Measurement 36 (2004) 53–66
www.elsevier.com/locate/measurement
Digital evaluation of interferograms
M. Hipp a,*, J. Woisetschl€
ager a, P. Reiterer b, T. Neger c
a
Institute for Thermal Turbomachinery and Machine Dynamics, Graz University of Technology, Inffeldgasse 25, 8010 Graz, Austria
b
Institute for Theoretical Physics, Graz University of Technology, Petersgasse 16, 8010 Graz, Austria
c
Institute for Experimental Physics, Graz University of Technology, Petersgasse 16, 8010 Graz, Austria
Received 3 July 2003; received in revised form 31 March 2004
Available online 19 May 2004
Abstract
For one and a half decade, software for interferometric fringe evaluation has been developed and used at Graz
University of Technology. Within the framework of an awarded grant, these software packages on interferometric
fringe evaluation were subsumed under Windows and Unix platforms and are accessibly under http://optics.tugraz.at.
The software focuses on high-resolution digital fringe evaluation including phase stepping, Fourier domain evaluation
as well as unwrapping techniques for regular and irregular fringe patterns. For phase objects, Abel-inversion and
optical tomography also has been included. New developments on digital holography were taken into account. This
paper describes the methods used and focuses on their implementation in such a universal software.
2004 Elsevier Ltd. All rights reserved.
Keywords: Interferometry; ESPI; DSPI; Digital holography; Phase unwrapping
1. Introduction
Interferometry is a trusted and widely used
optical technique for measurements of surface
deformation and displacements as well as of the
refractivity of objects, from which related quantities like density, temperature, or magnetic fields
can be determined [1,2]. In order to increase
accuracy of the phase evaluation of interference
fringe patterns beyond the early fringe scanning
technique, several procedures have been developed
which are now well established, e.g., the Fourier
Transform technique [3] or the temporal phase
*
Corresponding author. Fax: +43-316-873-7234.
E-mail address: idea@ttm.tu-graz.ac.at (M. Hipp).
stepping technique [4,5]. Especially in the field of
speckle pattern interferometer, the inherent noise
puts high demands on evaluation algorithms.
Additionally, as the phase is usually calculated
from the interferograms consisting of bright and
dark fringes, sign ambiguity has to be removed
and unwrapping of modulo 2p data is always required to obtain continuous maps of phase data
[6].
At Graz University of Technology, interferometry has been used in a number of applications
for nearly 20 years [7–14]. Harsh experimental
conditions put high demands on the accuracy of
the evaluated phase, forcing the researchers to
keep track with actual developments in phase
evaluation procedures all the time. With experience and programming codes accumulated over
0263-2241/$ - see front matter 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.measurement.2004.04.003
54
M. Hipp et al. / Measurement 36 (2004) 53–66
this period, a free software package has been
developed dedicated to phase evaluation and postprocessing of interferogram data, e.g. Abel inversion and tomography for plasma and fluid flow
applications [8,15].
The motivation to distribute the software freely
via internet at http://optics.tugraz.at was to give
newcomers the possibility to apply interferometry
without spending time to implement phase evaluation algorithms. On the other hand, as the
mathematical procedures of this interferogram
evaluation software are written in pure C and
C++, it can serve as a benchmark for self written
software.
Altogether, it is quite an effort to write a data
processing software for fringe analysis when the
provided accuracy in phase evaluation should be
better than p=10 rad. The purpose of this paper is
to outline the structure of the most common phase
evaluation procedures for both regular and irregular fringe patterns, and to point out features a
versatile fringe processing software should include.
superimposed forming the interferogram. The
fringes in these holographic interferograms (secondary interferograms) indicates iso-lines of
e.g. surface displacements or changes in density
[2,17].
In electronic speckle pattern interferometry
(ESPI) the primary interferograms are formed as
speckle patterns on a CCD-array. Two of these
subtracted provide the subtraction fringe interferogram that indicates changes in the object between
the two recordings [18], though the high noise in
these secondary interferograms lead to other
techniques to obtain the phase information.
Any of these interferograms in DH, HI or ESPI
need an evaluation of high sensitivity to obtain a
detailed map of the phase difference between the
superimposed light waves. The spatial intensity
distribution of the fringe pattern is determined by
the local phase shift /ðx; yÞ between the two waves,
by the background intensity distribution i0 ðx; yÞ
the and contrast function mðx; yÞ:
iðx; yÞ ¼ i0 ðx; yÞ þ mðx; yÞ cosð/ðx; yÞÞ;
ð1Þ
where x and y is the position in the image.
2. Basic principle
In optical interferometry the phase difference
between two coherent and overlapping light waves
is visualized as a periodic intensity pattern in space
(fringe pattern). Such, the phase difference between
two light waves is made accessible to intensity
measurement. With the availability of affordable
high resolution CCD-cameras up to 3 k · 3 k pixels
and powerful computers, experimental limitations
are reduced significantly and the computing power
keeps pace with the increased amount of information to be evaluated. This development also led
to digital holography (DH), introduced in the
early 90’s [16]. With the hologram recorded by a
CCD-chip, the reconstruction can be done
numerically by the Fresnel transform. However,
with imaging of the object to the CCD, the evaluation processes get closely related to conventional
methods.
In classical holographic interferometry (HI) a
primary interferogram (hologram) is first formed
to store the light wave from or through the object
under investigation. Two of such object waves are
3. Procedures for analysis of regular fringe pattern
3.1. Fourier transform method
Various approaches exist to calculate the phase
shift / from Eq. (1). Most important are the twodimensional Fast Fourier Transformation (2DFFT) and phase-shift techniques. For 2D-FFT,
experimentally, a parallel fringe pattern has to be
superimposed to the original interferogram usually
by tilting a mirror within one arm of the interferometer. The corresponding spatial carrier frequency is modified by the phase shift / of interest.
First proposed for evaluation of one-dimensional
phase distributions [3], the method has soon be
adapted to obtained two-dimensional phase maps
[19].
With such a spatial carrier frequency in xdirection, Eq. (1) writes
iðx; yÞ ¼ i0 ðx; yÞ þ mðx; yÞ cosð2pm0 x þ /ðx; yÞÞ
ð2Þ
M. Hipp et al. / Measurement 36 (2004) 53–66
where m0 is the carrier frequency vector. Switching
to complex notation, relation (2) can be written as
iðx; yÞ ¼ i0 ðx; yÞ þ cðx; yÞ e2pm0 x þ c ðx; yÞ e2pm0 x
ð3Þ
where
1
cðx; yÞ ¼ mðx; yÞei/ðx;yÞ
2
ð4Þ
is referred to as complex amplitude. The asterisk
superscript (*) denotes a complex conjugate.
Fourier transforming iðx; yÞ yields
^iðx; yÞ ¼ ^i0 ðmÞ þ ^cðm m0 Þ þ ^c ðm þ m0 Þ
ð5Þ
In a next step a filter is applied in the Fourier space
which sets all frequencies except those belonging to
^cðm m0 Þ to zero. Translating the distribution so
that the center m0 is located at the origin eliminates
the carrier, and subsequent backtransformation
yields the complex amplitude cðx; yÞ, from which
the phase / is obtained by
Imðcðx; yÞÞ
/ðx; yÞ ¼ arctan
;
ð6Þ
Reðcðx; yÞÞ
where Imð::Þ and Reð::Þ denote the imaginary and
real part of a complex number. The purpose of
applying a fringe carrier system is to eliminate low
frequency background noise. Basic assumption is
that background i0 and contrast function m are
varying slowly when compared to to the carrier
frequency.
This procedure assumes a completely parallel
carrier fringe system which is hard to obtain. In
general it is recommended to skip the transfer of m0
to the origin, yielding /0 ðx; yÞ ¼ /ðx; yÞ þ wðx; yÞ
after calculations of phase, with w denoting carrier
fringe phase. If wðx; yÞ corresponds to a perfectly
constant carrier frequency m0 as assumed in previous explanations, then /ðx; yÞ can be obtained by
1
mðx; yÞ ¼
3
reference carrier fringe system must be evaluated
for phase in a separate step (recording of a reference interferogram with the carrier frequency
only). Subsequently, this phase distribution has to
be subtracted from the result obtained with the
fringe pattern recorded after a change of the object. The achieved accuracy is typically around
2p=50, but can be as high as 2p=100. The overall
procedure is shown in the block diagram of Fig. 1.
3.2. Temporal phase shifting
The Phase Shifting technique is another method
to evaluate phase distribution from interferograms. Instead of sampling the interferogram in
the spatial domain to determine the phase (2DFFT), the sampling is done in the time domain by
adding constant phase values to the interferogram
[4]. This is done in several steps (e.g. by shifting a
piezo driven mirror in one beam) and at each step
an interferogram is recorded.
In case the minimum of three interferograms
I1 ðx; yÞ, I2 ðx; yÞ and I3 ðx; yÞ is recorded at relative
phases of 0, 2p=3 and 4p=3 rad, Eq. (1) is then
i1 ðx; yÞ ¼ i0 ðx; yÞ þ mðx; yÞ cosð/ðx; yÞÞ
2p
i2 ðx; yÞ ¼ i0 ðx; yÞ þ mðx; yÞ cos /ðx; yÞ þ
3
4p
i3 ðx; yÞ ¼ i0 ðx; yÞ þ mðx; yÞ cos /ðx; yÞ þ
3
ð7Þ
In this case the phase / is given by the well known
3-frame algorithm
pffiffiffi
I1 ðx; yÞ I3 ðx; yÞ
/ðx; yÞ ¼ arctan
3
2I2 ðx; yÞ I1 ðx; yÞ I3 ðx; yÞ
ð8Þ
Additionally, the modulation can be calculated by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½I1 ðx; yÞ I3 ðx; yÞ 2 þ ½2I2 ðx; yÞ I1 ðx; yÞ I3 ðx; yÞ 2
subtracting the linear tilt corresponding to the
carrier phase. Otherwise, the slightly disturbed
55
ð9Þ
The most popular methods are the 3-frame, the 4frame (p=2 phase shift) and the Carre-method, a
56
M. Hipp et al. / Measurement 36 (2004) 53–66
Fig. 1. Phase evaluation procedures for regular fringe patterns, as obtained by classic interferometry. Images with white borders are
modulo 2p phase maps. There are two possibilities for the Fast Fourier Technique, namely with and without recording of the carrier
fringe system. Without a reference the phase ramp introduced by the carrier fringes has to be subtracted subsequently to phase
unwrapping.
4-frame method where the phase shift must not be
know. In [4], an error analysis can be found
investigating influences of linear and quadratic
phase-shifting and detector errors. Though the
standard methods provide a sufficient accuracy in
most cases, a high number of more sophisticated
M. Hipp et al. / Measurement 36 (2004) 53–66
phase shifting techniques and analysis of algorithms can be found in literature [20–28].
4. Procedures for analysis of irregular interference
patterns
If one or both of the interfering wavefronts are
diffuse, e.g. if reflected from a rough surface, the
recorded interference pattern does not consist of
fringes with a narrow bandwidth of spatial frequency, but is ‘speckled’. In principle, there are
two techniques in interferometry leading to irregular fringe patterns: Speckle interferometry, where
the object under investigation has to be imaged
onto the detector, and digital holography, where
the interference pattern is usually recorded without
any camera objective. The regular fringe evaluation procedures can not be applied, as the change
in object phase is superimposed to the random
speckle phase. Therefore, any phase evaluation
procedure requires the recording of reference interferograms.
For example, by recording a set of temporally
phase shifted speckle interferograms as reference,
the initial random speckle phase aðx; yÞ can be
obtained (e.g. by Eq. (8)). Repeating the procedure
after alteration of the object gives aðx; yÞþ
D/ðx; yÞ, and the corresponding phase shift
D/ðx; yÞ caused by the object can be calculated by
a subtraction modulo 2p. This is the standard
speckle phase shifting procedure [27], often
referred to as Difference-of-Phase method.
It is shown left in Fig. 2. Suffering from the
requirement of object stability during the phase
shifting process, it tends to fail in the presence of
external disturbances like vibration, rigid body
motion, or temperature and flow fluctuations
(turbulence) for phase objects.
For such objects, the spatial phase shifting
method is recommended [28–30]. Experimentally,
a (quasi-) plane reference wave has to be guided to
the CCD, while the object is imaged onto it. The
tilt of the reference plane has to be adjusted in a
way that the average phase difference of neighbouring pixels in one direction, let’s say x, is less or
equal to 2p=3 rad. By stretching the speckle size in
x-direction to a corresponding number of pixels
57
using an aperture, a phase-shifted set of intensities
can be obtained from neighbouring pixels by
applying a linear phase ramp in the reference
beam. The intensities of each group can then be
fed to the phase shifting algorithm, which is for
this example Eq. (9). Obviously, this method has
also to be applied in two steps, where interferograms of the initial and final state are recorded and
the calculated phases are subtracted afterwards.
The requirement of the rather large speckle sizes of
at least three pixels, however, is inevitable tied to a
loss of spatial resolution. The procedure is shown
as the middle branch in Fig. 2.
Other methods utilize a temporally phase shifted set of speckle interferograms in a stable reference state of the object under investigation.
During fast object alteration, one or more interferograms are recorded where each of it is correlated to the reference state. For example, with
the Phase-of-Difference method [31,32], each
of these interferograms is subtracted from all
reference interferograms. The resulting subtraction fringe interferograms are low pass filtered and form new sets of phase shifted
interferograms.
Another approach utilizes the stochastic nature
of the speckle phase [33]. With the modulation for
each pixel determined from the phase shifted
intensities in the reference state, e.g. by Eq. (10),
the cosine of the pixel phase corresponding to the
intensity in the other interferogram can easily be
determined. The ambiguity of the arcos-function is
then removed by taking into account the neighbourhood of the pixel in a least squares fit on the
assumption that the phase shift caused by the
object is constant within this region. The accuracy
of the evaluated phase are between 2p=15 for the
Phase-of-Difference method up to 2p=30 for the
fitting method.
Digital holography is used both for image
reconstruction and phase determination. Avoiding
the impractical wet photo processing of classical
holography, this method enjoys growing popularity, pushed by availability of CCDs with high
resolution.
There are four main ways to produce digital
holograms, each of it requiring different ways of
digital reconstruction (see Fig. 3):
58
M. Hipp et al. / Measurement 36 (2004) 53–66
Fig. 2. Common procedures for phase evaluation of speckle interferograms. All methods require acquisition of interferogram(s) at two
different states of the object in order to determine the change of phase between these states. Phase maps modulo 2p have a broad white
border.
1. Image Plane holograms are equivalent to
speckle interferograms with spatial phase shift,
but here usually the FFT-method is used for
phase evaluation [34]. This way, the plane refer-
M. Hipp et al. / Measurement 36 (2004) 53–66
ence wave might have arbitrary tilt to produce
the carrier fringe system.
2. Fresnel holograms are produced by recording
the free interference of a plane or spherical
wavefront with the object wave without imaging
of the object. With known pixel size of the CCD
and the reference wave geometry, the reconstructed complex amplitude can be calculated
at any distance to the hologram by the Fresnel
diffraction formula, referred to as Fresnel transformation (see Eq. (10)).
3. Quasi Fourier holograms are recorded by placing the source of the spherical reference wave
to the same distance to the hologram as the object. This is the condition for lensless Fourier
Transformation [35], therefore a single FFT
is sufficient to obtain the complex reconstruction.
4. Phase shifting digital holography [36] addresses
the problem of the inevitable occurrence of the
conjugate- and zero order reconstruction, which
superimpose on the ‘true’ ‘reconstruction’ with
a given angular separation. These disturbances
do not occur if the complex object wave is Fresnel transformed instead of the product of hologram intensity and complex reference wave.
This requires at one hand measurement of the
object phase relative to the reference wave by
a phase shifting technique, and on the other
hand recording of the wave front intensity coming from the object only. This method is mainly
used for image reconstruction, especially in
holographic microscopy, where the ability to
‘digitally’ focus to different planes is taken
advantage of. The minimum number of 8 holograms required for phase determination (3
phase shifted holograms plus 1 image of object
intensities for two states compared interferometrically) prevents this method to be popular
for phase measurements.
For Fresnel holograms, the reconstruction is
calculated from the recorded intensity distribution
(hologram) by the Fresnel diffraction formula,
which is [37]
Z Z
i
eikq
U ðx; yÞ ¼
dn dg
ð10Þ
rðn; gÞH ðn; gÞ
k
q
H
59
where U is the complex amplitude at point ðx; yÞ in
the observation plane (reconstruction plane), r is
the complex reference wave, H the hologram and
q the distance of point ðn; gÞ in the hologram to
ðx; yÞ. The angles from the normal vector of the
hologram plane to the source point of the reference, e.g. to the observation point are assumed to
be p, or 0, respectively.
There are different ways to numerically calculate the Fresnel transformation Eq. (10), but all of
them utilize Fast Fourier Transformation (FFT).
In the full form, three FFTs are required:
U ðx; yÞ ¼ FT 1 fFT ðH rÞ FT ðgÞg
ð11Þ
with
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
i eik n þg þz
gðn; gÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
k
n2 þ g2 þ z2
ð12Þ
where k is the wave number, z is the distance to the
reconstruction. In most cases it is justified to
approximate the square root by series expansion.
Using this in the exponent, and approximating the
square root in the denominator by z, a single FFT
is sufficient to yield the reconstruction [16]:
U ðx; yÞ ¼ FT ðH r wÞ;
ð13Þ
with the function
wðn; gÞ ¼
eik 2
ðn þ g2 Þ:
2z
ð14Þ
A factor with amplitude 1 and constant phase is
neglected for the result in Eqs. (11) and (13). To
suppress occurrence of the zero order reconstruction, one approach is a subtraction of the mean
intensity value from the hologram preliminary to
Fresnel transformation [38].
For phase shifting digital holography, the
complex object wave amplitude is transformed
instead of H r. From the reconstructed complex
waves, the phase is calculated according to Eq. (6).
5. Phase unwrapping procedures
Both Fourier Transform and phase shifting
methods yield phase data modulo 2p, since the
arctan-function has to be used for the final step in
phase calculation (Eqs. (6) and (8)). So the phase
60
M. Hipp et al. / Measurement 36 (2004) 53–66
data related to measured property is ‘wrapped’
upon itself with discontinuities in phase of height
2p. The procedure to recover the absolute phase is
called ‘Phase Unwrapping’ and belongs to those
problems which are easily formulated but arbitrary complex to solve for noisy phase data by a
software package (Fig. 3).
The simplest approach is done by starting at a
pixel with well defined neighbourhood, assuming
error free phase there. By following a defined path
in the wrapped phase map that visits all the pixles,
e.g. a spiral path, the phases at the visited pixels are
compared to previously validated neighbours.
Differences with absolute values higher than p
(according to Shannon, e.g. Nyquist sampling
theorem) suggest that a regular phase discontinuity
is detected. Taking into account the sign of the
difference, the information about the number of
periods, e.g. 2p, to be added in order to smooth out
the discontinuity is obtained. Ideally, comparison
to two neighbours would be sufficient to determine
two-dimensional unwrapped phase from well behaved phase maps. In presence of noise, however, it
is necessary to take into account as much validated
neighbours as possible (see Fig. 4).
The error susceptibility, e.g. the dependence of
the result on the scanning path can be eliminated
by a preliminary step. The unwrapping procedure
is then referred to as branchcut unwrapping. Its
principle is based on the fact that sources of phase
unwrapping errors can be identified by calculating
the integral of changes in phase order (discontinuities counted according to their sign by +1 or
)1) along closed paths in the phase map [39]. If an
error source is encircled by the path, the integral
results either in +1 or in )1 instead of the valid
zero value, as shown in Fig. 5. Accordingly, the
phase inconsistency is referred to as +residuum or
–residuum. However, if an equal number of residues of different signs are enclosed, the path integral is zero and therefore valid. To ensure this for
any path taken with scanning methods, one has to
connect pairs of residues of different signs and
prevent the unwrap scanning path to cross this
‘branchcuts’. Different ways how to set the
branchcuts have been proposed, distinguished by
the approach to minimize the overall length of
branchcuts in the phase map (see Section 7).
In contrast to the rigorous approach to set
barriers within the phase map for subsequent
spatial phase unwrapping, a different technique
has been found which are referred to as minimumnorm methods [40,41]. The unwrapped phase differences between neighbouring pixels are matched
to the corresponding wrapped differences (gradients) from the measured data in a least squares
sense. Based on the Lp -norm minimization, the
problem is formulated as a discrete Poisson
equation, which can be efficiently solved by utilizing the discrete cosine transform. This has some
spatial smoothing effect on the unwrapped data in
contrast to the previously mentioned methods
where just a multiple of 2p is added. As a drawback, local noisy or shaded data might influence
undisturbed areas, resulting in systematic underestimation of the unwrapped phase differences.
This can be addressed by introducing weighting
factors for reliability of phase data, which then
requires an iterative implementation of the procedure.
6. Experimental limitations
More than on the algorithms used for phase
evaluation, the quality in terms of accuracy and
spatial resolution of the phase depends on experimental conditions. First of all, the type of interferometer used sets limits. In case of differential
interferometry [9], e.g. shearography [18], the
spatial resolution is limited by the shear of the
identical interfering wavefronts. The same is true
for spatial phase shifting due to the large speckle
size used for recording. This disadvantage is
compensated by real time capability, and in case of
shearography, by reduced susceptibility to environmental disturbances, though the reduced phase
sensitivity should meet the demands of the measurement.
Interferograms featuring closed fringes can not
be evaluated by the FFT-method, as a monotone
phase gradient is required. Therefore, the carrier
fringe system has to be sufficiently dense, otherwise
a more complex variation of the FFT-method has
to be used [42].
M. Hipp et al. / Measurement 36 (2004) 53–66
61
Fig. 3. Phase evaluation procedures for Digital Holography. Fresnel holograms are recorded without any imaging by superimposing
the object wave with an arbitrary, off-axis reference beam of known complex amplitude. Image plane holograms, recorded with
imaging of the object to the CCD, are evaluated by FFT-method (see Fig. 1). Phase shifting holography requires a large number of
recordings and is therefore not very popular, but the reconstruction of the complex amplitude is free from zero diffraction and
conjugate image.
62
M. Hipp et al. / Measurement 36 (2004) 53–66
Fig. 4. Path dependent phase unwrapping by the spiral scan
method. Starting at the center pixel ðx; yÞ, a rectangular spiral
path is taken around the center. Here, the path has reached the
pixel p, where the step function should be evaluated. The difference of the modulo 2p data in p and the four neighbours is
calculated. Differences with absolute values higher than p suggest a phase discontinuity, and the sign gives the direction of the
discontinuity. The best estimate is determined by a ‘vote’ of all
results from the neighbouring unwrapped pixels.
shifting errors, non-uniform phase shifts across the
beam and non-linearities of the detector might
impose relevant errors depending on the algorithm
used for phase evaluation. Other experimental
conditions which heavily influence phase accuracy
are stability of light intensity (problematic with
solid-state lasers) and the object.
The inherent noise in speckle interferometry due
to the stochastic nature of speckles sets significant
limits to phase accuracy. Even under ideal experimental conditions, saturated as well as ‘dark’
speckles which do not show any modulation
propagate the noise into the phase result. Additionally, there are inevitable decorrelation effects.
For example, for surface measurements, the displacement to be measured causes slight changes in
the speckle structure which leads to statistically
ascertainable phase errors [43,44].
As a consequence, it is common practice to
integrate low-pass filters in the phase evaluation
process. Usually, this is either done after calculation of the wrapped phase, or, if unwrapping
algorithms are capable to deal with the noisy data
in a reasonable time, as a final step.
7. Implementation of algorithms
Fig. 5. Phase map showing phase inconsistencies that lead to a
path-dependent phase result when unwrapped with scanning
methods. Following path 1 from point A to B, three phase
discontinuities corresponding to an overall change of phase
order +3 are encountered. Scanning from B downwards to A
along path 2, however, will reveal just two phase discontinuities
with reversed sign. Therefore, the change in phase order is not
zero when the circle path is closed, but +1. This shows that the
path includes a positive residuum, marked by a circle, which is
the source of a phase error. A second residuum can be detected
when integrating the changes of phase orders from A to B,
following path 2 in reversed direction, and closing the circle by
scanning along path 3. The overall change in phase order is then
)1, so a negative residuum is found. Note that the direction in
which the circle path is scanned must be consistent for each
integration. The closed path formed by path 1 and 3 encloses
both residues, the integral is zero and therefore valid.
In principle, the phase shifting method is the
most accurate one, as it does not influence spatial
resolution. However, linear and quadratic phase
The common algorithms for phase evaluation
have been implemented in the software package
IDEA. It features a graphical user interface provided by the open source C++ GUI framework
wxWidgets (http://www.wxwidgets.org) for crossplatform programming. The core routines are in
pure C and C++, compiled with Borland’s speed
optimization level 2, memory accessing optimizations that improve cache hits, and Pentium
instruction scheduling. To avoid exceptions due to
mathematical operations leading to non-defined
values such as infinity (1/0) or not-a-number (0/0),
the floating point unit is forced to write ‘invalid
values’ instead, as defined in the IEEE-floating
point specification. Image data is taken in 8-bit
depth, floating point data in the 8-byte double
precision format. When benchmarking with
IDEA, the time used for the calculation of an 256colour-palette bitmap for visualization of data
fields must be taken into account.
M. Hipp et al. / Measurement 36 (2004) 53–66
For the Fourier Transform method, the 2dimensional FFT-algorithm for real data in [45]
‘rlft3’ has been implemented. This algorithm is
restricted to data fields with dimensions of 2n 2m ,
so zero padding (embedding the original field in a
frame of zero-values) of arbitrary sized fields has
been added. The effort here was in the visualization of data to enable interactive masking, e.g.
filtering in the Fourier domain, due the ‘packed’
order of the transformed field. If recording of a
reference fringe system is not required, an additional routine to subtract the unperturbed plane
phase front was required. The simplest way was to
take three points from regions where the phase
difference is known to be zero (border regions) and
to calculate a plane from the phase values at these
coordinates, with the plane phase front then subtracted from the phase map. To evade influences
from local noise on the result, an algorithm that
calculates the plane by planar regression from selected regions is required.
Implementing phase shifting algorithms is
straight forward. It is convenient to set a value for
minimum modulation to mark unreliable phase results. Invalid values occur where modulation is zero.
For unwrapping, 2-dimensional scan methods
are simple to code. Good results can be achieved
by the spiral scan method [6]. Nevertheless, for
noisy data, especially when evaluating speckle interferograms or digital holograms, it is very likely
to encounter spatially propagating errors originating from residues. If there are only a few of
them, the implementation of manually directed
‘subscans’ to locally clean out these errors in
smaller sized windows solves this problem. However, if there are more residues, or if unwrapping
should be accomplished without human interaction, implementation of a branchcut method is
inevitable.
For this, there are three consecutive steps to be
accomplished:
1. Definition of the borders of valid regions in the
modulo 2p phase map.
2. Location of residues by probing all 2 · 2 pixel
squares for residues within these regions.
3. Connecting pairs of + and ) residuum by
branchcuts, or single residues to the border.
63
Eventual neighbourhood scans are efficiently
made in square spirals. To ensure to find closer
neighbours first at such poorly approximated
circles, it is recommended to start the scan in
the centers of the squares, proceeding outwards
to the corners.
4. Unwrap with a simple unwrapping procedure.
Recommended is an adapted flood-fill algorithm known from computer graphics. The usually recursive implementation might have to be
avoided by introducing a stack operation, as is
the case when using C++ or Fortran.
For connecting residuals, the simple nearest
neighbour method is sufficient in many cases. For
more demanding wrapped phase maps, a procedure to optimise for a minimal overall branchcut
length has been added in step 2. For the software
described here, the minimum-cost-matching approach has been chosen [46]. Solved by the iterative ‘Hungarian algorithm’, the global minimum is
guaranteed to be obtained.
However, high-grade optimising methods such
as minimum-cost-matching or simulated annealing
[47] are quite time- and memory consuming when
a high number of residues is encountered. To reduce computation times, it is possible to reduce the
number of residues by preliminary low-pass filtering of the wrapped phase map. Several filter
methods have been found that preserve the phase
discontinuities, e.g. the trigonometric filter or
adaptive medians [48,49]. Additionally, the number of border pixels to take into account can be
reduced, for example by only taking these into
account which are within a defineable number of
nearest partners to any residuum.
In general, it is a good idea to provide also a
branchcut-setting routine that is in between the
speed of the nearest-neighbour and a high-grade
optimising routine. Introducing additional conditions to the nearest neighbourhood principle adds
much of efficiency, as shown in [47]. For the TUGraz software, a new hybrid algorithm that
includes minimum cost matching to the neighbourhood scan has been implemented. Instead of
just scanning for the nearest neighbour of a residuum, the scan is continued until a specific number
of potential partners has been found. Within the
64
M. Hipp et al. / Measurement 36 (2004) 53–66
region covered by the scan, the minimum cost
matching algorithm is applied to all enclosed residues, but only the shortest branchcut is set. The
routine has to be applied iteratively to the whole
phase map until all residues are connected.
Most of the unwrapping problems can be solved
by a kind of branchcut method. For such desperate cases where this approach fails there remains
the iterative minimum norm method due to its
robustness. The L0 -norm method provides the
advantage that the weighting matrix is obtained
during the iterative treatment of the phase map, so
insignificant regions are automatically masked.
It should be mentioned that not only speckle
interferograms lead to phase maps that require
more sophisticated unwrap procedures than the
spatial scan methods. Also phase maps from
smooth-fringe interferograms obtained by the
Fourier-method, which usually results in rather
noise free wrapped phase maps if properly filtered
in the frequency domain, might feature a high
number of residuals near to a shadow region where
fringes are disrupted, especially if diffraction fringes are superimposed.
8. Other helpful features
For a software dedicated to a variety of phase
evaluation problems, some features are highly
recommendable. First of all, the procedures should
be applicable in batch-operation, as in many cases
temporally resolved measurements are desired, for
example to monitor the deformation of a surface
or the behaviour of a flow. Here, this problem has
been solved by wrapping a loop around every
significant algorithm that reads a filename from a
user-defined list, opens the corresponding file and
feeds the data to the algorithm. The results are
immediately saved after each turn.
Another proven feature is to provide verification of results. For example, to ensure that the
minimum-norm method did not underestimate
phase gradients, the unwrapped phase obtained by
this method is wrapped again. By setting a common point to zero in all calculations, the result is
directly comparable to the input data. Accordingly, it is possible to calculate interferograms
from phase data obtained by evaluation of the real
interferograms.
To have an interface to other software, file import and export should be possible in ASCII-format. Therefore, data fields can be saved either in
binary or ASCII format with a single header line
including information about dimensions of the
field and type of data (wrapped phase, phase,
image . . .). This way, the graphical user interface
can disable (‘grey out’) procedures in the menu bar
which are not applicable to the type of the file just
opened, which enhances user’s orientation in the
software structure enormously.
Usually, phase determination is an intermediate
step resulting in a different, directly related quantity, calculated by postprocessing algorithms. For
example for measurements with phase objects (e.g.
fluid flows or plasma flows), the phase shift is
integrated with the path of light through to object.
From these integral phase data, inverse algorithms
provide spatially resolved phase shifts. In case the
objects has radial symmetry, and phase differences
are measured from side-on observations, the phase
distribution can be Abel inverted [8] to obtain the
radial phase distribution, e.g. refraction index and
further temperature.
For asymmetric objects, there is more than one
observation direction required. To get the phase
distribution within the common cross-section,
tomographic reconstruction [15] is required.
Both inverse algorithms have to be included in
software used for fluid flow investigations [14]. For
surface measurements, the local direction of the
sensitivity vector has to be taken into account to
obtain true deformation. This requires knowledge
about the 3D-shape of the surface [50].
9. Conclusion
In this paper, the common procedures to evaluate phase from regular and irregular interferograms have been outlined, showing the basic
structures of the experimental approach and software procedures. The Fast Fourier Transform
technique, phase shifting, methods of ESPI and
digital holography as well as procedures for phase
unwrapping have been illustrated. Algorithms and
M. Hipp et al. / Measurement 36 (2004) 53–66
features found to be useful with a software package covering these fields of interferometry are explained.
Acknowledgements
This work was made possible by the Austrian
Science Foundation (FWF) and the Austrian
Ministry for Education, Science and Culture
(BMBWK) within the grant Y57-TEC (Nonintrusive measurement of turbulence in turbomachinery).
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