In this article, we show how the variation of sampling rate in a dynamic speckle analysis affects the value of dynamic speckle indexes concerning the absolute value of the differences index, the temporal speckle standard deviation index, and the temporal speckle mean index. We show that the dynamic speckle index value changes its maximum excursion with the variation of sampling rate, affected directly by the camera's time integration (time of exposure). We highlight the importance of knowing the frequency band of the analysed phenomenon and its signal to choose the appropriate sampling rate, with the recommendation of using the lowest sampling rate possible – without compromise the speckle grains - to obtain an acceptable maximum excursion and an illumination level with a good signal-noise ratio.
Research Article
Sampling Rate in the Dynamic Speckle Analysis
https://doi.org/10.21203/rs.3.rs-1768875/v1
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Sampling rate
Dynamic speckle index
Dynamic speckle analysis
Dynamic laser speckle (DLS) has been used in many applications in medicine and agriculture. This non-invasive optical technique, through the temporal variation of the speckle pattern, evaluates the activity of cells, bacteria, and other biological fluids [1], [2], [3], [4].
The influence of the relationship between the size of the speckle and the size of the pixel is an important aspect that must be considered in all these applications. It was verified that the signal obtained by this technique varies in frequency composition regarding the biological activity. Despite considering the well-known Nyquist theorem (sampling frequency) during the assembling of the signal, it is not enough since the configuration of the optical camera with its attached photographic objective and its numerical aperture (f-number) must also be guaranteed according to the frequency response [5].
Another important aspect is the stability of the laser illumination when using laser diodes or a He-Ne laser to ensure that the speckled grains fluctuate only due to biological changes and not because of changes in illumination. The experimental results show that the stability of the diode laser is greater than that of the He-Ne laser in all cases, breaking the paradigm of the stability of He-Ne devices [6].
Thus, it is necessary to analyse the effect of technical parameters that affect the perceived illumination level. In this sense, we studied the effect of sampling rate and, consequently, the time of exposure in a dynamic laser speckle analysis considering the absolute value of the differences index, the temporal speckle standard deviation index, and the temporal speckle mean index [7], [8], [9].
Knowing the effect of sampling rate over an index value, a researcher will be able to establish criteria to choose the appropriate sampling rate for a specific monitored phenomenon [10], [11], [12].
For this purpose, we show that, given a dynamic speckle test, there is an appropriate frequency band to the sampling rate. High sampling rate values may cause unintended consequences on the index value, such as the decrease of the excursion between two activity levels, given a determinate index value and illumination level. In turn, decreasing the sampling rate promoted by increasing the time of exposure will cause a reduction of the temporal speckle contrast and, consequently, limit the possibility of obtaining information from the sample.
To demonstrate the existence of this appropriate frequency band, we performed two types of tests; in the first, we analysed an ink drying process over time; in the second, we tested the activity state of a corn seed with three days of germination. In both cases, we used four different sampling rates and compared the behaviour of speckle indexes.
The next section describes how the speckle images were acquired and the setup used to analyse them. All theoretical definitions necessary to understand the analysis are presented in Section 3. The numerical results of the analysis are presented in Section 4, and an analysis of the results in Section 5. Finally, we present our conclusion in Section 6.
2.1 Time of exposure of the camera
The sampling rate (Fs) or acquisition time in frames per second (fps), using the camera Marlin F-033, is obtained according to Table 1, where we can see the four sampling frequencies used in the tests. These values were obtained using the shutter register value (Shutter), the time base register value (Base), the offset time of exposure (Offset), and the effective time of exposure (E) so that the time of exposure (Exposure),
Exposure = Shutter × Base (1)
\(\frac{1}{{F}_{s}}=E=\) Exposure + Offset (2)
| Shutter | Base [µs] | Offset [µs] | E [ms] | Fs [fps] |
|---|---|---|---|---|
| 3332 | 20 | 12 | 66.652 | 15.003 |
| 1665 | 20 | 12 | 33.312 | 30.019 |
| 1110 | 20 | 12 | 22.212 | 45.021 |
| 832 | 20 | 12 | 16.652 | 60.053 |
Where Fs is calculated in relation to E, considering that the Exposure represents the photography integration time and E the effective time between photographs, the difference between these two exposures arises from the Offset time, e.i., the time between the end and the beginning of a photograph. In all cases, the time of exposure is much longer than the Offset; therefore, we can approximate that Fs ≈ 1/Exposure.
2.2 Data packages of the ink drying process
This data set originates from an ink drying process, where groups of images (data package) are taken at times {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} min. At each time, an acquired package has 512 images of 147 pixels of height and 166 pixels of width. Four different sampling rates were used to take the images, using the rates 15, 30, 45, and 60 Hz.
2.3 Data package of the corn seed germination
This data package has the speckle pattern of a corn seed with three days of germination, where four data packages with different sampling rates (15,30, 45, and 60 Hz) were built. Each package has 512 images of 15 pixels of height and 15 pixels of width.
2.4 Test 1: ink drying process
Figure 1 represents the data analysis method in an ink drying process to an Fs sampling rate. Where P (t) is a data package with images at t minutes, this package has N images with M pixels each. Pn,m(t) defines the n-th image and m-th pixel considering 1 ≤ n ≤ N, 1 ≤ m ≤ M.
The MEAN block represents the calculation of the temporal speckle mean index on the package P (t), giving the value MEAN (t). The AVD block represents the calculation of the absolute values of the differences index on the package P(t), giving the value AVD(t). The STD block represents the calculation of the temporal speckle standard deviation index on the package P(t) or Q(t), giving the value STD(t) or STDF(t), respectively. And finally, the block HPF represents a digital finite impulse response "high-pass filter" with order 40 and cut-off at 0.25Fs, calculated by the P (t) package and return Q(t), leading to the STD block and the STDF(t) index value. According to the information in the data packages, we will have speckle indexes values for each minute for 10 minutes.
2.5 Test 2: Corn seed germination
The speckle activity of a corn seed was analyzed similarly to the ink drying, seen in Section 2.4, with the difference that only one data package was considered at time t, equal to three days of germination.
2.6 Test 3: Frequency band activity analysis
Figure 2 represents the frequency band analysis method of data package P, acquired with a sampling rate of Fs. This package is the input of many band-pass filter (BPF) blocks, described in Section 3.5. These band-pass filter blocks have cut-off frequencies between f1(l) and f2(l), where l indicates the current band, so that 1 ≤ l ≤ L; with L as the total number of analysis frequency bands. Thus, we obtained the Rl package (a filtered version of P). Rl was processed by a σ block, where the standard speckle value is calculated for each pixel in the package Rl, as described in Section 3.3. Finally, an image is obtained, named variable STDBl.
In the following subsections, we use the variable P to define P(t) at any time t; thus, P represents a data collection of images at any time.
3.1 MEAN index
The temporal speckle mean index (µm) [8] gives the mean value of the illumination level to the m-th pixel in the package P, implemented as presented in Eq. 3, where N is the number of images in the package
$${\mu }_{m}=\sum _{n=1}^{N}\frac{{P}_{n,m}}{N} \left(3\right)$$
Finally, the MEAN (t) index is the spatial mean value of all µm results, as demonstrated in Eq. 4,
$$MEAN\left(t\right)=\sum _{m=1}^{M}\frac{{\mu }_{m}}{M} \left(4\right)$$
3.2 AVD index
The Absolute Values of the Differences (AVDm) [6],[7] gives the mean value of the absolute differences in the illumination level of the m-th pixel in the package P, implemented by Eq. 5,
$${AVD}_{m}=\sum _{n=2}^{N}\frac{\left|{P}_{n,m}\right.-\left.{P}_{n-1,m}\right|}{N} \left(5\right)$$
Finally, the AVD(t) index is the spatial mean value of all AVDm results, as indicated in Eq. 6,
$$AVD\left(t\right)=\sum _{m=1}^{M}\frac{{AVD}_{m}}{M} \left(6\right)$$
3.3 STD index
The temporal speckle standard deviation index (σm) [8] gives the standard deviation value of the illumination level to the m-th pixel in the package P, implemented with Eq. 7,
$${\sigma }_{m}^{2}=\sum _{n=1}^{N}\frac{{({P}_{n,m}-{\mu }_{m})}^{2}}{N} \left(7\right)$$
Finally, the STD(t) index is the spatial mean value of all σm results, as presented in Eq. 8,
$$STD\left(t\right)=\sum _{m=1}^{M}\frac{{}_{m}}{M} \left(8\right)$$
3.4 HPF block
The high-pass filter (HPF) block was implemented (Fig. 1) with a finite impulse response (FIR) [13] filter of order 40 and cut-off at 0.25Fs. The 41 values in the filter are represented with h(i) for all 0 ≤ i ≤ 40, with zero in other cases. Thus, a data package P as input of the HPF block gives us a data package Q as a result, as indicated in Eq. 9,
$${Q}_{n,m}=\sum _{k=1}^{N} {P}_{k,m} h\left(n-k+20\right) \left(9\right)$$
3.5 BPF block
The band-pass filter (BPF) block is implemented (Fig. 2) similarly to the
HPF block, with an FIR filter of order 40 but with a cut-off in f1(l) and f2(l),
$$\left|{f}_{1}\left(l\right),{f}_{2}\left(l\right)\right|=\left[\frac{\left(l-1\right)}{L},\left.\frac{l}{L}\right]\right.\frac{{F}_{s}}{2} \left(10\right)$$
representing l, for all 1 ≤ l ≤ L, the l-th band of L bands; each band has FS/2L Hz. The filter is represented with g(i) for all 0 ≤ i ≤ 40, with zero in other cases. Thus, a data package P as input of the BPF block gives us a data package R as a result, as indicated in Eq. 11,
$${R}_{n,m}=\sum _{k=1}^{N} {P}_{k,m} g\left(n-k+20\right) \left(11\right)$$
4.1 Test 1: ink drying process
This test shows the analysis result of an ink drying process for 10 minutes, using the sampling rates: 15, 30, 45, and 60 Hz.
Figure 3 presents the MEAN (t) index in the test shown in Section 2.4 for each time t for the four sampling rates. It can be seen that the value of the index has a monotonous behavior over time. On the other hand, the values in the curves decrease in proportion to the growing sampling rate.
Figure 4 shows the analysis result explained in Section 2.4 regarding the AVD(t) index. Figure 4a shows the AVD(t) index, in each time t, of four sampling rates, showing a different behavior in time of each sampling rate so that the value of the index in all curves decreases in proportion with the increase in the sampling rate. On the other hand, Fig. 4b shows a normalized version of the AVD(t) index so that the maximum value of curves has a unit value; thus, the maximum excursion of the curve is most significant when the sampling rate decreases. It is worth noting that this index uses information from a frequency band between Fs/4 and Fs/2 Hz [7].
Figure 5 presents the STD(t) index in the test presented in Section 2.4. Figure 5a shows the behavior of the STD(t) index, in each time t, for four different sampling rates. This index uses information in the entire frequency band (0 until Fs/2 Hz), as seen in Section 3.3. It shows different behavior in time to each sampling rate so that the value of the index in each time decreases in proportion to the increase in sampling rate. On the other hand, Fig. 5b shows a normalized version of the STD(t) index, with the unit as the maximum value of curves. Therefore, it can be seen that there is a slight difference between the maximum excursion of the curves with different sampling rates. Despite this, it is possible to observe a slight decrease in the maximum excursion in the curve with the increase in the sampling rate, unlike the AVD(t) index.
Figure 6 presents the STDF(t) index in the test presented in Section 2.4. Figure 6a shows the behavior of the STDF(t) index, in each time t, for four different sampling rates. This index uses filtered information from the datapack so that the frequency band is between Fs/4 and Fs/2 Hz. It shows monotone decreasing behavior in time, with a different behavior in time to each different sampling rate so that the value of the index at each time decreases in proportion to the increase in sampling rate. In contrast, Fig. 6b shows a normalized version of the STDF(t) index with the unit as the maximum value of the curves. Therefore, there is a considerable difference between the maximum excursion of the curves using the sampling rate, allowing us to observe a rise in the maximum excursion when the sampling rate decrease, like the AVD(t) index.
4.2 Test 2: corn seed germination
This test shows the result of the analysis of a corn seed at three days of germination. The images were taken using the sampling rates 15, 30, 45, and 60Hz.
Figure 7 shows the MEAN (t) index with a decreasing value as the sampling rate increases, where the index value reached 20.1% of its value from 15 Hz to 60 Hz.
Figure 8 shows the AVD(t), STD(t), and STDF(t) indexes with the same behavior presented by the MEAN index. The decrease in the activity observed is related to the increase of Fs where STD, AVD, and STDF presented decreasing values of 22.2%, 19.3%, and 20.4%, respectively.
4.3 Test 3: frequency band activity analysis
Table 2 shows the STDBl image in different frequency bands of the corn seed, using four different sampling rates of 15, 30, 45, and 60 Hz, as indicated in the first column.
Table 2: Frequency band analysis
The other columns show the results of eight frequency bands of package P, represented in the first line of the Table, where l indicates the position of the frequency band in crescent order, relative to its frequency components, with limits assigned at the frequencies of 0, 3.75, 7.5, 11.25, 15, 18.75, 22.5, 26.25, and 30 Hz, as can be seen in the second line of Table. The package sampled at 15hz was divided into L = 2 frequency bands, the package sampled at 30 Hz was divided into L = 4 frequency bands, the package sampled at 45 Hz into L = 6 frequency bands, and the package sampled at 60 Hz into L = 8 frequency bands.
The matrices STDBl are represented using a color palette from dark blue (low values) to dark red (high values), linked to the activity values. As shown, the index values decrease with the increase of sampling rate Fs. The index values also decrease with the increase of the position of the frequency band between 0 Hz and 3.75 Hz, with a sampling rate of 15 Hz, the better analysis case: in the sense of better differentiating places of lower and higher index value.
Figures 3 and 7 show a relationship between the profile of the mean index curve and the sampling rate of the datapack. The temporal speckle mean index (Fig. 3) is related to the observed illumination level on the surface of the material [8] and corresponds to the analysis of zero frequency components of the signal. Thus, we can conclude that the level of illumination perceived by the camera decreases with the sampling rate increment. This occurs due to the modification of the time of exposure with the alteration of sampling frequency to commercial cameras so that less lighting is used to take the picture. Consequently, the value of the temporal speckle mean index decreases. In this sense, it is important to choose a sampling rate that gives us an index value superior to the noise level of the test or the quantization level of the camera.
The modification of the time of exposure also affects and limits other indexes. We can see this interference in Figs. 4, 5, 6, and 8, where the values of the AVD(t), STD(t), and STDF(t) indexes decrease according to the decrease in time of exposure. Another way the sampling rate affects the values of indexes is in the limitation of the frequency band of an analyzed signal. For example, a sampling rate Fs, by the Nyquist theorem [14],[15], causes the limitation of the frequency band between 0 Hz and Fs/2 Hz. Thus, we have an index, such as the STD(t), that uses information between [0, Fs/2] Hz, while other indexes, such as the AVD(t) and STDF (t), that use the information of a half frequency band, between [Fs/4, Fs/2] Hz. In the comparison between STD(t) vs {STDF(t) and AVD(t)}, we can see how the use of half of a frequency band causes the decrease in the values of the curves but give us considerably good values in the maximum excursion of the curve over time. In contrast, using a complete frequency band in the ink drying process (STD(t) index) returns low values of maximum excursion in the curves over time. The importance of excursion in this test is due to significant differences between the values of two states during the drying process.
It is necessary to highlight the importance of choosing the best value of sampling rate Fs so that the frequency band of a signal contains the frequency components with the information necessary to analyze and that the index values have the greatest values and a good excursion compared with an inert part of the sample. For example, in Table 2, a corn seed is analyzed, where we can see the significant index values obtained from 0 Hz and 7.5 Hz when Fs = 15 Hz.
Thus, at this point, a question is evident: What is the best frequency band? According to the corn seed test, the best range for all frequency bands is the one with the components with the lowest frequencies. However, we must consider that this index extracts information from inner parts of material [8], and this information decreases along with the decrease in a frequency band, given an STD(t) analysis. This is demonstrated when comparing the result in the excursion of STD(t) with components in low frequencies and the excursion of {STDF(t) and AVD(t)} with components in high frequencies. Therefore, we must compromise between the desired quantity of inner information in the sample, the perceived illumination level, and the sampling rate.
This work compared three dynamic laser speckle indexes, subject to different sampling rate values. We conclude that it is essential to choose an appropriate sampling rate, recommending the minimal sampling rate possible to obtain an acceptable maximum excursion for indexed such as the AVD and STDF and an illumination level in the images with a good signal-noise ratio so that the phenomenon under study can be collected in the chosen analysis frequency band. We showed that the digitization of a speckle signal implies a restriction of the signal's frequency band, and, consequently, this affects the result of a speckle analysis. Additionally, the digital camera user must be aware that the increase in the frame rate (fps) forces commercial cameras to change the perceived illumination level in the camera sensor.
Acknowledgment
We wish to acknowledge the partial financial support for this study provided by the CAPES scholarship PNPD Program, FAPEMIG, and CNPQ.
Biographies
Fernando Pujaico Rivera received a Diploma in electronic engineering from the National University of Engineering (Perú), an M.Sc. degree in electric engineering from the University of Campinas, and a Ph.D. in electric engineering from the University of Campinas. He is currently a Post-Doctoral Fellow with the CEDIA, University of Lavras, Brazil, in 2017. His current research interests include digital signal processing, particle image velocimetry, biospeckle lasers, programming, error-correcting codes, and hardware design.
Rolando J. González-Peña received his Master of Science degree in optics from the High Technology Institute of Havana in 1998 and his PhD in technical science from the High Technology Institute of Havana in 2001. His current research is focused on digital speckle interferometry and dynamic speckle lasers. Currently, he is a professor in the Physiology Department in the Faculty of Medicine and Odontology at the University of Valencia.
Roberto A. Braga Jr. has a BA and Master's degree in electrical engineering from UFMG, Brazil, and has a doctoral degree in agricultural engineering from UNICAMP, Brazil. He has been a full professor at UFLA, Brazil, since 1996. He teaches electricity and research optical instrumentation. His sabbatical leaves include BIOSS Scotland 2005 and 2008 and senior stage in JHI Scotland in 2011. He was a visiting professor at the University of Valencia, Spain, in 2014. He has more than 50 papers, most in international journals; he has some patents deposited, software registered, book chapters, and books published. He is the coordinator of a master's degree in automation and system engineering.
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No competing interests reported.