APPLICATIONS OF THE EMPIRICAL MODE DECOMPOSITION METHOD IN
SPECKLE METROLOGY
Alejandro Federico
Electrónica e Informática, Instituto Nacional de Tecnología Industrial
P.O. Box B1650WAB, B1650KNA San Martín, Argentina
María B. Bernini, Gustavo E. Galizzi, Guillermo H. Kaufmann
Instituto de Física Rosario (CONICET – UNR)
Blvd. 27 de Febrero 210 bis, S2000EZP Rosario, Argentina
[email protected]
ABSTRACT
This paper shows the application of the Empirical Mode Decomposition (EMD) method to the reduction of speckle noise in
Digital Speckle Pattern Interferometry (DSPI) fringes and to the time–frequency analysis of the activities of dynamic speckle
series. A discussion of the results obtained with simulated and experimental data is also presented.
Introduction
In the last decade, Huang et al. [1] introduced a non linear method called Empirical Mode Decomposition (EMD) for adaptively
representing a non stationary signal as a sum of zero-mean well behaved fast and slow oscillation empirical modes, which are
known as Intrinsic Mode Functions (IMFs), and a residue. An IMF is a function that satisfies two conditions: (1) in the whole
data set, the number of extrema and the number of zero crossings must be either equal or differ at most by one; (2) at any
point, the mean values of the envelopes defined by the local maxima and minima are zero. This decomposition is carried out
through a sifting algorithm that generates a fully data-driven method, so that no basis functions are fixed in the analysis
process. Therefore, the frequency discrimination does not correspond to a predetermined sub-band filtering and the mode
selection corresponds to an automatic and adaptive filtering. Consequently, the evaluation of the performance of the EMD
method in a given field must be studied by means of numerical analysis. In addition, the sifting process facilitates the
application of the Hilbert transform to each IMF, so that it can be associated to an instantaneous amplitude and frequency
obtained by means of the implementation of its analytic signal.
This paper shows the application of the EMD method to the reduction of speckle noise in digital speckle pattern interferometry
(DSPI) fringes and to the analysis of the activities of dynamic speckle series.
The removal of speckle noise in DSPI is a complex problem that has generated an active area of research. During the last two
decades, various methods have been applied to reduce speckle noise in DSPI, although they were only partially successful.
For example, various filters that were originally developed to reduce speckle noise in synthetic aperture radar (SAR) images
were also applied to smooth DSPI fringes [2]. However, their performance is quite low because SAR images have a high
spatial content while most test objects and DSPI fringes have low spatial content. Limited success was only obtained with
scale space filtering in which the smoothing is adapted to the local fringe density while preserving high frequency object details
[3,4]. In the last decade, wavelet based approaches have had great success in the field of signal denoising [5]. The wavelet
transform can effectively be used to analyze DSPI fringes and this property can be applied to filter speckle noise from the
shrinkage of the wavelet coefficients or from the removal of a given sub-band [6,7]. More recently, a method based on the use
of cubic splines providing local treatment of the smoothness was also successfully applied to reduce speckle noise [8].
However, these approaches depend on fixed basis functions so that the performance of the denoising method strongly
depends on them.
On the other hand, the EMD method is an adaptive approach that does not require the use of fixed basis functions in the
analysis process. Consequently, it is expected that the performance of the EMD method in DSPI image denoising would not be
largely influenced by the decomposition. In this paper, the effectiveness and limitations of an EMD-based speckle denosing
technique in DSPI are evaluated.
Dynamic speckle is the phenomenon generated by the interference between the coherent optical fields from a large numbers
of scattering centres that present some type of activity [9]. This phenomenon can be due to the random movement of the
scattering centres and also to other more complex phenomena present in the sample. As these phenomena are quite difficult
to be characterised, the evaluation of the speckle activity can be a promising approach to understand the processes produced
in the sample. Dynamic speckle can be used to study several industrial processes such as steel corrosion and paint drying, as
well as some biological phenomena such as fruit bruising and seed viability.
There are several methods that can be used to segment the different activity regions produced by a sequence of dynamic
speckle patterns. Methods based on the evaluation of the time history of the speckle patterns are the most frequently used [1014], and its characterization is made by means of different descriptors that give quantitative measurements of the activity.
Recently, another approach based on the decomposition of dynamic speckle patterns in temporal spectral bands using
Butterworth filters was presented [15]. Even though this approach is promising, it cannot produce an energy–time–frequency
analysis of the dynamic speckle patterns for each selected temporal scale. In this work, we use the EMD method as an
energy–time–frequency analysis tool in order to find different activity levels of the sample and also to characterise them in the
time–frequency plane.
The 1D EMD method
Given the one-dimensional signal I(k), the algorithm to carry out the 1D EMD method can be briefly summarised as follows [1]:
1. Identification of all extremes of I(k).
2. Creation of the upper envelope Emax(k) by spline interpolation of the local maxima and the lower envelope Emin(k) using the
same procedure for the local minima.
3. Computation of the mean M(k)=[Emax(k)+Emin(k)]/2
4. Evaluation of the difference D(k)=I(k)-M(k). The steps 1–4 are iterated on D(k) until this latter can be considered as zero
mean. Once this is achieved, the difference is referred to as oscillation mode or the first IMF c1(k).
5. Determination of the residue R1(k)=I(k)-c1(k).
6. Iteration on R1(k) from step 1 in order to obtain the next IMF.
7. The decomposition process is stopped when the residue Rn(k) of the n-th iteration becomes a monotonic function.
Finally, the decomposition of I(k) can be expressed as
n
I (k ) =
∑ c j (k ) + Rn (k )
(1)
j =1
or equivalently
n
I (k ) =
∑ a j (k )
cos φ j (k ) + Rn (k )
(2)
j =1
where
a j (k ) and φ j (k ) are respectively the instantaneous amplitude and the phase defined by using the Hilbert transform on
the j-IMF component, respectively. Note that this representation is complete and the number n of modes is empirically
determined.
Speckle noise reduction in DSPI: the proposed method
DSPI fringe images are generated by digitally correlating two speckle interferograms. On the contrary of what happens with
Synthetic Aperture Radar (SAR) images, the fringe information in DSPI is held in the lower spatial frequencies. Consequently,
the denoising method should preserve the low frequency content related to the fringes and also the high frequency details of
the object.
The proposed denoising approach is based on the decomposition of the image into the first IMFs, which carry most of the
speckle noise power, and its corresponding residue. The method was implemented applying the 1D EMD decomposition along
four directions over the DSPI image: horizontal, vertical, left diagonal and right diagonal, respectively. Afterwards, the obtained
fast oscillation modes are removed and an image is generated using the corresponding residue. Finally, the four resulting
images generated along each direction are averaged to compute the filtered fringe pattern associated to the appropriate
decomposition level. To remove the spiky residual texture, a 5×5 median filter was applied in the spatial domain to the filtered
image.
The performance of the EMD denoising method to remove speckle noise was determined using computer simulated DSPI
fringes, as this approach allows us to know precisely the noise free image. For this purpose, two figures of merit were
calculated [4]. The image fidelity f is a parameter that quantifies how good image details are preserved after noise removal. A
fidelity value close to 1 will indicate that the filtered image is very similar to the noiseless one. The image fidelity is defined as
∑ [I 0 ( P) − I ( P)] 2
f = 1−
P
(3)
∑ I 02 ( P)
P
where I0 is the noiseless image, I is the filtered fringe pattern and P is a given pixel.
The second figure of merit s, called the speckle index, was used to quantify the local smoothness of the filtered fringe patterns.
This parameter was calculated as the sum of the ratios of the local standard deviation to its mean for windows of 5 × 5 pixels.
Low values of the speckle index are considered as an indication of local smoothness.
Speckle noise reduction in DSPI: results
The performance of the proposed denoising method was evaluated using computer simulated DSPI fringes. For this purpose,
we have used the numerical method that simulates the process of producing DSPI fringes in which the coded phase
distribution is known [4]. The correlation fringes were simulated with a resolution of 512×512 pixels and 256 grey levels, and
for an average speckle size of 1 and 2 pixels. The EMD denoising method was evaluated using DSPI patterns of equispaced
circular fringes with different fringe densities.
The numerical tests showed that best results were obtained by removing the first three IMFs c1, c2 and c3, which means that
each speckle patterns should be decomposed until the level j =3. It should be pointed out that the residue generated by a
decomposition level higher than 3 introduces an excessive blurring in the filtered image and destroys the fringe pattern. These
results suggest that most of the speckle noise power is present in the first three modes. The numerical tests also showed that
the performance of the EMD denoising method slightly decreased as the fringe density was increased. Another important
observation is that the image fidelity decreases when the speckle size of the DSPI fringes increased from 1 to 2 pixels.
As a typical example, Fig. 1(a) shows the computer simulated DSPI fringes generated for a pattern of 4 circular fringes with an
average speckle size of 1 pixel. The filtered image given by the residue R3 of the decomposition until the second level j =3 is
shown in Fig. 1(b). This last figure clearly shows the effectiveness of the proposed denoising approach for speckle noise
removal. In this case, the figures of merit resulted f =0.946 and s =0.09.
The results obtained with the EMD denoising method were also compared with those determined by the application of a
wavelet sub-band removal technique [7]. The wavelet denoising method was based on the use of a Daubechies filter with 20
coefficients using the removal of the two first levels of the decomposition. Figure 1(c) shows the filtered image obtained when
the fringe pattern depicted in Fig. 1(a) was denoised with the previously mentioned wavelet approach.
(a)
(b)
(c)
Figure 1. Computer simulated DSPI fringes generated for a fringe density of 4 circular fringes with a speckle size of 1 pixel:
(a) original image; (b) image filtered with IMFs c1, c2 and c3 removed; (c) image filtered with a wavelet sub-band removal
technique.
When the filtered fringe pattern obtained with the wavelet method was analyzed, values of f = 0.971 and s = 0.09 were
determined. Comparing these last values with those obtained with the proposed denoising scheme for the same case, it was
found that the performance of the EMD denoising method was quite similar to the one given by the wavelet approach.
However, it is seen that the filtered fringe pattern obtained with the EMD denoising method displays a residual texture which is
visually coarser than the one determined with the wavelet approach. This result is due to the fact that speckle noise is not
totally decoupled from the fringe structure. We also think that this dependence is the result of approximating the bidimensional
IMFs with the average of the 1D IMFs obtained along the four previously mentioned directions.
Analysis of dynamic speckle activity: the method
The EMD method was applied to analyse the intensity signal I(t) along the time axis for each pixel of the sequence of dynamic
speckle patterns. Thus, each IMF is a 3D array where two coordinates are associated with the analysed pixel of the speckle
pattern and the third coordinate corresponds to the temporal axis. Therefore, temporal and spatial activities of the speckles
can be investigated.
To generate dynamic speckle activity images, the following parameters were defined. The energy EP(j) corresponding to a
given pixel P and j-IMF component was evaluated as
E p ( j ) = ∑ a 2j ( P, t ) cos 2φ j ( P, t )
(4)
t
The average instantaneous energy 〈E(j)〉(t) and the average instantaneous frequency 〈ω (j)〉 over a window containing N pixels
centred on pixel P were computed by
E ( j ) (t ) = ∑ a 2j ( P, t ) cos 2φ j ( P, t ) / N
(5)
P
and
[
]
ω ( j ) (t ) = ∑ φ j ( P, t ) − φ j ( P, t − 1) / N
(6)
P
respectively. As it will be discussed later, we will also name 〈E(j)〉(t) and 〈ω (j)〉 as the instantaneous activity level and the
instantaneous activity degree.
Analysis of dynamic speckle activity: results
The EMD method was used to analyse a temporal sequence of 500 dynamic speckle patterns of 300 × 300 pixels recorded
from a bruised apple experiment [12]. The sample was illuminated by an expanded low power He–Ne laser and a CCD camera
was used as the detector. During the acquisition of the dynamic speckle patterns, the laser intensity was adjusted to keep
constant the mean intensity in the images. In addition, experimental considerations were taken into account to assure that the
size of the speckle grains was well resolved by the CCD sensor. The apple was bruised by dropping a small and light steel
sphere on its surface from a given height. The analysed temporal sequence of dynamic speckle patterns was recorded just
after the impact. The bruised region of the apple could not be distinguished from the unbruised area by visual inspection. A
rigid surface was included as a reference at the lower left corner of the images.
(a)
(c)
(b)
(d)
Figure 2. Energy EP for different empirical modes: (a) EP(1), (b) EP(2), (c) EP(4), (d) EP(6).
Figures 2(a)–2(d) show the results in grey levels obtained for the energy EP(j) for every pixel, where the lowest energies are
represented in black colour. In these figures, the rigid surface is shown as the darkest region. The almost circular shape
centred in the bruised region can be clearly distinguished at the right side of these images from the undamaged surroundings.
Figures 2(a)–2(b) show the energy EP(1) and EP(2) respectively. It can be noticed a higher speckle activity in the bruised
region than in the healthy one. Figure 2(c) depicts the energy EP(4), where the transition between the bruised and the
unbruised regions results smoother. Finally, Fig. 2(d) shows the energy EP(6). It is seen that the speckle activity in the bruised
region is lower than the unbruised region although higher than in the reference rigid surface. It is important to point out that
only six frequency bands empirically determined by the EMD method were necessary to show the speckle activity related to
the energy EP.
(a)
(b)
(c)
(d)
Figure 3. Average instantaneous energy 〈E(j)〉 for a group of 40 × 40 pixels located in the unbruised (dash line) and the bruised
(continuous line) regions for different empirical modes: (a) 〈E(1)〉, (b) 〈E(2)〉, (c) 〈E(4)〉, (d) 〈E(6)〉.
Figures 3(a)–3(d) depict the average instantaneous energy 〈E(j)〉(t) in arbitrary units, for a window of 40 × 40 pixels located in
the unbruised (dash line) and the bruised (continuous line) regions. Note that the activity level of the first two empirical modes
evaluated in the bruised region are higher than those obtained in the unbruised one, and that both levels are characterised by
decreasing functions of time (see Figs. 3(a)–3(b)). Considering the bruised region, the decrease in the instantaneous activity
level with time is directly related with the instantaneous amplitude of its vibration modes in the representation given by Eq. (4).
Clearly, the activity level shown in Figs. 3(a)–3(b) disappears after the impact and no differences can be observed between the
bruised and unbruised regions.
On the other hand, in Figs 3(c)–3(d) the activity levels in the bruised region are described by increasing functions of time.
Furthermore, lower activity levels than those obtained in the unbruised region are displayed by the bruised one.
To summarise, when the higher modes of the decomposition (high frequency bands) are considered, the temporal
dependence of the speckle activity of a bruised apple after the impact is characterised by a sudden increment of the
instantaneous activity level followed by its decrease. For the lower modes of the decomposition (low frequency bands), it goes
in the other way. It is necessary to point out that the strong peaks displayed by the graphs at the beginning and at the end of
the horizontal axis shown in Figs. 3(a)–3(d) are due to computation artefacts introduced by the spline interpolation used by the
sifting algorithm.
Using these results, a physical interpretation of the dynamics of the scattering centres can be proposed provided some
previous considerations are assumed to be valid. Although the speckle activity can be due to both volume and surface effects,
it will be supposed that the speckle field is generated by the superposition of the elementary waves originated from the
scattering centres located at the surface of the sample. It will also be assumed that the changes in the height profiles
corresponding to these scattering centres are directly related to the optical phase changes of the scattered light, even though
in-plane strains could also have an influence on the speckle activity. Therefore, the speckle activity can be associated to the
changes of height that presents the surface, which can be defined as a height activity.
Two typical height distributions can be defined over the apple surface, one corresponds to the bruised region and the other to
the healthy one. The bruised region can be thought as a surface with two characteristic scales of roughness which vary with
time. One scale corresponds to the roughness given by its microbiologic activity and the other is related to the larger
deformations that are mainly produced at the surface by the impact. When the bruised region is compared with the healthy
one, we expect that in the higher modes of the height dynamics (i.e. modes that exhibits fast changes in the height
distribution), will decrease its oscillation velocity while increase its excursions in height as consequence of the impact. These
excursions will decrease as function of time, thus transferring the local energy to the lower modes of the movement (i.e. modes
that exhibit slow changes in the height distribution). These lower modes will increase its oscillation velocity, although they
cannot increase its excursions in height so quickly. This means that the energy of the higher modes will be mainly transferred
to the lower ones. Therefore, these excursions will be lower than those observed in the healthy region, which can only shows
microbiologic activity, although the difference between both regions will tend to decrease as a function of time such as it is
shown in Figs. 3(c)–(d).
It is seen that the EMD method can clearly identify the bruised region and the inert object, and these results agree quite well
with those obtained in Ref. [15] through the application of frequency filter banks. However, one of the limitations of this last
approach is that the temporal evolution of the energy for each frequency band cannot be retrieved. Another limitation is that
the characteristics associated with speckle activities corresponding to a given frequency band cannot be discriminated as a
function of time. Both limitations can be overcome through the application of the EMD method, and therefore more selective
information about the characteristics of the speckle activity in the sample can be obtained.
Conclusions
This paper presents an evaluation of the performance of the 1D EMD method when it is used to reduce speckle noise in DSPI
fringes and to analyse dynamic speckle activity images. The major advantage of the EMD method is that no basis functions
must be fixed, so that they are derived from the signal itself by the sifting process. Therefore, the signal analysis is adapted. It
should also be pointed out that the sifting process is not computationally time consuming.
The denoising approach was tested on computer simulated DSPI images presenting different fringe densities and average
speckle sizes. The performance of the denoising scheme was evaluated using two parameters, namely the image fidelity and
the speckle index. Results obtained from the numerical analysis showed that the 1D EMD method has a similar performance
as the one given by a wavelet-based noise reduction technique. This result is due to the fact that speckle noise is not totally
decoupled from the fringe structure. We think that this dependence can be weakened using a 2D approach for the
implementation of the sifting process. Work in this area is currently under development.
Additionally, we have shown that the application of the EMD method can improve the analysis of dynamic speckle activity
images because it gives an energy–time–frequency distribution of the scattered light intensity. Therefore, more insight can be
obtained to analyse the dynamical processes involved in the sample when this novel approach is used. As it was illustrated
here, regions with different levels of activity can be segmented and also characterised as a function of time.
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