PHASE EXTRACTION IN DYNAMIC SPECKLE INTERFEROMETRY BY
EMPIRICAL MODE DECOMPOSITION
a
Sébastien Equisa, Antonio Baldib, and Pierre Jacquota
Nanophotonics and Metrology Laboratory, Swiss Federal Institute of Technology Lausanne
EPFL-STI-NAM, Station 11, CH-1015 Lausanne, Switzerland
[email protected], [email protected]
b
Dipartimento di Ingegneria Meccanica, Università degli Studi di Cagliari,
Piazza d’Armi, 09123 Cagliari, Italy
[email protected]
ABSTRACT
In many respects, speckle interferometry techniques are now considered as mature tools in the experimental mechanics
circles. These techniques have enlarged considerably the field of optical metrology, featuring nanometric sensitivities in wholefield measurements of profile, shape, and deformation of mechanical rough surfaces. Nonetheless, the phase extraction of
speckle interferometry patterns is still computationally intensive, preventing a more widespread use of this technique especially
in dynamic experiments. A promising approach lies in the phase extraction of the temporal pixel signals of photodetector
arrays. We propose here to use the Empirical Mode Decomposition algorithm to put the interferometric 1D signal in an
appropriate shape for phase computation.
Introduction
One of the main advantages of whole-field interferometric techniques – at the same time a major source of difficulty – is that a
huge amount of information, coded as a grey-level modulation, is usually readily available in each frame of a recorded
sequence. Each pixel of the photo-detector array acts a priori like an independent sensor, as if the object surface were
covered by, say, one million of point detectors. This is a well-known characteristic not only of interferometric techniques, but
also of all fringe-based methods, like Moiré, fringe projection, photoelasticimetry… In addition, in a dynamic experiment, the
phase is most likely to vary at a different rate from point to point. Hence, the processing task consists in handling in parallel
that million of non-stationary signals at a temporal sampling rate given by the frame frequency of the camera. However to
really take advantage of this large amount of data, efficient signal processing procedures, whose input is either the 2D pattern
or the 1D pixel signal, are absolutely mandatory. Clearly, experimental mechanics can greatly benefit from such dense spatial
and temporal information about the deformational behavior of solid objects, for whatever purpose this information may serve,
either for structural analyses, material properties determination, or nondestructive testing.
The focus here is on Speckle Interferometry (SI) signals, because they exhibit substantial intensity and phase fluctuations, and
thus represent the worst case to be treated. Actually, there are many techniques to process fringes patterns in dynamic
situations [1]. They are essentially based on phase-shifting, morphological or mathematical transforms approaches. Among
them, we can mention the object-deformation induced phase shifting technique [2], the local interpolation of the phase using a
polynomial [3], the Fourier transform (FT) method combined with spatial carrier fringes [4], and the spiral phase quadrature
transform method [5] (basically, a 2D modified Hilbert transform (HT)). The FT technique has been extensively and
successfully employed and implemented through different variants [6-7]. Another technique, based on the Morlet wavelet
transform, has been previously developed [8]. This method, although effective, remains computationally intensive. We thus
need more powerful processing tools to extract the phase from dynamic SI signals.
This paper introduces the Empirical Mode Decomposition (EMD) [9] method as a new way to pre-process pixel signals. This
method has been successfully used in several domains, dealing with strongly non stationary signals [9-12], and shows some
promising features in our field of interest. As far as the authors know, this technique has not been adapted to interferometry
experiments yet. In this contribution, we will limit ourselves to a quite standard approach of EMD, just to introduce the concept
and asset its possibilities. Some robustness improvements should be presented in a further work.
The paper is organized as follows: in the first part, EMD will be briefly described and this particular choice will be more
extensively vindicated. The second part is dedicated to the phase extraction of a real-valued signal. We will rapidly review
which requirements the signal has to fulfill, so as to extract a physically meaningful phase. In a third part, the basic principles
of the EMD are exposed in further details, and its usefulness to put a given non stationary signal in an appropriate shape for
phase extraction will be shown. It will lead us, finally, to the computer implementation and the first simulation results.
Necessity of pre-processing for SI signals
Dynamic SI signals are likely to be strongly non-stationary. Wavelet analysis is well-known to be efficient with such signals, but
it would be valuable to have a tool able to gather in a single component, the phase and the amplitude modulations along all the
measurement duration, and to get rid of the varying mean quantity.
EMD is essentially an algorithm which decomposes any adequately sampled signal into intrinsic oscillation modes. It extracts
first higher frequencies and proceeds then to lower ones. Moreover, by definition, those intrinsic modes are perfectly wellshaped for further phase computation. The technique is strongly signal dependant, so it is something really different from a
filtering process with predetermined band-pass filters. We will go deeper into details in the section dedicated to EMD principles
and implementation, but so far, the key thing to retain is the ability of EMD to decompose in a very sparse way any non
stationary signal, insuring at the same time a meaningful phase extraction for each component. EMD is used here as an
adaptive filtering operation to remove locally the trend, i.e. the local mean intensity. Hence, the EMD method seems to be an
efficient candidate to make the SI signals amenable to phase extraction methods.
Phase extraction from a real-valued signal: the analytic method
The analytic method is probably the most commonly used technique of phase extraction. The analytic signal z(t) (see equation
(1)) is built from the zero mean real valued signal s(t), using the HT (designated by H in equations):
s( t ) = α( t ) ⋅ cos(ψ( t )) ,
z( t ) = s( t ) + i ⋅ H [s( t )] , and so,
z( t ) = α( t ) ⋅ exp(i ⋅ ψ( t ))
(1)
As a reminder, the Hilbert transform of a function s(t) is defined by the following convolution product (designated by ⊗):
1
s( x )
1
H [s( t )] =
dx =
⊗ s( t ) , where ℜ designates the real numbers set.
(2)
π x∈ℜ t − x
πt
It is well-known, that it is much more efficient to compute the convolution product in the reciprocal space, thanks to the Fast
Fourier Transform algorithm. Taking the FT (designated by FT in equations) of equation (2) yields:
∫
⎡1⎤
FT [H [s( t )]] =FT ⎢ ⎥ ⋅ FT [s( t )] = −i ⋅ sign( ν ) ⋅ S(ν ) , where ν is the frequency and S(ν) is the FT of s(t).
⎣πt ⎦
Going further into details with the HT is out of the scope of this paper, but it is worth mentioning the following two results:
H [cos( t )] = sin( t )
(3)
(4)
H [α ] = 0 , ∀α ∈ ℜ
(5)
The phase ψ(t) is then extracted, modulo 2π, from the complex valued signal as following:
H [s( t )]
ψ( t ) = tan −1(
(6)
)
s( t )
In fact, there are some restrictive conditions in a meaningful use of the Hilbert transform to get the corresponding analytic
signal, or in other words, to extract a phase representative of the physical phenomena: i) amplitude and phase modulations
spectra have to be well separated, ii) the mean has to be locally zero and iii) the signal has to be narrow-band.
i) Amplitude and Frequency spectra well separated
The spectra of the amplitude α(t) and of the phase ψ(t) have to be well separated (amplitude modulation restricted to low
frequencies range, and phase modulation to high frequencies range), otherwise, the computed phase would depend on both,
loosing so all physical sense. By using the Bedrosian’s product theorem [13] (designated by ∆ in (7)) with the signal s(t)
defined in (1) and with relation (4), we obtain:
∆
H [s( t )] = H [α( t ) ⋅ cos(ψ( t ))] = α( t ) ⋅ H [cos(ψ( t ) ] = α( t ) ⋅ sin(ψ( t ))
(7)
With (6) (α(t) is supposed to be non-zero everywhere), we compute the phase ψ h ( t ) , equal to ψ( t ) modulo 2π:
sin(ψ( t ))
ψ h ( t ) = tan −1(
) = ψ( t ) [2π]
(8)
cos(ψ( t ))
So, when a product of functions is considered, the phase term found through HT use is essentially brought by the highest
frequency term. We will have to pay attention to what it means in SI experiments in terms of temporal sampling and optical setup: indeed, the correlation length of the speckle field, which gives the amplitude modulation spectral range, the camera frame
rate, and the evolution velocity of the phenomena under investigation, will all have great influence.
ii) Null local mean.
The analytical method fails in the case of real-valued signals with non zero mean. Indeed the HT of such a signal is given by:
H [α. cos( t ) + β] = α. sin( t )
(9)
The extracted phase is then (see Fig.1 representing z(t) in the complex plane):
α ⋅ sin( t )
ψ h ( t ) = tan −1(
) ≠ ψ( t )
α ⋅ cos( t ) + β
iℜ
β=0
ψmax
α
ψ(t)
(10)
β=α
β>α
ψh(t)
ψ(t)
ℜ
ψmin
Figure 1. Representation of the analytic signal in the complex plane
For the dashed curve, the phase moves fast near the origin, with a π jump in the limit case where modulation depth equals
mean, leading to a singularity. As soon as the modulation depth is lower than the mean (dotted line), the phase computed with
the HT, ψh(t), is distributed in a narrow range, namely [ψmin,ψmax], and is indeed very different from the meaningful quantity of
interest ψ(t).
iii) Narrow-band condition.
This last condition will not be discussed in details here, but in few words, the narrower is the signal spectrum, the more
accurate is the computation of the analytic signal with the HT and thus the phase extraction (see e.g. [14]).
At this point, we can make the connection with a very commonly used physical quantity in signal processing techniques, i.e.
the instantaneous frequency (IF) (see e.g. [14]). The IF is actually the derivative of the instantaneous phase. By putting some
strong safeguards to guarantee a meaningful phase extraction, we insure at the same time a meaningful IF extraction. We can
then think to use the many existing IF estimation and tracking methods.
Empirical Mode Decomposition: principles
i) Standard algorithm.
Huang et al proposed in [9] the Empirical Mode Decomposition, which decomposes any non-stationary real-valued signal into
its intrinsic oscillation modes, namely the Intrinsic Modes Functions (IMF). The IMFs, that could be non-stationary as well,
have to satisfy two conditions: i) in the whole data set, the number of extrema and the number of zeros differ from each other
at most by one, and ii) the local mean is zero. Actually the first condition is equivalent to the narrow-band condition, and the
second one is a good approximation of the rigorous zero mean condition, and much less constraining as it does not need a
definition of a local time scale. By construction, the IMFs have a well-behaved Hilbert Transform, allowing thus good phase
extraction. Basically, we want to split the signal into a detail part (local higher frequency) and a residue part (local lower
frequency): s( t ) = d( t ) + m( t ) , d and m being respectively the local high and low frequency parts. To this aim, the mean has to
be estimated through the following procedure:
1- Identify all extrema of s(t).
2- Interpolate between maxima (resp. minima) to get an upper (resp. lower) envelope envmax(t) (resp. envmin(t)).
3- Compute the mean: m(t) = (envmax(t)+envmin(t))/2.
4- Extract the detail part: d(t) = s(t)-m(t).
nd
Then, the so-obtained component d(t) is the first IMF and the same procedure can be applied to the residue m(t), to extract 2
rd
IMF, then the 3 one and so on. Actually, the decomposition is not that straightforward and needs an iterative process. Indeed,
we do not catch the real extrema in once, but rather the upper and the lower points within a local oscillation period, and that is
why some mistakes are made into the mean computation. Thus, this iterative procedure, namely the sifting process, has two
purposes: eliminate ridding waves and make the IMF as symmetric as possible (to fulfill the two previously mentioned
conditions). The previous procedure 1-4 is injected into an iterative algorithm, giving the standard EMD algorithm [8]:
5- If d(t) is an IMF go to 6 (sifting process ending criterion), else go to 1 and proceed with d(t) instead of s(t).
6- Iterate on the residue m(t) = s(t)-d(t) until the final residue has no extrema or is too small (EMD ending criterion).
This algorithm is illustrated on a simple example (sum of two harmonic signals with a linearly growing trend) in Fig.2. We can
see the original simulated signal with its upper and lower envelopes and its computed mean (Fig.2 a)), the first extracted mode
(Fig.2 b)), the first residue (Fig.2 c)), the second mode with its envelopes and its mean as well (Fig.2 d)). The last picture (Fig.2
e)) depicts the final residue. We get, in fine, the following final decomposition at the rank K:
K
s( t ) =
∑ d (t) + m
k
k =1
K (t)
, where the dk are the IMFs and mK is the final residue.
(11)
a)
b)
c)
Original signal with upper,
lower envelopes, and mean
st
st
1 residue
1 IMF
e)
d)
nd
2
IMF
Figure 2. Standard EMD process.
Final residue
ii) Issues in basic EMD implementation.
One of the main assets of the EMD is its sparseness: the decomposition necessitates indeed much less components to
characterize an arbitrary signal than classical Fourier-based or even wavelet-based analysis, especially for non-stationary
signals. But one of its main drawbacks is its lack of theoretical basis, making the final decomposition strongly dependent on
the different algorithm parameters, like the sifting ending criterion, the boundaries ending technique (signal continuation), and
the interpolation method…
Indeed, applying the sifting process too many times will over-smooth the mode, resulting in the loss of precious information
and physical meaning. This information will be caught by one or several successive IMF(s), leading to leakage between modes
(information contained in a given frequency band is spread over several modes) and even to over-decomposition (see Fig 3).
So a trade-off has to be found thanks to a judicious sifting stopping criterion.
The boundary ending is a sensitive part of the mean estimation. An interpolation kernel is chosen to link the extrema with
smooth curves, and some extrapolation is needed near the edges to process the whole dataset. It is clear that leaving the
extrapolant without safeguards near the ends is dangerous. We choose to keep the values of the first (last) extrema at the
beginning (end) of the dataset, keeping in mind that this simple choice introduces problems, as shown in Fig. 2 e).
In practice the extrema finding and the interpolation steps on discrete-time signals are quite sensitive, and EMD requires a
certain amount of over-sampling. Bad choices for the aforementioned issues could lead to severe errors in the entire
decomposition, like over-decomposition but also mode-mixing (for instance, higher frequencies oscillations are not caught
locally by a given mode but by a successive one that should contain lower frequencies oscillations). To avoid this latter effect,
Huang ([11]) proposes to fix an upper limit of distance between consecutive maxima or minima. Beyond this limit, the data
point is replaced by the local mean. We can imagine a lower limit between consecutive extrema as well, as the case may be.
Attention should be paid for low modulation pixels in real experiments. The counterpart is that we will not be able to separate
the noise and the first IMF, and more disturbing, fixing an absolute value hinders the crucial adaptability of the method.
iii) The orthogonality of the decomposition.
At this point, we shall discuss the orthogonality and the completeness of the EMD. Completeness is actually straightforward
from the decomposition itself. Orthogonality is a little bit trickier. Let’s first rewrite the decomposition in (10) [9]:
K +1
x( t ) =
∑ C (t)
(12)
k
k =1
where we consider the final residue as a component. Taking the square of equation (12) yields:
K +1
x 2 (t) =
K +1 K +1
∑ C (t) + 2∑∑ C (t) ⋅ C (t)
2
k
k =1
i
i
j
j
(13)
If the basis vectors form an orthogonal set, the second term of the right member is null. So, we can assess the obtained
decomposition orthogonality by estimating the following global index:
T ⎛ K +1 K +1
⎞
T
⎜
Ci ( t ) ⋅ C j ( t ) ⎟
(C1( t ) ⋅ C 2 ( t ))
⎟
⎜
t =0 ⎝ i
j
⎠ , which, for 2 modes, becomes: IO = t =0
(14)
IO =
T
T
∑ ∑∑
∑
∑ x (t)
∑ x (t)
2
2
t =0
t =0
We can even build (K+1)×(K+1) matrices defined by:
T
OM(i, j) =
T
∑ (Ci (t) ⋅ C j (t))
OMn(i, j) =
T
∑
∑ (C (t) ⋅ C (t))
i
t =0
x 2 (t)
j
t =0
T
∑ (Ci (t)2 + C j (t)2 )
(15)
t =0
t =0
These matrices of orthogonality are symmetric, and even diagonal if the decomposition is orthogonal. On one hand, the matrix
OM allows a quick identification of the number of modes of significant energy, and on the other hand, OMn is handy to see
qualitatively the leakage between modes.
EMD: simulation results
In this section, some simulations are provided firstly based on pure mathematical signals, which are perfect candidates to
highlight the assets and the limits of the implementation choices, and then on interferometric signals.
i) Chirps and harmonic signals.
We add two linear chirps, with frequencies ranges [2π/64, 2π/16] and [2π/32, 2π/8], with a centered harmonic signal. We
already briefly touched on the problem of over-decomposition and this trivial example shows how the matrices of orthogonality
can help to rebuild a posteriori an orthogonal decomposition.
a)
b)
d)
c)
rd
Figure 3. a) IMFs, b) IO matrices, c) 3 mode reconstructed, d) original and reconstructed signal
We observe a posteriori that except for the 2 first modes, there are also 3 other important contributions, i.e. the IMFs number
rd
3, 4 and 5, that are correlated. Without a priori knowledge on the signal, we guess that the 3 main contribution from an
energetic point of view is given by the sum of the IMFs n° 3, 4 and 5. It is quite well verified in Fig.3 c), and permits a quite
good reconstruction of the original signal (Fig.3 d)). We observe readily the impact of the edges ending choice on the modes
shape.
ii) Simulated interferometric signal.
We consider now a simulated interferometric signal. Its computation is explained in [15]. Basically, in an imaging set-up, the
object field phase is a random variable uniformly distributed in the [-π,π] range. The coherent transfer function of the objective
is a classical zero-phase low-pass filter (Disc function) for the focused case, but a phase term can be easily added to simulate
defocusing and optical aberrations. The cut-off frequency of this low-pass filter is a tunable parameter, which allows to control
the speckle field characteristic size. Hence, the image field is obtained by a simple linear filtering operation in the Fourier
domain. The reference beam is a plane or a speckle wave. A phase step ∆φ may be added in the reference beam, to simulate
a temporal carrier. In this latter situation, the reference beam phase follows the repetitive cycle (0, ∆φ, 2∆φ,…,2π-∆φ). The
intensity pattern is then normalized and quantized on 8 bits without saturation to simulate the acquisition process. There is no
noise added in this present case. We compare in Fig. 4, a simulated signal and an experimental one: the latter one was
obtained in a dynamic but slow experiment, where phase exhibits a pretty linear growth and where a π/2 phase-stepping
device is used. The difference of scale is only here for clarity sake. We will not carry on the comparison too far, but the global
shapes are very similar, and as for the simulated signal the phase is perfectly under control, we will process only this one.
a)
b)
Figure 4. a) simulated interferometric signal and b) experimental signal.
Phase Extraction
i) Comparison between phase extraction with the EMD and the analytical method, and with the Morlet transform:
We present here the first simulation results. We will compute the phase of the signal depicted in Fig. 4 a) (where the global
mean has been removed) with the ridge extraction algorithm. This method, derived from the Morlet wavelet transform, was
successfully used in interferometry in [8]. We will also compute the phase of the first IMF obtained by the EMD of the same
signal with the analytic method (the combination of the two processing steps is known as the Huang-Hilbert transform).
ii) Simulations results:
The first IMF extracted from the previous simulated interferometric signal shown in Fig.4 a) is depicted below in Fig.5:
st
Figure 5. 1 IMF of the simulated interferometric signal of Fig.4 a)
The unwrapped extracted phases are depicted in Fig.6. The phases computed with the analytic method and with the ridge
extraction algorithm are very close to the theoretical one.
Theoretical phase
Ridge extraction
HT
Figure 6. phase extracted with RE algorithm and EMD-HT method
The error in the phase computation with respect to the theoretical phase is depicted in Fig.7.
π/2
π/4
Ridge extraction
HT
0
-π/4
-π/2
Figure 7. phase error.
The analytic method is much more noisy than the ridge extraction method, while this latter one is known not to behave well
when rapid changes of phase occur. For both methods, the error is almost always below π/2 in absolute value, and does not
propagate, leading to a good characterization of the mechanical quantity of interest. Clearly, the meaningful quantity of interest
is well caught by the first IMF obtained with the EMD method applied to the simulated SI signal.
Conclusion and outlooks
This paper aims at introducing the Empirical Mode Decomposition as a new efficient and flexible processing tool to handle
temporal pixel signals in SI, and in all whole-field measurements techniques as well. We limited ourselves in presenting here a
quite standard implementation, but further results to make the decomposition more robust and faster have been obtained. Of
course, it still lacks a complete assessment with experimental signals, and also a quantitative study of the estimated phase
error, especially with noise. But, those promising preliminary results open widely the door to many phase extraction methods,
and might allow a more widespread use of optics in the characterisation of dynamic behaviours of mechanical structures.
Acknowledgments
This work is supported by the Swiss National Science Foundation.
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