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Error analysis for polychromatic speckle contrast measurements
Optics and Lasers in Engineering 49 (2011) 1397–1401
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Error analysis for polychromatic speckle contrast measurements L. Tchvialeva a, I. Markhvida a, T.K. Lee a,b,c,n a b c
Photomedicine Institute, Department of Dermatology and Skin Science, Vancouver Coastal Health Research Institute and University of British Columbia, Vancouver, Canada Cancer Control Research, BC Cancer Research Centre, Vancouver, Canada School of Computing Science, Simon Fraser University, Burnaby, Canada
a r t i c l e i n f o
abstract
Article history: Received 27 April 2011 Received in revised form 4 July 2011 Accepted 14 July 2011 Available online 31 July 2011
Revival of interest in speckle technologies raises a curial question on the accuracy calculation of speckle measurements. In particular, the accuracy calculation of speckle contrast, an important metrics of a stochastic process, is quite different from the accuracy calculation of a typical intensity measurement. Speckle contrast depends more on stochastic characteristics of a process rather than on hardware characteristics. In this article, we consider errors introduced by the limited number of available speckle and by intensity saturation for monochromatic and polychromatic speckles. Equations for these types of errors were derived. Particularly, we show that the error due to limited number of speckles is inversely proportional to the square root of the speckle number, in the similar way as the average intensity error. For the error due to limited dynamic range of a recording device, the truncation effect increases the error as the mean intensity increased. Monochromatic speckles are more sensitive to such truncation than polychromatic speckles. We report the optimal mean intensity in terms of the pixel depth of a recording device and the total number of speckle recorded. In addition, a recommendation to minimize the saturation error for a typical camera is included. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Speckle contrast Error analysis Polychromatic speckle
1. Introduction Laser speckle is an important technique in surface metrology and engineering [1], astronomy [2], and biological tissue evaluation [3]. Comparing to other common optical methods, speckle techniques are faster, cheaper and simpler due to their simple setup: mainly a laser and a registration device are required. A typical experimental setup for speckle techniques is shown in Fig. 1a and an example of a speckle pattern is shown in Fig. 1b. Despite the speckle theory was established a couple of decades ago [1], the theoretical formulations had not been developed into practical instruments in the early years due to technical limitations. Recent advances in light sources and registration devices have now revived interest in speckle techniques [4–11]. Typically speckle patterns are usually characterized by statistical moments of intensities. In this paper, we will focus on speckle contrast and address some factors influencing its measurement. Speckle contrast is defined as the ratio of the standard deviation sI of light intensity over the average intensity /IS [2]. C¼
sI /IS
ð1Þ
n Corresponding author at: Cancer Control Research, BC Cancer Research Centre Program, 675 West 10th Avenue, Vancouver, BC, Canada V5Z 1L3. Tel.: 1 604 675 8053; fax: 1 604 675 8180. E-mail address: [email protected] (T.K. Lee).
0143-8166/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2011.07.008
where /yS designates averaging over ensemble. It is a widely used metrics applied for pavement surface analysis [5], estimation of tissue optical properties [6], monitoring of thermal tissue conditions [7], internal inhomogeneities visualization [8], viscoelasticity of atherosclerotic plaques measurements [9], blood flow evaluation [10] and in-vivo skin surface roughness assessment [11]. Contrast can be affected by the light source, the targeted object and the optical system [2] in various ways; these factors make accurate contrast estimations a challenge in terms of experiment design. Thus, a design should be thoroughly analyzed and all secondary factors affecting contrast must be removed or properly accounted for. The apparatus factors such as temporal and spatial averaging, the geometry of setup has been already considered [10,12–14]. In this paper, we further examine the errors introduced by the limited number of speckles available for contrast calculation, and by intensity saturation. The paper is organized in a following way. In Section 2, we analyze the speckle contrast error due to a finite number of available speckles for registration. Section 3 deals with contrast reduction caused by limiting the dynamic range of measured intensities. Also, the optimal mean intensity in such a situation is derived. A short conclusion is followed in Section 4.
2. Error due to limited amount of speckles Despite the simplicity of Eq. (1), estimating the contrast accuracy in experiments is not trivial because the accuracy of /IS is not the
1398
L. Tchvialeva et al. / Optics and Lasers in Engineering 49 (2011) 1397–1401
Fig. 1. Typical speckle setup geometry and an example of a speckle pattern.
Fig. 2. Intensity distributions for monochromatic and polychromatic speckle patterns.
same as the accuracy of I. The former variable depends not only on the accuracy of intensity measurements, but also on the stochastic nature of speckle. In order to estimate /IS, we compute /ISn, where n denotes the number of independent intensity measurements available for estimating the mean value. A similar procedure is applied for the estimation of sI: we calculate n independent standard deviations sIn. Both /ISn and sIn are stochastic variables, whose behavior depend on the distribution of I and the sample size n. We evaluate the above estimations using monochromatic (Gaussian) speckle and polychromatic speckle patterns. The former pattern is well studied and has an exponential probability density function for intensity [2]. Note that the term Gaussian originates from speckle amplitude that has the Gaussian statistics and the intensity is proportional to the absolute square of the amplitude. On the other hand, polychromatic speckle patterns, which can be obtained by summarizing several independent monochromatic speckle patterns, are described by gamma distribution G(k), where k is a number of independent monochromatic speckle patterns. A monochromatic speckle pattern is a special case of G(k) where k¼1. Fig. 2 demonstrates the probability density functions of the intensity for speckle patterns that we will examine in the paper. Suppose we have n independent intensity measurements (in our case n can be considered as the number of independent speckles N where intensity is not correlated; we also suppose that speckle is an ergodic process and the assemble averaging can be replaced by a spatial averaging). According to the central limit theorem, we can estimate the distribution of /ISn (do not confuse with the distribution of I) as normal for a large n and the difference between /ISn and /IS (i.e. the error of mean as denoted by e/IS) is in the order of sI/On [15]
e/IS ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sI //IS2n S/IS2 ¼ pffiffiffi n
ð2Þ
Fig. 3. Error of standard deviation vs. sample size n.
The central limit theorem does not specify the distribution of intensity I, so Eq. (2) is valid for monochromatic and polychromatic speckles; only the coefficient sI is different. If a polychromatic and a monochromatic speckle have the same /IS shown in Fig. 2, the standard deviation for the polychromatic speckle is Ok smaller than the corresponding standard deviation of the monochromatic speckle. This phenomenon is expected because polychromatic speckle is the sum of all monochromatic speckles. This summation is kind of averaging over k monochromatic speckles, and, hence, standard deviation decreases. The difference between sI and sIn is not governed by the central limit theorem and is more difficult to estimate. We use numerical simulation because its simplicity outweighs the advantage of an analytical calculation. In the simulation, n intensity values are generated according to the given exponential distribution and then the standard deviation sIn is calculated. Then the process is repeated m times for a large m (we set m to 10,000) to provide an average /ISm that is closed to the ideal mean value /IS, i.e. /ISm E/IS. All m values of sIn were used to estimate its distribution and statistics (including standard deviation). We characterize the error of standard deviation by esI ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /s2In Sm /sIn S2m , which depends on n as shown in Fig. 3. The graph provides a good idea how the accuracy depends on the speckle number. As expected, the error decreases while n increases. It has the same pattern as the error of mean intensity. Also one can see that the error is smaller for polychromatic speckles. This fact is not surprising because the process of the polychromatic speckle formation is kind of averaging over several k monochromatic speckle patterns as was mentioned above.
L. Tchvialeva et al. / Optics and Lasers in Engineering 49 (2011) 1397–1401
0.4
0.4 Monte-Carlo Linear Fit Quadratic Fit
0.2 0.0 0.0
0.4 0.2 0.6 Error of Mean
0.8
Monte-Carlo Linear Fit Quadratic Fit
Error of StDev
0.6
Error of StDev
Error of StDev
0.8
0.2
0.0 0.0
0.2 Error of Mean
Monte-Carlo Linear Fit Quadratic Fit
0.2
0.0 0.0
0.4
1399
0.2 Error of Mean
Fig. 4. Error of standard deviation vs. error of mean intensity for speckle patterns with different intensity distribution: (a) exponential or G(1), (b) gamma distribution G(k) with k¼ 4; (c) gamma G(k) with k¼10. The error values generated by Monte-Carlo simulations are represented by circles. The liner and quadratic regression fits are indicated by the solid and dashed lines, respectively.
Table 1 Slopes and correlation coefficients for the standard deviation error and mean intensity error. Intensity distribution
G(1)
G(4)
G(10)
Slope a Correlation coefficient r
1.27 0.70
0.91 0.53
0.81 0.38
It is worth the effort to find the correlation between the error of standard deviation and the error of mean. If this correlation exists we could easily express esI through e/IS and estimate the total contrast error. The dependence of esI over e/IS is shown in Fig. 4. Regression can be very well approximated by a second order polynomial function. But for n410, which is common in all practical speckle measurements, the linear regression model also works well. For n 410, the error of standard deviation can be approximated by the function esI ¼asI/On, where a, for different distributions, is given in Table 1. As one can see from Table 1, the uncertainness of sI has the same order as the uncertainness of /IS. For exponential distribution, the correlation between standard deviation error and mean error can be fitted by a linear function whose slope equals to 1.27. In other words, esI E1.27e/IS ¼1.27sI/On ¼1.27/IS/On. Finally, we can evaluate the error of contrast eC due to the limited amount of speckles as e 2 C
C
¼
esI sI
2 þ
e/IS
2
/IS
2r
esI e/IS sI /IS
ð3Þ
3. Error due to inappropriate level of mean intensity One of the challenges in measuring speckle patterns is that one never knows the maximum level of intensity appearing in a stochastic intensity pattern. This maximum value should be consistent with the dynamic range of the camera used in the experiment. All that we can use is an estimation of the average intensity. Therefore, choosing the appropriate gain becomes an important issue: if gain is too small, the dynamic range of the CCD camera is underutilized and we lose accuracy of the measurements (Fig. 5a); if gain is too large, signal will be saturated that distorts the calculation of average intensity, and, hence, the finally contrast (Fig. 5c). In this section, we discuss errors due to intensity saturation and we present the optimal gain for speckle measurements (Fig. 5b). Let us suppose that the dynamic range of a camera is limited by the maximum value Icut. It means that all pixels where I4Icut will have the intensity Icut instead of I. How will this truncation affect the estimation of /IS? What should the relationship be between /IS and Icut, in order to avoid contrast reduction? Fig. 6 illustrates the concept of the problem. Before registration, the speckle pattern is described by the probability density function (pdf) p(I) of the intensity I, that provides the average intensity /IS. After registration, all intensity values higher than Icut are truncated to Icut. The new pdf is now described as ( pðIÞ if I oIcut pcut ðIÞ ¼ ð5Þ AdðIIcut Þ if I ZIcut R1 where A ¼ Icut pðIÞdI and d(I Icut) is the delta-function. The mean intensities of the two pdfs can be expressed as Z 1 /IS ¼ IpðIÞdI ð6Þ 0
where r is a correlation coefficient between /IS and sI. We found that this correlation really exists and some of its values are shown in Table 1. The positive correlation actually reduces the relative contrast error according to the third term of Eq. (3). Combining the Eqs. (3) and (2) with the coefficients from Table 1 allow us to calculate the error for polychromatic speckle pattern contrast. For a rough estimation, one can neglect the correlation between errors and suppose that esI Eae/IS. From (3) and (2) we can obtain
eC C
¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þC 2 n
ð4Þ
For example, if we assume that C equals to 1, for a typical 8-bit camera with 1600 1200 pixels and speckle size about 10 pixels in both directions, the contrast error for monochromatic speckles is about 1%.
/Icut S ¼
Z
1
Ipcut ðIÞdI
ð7Þ
0
The difference between the actual /IS and the measured /IcutS represents the absolute error due to saturation effect Z 1 D o I 4 ¼ /IS/Icut S ¼ I½pðIÞpcut ðIÞdI ð8Þ 0
Substituting (5) in (8), we can evaluate the error as Z 1 D/IS ¼ pðIÞðIIcut ÞdI
ð9Þ
Icut
Let us analyze monochromatic speckles, which have an analytic formulation for p(I) as pðIÞ ¼
1 I=/IS e /IS
ð10Þ
1400
L. Tchvialeva et al. / Optics and Lasers in Engineering 49 (2011) 1397–1401
300
300
250
250
200
200
150
150
100
100
50
50 0
0 1
51
101 151 201 251 301 351 401 451
1
51
101 151 201 251 301 351 401 451
300 250 200 150 100 50 0 1
51
101 151 201 251 301 351 401 451
Fig. 5. Speckle intensity over the camera pixels for low (a), optimal (b), and saturated (c) signals.
Fig. 7. Error of mean intensity vs. the mean intensity /IS due to saturation. Fig. 6. Probability density function of intensity for original speckle pattern (thick dashed line) and after registration (thin solid line). Exponential distribution was selected for simplicity.
is given below:
D/IS ¼ After substitution (10) in (9) and solving the integral, we obtain
D/IS ¼ /ISeIcut =/IS
ð11Þ
Generally speaking, similar calculations can be performed for any G distribution as shown in Fig. 1. The result will be an exponent function multiplied by a polynomial of degree k of Icut or a rational function of /IS. An example for G(4) distribution
o eIcut =B n 4 BIcut þð4BIcut Þ½BI3cut þ 3B2 I2cut þ6B3 Icut þ 6B4 4 B 3! ð12Þ
where B ¼4/IS. Dependence of D/IS from the mean intensity /IS is shown in Fig. 7. The smooth line is calculated according to Eq. (12); the wavy line is a Monte-Carlo simulation. Both lines overlay nicely. The saturation level is selected as Icut ¼255 to simulate the maximum intensity of an 8-bit camera.
L. Tchvialeva et al. / Optics and Lasers in Engineering 49 (2011) 1397–1401
Error
100
1401
Mean Γ (1) StDev Γ (1) Mean Γ (4) StDev Γ (4) Mean Γ (10) StDev Γ (10)
50
0 0
50
100 150 Mean Intensity
200
250
Fig. 8. Comparison of mean intensity and standard deviation errors due to saturation for different distribution of intensity using Monte-Carlo simulations.
Standard deviation errors are also calculated. Fig. 8 compares such errors for three different intensity distributions G(1), G(4), and G(10). Monochromatic speckles are found to have the largest level of error. If we do not want the error due to saturation seriously affect our measurements, we should demand that the error D/IS does not exceed other experimental errors such as D/IS r e. Let us compare D/IS with another factor that could limit the accuracy of /IS: that is the number of available speckles discussed in the previous section. Let us assume that there are N numbers of speckles. If the standard deviation of the intensity I is known, the standard error e of /IS is 1/ON of the standard deviation for monochromatic speckle; that is e ¼/IS/ON. Taking into the consideration of the depth of a camera pixel and set Icut to the maximum intensity 2m 1, we obtain an estimation for the optimal average intensity as /ISr 2ðm þ 1Þ =ln N
ð13Þ
For a typical 8-bit camera with 1600 1200 pixels and speckle size about 10 pixels in each direction, the estimated optimal average intensity is about 50 units. This estimation is consistent with the exponential distribution G(1) shown in Fig. 8. Average level for G(k), kZ4 distributed intensity can be selected even higher, up to 100–120 units for a registration device with a dynamic range of 256 levels.
4. Conclusion In this paper, we examined two factors, which may reduce the accuracy of speckle contrast measurements. Using Monte-Carlo simulation, we showed that the error of the intensity standard deviation is of the same order as the error of the mean intensity measurements. Therefore the total contrast error is inversely proportional to the square root of the speckle number—the same rule that the average intensity obeys. Another factor addressed was the mean speckle intensity. Although a higher intensity provides a better signal to noise ratio, the optimal intensity value is restricted due to the limited dynamic range of registration device. The derived formula of the optimal mean intensity depends on the pixel depth of a CCD camera and the total number of speckle recorded. Analyzing these two factors allows one to optimize the experimental setup for speckle measurements.
Acknowledgments This work was supported in part by grants from the Canadian Institutes of Health Research, Natural Sciences and Engineering Research Council of Canada, Canadian Dermatology Foundation, and UBC Faculty of Medicine. References [1] Briers JD. Surface roughness evaluation. In: Sirohi RS, editor. Speckle metrology. CRC Press; 1993. [2] Goodman JW. Speckle phenomena in optics: theory and application. Roberts and Company Publishers; 2006. [3] Tuchin VV. Handbook of optical biomedical diagnostics. Bellingham, WA: SPIE Optical Engineering Press; 2002. [4] Gossage KW, Smith CM, Kanter EM, Hariri LP, Stone AL, Rodriguez JJ, et al. Texture analysis of speckle in optical coherence tomography images of tissue phantoms. Physics in Medicine and Biology 2006;51:1563–75. [5] Hun C, Caussignac J-M, Bruynooghe MM. Speckle techniques for pavement surface analysis. Proceedings of the SPIE 2004;4933. [6] McKinney JD, Webster KJ, Webb KJ, Weiner AM. Characterization and imaging in optically scattering media by use of laser speckle and a variable-coherence source. Optics Letters 2000;25:4–6. [7] Zimnyakov DA, Agafonov DN, Sviridov AP. Speckle-contrast monitoring of tissue thermal modification. Applied Optics 2002;41:5989–96. [8] Nothdurft R, Yao G. Imaging obscured subsurface inhomogeneity using laser speckle. Optics Express 2005;13:10034–9. [9] Nadkarni SK, Bouma BE, Yelin D, Gulati A, Tearney GJ. Laser speckle imaging of atherosclerotic plaques through optical fiber bundles. Journal of Biomedical Optics. 2008;13:054016. [10] Boas DA, Dunn AK. Laser speckle contrast imaging in biomedical optics. Journal of Biomedical Optics 2010;15:011109. [11] Tchvialeva L, Zeng H, Markhvida I, McLean DI, Lui H, Lee TK. Skin roughness assessment. In: Campolo D, editor. New developments in biomedical engineering. INTECH. /http://www.sciyo.com/articles/show/title/skin-roughnessassessmentS. [12] Kirkpatrick SJ, Duncan DD, Wells-Gray EM. Detrimental effects of specklepixel size matching in laser speckle contrast imaging. Optics Letters 2008;33: 2886–8. [13] Markhvida I, Tchvialeva L, Lee TK, Zeng H. Influence of geometry on polychromatic speckle contrast. Journal of the Optical Society of America A, Optics Image Science and Vision 2007;24:93–7. [14] Tchvialeva L, Markhvida I, Zeng H, McLean DI, Lui H, Lee TK. Surface roughness measurement by speckle contrast under the illumination of light with arbitrary spectral profile. Optics and Lasers in Engineering 2010;48: 774–8. [15] Rice JA. Mathematical statistics and data analysis. 3rd ed. Australia, Thompson: Brooks/Cole; 2007.
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Error analysis for polychromatic speckle contrast measurements L. Tchvialeva a, I. Markhvida a, T.K. Lee a,b,c,n a b c
Photomedicine Institute, Department of Dermatology and Skin Science, Vancouver Coastal Health Research Institute and University of British Columbia, Vancouver, Canada Cancer Control Research, BC Cancer Research Centre, Vancouver, Canada School of Computing Science, Simon Fraser University, Burnaby, Canada
a r t i c l e i n f o
abstract
Article history: Received 27 April 2011 Received in revised form 4 July 2011 Accepted 14 July 2011 Available online 31 July 2011
Revival of interest in speckle technologies raises a curial question on the accuracy calculation of speckle measurements. In particular, the accuracy calculation of speckle contrast, an important metrics of a stochastic process, is quite different from the accuracy calculation of a typical intensity measurement. Speckle contrast depends more on stochastic characteristics of a process rather than on hardware characteristics. In this article, we consider errors introduced by the limited number of available speckle and by intensity saturation for monochromatic and polychromatic speckles. Equations for these types of errors were derived. Particularly, we show that the error due to limited number of speckles is inversely proportional to the square root of the speckle number, in the similar way as the average intensity error. For the error due to limited dynamic range of a recording device, the truncation effect increases the error as the mean intensity increased. Monochromatic speckles are more sensitive to such truncation than polychromatic speckles. We report the optimal mean intensity in terms of the pixel depth of a recording device and the total number of speckle recorded. In addition, a recommendation to minimize the saturation error for a typical camera is included. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Speckle contrast Error analysis Polychromatic speckle
1. Introduction Laser speckle is an important technique in surface metrology and engineering [1], astronomy [2], and biological tissue evaluation [3]. Comparing to other common optical methods, speckle techniques are faster, cheaper and simpler due to their simple setup: mainly a laser and a registration device are required. A typical experimental setup for speckle techniques is shown in Fig. 1a and an example of a speckle pattern is shown in Fig. 1b. Despite the speckle theory was established a couple of decades ago [1], the theoretical formulations had not been developed into practical instruments in the early years due to technical limitations. Recent advances in light sources and registration devices have now revived interest in speckle techniques [4–11]. Typically speckle patterns are usually characterized by statistical moments of intensities. In this paper, we will focus on speckle contrast and address some factors influencing its measurement. Speckle contrast is defined as the ratio of the standard deviation sI of light intensity over the average intensity /IS [2]. C¼
sI /IS
ð1Þ
n Corresponding author at: Cancer Control Research, BC Cancer Research Centre Program, 675 West 10th Avenue, Vancouver, BC, Canada V5Z 1L3. Tel.: 1 604 675 8053; fax: 1 604 675 8180. E-mail address: [email protected] (T.K. Lee).
0143-8166/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2011.07.008
where /yS designates averaging over ensemble. It is a widely used metrics applied for pavement surface analysis [5], estimation of tissue optical properties [6], monitoring of thermal tissue conditions [7], internal inhomogeneities visualization [8], viscoelasticity of atherosclerotic plaques measurements [9], blood flow evaluation [10] and in-vivo skin surface roughness assessment [11]. Contrast can be affected by the light source, the targeted object and the optical system [2] in various ways; these factors make accurate contrast estimations a challenge in terms of experiment design. Thus, a design should be thoroughly analyzed and all secondary factors affecting contrast must be removed or properly accounted for. The apparatus factors such as temporal and spatial averaging, the geometry of setup has been already considered [10,12–14]. In this paper, we further examine the errors introduced by the limited number of speckles available for contrast calculation, and by intensity saturation. The paper is organized in a following way. In Section 2, we analyze the speckle contrast error due to a finite number of available speckles for registration. Section 3 deals with contrast reduction caused by limiting the dynamic range of measured intensities. Also, the optimal mean intensity in such a situation is derived. A short conclusion is followed in Section 4.
2. Error due to limited amount of speckles Despite the simplicity of Eq. (1), estimating the contrast accuracy in experiments is not trivial because the accuracy of /IS is not the
1398
L. Tchvialeva et al. / Optics and Lasers in Engineering 49 (2011) 1397–1401
Fig. 1. Typical speckle setup geometry and an example of a speckle pattern.
Fig. 2. Intensity distributions for monochromatic and polychromatic speckle patterns.
same as the accuracy of I. The former variable depends not only on the accuracy of intensity measurements, but also on the stochastic nature of speckle. In order to estimate /IS, we compute /ISn, where n denotes the number of independent intensity measurements available for estimating the mean value. A similar procedure is applied for the estimation of sI: we calculate n independent standard deviations sIn. Both /ISn and sIn are stochastic variables, whose behavior depend on the distribution of I and the sample size n. We evaluate the above estimations using monochromatic (Gaussian) speckle and polychromatic speckle patterns. The former pattern is well studied and has an exponential probability density function for intensity [2]. Note that the term Gaussian originates from speckle amplitude that has the Gaussian statistics and the intensity is proportional to the absolute square of the amplitude. On the other hand, polychromatic speckle patterns, which can be obtained by summarizing several independent monochromatic speckle patterns, are described by gamma distribution G(k), where k is a number of independent monochromatic speckle patterns. A monochromatic speckle pattern is a special case of G(k) where k¼1. Fig. 2 demonstrates the probability density functions of the intensity for speckle patterns that we will examine in the paper. Suppose we have n independent intensity measurements (in our case n can be considered as the number of independent speckles N where intensity is not correlated; we also suppose that speckle is an ergodic process and the assemble averaging can be replaced by a spatial averaging). According to the central limit theorem, we can estimate the distribution of /ISn (do not confuse with the distribution of I) as normal for a large n and the difference between /ISn and /IS (i.e. the error of mean as denoted by e/IS) is in the order of sI/On [15]
e/IS ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sI //IS2n S/IS2 ¼ pffiffiffi n
ð2Þ
Fig. 3. Error of standard deviation vs. sample size n.
The central limit theorem does not specify the distribution of intensity I, so Eq. (2) is valid for monochromatic and polychromatic speckles; only the coefficient sI is different. If a polychromatic and a monochromatic speckle have the same /IS shown in Fig. 2, the standard deviation for the polychromatic speckle is Ok smaller than the corresponding standard deviation of the monochromatic speckle. This phenomenon is expected because polychromatic speckle is the sum of all monochromatic speckles. This summation is kind of averaging over k monochromatic speckles, and, hence, standard deviation decreases. The difference between sI and sIn is not governed by the central limit theorem and is more difficult to estimate. We use numerical simulation because its simplicity outweighs the advantage of an analytical calculation. In the simulation, n intensity values are generated according to the given exponential distribution and then the standard deviation sIn is calculated. Then the process is repeated m times for a large m (we set m to 10,000) to provide an average /ISm that is closed to the ideal mean value /IS, i.e. /ISm E/IS. All m values of sIn were used to estimate its distribution and statistics (including standard deviation). We characterize the error of standard deviation by esI ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /s2In Sm /sIn S2m , which depends on n as shown in Fig. 3. The graph provides a good idea how the accuracy depends on the speckle number. As expected, the error decreases while n increases. It has the same pattern as the error of mean intensity. Also one can see that the error is smaller for polychromatic speckles. This fact is not surprising because the process of the polychromatic speckle formation is kind of averaging over several k monochromatic speckle patterns as was mentioned above.
L. Tchvialeva et al. / Optics and Lasers in Engineering 49 (2011) 1397–1401
0.4
0.4 Monte-Carlo Linear Fit Quadratic Fit
0.2 0.0 0.0
0.4 0.2 0.6 Error of Mean
0.8
Monte-Carlo Linear Fit Quadratic Fit
Error of StDev
0.6
Error of StDev
Error of StDev
0.8
0.2
0.0 0.0
0.2 Error of Mean
Monte-Carlo Linear Fit Quadratic Fit
0.2
0.0 0.0
0.4
1399
0.2 Error of Mean
Fig. 4. Error of standard deviation vs. error of mean intensity for speckle patterns with different intensity distribution: (a) exponential or G(1), (b) gamma distribution G(k) with k¼ 4; (c) gamma G(k) with k¼10. The error values generated by Monte-Carlo simulations are represented by circles. The liner and quadratic regression fits are indicated by the solid and dashed lines, respectively.
Table 1 Slopes and correlation coefficients for the standard deviation error and mean intensity error. Intensity distribution
G(1)
G(4)
G(10)
Slope a Correlation coefficient r
1.27 0.70
0.91 0.53
0.81 0.38
It is worth the effort to find the correlation between the error of standard deviation and the error of mean. If this correlation exists we could easily express esI through e/IS and estimate the total contrast error. The dependence of esI over e/IS is shown in Fig. 4. Regression can be very well approximated by a second order polynomial function. But for n410, which is common in all practical speckle measurements, the linear regression model also works well. For n 410, the error of standard deviation can be approximated by the function esI ¼asI/On, where a, for different distributions, is given in Table 1. As one can see from Table 1, the uncertainness of sI has the same order as the uncertainness of /IS. For exponential distribution, the correlation between standard deviation error and mean error can be fitted by a linear function whose slope equals to 1.27. In other words, esI E1.27e/IS ¼1.27sI/On ¼1.27/IS/On. Finally, we can evaluate the error of contrast eC due to the limited amount of speckles as e 2 C
C
¼
esI sI
2 þ
e/IS
2
/IS
2r
esI e/IS sI /IS
ð3Þ
3. Error due to inappropriate level of mean intensity One of the challenges in measuring speckle patterns is that one never knows the maximum level of intensity appearing in a stochastic intensity pattern. This maximum value should be consistent with the dynamic range of the camera used in the experiment. All that we can use is an estimation of the average intensity. Therefore, choosing the appropriate gain becomes an important issue: if gain is too small, the dynamic range of the CCD camera is underutilized and we lose accuracy of the measurements (Fig. 5a); if gain is too large, signal will be saturated that distorts the calculation of average intensity, and, hence, the finally contrast (Fig. 5c). In this section, we discuss errors due to intensity saturation and we present the optimal gain for speckle measurements (Fig. 5b). Let us suppose that the dynamic range of a camera is limited by the maximum value Icut. It means that all pixels where I4Icut will have the intensity Icut instead of I. How will this truncation affect the estimation of /IS? What should the relationship be between /IS and Icut, in order to avoid contrast reduction? Fig. 6 illustrates the concept of the problem. Before registration, the speckle pattern is described by the probability density function (pdf) p(I) of the intensity I, that provides the average intensity /IS. After registration, all intensity values higher than Icut are truncated to Icut. The new pdf is now described as ( pðIÞ if I oIcut pcut ðIÞ ¼ ð5Þ AdðIIcut Þ if I ZIcut R1 where A ¼ Icut pðIÞdI and d(I Icut) is the delta-function. The mean intensities of the two pdfs can be expressed as Z 1 /IS ¼ IpðIÞdI ð6Þ 0
where r is a correlation coefficient between /IS and sI. We found that this correlation really exists and some of its values are shown in Table 1. The positive correlation actually reduces the relative contrast error according to the third term of Eq. (3). Combining the Eqs. (3) and (2) with the coefficients from Table 1 allow us to calculate the error for polychromatic speckle pattern contrast. For a rough estimation, one can neglect the correlation between errors and suppose that esI Eae/IS. From (3) and (2) we can obtain
eC C
¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þC 2 n
ð4Þ
For example, if we assume that C equals to 1, for a typical 8-bit camera with 1600 1200 pixels and speckle size about 10 pixels in both directions, the contrast error for monochromatic speckles is about 1%.
/Icut S ¼
Z
1
Ipcut ðIÞdI
ð7Þ
0
The difference between the actual /IS and the measured /IcutS represents the absolute error due to saturation effect Z 1 D o I 4 ¼ /IS/Icut S ¼ I½pðIÞpcut ðIÞdI ð8Þ 0
Substituting (5) in (8), we can evaluate the error as Z 1 D/IS ¼ pðIÞðIIcut ÞdI
ð9Þ
Icut
Let us analyze monochromatic speckles, which have an analytic formulation for p(I) as pðIÞ ¼
1 I=/IS e /IS
ð10Þ
1400
L. Tchvialeva et al. / Optics and Lasers in Engineering 49 (2011) 1397–1401
300
300
250
250
200
200
150
150
100
100
50
50 0
0 1
51
101 151 201 251 301 351 401 451
1
51
101 151 201 251 301 351 401 451
300 250 200 150 100 50 0 1
51
101 151 201 251 301 351 401 451
Fig. 5. Speckle intensity over the camera pixels for low (a), optimal (b), and saturated (c) signals.
Fig. 7. Error of mean intensity vs. the mean intensity /IS due to saturation. Fig. 6. Probability density function of intensity for original speckle pattern (thick dashed line) and after registration (thin solid line). Exponential distribution was selected for simplicity.
is given below:
D/IS ¼ After substitution (10) in (9) and solving the integral, we obtain
D/IS ¼ /ISeIcut =/IS
ð11Þ
Generally speaking, similar calculations can be performed for any G distribution as shown in Fig. 1. The result will be an exponent function multiplied by a polynomial of degree k of Icut or a rational function of /IS. An example for G(4) distribution
o eIcut =B n 4 BIcut þð4BIcut Þ½BI3cut þ 3B2 I2cut þ6B3 Icut þ 6B4 4 B 3! ð12Þ
where B ¼4/IS. Dependence of D/IS from the mean intensity /IS is shown in Fig. 7. The smooth line is calculated according to Eq. (12); the wavy line is a Monte-Carlo simulation. Both lines overlay nicely. The saturation level is selected as Icut ¼255 to simulate the maximum intensity of an 8-bit camera.
L. Tchvialeva et al. / Optics and Lasers in Engineering 49 (2011) 1397–1401
Error
100
1401
Mean Γ (1) StDev Γ (1) Mean Γ (4) StDev Γ (4) Mean Γ (10) StDev Γ (10)
50
0 0
50
100 150 Mean Intensity
200
250
Fig. 8. Comparison of mean intensity and standard deviation errors due to saturation for different distribution of intensity using Monte-Carlo simulations.
Standard deviation errors are also calculated. Fig. 8 compares such errors for three different intensity distributions G(1), G(4), and G(10). Monochromatic speckles are found to have the largest level of error. If we do not want the error due to saturation seriously affect our measurements, we should demand that the error D/IS does not exceed other experimental errors such as D/IS r e. Let us compare D/IS with another factor that could limit the accuracy of /IS: that is the number of available speckles discussed in the previous section. Let us assume that there are N numbers of speckles. If the standard deviation of the intensity I is known, the standard error e of /IS is 1/ON of the standard deviation for monochromatic speckle; that is e ¼/IS/ON. Taking into the consideration of the depth of a camera pixel and set Icut to the maximum intensity 2m 1, we obtain an estimation for the optimal average intensity as /ISr 2ðm þ 1Þ =ln N
ð13Þ
For a typical 8-bit camera with 1600 1200 pixels and speckle size about 10 pixels in each direction, the estimated optimal average intensity is about 50 units. This estimation is consistent with the exponential distribution G(1) shown in Fig. 8. Average level for G(k), kZ4 distributed intensity can be selected even higher, up to 100–120 units for a registration device with a dynamic range of 256 levels.
4. Conclusion In this paper, we examined two factors, which may reduce the accuracy of speckle contrast measurements. Using Monte-Carlo simulation, we showed that the error of the intensity standard deviation is of the same order as the error of the mean intensity measurements. Therefore the total contrast error is inversely proportional to the square root of the speckle number—the same rule that the average intensity obeys. Another factor addressed was the mean speckle intensity. Although a higher intensity provides a better signal to noise ratio, the optimal intensity value is restricted due to the limited dynamic range of registration device. The derived formula of the optimal mean intensity depends on the pixel depth of a CCD camera and the total number of speckle recorded. Analyzing these two factors allows one to optimize the experimental setup for speckle measurements.
Acknowledgments This work was supported in part by grants from the Canadian Institutes of Health Research, Natural Sciences and Engineering Research Council of Canada, Canadian Dermatology Foundation, and UBC Faculty of Medicine. References [1] Briers JD. Surface roughness evaluation. In: Sirohi RS, editor. Speckle metrology. CRC Press; 1993. [2] Goodman JW. Speckle phenomena in optics: theory and application. Roberts and Company Publishers; 2006. [3] Tuchin VV. Handbook of optical biomedical diagnostics. Bellingham, WA: SPIE Optical Engineering Press; 2002. [4] Gossage KW, Smith CM, Kanter EM, Hariri LP, Stone AL, Rodriguez JJ, et al. Texture analysis of speckle in optical coherence tomography images of tissue phantoms. Physics in Medicine and Biology 2006;51:1563–75. [5] Hun C, Caussignac J-M, Bruynooghe MM. Speckle techniques for pavement surface analysis. Proceedings of the SPIE 2004;4933. [6] McKinney JD, Webster KJ, Webb KJ, Weiner AM. Characterization and imaging in optically scattering media by use of laser speckle and a variable-coherence source. Optics Letters 2000;25:4–6. [7] Zimnyakov DA, Agafonov DN, Sviridov AP. Speckle-contrast monitoring of tissue thermal modification. Applied Optics 2002;41:5989–96. [8] Nothdurft R, Yao G. Imaging obscured subsurface inhomogeneity using laser speckle. Optics Express 2005;13:10034–9. [9] Nadkarni SK, Bouma BE, Yelin D, Gulati A, Tearney GJ. Laser speckle imaging of atherosclerotic plaques through optical fiber bundles. Journal of Biomedical Optics. 2008;13:054016. [10] Boas DA, Dunn AK. Laser speckle contrast imaging in biomedical optics. Journal of Biomedical Optics 2010;15:011109. [11] Tchvialeva L, Zeng H, Markhvida I, McLean DI, Lui H, Lee TK. Skin roughness assessment. In: Campolo D, editor. New developments in biomedical engineering. INTECH. /http://www.sciyo.com/articles/show/title/skin-roughnessassessmentS. [12] Kirkpatrick SJ, Duncan DD, Wells-Gray EM. Detrimental effects of specklepixel size matching in laser speckle contrast imaging. Optics Letters 2008;33: 2886–8. [13] Markhvida I, Tchvialeva L, Lee TK, Zeng H. Influence of geometry on polychromatic speckle contrast. Journal of the Optical Society of America A, Optics Image Science and Vision 2007;24:93–7. [14] Tchvialeva L, Markhvida I, Zeng H, McLean DI, Lui H, Lee TK. Surface roughness measurement by speckle contrast under the illumination of light with arbitrary spectral profile. Optics and Lasers in Engineering 2010;48: 774–8. [15] Rice JA. Mathematical statistics and data analysis. 3rd ed. Australia, Thompson: Brooks/Cole; 2007.