- Email: [email protected]
Hurst exponent determination for digital speckle patterns in roughness control of metallic surfaces
Optics and Lasers in Engineering 49 (2011) 32–35
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Hurst exponent determination for digital speckle patterns in roughness control of metallic surfaces A.L. Sampaio a, D.C. Loba~ o a, L.C.S. Nunes c, P.A.M. dos Santos d, L. Silva a,b, J.A.O. Huguenin a,b,n a
´rio de Volta Redonda—UFF, Brazil ´lo Universita Departamento de Ciˆencias Exatas, Po ´rio de Volta Redonda—UFF, Brazil ´lo Universita Departamento de Fı´sica, ICEx, Po c Departamento de Engenharia Mecˆ anica(TEM/PGMEC), Escola de Engenharia—UFF, Brazil d Instituto de Fı´sica—UFF, Brazil b
a r t i c l e in f o
a b s t r a c t
Article history: Received 20 July 2010 Received in revised form 3 September 2010 Accepted 3 September 2010 Available online 25 September 2010
In this paper, we present a study of metallic surface roughness using the Hurst exponent calculated from speckle pattern. A set of samples was prepared using polishing techniques and the roughness was directly measured by means of an optical profilometer. To study the H exponent, an experiment was performed by illuminating the samples using an expanded laser beam and the surface image was captured by a CCD camera. We applied techniques of the Hurst exponent calculation, traditionally calculated from surface profile, in the digitalized speckle patterns generated by the rough surfaces. We showed a clear dependence of the H exponent on roughness of the samples. We demonstrated that this tool is very sensitive to defects in the surfaces and can be used for roughness control. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Roughness surfaces Speckle patterns Hurst exponent Digital image
1. Introduction Nowadays, material surface properties are very important for industry and technological and scientific development in general. Particularly, surface roughness is one of the most widely studied property. There are several methods to investigate it. Optical methods, for instance, offer many significant advantages because they are non-destructive, of non-contact type and accurate; in some cases they are the simplest techniques to use. Even today, decades after their discovery, laser speckle patterns are one of the powerful tools used to analyze and characterize the surface roughness parameters. Laser speckle patterns are produced by highly coherent light scattering from the roughness surface topography. The standard technique used to determine the properties of rough surfaces is based on a mechanical profilometer. The resolution of this instrument depends on physical characteristics of the needle tip. In addition, this technique presents disadvantages such as contact measurement and large time. However, in industrial application a non-contact technique with real-time measurement is required. In this way, several optical techniques,
n Corresponding author at: Departamento de Fu´sica, ICEx, Polo Universita´rio de Volta Redonda, UFF, 27255-125 Volta Redonda, Rio de Janeiro, Brazil. Tel.: + 55 24 21073576; fax: +55 24 33443019. E-mail address: [email protected] (J.A.O. Huguenin).
0143-8166/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2010.09.005
based on speckle effect, have been developed. Optical interferometry technique is usually employed for roughness measuring range of less than 0.1 mm. In the range between 0.1 and 3 mm, generally, scattered technique is used. Finally, in the range up to 3 mm the speckle contrast technique is applied. In the last few years one can identify a promise revival in the optical literature concerning speckle patterns and their applications. Surface profiles of the machined surface (ground) are measured using the fringe projection technique [1]. A method based on spatial-average analysis of the objective speckle pattern in the specular direction has been proposed [2]. This method is simply called the spatial-average method (SAM). Furthermore, similarly, there are the speckle contrast method (SCM) and the light scattering method (LSM). In another interesting work [3], an experimental approach for surface roughness measurement based on the coherent speckle scattering pattern caused by a laser beam on machined surfaces (grinding and milling) was developed. Studies of metallic rough surfaces were made using digitalized speckle pattern [4]. A correlation technique is also provided [5] to analyze full-field surface roughness; in this case the measurement is made by means of laser speckle determination. Measurement of surface roughness using angular speckle correlation [6] on machined surfaces is also possible. The traditional speckle contrast technique has been used with different approaches like utilization of light with arbitrary spectral profile [7] in order to measure surface roughness. Concerning the Hurst exponent [8], its use in surface science is common [9] in the context of
A.L. Sampaio et al. / Optics and Lasers in Engineering 49 (2011) 32–35
roughness measurement. In these cases the surface profile is used to calculate the H exponent [10]. In the study of time series, stochastic properties are analyzed by means of the Hurst exponent [11]. In the present paper we use the well-known Hurst exponent applied to digital image of a speckle pattern. Calculation of the H exponent for a digital image is applied in the speckle pattern obtained from the scattering of a laser beam onto a rough surface. We used this approach to study the dependence of H exponent calculated on roughness of metallic surfaces and its use as a roughness control parameter. In Section 2 we discuss the method for calculation of the H exponent and define how to proceed in order to calculate it from a digital image. In Section 3, the experimental apparatus is presented. Results and discussions are presented in Section 4. Finally, conclusions are presented in Section 5.
2. Theory The rescaled range analysis (R/S), a tool for studying long-term memory and fractality of a time series, was first introduced by Hurst [8] in hydrology for studying the Nile River and water storage [11]. Mandelbrot [12] discussed that (R/S) analysis is a more powerful tool in detecting long range dependence when compared with the more conventional analyses such as autocorrelation analysis, variance ratios and spectral analysis. Surface roughness description has two basic geometrical features: random aspect and structural one. The random aspect takes into account the fact that the rough surface can vary considerably in space in a random manner, and subsequently there is no spatial function being able to describe the geometrical form. The structural aspect is based on the fact that variances of roughness are not completely independent with respect to their spatial positions, but their correlation depends on distance. The classical roughness characteristics such as RMS roughness indicate that the density of summits and the mean absolute surface slope are functions of fractal dimension and cut-off frequencies only [10,11]. These roughness parameters are not intrinsic properties of a surface and vary with the conditions of measurement. Selfsimilar fractal curves and surfaces are described completely by a single parameter, known as the fractal dimension D, which is an intrinsic property of the surface and does not change with the scale of measurement. It has been shown that metal surfaces obey statistical scale invariance, known as self-affinity [11,12]. Such surfaces, which are assumed to be on average parallel to the (x,y) plane, are described here through a single valued function z ¼h(x,y). The self-affinity property means that the surface remains statistically invariant under the scaling transformation (x,y) to (lx, ly) and h(lx, ly)¼ lHh(x,y). The equal sign for h function is to be understood in the statistical sense as it gives rise to the same expected value for any observable computed through an average over space. H is the characteristic roughness exponent called Hurst’s exponent. Let x(p) be the pixel intensity of a speckle image at position p. Let X(p) be the accumulated values of x(p) from the mean o x 4 t (t ¼image rows), given by Xðp, tÞ ¼
p X
fxðuÞ- o x 4 t g
33
The standard deviation S is written as " #!1=2 p X 2 S ¼ 1=t fxðuÞ- o x 4 t g
ð3Þ
u¼1
Following the idea of Tchvialeva et al. [7], the data of speckle image intensity and surface roughness are related to the rescaled range, R/S, and to an exponent H, such that R=S ¼ ðtÞH
ð4Þ
The Hurst exponent H expresses the tendency of dh ¼[dh(x)/ dx]dx to change sign. Hurst exponent H is limited to the range 0oHo1. In order to calculate the H exponent of digital image, the algorithm presented in this section is implemented by taking the image data by column and the then averaging over the number of columns.
3. Experiment In this section speckle patterns produced by rough metallic surfaces were explored. We will show how the Hurst exponent responds to pattern changes due to surfaces with different roughness. In order to observe the dependence of H exponent of digital image speckle pattern on roughness of metallic surfaces, commercial aluminum was used to prepare 5 samples with different roughnesses. We used polishing techniques, taking sandpaper with grain size of 80, 220, 320, 800 and 1200 MESH. The procedure starts with polishing using the higher grain size sandpaper for all samples and using smaller grain size sandpaper successively. An optical profilometer (MahrR) was used to characterize the samples according to their roughness properties. The results of the arithmetic average roughness (Ra), mean roughness depth (Rz) and root mean square roughness (Rq) of each sample are shown in Table 1. Speckle at image plane is obtained using these samples, as sketched in Fig. 1. A 50 mW green laser beam is expanded using two lenses L1 and L2 in order to illuminate an area of approximately 1 cm2 of sample S. The illuminated area is Table 1 Result for direct roughness measurement, where Ra is arithmetic average roughness, Rz is roughness depth and Rq is root mean square roughness. Roughness/sandpaper (Mesh)
Ra (lm) Rq (lm) Rz (lm)
80
220
320
800
1200
0,39 7 0,02 0,49 7 0,01 2,06 7 0,07
0,28 70,01 0,35 70,02 1,36 70,05
0,207 01 0,257 0,01 1,147 0,08
0,14 70,01 0,18 70,01 0,89 70,06
0,127 0,01 0,167 0,01 0,757 0,03
ð1Þ
u¼1
The range R is the difference between the maximum and minimum amounts of intensity contained in a speckle image. This can be written as R ¼ maxXðp, tÞ-min Xðp, tÞ,
1 rp r t
ð2Þ
Fig. 1. Schematic diagram for experimental setup. L1 and L2: lens; S: sample.
34
A.L. Sampaio et al. / Optics and Lasers in Engineering 49 (2011) 32–35
Fig. 2. Speckle patterns produced by rough surface with Ra of 0.39 mm (A), 0.20 mm (B) and 0.12 mm (C).
captured by a monochromatic CCD camera with a resolution of 640(H) 480(V) and a pixel pitch of 11.0(H) 13.0(V) mm2. The pixels are sensitized in the visible spectrum. Normal direction is chosen to photograph the surface. The samples are placed on a holder in order to keep the angle of incidence constant. The reflected light in the specular direction can be used to monitor the incident angle. The experiment is performed by acquiring the surface image of the samples, keeping the laser intensity, the polarization and the angle of incidence fixed. In this way, roughness surface is the unique factor responsible for changes in the speckle patterns. Fig. 2 presents three typical image results of the samples with Ra equal to 0.39, 0.20 and 0.12 mm.
4. Results and discussion The mean H exponent, defined in Section 2, is calculated for the complete set of samples. A dependence of the H exponent on roughness is observed. Fig. 3 shows the mean H exponent as a function of arithmetic average roughness (Ra). A clear relation is obtained. It is important to stress that in our paper H exponent is calculated from the speckle pattern generated by rough surfaces and not by the profile function of these surfaces, as in the classical application. In the case of roughness measurement where Ra has large variations, the H exponent can be used as an indirect and noncontact measurement. Let us now study the case of a long sample produced using the 80 MESH sandpaper. We used a 300 mm long bar of the same material (aluminum) previously used. We polished the bar in the same direction. In order to investigate the responses of the H coefficient due to changes of roughness along the bar we introduced damages at different regions of the bar. This procedure is carried out using the same experimental arrangement illustrated in Fig. 1. The results for the H exponents at different positions of the bar are shown in Fig. 4. The x-axis indicates images of different regions of the bar. The y-axis shows the H exponent calculated from the speckle pattern of each image. In the images 1–7, the speckle pattern photographed is generated by regions of regular polished surface. The calculated H exponent does not vary significantly. Images 8–10 of the bar were obtained from a region where 3 slim risks were introduced orthogonally to the polishing direction. Picture 9 contains 3 damages. Images 8 and 10 contain only one of them. The H exponent grows in the presence of damages. It grows more significantly for image 9 because it contains all damaged regions, as mentioned before. Images 13 and 14 were re-polished using the 120 MESH sandpaper in an irregular manner. The irregular polishing process used here gave rise to imperfections, which produce a small growth of the Hurst exponent. Finally, after image 16 we started polishing in different directions and, at the end of the bar (images 19–21), we polished in a direction orthogonal to the original one. The H exponent exhibited a significant improvement for this change in the surface. The H
Fig. 3. Mean H exponent as a function of mean roughness Ra. Solid line: guide to the eye; dotted line: linear regression (RL).
Fig. 4. H mean exponent for different regions of bar polished with a 80 MESH sandpaper, where different damages were introduced at different points. The x-axis indicates captured images along the bar.
exponent grows in the images from regions that contain damages. We calculated the mean Hurst exponent for the images of the undamaged regions of the bar and it resulted in H¼0.530170.0034. The standard deviation is smaller than the increment in the Hurst exponent due to damages on the bar. It is interesting to note that the damages increase scattered light from the surfaces, giving rise to H improvement. This result can be understood as a consequence of scattered light intensity.
A.L. Sampaio et al. / Optics and Lasers in Engineering 49 (2011) 32–35
35
5. Conclusion
References
In conclusion, we have introduced the calculation of the mean Hurst exponent from digital images of speckle patterns produced by metallic rough surfaces. A clear dependence of the H exponent calculated on surface roughness was demonstrated. The H exponent was not calculated from the surface profile, but from the speckle patterns generated by the rough surfaces. For a long range of Ra the mean H exponent can be used as an indirect and non-contact roughness measurement. This technique can also be used to monitor rough surfaces. We have shown that for a long bar prepared using a unique sandpaper the mean H exponent is almost constant, with an exception of the ones calculated from a damaged region of the bar. These results can be used as a tool for roughness control.
[1] Dhanasekar B, Ramamorthy B. Digital speckle interferometry for assessment of surface roughness. Optics and Lasers in Engineering 2008;46:272–80. [2] Zhao X, Gao Z. Surface roughness measurement using spatial-average analysis of objective speckle pattern in specular direction. Optics and Lasers in Engineering 2009;47:1307–16. [3] Dhanasekar B, Mohan NK, Bhaduri B, Ramamorthy B. Evaluation of surface roughness based on monochromatic speckle correlation using image processing. Precision Engineering 2008;32:196–206. [4] Hamed AM, El-Ghandoor H, El-Diasty F, Saudy M. Analysis of speckle images to assess surface roughness. Optics & Laser Technology 2004;36:249–53. [5] Toh SL, Quan C, Woo KC, Tay CJ, Shang HM. Whole field surface roughness measurement by laser speckle correlation technique. Optics & Laser Technology 2001;33:427–34. [6] Persson Ulf. Surface roughness measurement on machined surfaces using angular speckle correlation. Journal of Materials Processing Technology 2006;180:233–8. [7] 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(7-8):774–8. [8] Hurst HE. Long-term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers 1965;116:770–9. [9] Baraba´si A-L, Stanley HE. Fractal concepts in surface growth. Cambridge: Cambridge University Press; 1995. [10] Wawszczak J. Methods for estimating the Hurst exponent. The analysis of its value for fracture surface research. Materials Science—Poland 2005;23: 585–91. [11] Fender J. Fractals. New York: Plenum Press; 1988. [12] Mandelbrot BB. The fractal geometry of nature. New York: W.H. Freeman; 1983.
Acknowledgements The authors thank Dr. B. Coutinho dos Santos for fruitful discussions. They thank Brazilian Founding Agencies FAPERJ (Fundac- a~ o Carlos Chagas Filho de Apoio a Pesquisa do Estado do Rio de Janeiro and CNPq (Conselho Nacionional Desenvolvimento Cientı´fico e Tecnolo´gico) for financial support.
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Hurst exponent determination for digital speckle patterns in roughness control of metallic surfaces A.L. Sampaio a, D.C. Loba~ o a, L.C.S. Nunes c, P.A.M. dos Santos d, L. Silva a,b, J.A.O. Huguenin a,b,n a
´rio de Volta Redonda—UFF, Brazil ´lo Universita Departamento de Ciˆencias Exatas, Po ´rio de Volta Redonda—UFF, Brazil ´lo Universita Departamento de Fı´sica, ICEx, Po c Departamento de Engenharia Mecˆ anica(TEM/PGMEC), Escola de Engenharia—UFF, Brazil d Instituto de Fı´sica—UFF, Brazil b
a r t i c l e in f o
a b s t r a c t
Article history: Received 20 July 2010 Received in revised form 3 September 2010 Accepted 3 September 2010 Available online 25 September 2010
In this paper, we present a study of metallic surface roughness using the Hurst exponent calculated from speckle pattern. A set of samples was prepared using polishing techniques and the roughness was directly measured by means of an optical profilometer. To study the H exponent, an experiment was performed by illuminating the samples using an expanded laser beam and the surface image was captured by a CCD camera. We applied techniques of the Hurst exponent calculation, traditionally calculated from surface profile, in the digitalized speckle patterns generated by the rough surfaces. We showed a clear dependence of the H exponent on roughness of the samples. We demonstrated that this tool is very sensitive to defects in the surfaces and can be used for roughness control. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Roughness surfaces Speckle patterns Hurst exponent Digital image
1. Introduction Nowadays, material surface properties are very important for industry and technological and scientific development in general. Particularly, surface roughness is one of the most widely studied property. There are several methods to investigate it. Optical methods, for instance, offer many significant advantages because they are non-destructive, of non-contact type and accurate; in some cases they are the simplest techniques to use. Even today, decades after their discovery, laser speckle patterns are one of the powerful tools used to analyze and characterize the surface roughness parameters. Laser speckle patterns are produced by highly coherent light scattering from the roughness surface topography. The standard technique used to determine the properties of rough surfaces is based on a mechanical profilometer. The resolution of this instrument depends on physical characteristics of the needle tip. In addition, this technique presents disadvantages such as contact measurement and large time. However, in industrial application a non-contact technique with real-time measurement is required. In this way, several optical techniques,
n Corresponding author at: Departamento de Fu´sica, ICEx, Polo Universita´rio de Volta Redonda, UFF, 27255-125 Volta Redonda, Rio de Janeiro, Brazil. Tel.: + 55 24 21073576; fax: +55 24 33443019. E-mail address: [email protected] (J.A.O. Huguenin).
0143-8166/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2010.09.005
based on speckle effect, have been developed. Optical interferometry technique is usually employed for roughness measuring range of less than 0.1 mm. In the range between 0.1 and 3 mm, generally, scattered technique is used. Finally, in the range up to 3 mm the speckle contrast technique is applied. In the last few years one can identify a promise revival in the optical literature concerning speckle patterns and their applications. Surface profiles of the machined surface (ground) are measured using the fringe projection technique [1]. A method based on spatial-average analysis of the objective speckle pattern in the specular direction has been proposed [2]. This method is simply called the spatial-average method (SAM). Furthermore, similarly, there are the speckle contrast method (SCM) and the light scattering method (LSM). In another interesting work [3], an experimental approach for surface roughness measurement based on the coherent speckle scattering pattern caused by a laser beam on machined surfaces (grinding and milling) was developed. Studies of metallic rough surfaces were made using digitalized speckle pattern [4]. A correlation technique is also provided [5] to analyze full-field surface roughness; in this case the measurement is made by means of laser speckle determination. Measurement of surface roughness using angular speckle correlation [6] on machined surfaces is also possible. The traditional speckle contrast technique has been used with different approaches like utilization of light with arbitrary spectral profile [7] in order to measure surface roughness. Concerning the Hurst exponent [8], its use in surface science is common [9] in the context of
A.L. Sampaio et al. / Optics and Lasers in Engineering 49 (2011) 32–35
roughness measurement. In these cases the surface profile is used to calculate the H exponent [10]. In the study of time series, stochastic properties are analyzed by means of the Hurst exponent [11]. In the present paper we use the well-known Hurst exponent applied to digital image of a speckle pattern. Calculation of the H exponent for a digital image is applied in the speckle pattern obtained from the scattering of a laser beam onto a rough surface. We used this approach to study the dependence of H exponent calculated on roughness of metallic surfaces and its use as a roughness control parameter. In Section 2 we discuss the method for calculation of the H exponent and define how to proceed in order to calculate it from a digital image. In Section 3, the experimental apparatus is presented. Results and discussions are presented in Section 4. Finally, conclusions are presented in Section 5.
2. Theory The rescaled range analysis (R/S), a tool for studying long-term memory and fractality of a time series, was first introduced by Hurst [8] in hydrology for studying the Nile River and water storage [11]. Mandelbrot [12] discussed that (R/S) analysis is a more powerful tool in detecting long range dependence when compared with the more conventional analyses such as autocorrelation analysis, variance ratios and spectral analysis. Surface roughness description has two basic geometrical features: random aspect and structural one. The random aspect takes into account the fact that the rough surface can vary considerably in space in a random manner, and subsequently there is no spatial function being able to describe the geometrical form. The structural aspect is based on the fact that variances of roughness are not completely independent with respect to their spatial positions, but their correlation depends on distance. The classical roughness characteristics such as RMS roughness indicate that the density of summits and the mean absolute surface slope are functions of fractal dimension and cut-off frequencies only [10,11]. These roughness parameters are not intrinsic properties of a surface and vary with the conditions of measurement. Selfsimilar fractal curves and surfaces are described completely by a single parameter, known as the fractal dimension D, which is an intrinsic property of the surface and does not change with the scale of measurement. It has been shown that metal surfaces obey statistical scale invariance, known as self-affinity [11,12]. Such surfaces, which are assumed to be on average parallel to the (x,y) plane, are described here through a single valued function z ¼h(x,y). The self-affinity property means that the surface remains statistically invariant under the scaling transformation (x,y) to (lx, ly) and h(lx, ly)¼ lHh(x,y). The equal sign for h function is to be understood in the statistical sense as it gives rise to the same expected value for any observable computed through an average over space. H is the characteristic roughness exponent called Hurst’s exponent. Let x(p) be the pixel intensity of a speckle image at position p. Let X(p) be the accumulated values of x(p) from the mean o x 4 t (t ¼image rows), given by Xðp, tÞ ¼
p X
fxðuÞ- o x 4 t g
33
The standard deviation S is written as " #!1=2 p X 2 S ¼ 1=t fxðuÞ- o x 4 t g
ð3Þ
u¼1
Following the idea of Tchvialeva et al. [7], the data of speckle image intensity and surface roughness are related to the rescaled range, R/S, and to an exponent H, such that R=S ¼ ðtÞH
ð4Þ
The Hurst exponent H expresses the tendency of dh ¼[dh(x)/ dx]dx to change sign. Hurst exponent H is limited to the range 0oHo1. In order to calculate the H exponent of digital image, the algorithm presented in this section is implemented by taking the image data by column and the then averaging over the number of columns.
3. Experiment In this section speckle patterns produced by rough metallic surfaces were explored. We will show how the Hurst exponent responds to pattern changes due to surfaces with different roughness. In order to observe the dependence of H exponent of digital image speckle pattern on roughness of metallic surfaces, commercial aluminum was used to prepare 5 samples with different roughnesses. We used polishing techniques, taking sandpaper with grain size of 80, 220, 320, 800 and 1200 MESH. The procedure starts with polishing using the higher grain size sandpaper for all samples and using smaller grain size sandpaper successively. An optical profilometer (MahrR) was used to characterize the samples according to their roughness properties. The results of the arithmetic average roughness (Ra), mean roughness depth (Rz) and root mean square roughness (Rq) of each sample are shown in Table 1. Speckle at image plane is obtained using these samples, as sketched in Fig. 1. A 50 mW green laser beam is expanded using two lenses L1 and L2 in order to illuminate an area of approximately 1 cm2 of sample S. The illuminated area is Table 1 Result for direct roughness measurement, where Ra is arithmetic average roughness, Rz is roughness depth and Rq is root mean square roughness. Roughness/sandpaper (Mesh)
Ra (lm) Rq (lm) Rz (lm)
80
220
320
800
1200
0,39 7 0,02 0,49 7 0,01 2,06 7 0,07
0,28 70,01 0,35 70,02 1,36 70,05
0,207 01 0,257 0,01 1,147 0,08
0,14 70,01 0,18 70,01 0,89 70,06
0,127 0,01 0,167 0,01 0,757 0,03
ð1Þ
u¼1
The range R is the difference between the maximum and minimum amounts of intensity contained in a speckle image. This can be written as R ¼ maxXðp, tÞ-min Xðp, tÞ,
1 rp r t
ð2Þ
Fig. 1. Schematic diagram for experimental setup. L1 and L2: lens; S: sample.
34
A.L. Sampaio et al. / Optics and Lasers in Engineering 49 (2011) 32–35
Fig. 2. Speckle patterns produced by rough surface with Ra of 0.39 mm (A), 0.20 mm (B) and 0.12 mm (C).
captured by a monochromatic CCD camera with a resolution of 640(H) 480(V) and a pixel pitch of 11.0(H) 13.0(V) mm2. The pixels are sensitized in the visible spectrum. Normal direction is chosen to photograph the surface. The samples are placed on a holder in order to keep the angle of incidence constant. The reflected light in the specular direction can be used to monitor the incident angle. The experiment is performed by acquiring the surface image of the samples, keeping the laser intensity, the polarization and the angle of incidence fixed. In this way, roughness surface is the unique factor responsible for changes in the speckle patterns. Fig. 2 presents three typical image results of the samples with Ra equal to 0.39, 0.20 and 0.12 mm.
4. Results and discussion The mean H exponent, defined in Section 2, is calculated for the complete set of samples. A dependence of the H exponent on roughness is observed. Fig. 3 shows the mean H exponent as a function of arithmetic average roughness (Ra). A clear relation is obtained. It is important to stress that in our paper H exponent is calculated from the speckle pattern generated by rough surfaces and not by the profile function of these surfaces, as in the classical application. In the case of roughness measurement where Ra has large variations, the H exponent can be used as an indirect and noncontact measurement. Let us now study the case of a long sample produced using the 80 MESH sandpaper. We used a 300 mm long bar of the same material (aluminum) previously used. We polished the bar in the same direction. In order to investigate the responses of the H coefficient due to changes of roughness along the bar we introduced damages at different regions of the bar. This procedure is carried out using the same experimental arrangement illustrated in Fig. 1. The results for the H exponents at different positions of the bar are shown in Fig. 4. The x-axis indicates images of different regions of the bar. The y-axis shows the H exponent calculated from the speckle pattern of each image. In the images 1–7, the speckle pattern photographed is generated by regions of regular polished surface. The calculated H exponent does not vary significantly. Images 8–10 of the bar were obtained from a region where 3 slim risks were introduced orthogonally to the polishing direction. Picture 9 contains 3 damages. Images 8 and 10 contain only one of them. The H exponent grows in the presence of damages. It grows more significantly for image 9 because it contains all damaged regions, as mentioned before. Images 13 and 14 were re-polished using the 120 MESH sandpaper in an irregular manner. The irregular polishing process used here gave rise to imperfections, which produce a small growth of the Hurst exponent. Finally, after image 16 we started polishing in different directions and, at the end of the bar (images 19–21), we polished in a direction orthogonal to the original one. The H exponent exhibited a significant improvement for this change in the surface. The H
Fig. 3. Mean H exponent as a function of mean roughness Ra. Solid line: guide to the eye; dotted line: linear regression (RL).
Fig. 4. H mean exponent for different regions of bar polished with a 80 MESH sandpaper, where different damages were introduced at different points. The x-axis indicates captured images along the bar.
exponent grows in the images from regions that contain damages. We calculated the mean Hurst exponent for the images of the undamaged regions of the bar and it resulted in H¼0.530170.0034. The standard deviation is smaller than the increment in the Hurst exponent due to damages on the bar. It is interesting to note that the damages increase scattered light from the surfaces, giving rise to H improvement. This result can be understood as a consequence of scattered light intensity.
A.L. Sampaio et al. / Optics and Lasers in Engineering 49 (2011) 32–35
35
5. Conclusion
References
In conclusion, we have introduced the calculation of the mean Hurst exponent from digital images of speckle patterns produced by metallic rough surfaces. A clear dependence of the H exponent calculated on surface roughness was demonstrated. The H exponent was not calculated from the surface profile, but from the speckle patterns generated by the rough surfaces. For a long range of Ra the mean H exponent can be used as an indirect and non-contact roughness measurement. This technique can also be used to monitor rough surfaces. We have shown that for a long bar prepared using a unique sandpaper the mean H exponent is almost constant, with an exception of the ones calculated from a damaged region of the bar. These results can be used as a tool for roughness control.
[1] Dhanasekar B, Ramamorthy B. Digital speckle interferometry for assessment of surface roughness. Optics and Lasers in Engineering 2008;46:272–80. [2] Zhao X, Gao Z. Surface roughness measurement using spatial-average analysis of objective speckle pattern in specular direction. Optics and Lasers in Engineering 2009;47:1307–16. [3] Dhanasekar B, Mohan NK, Bhaduri B, Ramamorthy B. Evaluation of surface roughness based on monochromatic speckle correlation using image processing. Precision Engineering 2008;32:196–206. [4] Hamed AM, El-Ghandoor H, El-Diasty F, Saudy M. Analysis of speckle images to assess surface roughness. Optics & Laser Technology 2004;36:249–53. [5] Toh SL, Quan C, Woo KC, Tay CJ, Shang HM. Whole field surface roughness measurement by laser speckle correlation technique. Optics & Laser Technology 2001;33:427–34. [6] Persson Ulf. Surface roughness measurement on machined surfaces using angular speckle correlation. Journal of Materials Processing Technology 2006;180:233–8. [7] 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(7-8):774–8. [8] Hurst HE. Long-term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers 1965;116:770–9. [9] Baraba´si A-L, Stanley HE. Fractal concepts in surface growth. Cambridge: Cambridge University Press; 1995. [10] Wawszczak J. Methods for estimating the Hurst exponent. The analysis of its value for fracture surface research. Materials Science—Poland 2005;23: 585–91. [11] Fender J. Fractals. New York: Plenum Press; 1988. [12] Mandelbrot BB. The fractal geometry of nature. New York: W.H. Freeman; 1983.
Acknowledgements The authors thank Dr. B. Coutinho dos Santos for fruitful discussions. They thank Brazilian Founding Agencies FAPERJ (Fundac- a~ o Carlos Chagas Filho de Apoio a Pesquisa do Estado do Rio de Janeiro and CNPq (Conselho Nacionional Desenvolvimento Cientı´fico e Tecnolo´gico) for financial support.