Algorithm for dynamic Speckle pattern processing. J. Cari˜ nea,b,c,∗, R. Guzm´anc,d,∗, F. A. Torres-Ruizc,d,∗∗ a
Departamento de Ingenier´ıa El´ectrica, Universidad de La Frontera, Temuco, Casilla 54-D, Chile. b Departamento de Ingenier´ıa El´ectrica, Universidad de Concepci´ on, 160-C, Concepci´ on 4070386, Chile. c Center for Optics and Photonics, Universidad de Concepci´ on, Concepci´ on 4070386, Chile d Departamento de Ciencias F´ısicas, Universidad de La Frontera, Temuco, Casilla 54-D, Chile.
Abstract In this article we present a new algorithm for determining surface activity by processing speckle pattern images recorded with a CCD camera. Surface activity can be produced by motility or small displacements among other causes, and is manifested as a change in the pattern recorded in the camera with reference to a static background pattern. This intensity variation is considered to be a small perturbation compared with the mean intensity. Based on a perturbative method we obtain an equation with which we can infer information about the dynamic behavior of the surface that generates the speckle pattern. We define an activity index based on our algorithm that can be easily compared with the outcomes from other algorithms. It is shown experimentally that this index evolves in time in the same way as the Corresponding author Principal corresponding author Email addresses:
[email protected] (J. Cari˜ ne),
[email protected] (R. Guzm´an),
[email protected] (F. A. Torres-Ruiz) ∗
∗∗
Preprint submitted to Optics and Lasers in Engineering
March 9, 2016
Inertia Moment method, however our algorithm is based on direct processing of speckle patterns without the need for other kinds of post-processes (like THSP and co-occurrence matrix), making it a viable real-time method. We also show how this algorithm compares with several other algorithms when applied to calibration experiments. From these results we conclude that our algorithm offer qualitative and quantitative advantages over current methods. Keywords: Laser Speckle, Processing Speckle Pattern, Determination of surface activity 1. Introduction When a rough surface is illuminated by coherent radiation, a random granular interference structure is observed. Each spot of the resulting pattern is a Speckle grain and the whole effect is known as Speckle. This effect can be observed both in the reflected and the transmitted field. The speckle pattern is produced by random interference [1] and can be present in radio waves, optical experiments, ultrasonic fields and any phenomenon related with coherent wave propagation [2]. In signal processing, speckle patterns are treated as unwanted noise and several methods have been designed to reduce and filter their effect [3, 4]. However, Optical laser speckle has been one of the most studied effects of coherent light and several applications have been developed using this phenomenon, particularly in metrology [1, 2] where it has been applied to study the deformation of materials, heating of surfaces and several other phenomena [5, 6, 7]. Dynamic Optical Speckle occurs when objects which present superficial activity are illuminated with a coherent laser beam. In these cases, speckle 2
patterns change over time, and different kinds of statistical studies can be performed with a set of patterns [8, 9]. From these analyses, a relation between the statistical and dynamic parameters of the sample can be constructed, in order to obtain qualitative or quantitative information about the activity of the object [8, 10, 11]. Dynamic Speckle applications are typically oriented towards characterizing biological activity in plant or animal tissues [8, 12, 13, 14] and other metrological applications like paint drying, corrosion, heat exchange, evolution of foams and hydro-adsorption in gels [15, 16, 17, 18]. Different quantitative and qualitative algorithms have been developed in order to obtain information from speckle patterns.The most accepted and successful quantitative procedure is the Time History Speckle Pattern (THSP) [19, 10] combined with the Inertia Moment (IM) obtained from the co-occurrence matrix (CCM) [15]. This algorithm requires three processing steps: the construction of THSP, the calculation of the CCM, and the evaluation of the IM. Qualitatively, several algorithms exist, however Laser Speckle Contrast Analysis (LASCA) [20], Generalized Differences (GD), Weighted Generalized Differences (WGD) [21] and Fujii [22] are the most widely used. Recently, algorithm for speckle processing has been revisited [23, 24, 25, 26]. From the experimental design, Speckle patterns are obtained with a CCD camera in two different schemes, one based on the free propagation of the electromagnetic field, known as objective speckles, and the other by producing the image of a target plane on the CCD array, known as subjective speckles. Quantitative algorithms are usually related with objective speckle, while qualitative ones are related with subjective speckle.
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However, subjective and objective speckle present two important and complementary problems. Objective speckle integrates all the spatial information of the CCD camera, providing a single number for the whole field of view. To obtain a number from each portion of the object, it is necessary to scan the surface of the object with the laser, point by point, and finally obtain a number for each position, increasing the time taken by the whole process. On the other hand, subjective speckle analysis provides a qualitative description of the field of view indicating where the most active zone is, but does not provide a parameter for evaluating activity; it is therefore not possible to compare two different samples easily. A single parameter that provides a quantitative result localy is needed. In this article we propose an algorithm that provides qualitative and quantitative information of surface activity from a sequence of images of speckle patterns. This algorithm allows us both to obtain qualitative information which is more reliable than that obtained using Fujii, LASCA, GD or WGD and to define a numerical index equivalent to IM. It also opens up the possibility of using this index as a parameter for local motility or bio-activity characterization, as well as other applications. 2. Material and methods In our experiments we used a 10mW He-Ne laser with λ = 632.8nm. We use a spatial filter, consisting of a 60x microscope objective and a 25µm diameter pinhole that produces a spatially filtered Gaussian beam in transverse mode (0, 0) (see Figure 1). We perform the beam collimation using a lens of 7.62cm in diameter with 10cm of focal length, which produces an output 4
Figure 1: Schematic diagram of the experimental setup used to produce different speckle patterns. A He-Ne laser is spatially filtered and expanded to obtain a 7cm width coherent beam. The filtered beam is re-oriented vertically to the sample and a detection system based on a CCD camera register the objective (with lens) or subjective (without lens) speckle patterns.
beam of approximately 7cm. The spatial filter and collimation system are horizontally aligned, and by using two first layer mirrors, the beam is vertically reoriented, allowing above illumination of the samples. The detection system is aligned close to the illumination axis, 40cm from the sample. The optical system of the camera for subjective speckle is a 18mm Lens Focusable Double Gauss, model 54857 from Edmund Optics company. The aperture of this lens ranges from f /12 to f /4 (1.5mm to 4.5mm approximately). The speckle size can be estimated by the expression D = λR/D, where λ is the wavelength of the coherent light, R is the separation between the sample and the detector, and D is the aperture of the detection system. In our case, the separation of the sample and the CCD camera is 0.4m and the aperture ranges from 1.5mm to 4.5mm. Therefore, the speckle size ranges
5
approximately from 30µm to 200µm. The cell size of the camera is 7.4µm, therefore the smallest speckle grain is around 4 times bigger than cell size of the camera. We will describe the three experimental setups implemented to compare the performance of the most important algorithms. The first experiment consisted in the acquisition of subjective speckle patterns obtained in the paint drying process. We prepared a circular aluminum plate divided into four quadrants and applied water-based paint on each quadrant with a delay of 20 minutes between successive quadrants. This plate was exposed to the expanded 10mW He-Ne laser beam. A sequence of 200 images was recorded with a Pike camera, model F-032B, at 15 frames per second and 640 × 480 pixels spatial resolution. The four layers of paint were of approximately the same thickness. The second experiment consisted in the acquisition of objective speckle patterns obtained in the paint drying process. In this case, we prepared a circular aluminum plate covered with a uniform layer of water-based paint. The aluminum plate was placed in a KERN digital scale, model ABJ 2204M, where the plate and the paint were weighed to an accuracy of 0.1mg. In this case, the variation of the weight provided an independent variable to evaluate the evaporated water as a function of time. A 10mW , single mode He-Ne laser was directed at the paint surface and a dynamic objective speckle pattern was recorded. The camera was set to record 200 pictures at 15 frames per second every 5 minutes with a spatial resolution of 640 × 480 pixels. For the last experiment, we used a 10mW , single mode He-Ne laser aimed
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at the center of a rotating disc made of Medium-density fiberboard (MDF), obtaining a boiling speckle pattern [27]. The reflected intensity shows a speckle pattern and the objective speckle is recorded. The disc is rotated with a digitally controlled stepper motor attached to a reduction gear, producing a controlled speed, time-varying continuous speckle pattern. 3. Theory/calculation For the mathematical description of the algorithm, we consider a set of T pictures separated temporally by 1/f seconds, where f is the rate of frames per second of the acquisition camera. We will consider a pixel placed at the position x, y in the detection array of the camera, where the coordinate (1, 1) is the lower corner at the left side of the detector viewed from the front, and the upper corner at the right is the coordinate (w, h), where w is the number of pixels in the width, and h is the number of pixels in the height of the CCD detector. We record the intensity at instant t in pixel (x, y) Ixy,t . In this case, as we have a finite number of pictures, the time parameter t is discrete and enumerates the pictures, ranging from t = 1 to t = T . Variations in the intensity can be decomposed in two domains: a regime that changes slowly, representing the background pattern and the low frequency noise produced by temperature fluctuations and other slow-varying quantities; and another regime which changes rapidly and possesses the signal that we want to measure. According to this, the intensity of pixel x, y at instant t can be expressed as Ixy,t = Pxy,t + Sxy,t ,
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(1)
where Pxy,t contains the constant background pattern and the slowly varying part of the speckle pattern, while Sxy,t is the rapidly varying signal at instant t. Under this mathematical description, we can consider that the rapidly varying signal Sxy,t is a perturbation of the Pxy,t signal, and we can use a perturbative treatment for the intensity. Below, we proceed with the perturbation method described in references [28, 29, 30] where we can express the intensity as (2)
Ixy,t = Pxy,t + ǫSxy,t + ǫ2 Sxy,t + ...,
(2)
where ǫ is the perturbation parameter. Similarly, at instant (t + 1), the intensity is given by (2)
Ixy,t+1 = Pxy,t+1 + ǫSxy,t+1 + ǫ2 Sxy,t+1 + ...,
(3)
From a mathematical point of view, the ǫ parameter is useful for tracking the order of the perturbation, and from the physical point of view, it enables a total control over the intensity of the perturbation. For ǫ = 0 there is no perturbation, and for ǫ = 1 the intensity of the perturbation is complete, therefore the ǫ parameter regulates the intensity of the perturbation. Multiplying equations (2) and (3), considering first order in ǫ, and considering ǫ = 1 we obtain Ixy,t Ixy,t+1 = Pxy,t Pxy,t+1 + Pxy,t Sxy,t+1 + Pxy,t+1 Sxy,t .
(4)
On the other hand, as T is the total number of images recorded, we can sum the intensity pixel by pixel over N images, starting at time t, assuming that N ≤ T . From equation (2), considering the same conditions of the 8
perturbative method, we obtain the expression N −1 X
Ixy,t+j =
j=0
N −1 X
(Pxy,t+j + Sxy,t+j ) .
(5)
j=0
To proceed, we need to assume two conditions: • The variation of Pxy,t is slow for short times (i. e. for a small N ), therefore we can assume that for N pictures Pxy,t remains constant (Pxy,t+j = Pxy,t ). • The absolute value of Sxy,t is much smaller than the mean value of Pxy,t , Sxy,t could even be positive or negative according to the variation of the intensity, and therefore the sum of the Sxy,t terms may be ignored P N −1 j=0 Sxy,t+j = 0 . Seemingly these conditions imply restrictions on the experimental setup. However we will show later that they are satisfied by almost any setup. With these considerations, we can write equations (4) and (5) as 2 Ixy,t Ixy,t+1 = Pxy,t + Pxy,t (Sxy,t+1 + Sxy,t )
(6)
N −1 X
(7)
Ixy,t+j = N Pxy,t ,
j=0
respectively. From equation (7) we obtain an expression for Pxy,t based on the sum of the intensities recorded in a subset of pictures (N frames) obtaining Pxy,t = PN −1 j=0 Ixy,t+j /N . Replacing this result in equation 6 we obtain Sxy,t + Sxy,t+1
P N −1 Ixy,t Ixy,t+1 − j=0 = PN −1 Ixy,t+j j=0
9
N
Ixy,t+j N
2
.
(8)
From this result, we define the parameter JCxy (N ) as JCxy (N ) =
TX −N
|(Sxy,t + Sxy,t+1 )|
t=1
=
TX −N t=1
P N −1 Ixy,t Ixy,t+1 − j=0 PN −1 Ixy,t+j j=0 N
Ixy,t+j N
2 ,
(9)
where we introduce the absolute value to avoid negative contributions. We observe that this parameter has a dimension of intensity, therefore it associates the intensity value for each pixel, providing a qualitative description of the picture and identifying high and low intensity variations in each pixel. For the quantitative case, we can assume that the speckle pattern is random but uniformly distributed over the whole picture (for objective speckle patterns for example), therefore we calculate the average value of JCxy (N ) over the sub region of the picture, obtaining the number ∆JC(N ) defined by
P N −1 W X L N −1 1 X I I − X xy,t xy,t+1 j=0 ∆JC(N ) = PN −1 Ixy,t+j W L j=0 x=1 y=1 t=1 N
Ixy,t+j N
2 ,
(10)
where W × L is the width×height pixels considered of the sub region, or the total number of pixels in the CCD array (when W = w and L = h). Experimental results show that the best performance for our algorithm is obtained when we consider two frames of difference (N = 2), therefore we will perform all the rest considering JC(2) and ∆JC(2).
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4. Results We carried out three experiments (one qualitative and two quantitative) as described in section 2. 4.1. Qualitative experiment: Paint drying results
a)
b) Figure 2: Comparison of pseudo-images obtained with different qualitative methods. a) The first line of the sequence corresponds to the initial state of the paint drying process, while the second line b) corresponds to a later time, when one quadrant is almost dry. From left to right in the two lines we have: Lasca, Generalized differences (GD), Fujii and our proposal JC(2).
In this experiment we compared three qualitative methods Lasca, Generalized differences (GD) and Fujii against the results obtained by JC(2). Figure 2 shows the results obtained at two different times. The first line of the figure a) shows the results for 200 pictures taken at 15 frames per second starting 60 minutes after the first paint is applied, while the second line b) shows the results obtained for pictures taken at 15 frames per second 20 minutes later (80 minutes from the beginning of the painting process). In row a), Lasca shows an almost homogeneous result in each quadrant, the GD method shows a reasonable result with intensities proportional to the elapsed time. The Fujii method shows a high variation in the corners,
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which means that there is high activity on the surface of the optical table and also produces a similar result to Lasca but with low activity in the last quadrant. Finally, JC(2) shows a similar result to GD but with lower intensities and one completely dry quadrant. On the other hand, row b) shows that Lasca exhibits low activity in each quadrant, and higher activity than at the beginning while GD shows activity in the first two quadrants but also shows low activity in the other two (the intensity does not reach zero). Fujii still presents the problem of the corners but reflects low activity in two quadrants. Finally JC(2) shows low activity on two quadrants and no activity in the other two, reflecting dry paint. 4.2. Quantitative experiment: Gravimetric measurements. a)
c)
e)
b)
0.5
0.5
0
0
−0.5
−0.5
d)
0.5
0.5
0
0
−0.5
−0.5
f)
0.5
0.5
0
0
−0.5
−0.5
0
1 2 Time (hours)
3
0
1 2 Time (hours)
3
Figure 3: Plot of normalized mobility versus time for the paint drying process using different methods. We subtract the gravimetric measurement on each curve plotting the difference for a) LASCA, b) Weighted Generalized Differences (WGD), c) Fujii, d) Generalized Differences (GD), e) ∆JC(2) and f) Inertia Moment (IM).
For the first of the objective speckles experiments we studied the drying process again but now with a single uniform layer applied on a metallic substrate as described in section 2. 12
Lasca
WGD
Fujii
GD
∆JC
IM
Corr.
0.704
0.988
0.992
0.992
0.995
0.996
Err.
45
23
16
20
16
14
Table 1: Correlation coefficient and percentage error for different methods, compared with gravimetric measurement.
We mentioned previously in section 3 that for objective speckle we defined an average number obtained from JC(N ) that we called ∆JC(N ) (see equation 10). We applied the same procedure to obtain an average number for the other methods comparing the evolution of these averages over time with the mass of the sample. Figure 3 shows the difference between the gravimetric measurement and each of the algorithms, including two additional ones: Weighted Generalized Differences (WGD) and Inertia Moment (IM). We observe that Lasca and WGD show significant differences with the mass of the paint. We calculated the percentage error (Err.) and the correlation number (Corr.) for each method taking the gravimetric curve as reference, obtaining the results shown in Table 1 where we can conclude that with Fujii, GD, ∆JC(2) and IM we obtain better results Figure 4 shows specifically the results of Fujii, ∆JC(2) and IM as a function of time. We include the weight or gravimetric measurement (black line) and the Inertia Moment (IM) obtained from the THSP method. In the inset figure, we can observe in detail the curves obtained by subtracting the mass. We observe that Fujii, at less than 15%, presents the biggest difference in a single point.
13
0.15
1 0.9
0.1
0.8 0.05 0.7 0 0
0.6
1
2
3
Time (hours)
0.5 0.4 0.3 0.2 0.1 0 0
Mass Inertia Moment DJC(2) Fujii 0.5
1
1.5 Time (hours)
2
2.5
3
Figure 4: Plot of normalized mobility versus time for different methods for the paint drying process. We observe the exponential reduction of mass as a function of time (black). The residual mass is subtracted from all the curves. Inertia Moment (IM) and ∆JC(2) are closer to the weight curve (gravimetric measurement). (Inset) Comparison between Inertia Moment, ∆JC(2) and Fujii, subtracting normalized mass. We observe that the biggest difference between these curves is less than 15%. The correlation between IM and ∆JC(2) is 0.9994
4.3. Quantitative experiment: Rotating disc. The last experiment performed was the rotating disc with controlled angular speed, described in section 2. In this case, we controlled the frequency of the rotating disc and recorded objective speckle pattern images. Figure 5 shows the result for the Lasca, IM , ∆JC(2), WGD and Fujii methods. We also include the identical function (ω vs ω) for reference. For the purposes of comparison, we normalized all the curves present in the figure. We can see that Inertia Moment, Fujii and ∆JC(2) are the closed curves at the angular frequency (see table 2 for the numerical comparison of each method.) Inset of figure 5 shows Inertia Moment, Fujii and ∆JC(2) subtracting the normalized ω versus ω curve.
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1 0.9 0.8 0.7 0.6 IM DJC(2) Fujii WGD Lasca
0.5 0.4 0.3 0.2
0.2
0
0.1 0 0
−0.2 0
0.2
0.2
0.4
0.4
0.6
0.6
0.8
ω/max(ω)
0.8
1
1
Figure 5: Plot of normalized mobility versus normalized frequency for IM (blue), ∆JC(2) (green), Fujii (red), WGD (cyan), Lasca (magenta) obtained with a MDF wood rotating disc. (Inset) Comparison between Inertia Moment, ∆JC(2) and Fujii, subtracting the angular velocity.
Lasca
WGD
Fujii
∆JC
IM
Corr.
0.399
0.883
0.981
0.993
0.993
Err. (%)
43
28
12
8
9
Table 2: Correlation and accumulated error for different methods, compared with normalized frequency curve.
5. Discussion The two conditions described on section 3 for the development of the algorithm are satisfied by almost any experimental configuration. The first condition claims that Pxy,t remains constant for short times. This can be adjusted reducing the sample interval by increasing the number of frames per second of the camera. The second condition claims that the sum of Sxy,t over a short period of time can be negligible compared with the medium value of Pxy,t . The latter condition is satisfied with a good illumination of 15
the sample, avoiding saturation of the detection system, but with enough signal to produce a high medium value of the background signal. This is equivalent to a good signal-to-noise ratio in the setup of the experiment. Therefore, we consider that this two conditions are easily satisfied in almost any experimental setup. As we can see from the qualitative and quantitative experiments, not all the algorithms presented optimal performance under different situations. Qualitatively, Fujii is one of the most used and cited algorithms but we observe that it presents inaccuracy under low intensity conditions. Other algorithms show a poor correlation with observed phenomena. In this sense, we show that JC(2) is an appropriate algorithm for processing images obtained from speckle patterns, with no additional processing difficulties or computational resources. From the quantitative perspective, we also show that ∆JC(2) provides an optimal images processing result which is of practical use, as good as that of Inertia Moment but with less computational resources and shorter processing time. 6. Conclusions We present the main results of the comparisons of several qualitative and quantitative algorithms for processing speckle images, both for subjective and objective speckle patterns. We propose a new algorithm that presents better results for qualitative processing than other algorithms, and similar results for quantitative processing to the best algorithm (Inertia Moment) but using less computational resources, providing a real time algorithm processing for quantitative speckle. 16
The experimental evidence presented supports the optimal behavior of JC(2) and ∆JC(2) over traditional algorithms. This work was financed by grants from Fondecyt N o 11110258 and N o 1151290, PIA-CONICYT PF0824, DIUFRO through project DI08-0063. J. Cari˜ ne acknowledges the financial support of CONICYT. 7. Vitae
Figure 6: Author, Jaime Cari˜ ne
Jaime Cari˜ ne is a PhD. Student of University of Concepci´on and student member of the Center for Optics and Photonics (CEFOP), both located in Concepci´on, Chile. He obtained his first degree in Electronic Engineering in 2010 at La Frontera University, Chile, and currently is developing experiments in Quantum Communications among other topics. Robert Guzman has been associate professor of La Frontera University since 2005 and Research Associate of the Center for Optics and Photonics (CEFOP), both located in Chile. He obtained his PhD in physics in 2004 at University of Santiago, Chile, and is currently developing experiments in Coherent Anti-Stokes Raman Spectroscopy, Speckle, and Quantum Optics.
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Figure 7: Author, Robert Guzm´an
Figure 8: Author, Fabi´an Torres
Fabi´an Torres-Ruiz has been assistant professor of La Frontera University since 2010 and Research Associate of the Center for Optics and Photonics (CEFOP), both located in Chile. He obtained his PhD in physics in 2009 at University of Concepci´on, Chile, and is currently developing experiments in Second Harmonic Generation and Sum frequency, Speckle, Polarimetry, Interferometry, Optical Twissers and Quantum Optics. References [1] Danty JC. Laser Speckle and Related Phenomena. Springer,Berl´ın; 1975.
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