase distributions 4(x) is done by checking differences between adjacent pixels
236
Thomas M. Kreis
4(x + 1) - 4(x). If this difference is less than -rc, 2n is added to 4 from x + 1 on; if the difference is greater than +z, 27~ is subtracted from C$ starting at pixel x + 1. Several of these 2x-terms may accumulate to integer multiples of 27~. If due to noise a wrong difference occurs, the resulting phase error expands up to the outermost pixel (if it is not neutralised by another error in the opposite direction). This one-dimensional demodulation procedure can be transferred to two dimensions along lines and columns of pixels.‘” The result is a path-dependent demodulation. Besides the possible expansion of erroneous phase, path-dependent demodulation may fail with images of complex forms, e.g. containing holes with no interference phase being defined. at all. To circumvent these difficulties, path-independent demodulation procedures are recommended. The following algorithm interpretes the interference phase image modulo 27~ as a graph, where the pixels are the nodes and the arcs are the connections between neighbouring pixels. 4-neighbourhoods or Sneighbourhoods may be used. With each arc a value d,,(p,, p2) is associated, defined by the phase values p, and pz of the two pixels it connects.
&&lyp2) = min{IP,
-A,
Ip, -p2
+ 24,
IpI -p2
-WI
(20)
The values d21rmay be interpreted as a distance modulo 27~ The demodulation now proceeds along paths where these distances are least. Along these paths the probability for an erroneous phase is least. Pixels with wrong phase are surrounded this way, the same is true for regions without interference phase at all. If a pixel possesses an erroneous phase, that cannot be reached correctly along any path, nevertheless the incorrect demodulated pixel remains isolated in the finally resulting interference phase distribution. The algorithm proceeds as follows: 1. 2.
3.
For a starting pixel all emanating arcs are recorded in a list together with their values dxn. The minimal value in the list is searched for. The demodulation term 0, -2n, or +27r for this arc is stored in an extra file. The arc together with its value is discarded from the list and marked to avoid repeated consideration. The final node of the just considered arc acts as a new starting pixel in Step 1. Only these arcs are considered as free, which have not been stored formerly in the extra file in Step 2. If no free arcs emanate from this node, proceed with Step 2 directly.
Computer aided evaluation of fringe patterns
4.
5. 6.
237
If the capacity of the list is exhausted, the list is checked for arcs which have already entered the extra file along another path. These arcs are deleted. Steps 1 to 4 are repeated until all pixels have been an end node of an arc in the extra file. The interference phase distribution is demodulated using the values in the extra file.
A modification of this algorithm, which shortens the computational effort, allows only arcs with values less than a prescribed threshold to be recorded in the list. Thus the number of comparisons for searching the minimum in Step 2 is drastically reduced. This demodulation procedure is demonstrated in the numerical example of Fig. 6. Figure 6(a) shows the phase values of the pixels of an interference phase distribution modulo 27~ in the squares. At the lines between the squares the distances, calculated by dzn, are given. Figure 6(b) shows how the path-independent algorithm detects the best track and circumvents the one bad spot. Figure 6(c) displays the unwrapped phase resulting from demodulation along predetermined horizontal paths and Fig. 6(d) gives the result after path-independent phase demodulation. The result of this demodulation procedure applied to the signcorrected interference phase distribution of Fig. 5(e) is shown in Figs 5(f) and 5(g). 5 CONCLUSIONS At first glance, phase shifting seems to yield less precision than temporal heterodyning. But the precision of phase shifting is valid even for phase differences of points far apart in the interference pattern. This is due to the simulataneous recording of the intensity at all points. In heterodyning small drifts during the evaluation may accumulate and reduce the precision of phase differences between distant points. Since the most important criteria in metrology are accuracy and precision, phase shifting should be chosen if possible. If no phase shifts or multiple interferograms can be recorded, e.g. when using a pulsed laser or using holographic recording in photorefractive crystals which destroy the information during readout, the Fourier transform evaluation is recommended. The conceptual difference between phase shifting and Fourier transform evaluation is the automatic evaluation by phase shifting without interference by the operator, while the Fourier transform method is a tool that allows manifold manipulations with the interference pattern by the user.
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Computer aided evaluation of fringe patterns
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ACKNOWLEDGEMENT The research work reported here was supported partially by the DFG, Deutsche Forschungsgemeinschaft, under grant Kr 953/2-l, which is gratefully acknowledged.
REFERENCES 1.
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10. 11.
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