Optics and Lasers in Engineering 40 (2003) 573–588
Heisenberg principle applied to the analysis of
speckle interferometry fringes
C.A. Sciammarella*, F.M. Sciammarella
Department of Mechanical Materials and Aerospace Chicago, Illinois Institute of Technology, Illinois 60616,
USA
Received 16 January 2002; received in revised form 17 April 2002; accepted 23 April 2002
Abstract
Optical techniques that are used to measure displacements utilize a carrier. When a load is
applied the displacement field modulates the carrier. The accuracy of the information that can
be recovered from the modulated carrier is limited by a number of factors. In this paper, these
factors are analyzed and conclusions concerning the limitations in information recovery are
illustrated with examples taken from experimental data. r 2002 Elsevier Science Ltd. All
rights reserved.
Keywords: Fringe pattern analysis; Whittaker–Shannon theorem; Heisenberg uncertainty principle;
Window Fourier transform
1. Introduction
In [1] a method was introduced to recover displacement information from moire!
patterns using Fourier transform (FT) methodology. The displacement information was
recovered by introducing what in current nomenclature is called Windowed Fourier
transform (WFT). A long time elapsed from [1] to the development of the mathematics
required to understand the process of reconstruction of the signal from its WFT [2].
2. Windowed Fourier transform
When the FT of a function is analyzed it is assumed that the frequency of this
function is reasonably smooth in the region under analysis. If it is not smooth the
*Corresponding author. Fax: +1-312-567-7230.
E-mail address: [email protected] (C.A. Sciammarella).
0143-8166/03/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 1 4 3 - 8 1 6 6 ( 0 2 ) 0 0 0 7 8 - 7
574 C.A. Sciammarella, F.M. Sciammarella / Optics and Lasers in Engineering 40 (2003) 573–588
obtained results will not be correct. Frequency shifts will introduce wrong
components in the frequency spectrum of the signal. To solve the problem of
sudden frequency changes tapering windows are introduced. In what follows we will
be using a single coordinate x to focus on the important aspects of signal processing
without the extra complexity that involves the presence of an additional coordinate.
The WFT is
Sf ðxp ; fq Þpf ðxÞgxp; fq X
Z
þN
f ðxÞgðx xp Þe2pifq x dx:
ð1Þ
N
In the above equation the symbol S stands for the WFT, the second term after the
equal sign is a symbolic representation of the WFT as the inner product of the
function f ðxÞ times a window function gðxÞ: The window function is normalized so
that
Z
þN
½gðxÞ2 dx ¼ 1:
ð2Þ
N
This is equivalent to making the window energy equal to 1. The effect of the window
in (2) is to restrict the influence of the values of the transform to the neighborhood of
the point under analysis. The problem that we face is further complicated by the
fact that we have a discrete raster of points that define the function. The question of
the recovery of a discrete function f ðxp ; fp Þ is a complex one and it is answered by the
frame theory [3]. This theory is connected with the existence of an orthonormal base
to recover the function by its projection in its base. This is the case for the WFT; at
this point it suffices to say that for a WFT there is no orthonormal base. However,
under very general conditions it is possible to prove that with an adequate selection
of the window function gðxÞ it is possible to find threshold values for sampling
distances and frequencies Dxs and Dfs ; so that f ðxÞ can be recovered with a given
accuracy by a stable numerical procedure.
3. Heisenberg boxes
The concept of a Heisenberg box (Fig. 1) is very important in the theory of WFT.
The Heisenberg box gives the space and the frequency extension of a window at a
given point x and for a given frequency f : In the literature there are several
definitions of the extension of a window in the x direction, the following definition is
commonly used:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
uR þN
u
½ðx xp Þ2 ½gðxÞ2 dx
Dx ¼ t N R þN
:
2
N ½gðxÞ dx
ð3Þ
The above definition provides the measure of a window size as a root-mean-square
duration of the window in the physical space. Similarly, the size of a window in the
C.A. Sciammarella, F.M. Sciammarella / Optics and Lasers in Engineering 40 (2003) 573–588 575
f
Fq
2∆f
Xp
x
2∆x
Fig. 1. Heisenberg boxes representing energy spread.
frequency space is given by
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
uR þN
u
½ðf fq Þ2 ½Gðf Þ2 df
:
Df ¼ t N R þN
2
N ½Gðf Þ df
The Heisenberg box provides a geometrical representation of the Heisenberg
uncertainty principle.
The energy of a signal is concentrated in the area of the box, and the Heisenberg
principle using the definitions given in (3) and (4) can be expressed as
Dx Df X
1
:
4p
ð5Þ
In (5) the equal sign applies when gðxÞ is a Gaussian function. In selecting the
window we can choose independently Dx or Df ; but according to the Heisenberg
principle, once one of them is selected the other is automatically defined. In an
application we can get, for example, a very accurate value of the strain
(instantaneous frequency of the signal) but simultaneously we will reduce the
accuracy in the determination of the location of the point where this strain is located.
4. Gratings, carriers of displacement and strain information
Optical techniques that measure displacements use gratings as carriers of
information. Fig. 2 provides a vector representation of the initial and of the
deformed carrier [4]:
1
fp ¼ ;
p
ð6Þ
576 C.A. Sciammarella, F.M. Sciammarella / Optics and Lasers in Engineering 40 (2003) 573–588
Fig. 2. Vectorial representation of carrier: (a) rotating vector; (b) unmodulated carrier; (c) phase of
unmodulated carrier; (d) phase modulating function; (e) phase of modulated carrier; and (f) modulated
carrier.
where p is the pitch of the grating. The phase of the deformed carrier is
yðxÞ ¼ 2pfp x þ cðxÞ;
ð7Þ
Where cðxÞ is the modulating function. The modulating function is related to the
projected displacement u:
p
uðxÞ ¼
cðxÞ:
ð8Þ
2p
The instantaneous frequency is given by
dyðxÞ
dcðxÞ
¼ 2pfp þ
:
ð9Þ
dx
dx
The moire! effect makes the modulation function visible and creates a system of
fringes whose phase is given by cðxÞ: In this case the instantaneous frequency is given
for small strains and rotations by
p dcðxÞ
:
ð10Þ
ex ¼
2p dx
C.A. Sciammarella, F.M. Sciammarella / Optics and Lasers in Engineering 40 (2003) 573–588 577
Fig. 3. Displacement field spectrum. Spectra of the fringe patterns for the given carrier frequencies (disk
under diametrical compression).
Introducing Bessel functions in the equation of a frequency modulated carrier, for
a frequency modulating function of the form ea cos ð2pfm xÞ; one obtains for the
bandwidth of the signal the approximate expression [4],
Df ¼
ea fc
;
fm
ð11Þ
where ea is the amplitude of the modulating strain field, fc ¼ 1=p; fm is the frequency
of the modulating function. Fig. 3 shows the spectrum of the displacement function
of a disk under diametrical compression, along the radius of the disk. In this figure
are plotted also the spectra of the moire! fringes corresponding to two values of the
carrier pitch. As predicted by Eq. (11) the width of the spectrum is proportional to
the grating pitch.
5. Carrier selection and recovery of the signal
To obtain displacement information we must select the frequency of the carrier to
be used. The Whittaker–Shannon theorem gives the necessary but not sufficient
condition in the selection of the carrier pitch. According to this theorem a bandlimited function [4] can be recovered from a sampled version if the sampling
frequency satisfies
Dxs ¼
1
;
2fmax
ð12Þ
where fmax is the maximum frequency present in the spectrum of the signal. To
extract information from a fringe pattern it is necessary to sample the fringes with a
578 C.A. Sciammarella, F.M. Sciammarella / Optics and Lasers in Engineering 40 (2003) 573–588
device that digitizes the intensity information. Again the Whittaker–Shannon
theorem must be applied to select the sampling rate of the fringes. The signal is
recovered from the WFT by the following equation:
f ðxÞ ¼
f ¼N1
1 X
F ðf ÞHðf Þei2pxf =N :
N f ¼0
ð13Þ
In (13) N is the total number of samples taken, F ðf Þ is the FT of f ðxÞ; Hðf Þ is the
window component of Gðf Þ at the frequency f ; Gðf Þ ifs the FT of gðxÞ and f denotes
the frequency variable. The selection of the window is not independent of the
selection of Dxs and Dfs : The Heisenberg boxes must have the correct shape ratio and
size to insure that inside them there is enough sample points, otherwise some
information will be lost if the window weights fall on empty points.
6. Heisenberg principle for the recovery of fringe pattern information
The proposed form of the Heisenberg principle is written as
Dfs DIq ¼ C:
ð14Þ
Eq. (14) relates the accurately determined gray levels DIq to the fractional orders
Dfs that can be determined accurately from a frequency modulated fringe, C is a
constant that will be analyzed later. For a given pattern Dfs is a sampling frequency
that yields accurate fractional pitch displacements. Making the fringe spacing d ¼ 1
normalizes the fractional pitch values. By using the signals-in-quadrature technique,
phase stepping or any other similar process of fringe analysis, one can determine the
minimum angle that one can detect in the rotating vector scheme of Fig. 3:
arctan DyDDy ¼
DIq
;
Id
ð15Þ
where DIq represents the minimum accurately detectable gray level in the imaginary
direction of the vector plot and Id is the maximum amplitude of the vector. The gray
levels in a CCD camera or similar devices are quantized and the maximum
theoretical dynamical range (Id amplitude of the vector) is one-half of the total
number of gray levels that in binary notation are represented by 2M where M defines
the number of gray levels (so for M ¼ 8; Id ¼ 128). The actual dynamic range is
smaller than this quantity. The practical question to be answered is: What is the
minimum displacement information that can be recovered within a fringe spacing d?
It is evident that there is a finite limit to the subdivision of the fringe spacing. The
quantity Dfs is defined as
p
;
ð16Þ
Dfs ¼
Dum
where Dum is the minimum displacement that can be accurately measured and p is
the pitch of the carrier used to determine the displacement field. From Eqs. (8) and
C.A. Sciammarella, F.M. Sciammarella / Optics and Lasers in Engineering 40 (2003) 573–588 579
(15) one can obtain
Dfs DIq ¼ 2pId :
ð17Þ
Eq. (17) is a limit form of Eq. (14). Eq. (14) tells us that the maximum average
sampling frequency multiplied by a minimum detectable gray level is a constant. In
other words to be measurable a given fractional order must occupy a certain gray
level range. The constant C reflects the whole process to obtain displacement
information. The constant C is a function of the optical system, the device used to
detect the fringes (CCD camera) and the algorithms used to get the displacement
information.
In its classical form in quantum mechanics the Heisenberg principle is formulated
as an uncertainty in the position and the momentum of a free particle. If one wants
to locate a particle in space with great precision the momentum information is lost in
Fourier analysis as shown in Section 3 the principle addresses the recovery of a signal
whose energy is localized in small region of the space and whose FT has its energy
concentrated in a small frequency neighborhood. The increase of information in
localization of the signal in the space reduces the information about the frequency of
the signal in the Fourier space. Like wise Eq. (14) addresses the problem of
displacement information of a signal that is localized in a small spatial neighborhood
and the level of energy of the signal. Smaller fractions of the pitch are associated with
higher energy levels.
There are many important practical consequences of the principle formulated in
(14). This equation is a valuable tool for planning experiments involving fringe
analysis. Once the Whittaker–Shannon theorem is applied and the required
minimum frequency of the carrier is computed, the next step is to select the carrier
that is going to be used. In order to obtain frequency and displacement information
it is necessary to maximize the amount of energy levels to encode this information.
This implies that we must use the largest portion of the dynamic range of the
encoding system to store useful information thus minimizing the amount of noise in
the signal. An immediate consequence is the need to increase the visibility of the
fringes within the range of options available. Consequently, when one selects
a carrier one must take into consideration the optical transfer function (OTF) of
the whole system used to encode the information. When we refer to fringes we
are talking of the actual fringes if the modulation function has been made visible
through the moire! effect or the carrier itself, if the carrier is directly detected. The
OTF plots the amplitude of oscillation of a signal with the frequency of the signal. In
the technical literature the OTF is also referred to as the modulation transfer
function (MTF) when the signal has a sinusoidal intensity distribution. The OTF or
the MTF can be determined by using standard targets. One should find the OTF of
the whole system. In general, the used system is composed of several pieces of
equipment that are put into a sequence (i.e. recording camera with its lens system,
frame grabber, processor system). Each stage has an effect in the frequency response
of the system. One should consider that as a general rule, the higher the spatial
frequency the lower the amplitude available for encoding information. The minimum
displacement that can be detected depends on the minimum energy level that
580 C.A. Sciammarella, F.M. Sciammarella / Optics and Lasers in Engineering 40 (2003) 573–588
contains useful information above the system noise. Hence the dynamic range of the
signal is of paramount importance to have enough energy levels to store information.
A very important consequence comes from Eq. (14). If one has a given image size
and a given CCD camera, the frequency recovery once the Whittaker–Shannon is
satisfied does not depend on the selected carrier. With a fixed region size and a fixed
sample distance decreasing, the carrier pitch increases the number of fringes;
however, the sampling of the fringes is reduced. A consequence of reducing the
sampling of the fringes is the reduction of the fraction of the pitch recovered as
shown by Eq. (14). The end result is that no gain is achieved and the minimum
measurable displacement information remains the same. This important consequence of (14) has been experimentally verified for speckle interferometry in Refs.
[4–6].
The retrieval of information by using a computer brings a fundamental change
to a commonly accepted practice of increasing the frequency of the carrier to
increase the sensitivity of the techniques that measure displacements. In order to
increase the sensitivity both the grating pitch and the sampling rate must be
increased. There are two methods to increase the sampling rate; the first is to increase
the sampling rate of the camera; the second is to change the magnification of the lens
system. Attention should still be given to the OTF of the system. An increase in the
frequency of the fringes may reduce the amplitude of the signal to a level that may
reduce the signal-to-noise ratio.
7. Experimental verification of the Heisenberg principle
In [4] the same displacement field was measured using in-plane electronic speckle
interferometry with increasing sampling frequency. The increasing sampling
frequency was achieved by changing the illumination beams inclination with respect
to the normal of the plane of the specimen. The measured field was the displacements
along the diameter of a circular disk under diametrical compression. The theoretical
aspects of the optimization of the amplitude of fringes in speckle interferometry are
presented in [7–9]. Arguments similar to those presented in the above mentioned
papers were utilized to get the best visibility feasible. Best speckle visibility possible,
removal of rigid body displacements by visually getting the best correlation possible
between the unloaded and loaded images. Using the rule of one speckle in one pixel
optimized the correlation fringes. As shown in [7] using this rule when enough
illumination energy is available one obtains the maximum visibility of the correlation
fringes. Pixel phases were determined using a four-phase algorithm with phase steps
of 901. The pixel phases were subtracted in real time by displacing the unloaded and
the loaded images with respect to each other until an optimum signal was visually
observed in the monitor of the system. The wrapped phase was converted into moire!
fringes. From the moire! fringes, using a WFT a band-pass filtered version of the
fringes was obtained. From the filtered version of the fringes the wrapped phase of
the fringes was recovered using filters-in-quadrature [1]. An unwrapping numerical
algorithm was used to get the displacement field. The fringes obtained from the WFT
C.A. Sciammarella, F.M. Sciammarella / Optics and Lasers in Engineering 40 (2003) 573–588 581
were used to compute the strains by applying differentiating band pass filters in the
frequency space [10]. The processing was done with an imaging system of 512 480
pixels. The size of the window was 73 harmonics. Table 1 gives the geometrical
parameters of the system.
The characteristic length L is the total length of the region observed in the
x-direction. According to the power spectrum of Fig. 3 computed for the theoretical
displacement of the tested disk, the maximum frequency of the displacement
function is around 2.5 1/mm. The required sampling distance computed by applying
the Whittaker–Shannon theorem is Dxs ¼ 0:2 mm. Table 2 gives ratios of required
sampling distances to actual sampling distances for the different carriers.
Table 3 shows the approximate bandwidths Df of the spectra of the fringes for the
different pitches, the required sampling distances Dxr and the ratios of the required
sampling distances to the actual sampling distances. Fig. 4 gives the displacement
field corresponding to the different carrier frequencies. The standard deviation of the
data around the regression curve of the displacements as a function of x is
0.0185 mm. Rounding up to two significant figures in all cases the displacements were
measured to an accuracy of 20 nm. In Fig 5 the strains in a region of the right-hand
Table 1
Geometrical parameters
D ¼ 60 mm
Dxs ¼ 0:1893 mm
L ¼ 96:92 mm
Disk diameter
Sampling distance
Characteristic length
Table 2
Ratios of required sampling distance to actual sampling distance
p (mm)
Dxs =p
1.22
0.925
0.613
0.492
0.413
0.365
164
216
326
406
484
548
Table 3
Bandwith and ratios of required and actual sampling distances
p (mm)
Df (1/mm)
Dxr (mm)
Dxr =Dx
1.22
0.925
0.613
0.492
0.413
0.365
0.3
0.41
0.59
0.75
0.9
1.02
1.7
1.22
0.84
0.666
0.55
0.42
9
6
4
3
2.9
2
582 C.A. Sciammarella, F.M. Sciammarella / Optics and Lasers in Engineering 40 (2003) 573–588
3.5
displacement (microns)
3
2.5
2
1.5
p=.365
p=.413
p=.492
p=.633
p=.925
p=1.22
1
0.5
0
0
0.2
0.4
0.6
0.8
1
1.2
coordinates(R)
Fig. 4. Measured displacements for different carrier frequencies.
170
168
Theo.
p=.365
p=.413
p=.492
p=.633
p=.925
p=1.22
166
strains
164
162
160
158
156
154
152
0.25
0.26
0.27
0.28
0.29
0.3
coordinates
Fig. 5. Strains corresponding to different carriers and resulting trend (left side).
C.A. Sciammarella, F.M. Sciammarella / Optics and Lasers in Engineering 40 (2003) 573–588 583
168
Theo.!
p=.365
p=.413
p=.492
p=.633
p=.925
p=1.22
166
164
strain
162
160
158
156
154
152
0.25
0.26
0.27
0.28
0.29
0.3
coordinates
Fig. 6. Strains corresponding to different carriers and resulting trend (right side).
side of the disk are plotted. Fig. 6 corresponds to the left-hand side. The enlarged
scales are used to show the small differences between the different pitches.
Fig. 7 shows the trends of both sides compared to the theoretical solution. The
theoretical values correspond to the elastic singular solution (concentrated force).
The experimental values correspond to a distributed force resulting from the contact
stresses between the disk and the device applying the force. The theoretical values are
larger in the center of the disk and smaller at the sides when compared to the
experimental values. Both values agree within a region that depends on the contact
stresses distribution.
In the above given graphs it is possible to see that the coarser virtual gratings give
values that are closer to the theoretical values. Table 4 gives the comparison of the
theoretical and experimental values within the plotted region of the disk.
8. Data analysis
Data resulting from the application of Eq. (14) are given in Table 5. The values of
DIq have been normalized using the theoretical maximum Id ¼ 128: According to the
data given in Table 5, the minimum displacement that can be accurately determined
584 C.A. Sciammarella, F.M. Sciammarella / Optics and Lasers in Engineering 40 (2003) 573–588
168
166
Theoretical
164
Left
microstrain
162
Right
160
158
156
154
152
0.25
0.26
0.27
0.28
0.29
0.3
coordinates
Fig. 7. Trend of the strains for the left- and right-hand side and theoretical strain values.
is 0.00193 mm. This figure agrees with the standard deviation of the displacements.
An additional experimental verification of Eq. (14) was done [11] using information
obtained from the grating projection method applied to surface contouring. A
211.52 mm pitch grating was projected on a high-quality plane surface and measured
level lines were compared to theoretical level lines obtained. Three additional points
are added to the previously determined points. The results are plotted in Fig. 8.
Basically the same algorithms utilized in the speckle interferometry data analysis
were used to process the contour data. The whole system used to obtain the data was
different, but it has the same 512 480 resolution. High-quality images were
obtained with a very high dynamic range. The equation plotted in Fig. 8 gives the
trend for all the data points. The value of the constant C changed 0.36% with respect
to the value of C obtained from the speckle interferometry data.
9. Application of the Heisenberg principle to the selection of grating pitch
Let us assume that from a preliminary analysis it is estimated that the Whittaker–
Shannon condition will be satisfied if Dum ¼ 100 nm. From Eq. (16) one gets
Dfs ¼ 0:01p:
ð18Þ
C.A. Sciammarella, F.M. Sciammarella / Optics and Lasers in Engineering 40 (2003) 573–588 585
Table 4
Strains along the disk diameter in the region where theoretical values and experimental values are very
close
x (left)
Theo.
p ¼ 0:365
p ¼ 0:413
p ¼ 0:492
p ¼ 0:633
p ¼ 0:925
p ¼ 1:22
STDEV
0.295
0.288
0.282
0.276
0.269
0.263
0.256
0.25
153.825
155.696
157.549
159.382
161.195
162.986
164.754
166.497
155
157
157.999
159.999
161.999
163
165
165.999
153.927
154.953
157.005
158.031
159.058
161.11
162.136
164.188
153.255
155.34
157.426
158.468
160.553
161.596
163.681
164.723
153.705
155.768
156.799
158.863
159.895
160.926
162.989
164.021
154.681
156.8
157.859
159.978
162.097
164.216
165.276
167.395
155.167
156.188
158.229
160.271
161.292
163.333
164.354
166.396
0.92347
0.87696
0.57176
0.94405
1.2758
1.50523
1.52404
1.77052
x (right)
Theo.
p ¼ 0:365
p ¼ 0:413
p ¼ 0:492
p ¼ 0:633
p ¼ 0:925
p ¼ 1:22
STDEV
0.295
0.288
0.282
0.276
0.269
0.263
0.256
0.25
153.825
155.696
157.549
159.382
161.195
162.986
164.754
166.497
153.999
155
157
159
159.999
161.999
163.999
165
152.901
153.927
155.979
157.005
159.058
160.084
161.11
163.162
154.298
155.34
157.426
158.468
159.511
161.596
162.638
163.681
153.705
154.737
156.799
157.832
159.895
160.926
161.958
164.021
156.8
157.859
158.919
161.038
162.097
163.157
165.276
166.335
153.125
155.167
157.208
159.25
160.271
162.313
164.354
166.396
1.44657
1.38698
1.03316
1.53582
1.56144
1.79185
2.30549
2.34351
Table 5
Relationship between gray levels and sampling frequency
p (mm)
N
Dfs
p=Dfs
Dy
DI
C
1.22
0.925
0.633
0.492
0.413
0.365
2.486
3.281
4.79
6.171
7.3535
8.3232
63.55
47.85
32.77
25.44
21.35
18.86
0.0191
0.0195
0.0193
0.0193
0.0193
0.0193
avg=0.0193
0.1026
0.1357
0.1964
0.2506
0.296
0.3322
13.13
17.37
25.14
32.08
37.89
42.5
834
831
824
834
809
802
avg=822+14
From the type of surface and conditions of observation it is known that DIq ¼ 10
gray levels and that for the utilized system C ¼ 820; then p ¼ 8:2 mm. With these
values of the pitch one can compute the angle of illumination of the surface.
10. Discussion and conclusions
The commonly held belief that by just using higher carrier frequencies one can
obtain more accurate displacement and strain values is not upheld by the
experimental evidence presented in this paper. When one increases the carrier
586 C.A. Sciammarella, F.M. Sciammarella / Optics and Lasers in Engineering 40 (2003) 573–588
50
45
gray levels
40
Heisenberg Principle
35
Experimental data
30
25
20
15
10
5
0
0
20
40
60
80
100 120 140 160 180 200 220 240
∆fs
Fig. 8. Heisenberg principle. Plot of experimental data, C ¼ 819:
frequency and keeps an invariable sampling distance for the fringes, one trades
accurately measured gray levels with fractional orders. In all cases the amount of
energy used to obtain information must be above the noise energy level. For a given
system Eq. (14) can only be experimentally determined. In the speckle interferometry
data that have been analyzed in this paper (Table 2), one can see that the studied
displacement field has been over sampled. It is necessary to remember that one wants
to determine a function and not isolated punctual values. In this case, from the
statistical point, over-sampling provides more accurate average results. It is also
necessary to point out that by numerical interpolation it is possible to obtain
fractional values of the quantized quantities: coordinates and gray levels. To get
further understanding between the energy of the signal and the accuracy of the
results we can look at Fig. 9. In this figure the strain information has been extracted
from the fringe pattern of the disk using an energy plot. If one plots [12,13] the power
of a signal in the space–frequency representation, the instantaneous frequency of the
signal corresponds to the points of maxima of the plotted function, called ridge
points. The accuracy of the instantaneous frequencies depends on the shape of the
power function in the space–frequency plot. In other words, it depends on the energy
content of the signal compared to the energy content of the noise. As a final practical
conclusion one can see that the signal-to-noise ratio as in all cases of measurements is
the most important quantity. Just increasing the frequency of the carrier does not
yield more accurate results. A noisier high-frequency carrier will not yield as good
results as a good visibility lower frequency carrier.
An accurate retrieval of strains requires the use of a WFT selected in such a way
that the results at a given point are not contaminated by spurious information
coming from regions away from the analyzed point. The WFT has a limited power of
resolution since the window size is the same throughout the frequency space. A more
accurate selective resolution can be obtained by using wavelets, thus using different
C.A. Sciammarella, F.M. Sciammarella / Optics and Lasers in Engineering 40 (2003) 573–588 587
350
300
microstrain
250
200
150
100
experimental
Theoretical
50
0
-1
-0.5
0
0.5
1
coordinates
Fig. 9. Strain distribution in a disk under diametrical compression obtained by using ridge points and
wavelet transform.
window sizes for different frequencies. Finally, for each system utilized to retrieve
displacement and frequency information, there is a maximum accuracy that can be
achieved and whether we increase the frequency of the carrier or use fractional
sampling the final results will be the same.
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