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Single frame digital fringe projection profilometry for 3-D surface shape measurement
Optik 124 (2013) 166–169
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Single frame digital fringe projection profilometry for 3-D surface shape measurement U. Paul Kumar, U. Somasundaram, M.P. Kothiyal, N. Krishna Mohan ∗ Applied Optics Laboratory, Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
a r t i c l e
i n f o
Article history: Received 21 June 2011 Accepted 15 November 2011
Keywords: Fringe projection Hilbert transformation 3-D shape Fringe analysis
a b s t r a c t Multiple frame digital fringe projection technique is widely used for measuring the 3-D surface shape. In dynamic situations single frame analysis techniques are desirable. In this paper we discuss a Hilbert transform based single-frame analysis. Hilbert transformation method requires only one fringe pattern for the extraction of phase reducing the calculation time. The method is easy to implement, and it is capable of conducting automated measurements at video frame rate. The application of the proposed method for curved surfaces is emphasized. A few experimental results are presented. © 2011 Elsevier GmbH. All rights reserved.
1. Introduction Digital fringe projection profilometry (DFPP) to retrieve the surface topography of the 3-D objects is one of the active research areas in optical metrology [1–5]. Its applications range from measuring the 3-D surface shape of small systems such as Micro Electro Mechanical Systems (MEMS) [6–8] to large engineering structures such as aircraft honeycomb panels [1–5]. A recent review article by Gorthi and Rastogi [2] covers the recent trends in fringe projection techniques. The technique is used in material testing, quality control, reverse engineering, biomedical engineering, on-line inspection, surface roughness measurements, MEMS characterization, etc. It is whole-field, non-contact, non-invasive, flexible, inexpensive, and capable of providing high resolution. Further, it is non-scanning and provides 3-D reconstruction at video frame rates. Multiple frame methods such as phase shifting techniques which require at least three images with known or unknown phase shifts between the subsequent images are used for extracting phase information [9,10]. Single fringe methods such as Fast Fourier Transform (FFT) [11] and Hilbert transform (HT) [12] have also been employed. In FFT method the image is processed in the frequency domain whereas HT method can be implemented in spatial domain. The HT method has several advantages: (a) it has simplicity in the calculation algorithms, (b) it can be used in wide range of interferometric applications, (c) it has relatively shorter calculation time compared to FFT method, and (d) the method can be made fully
∗ Corresponding author. E-mail address: [email protected] (N. Krishna Mohan). 0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.11.030
automatic [13–15]. However, not much work has been reported on fringe projection techniques using Hilbert transformation. We have used the inbuilt 1-D Hilbert transform function in MATLAB for our application. This function results in proper phase if the fringes are linear. The direct application of 1-D HT method on a closed fringe system (indicate a curved surface) results in an ambiguous phase which requires further rigorous image processing to remove the ambiguity [15–18]. So the phase evaluation procedure in case of curved surfaces using the 1-D HT method is not straight forward. But the digital fringe projection technique allows the projection of linear fringes on to the object (plane or curved) surface which makes it easier to implement HT method even on curved surfaces as well in a straight forward manner. In this paper we demonstrate the use of the Hilbert transformation method in fringe projection technique and address issues such as bias removal and phase error correction.
2. Theory: single frame fringe projection method Fig. 1 shows a typical experimental arrangement of the fringe projection method. If a sinusoidal fringe pattern with a spatial frequency f which is parallel to x-axis is projected on to the surface of a 3-D object, the intensity distribution of the distorted fringe pattern captured by the camera can be expressed as [1–5] I(x, y) = Io (x, y)(1 + V (x, y) cos(2fx + (x, y)))
(1)
where Io (x, y) is the bias intensity, V(x, y) is the fringe visibility, and (x, y) is the phase introduced due to height variations, H(x, y). To profile the height variations, (x, y) needs to be determined.
U. Paul Kumar et al. / Optik 124 (2013) 166–169
167
Fig. 2. (a) Original interferogram (Eq. (1)) and (b) central line scan profiles of the fringe pattern.
Fig. 1. Schematic of opto-electronic arrangement for fringe projection profilometry system.
2.1. Numerical phase shifting According to the signal theory, with each real wave function u(x), we may associate a complex wave function (x) = u(x) + iv(x) [13]. The imaginary part v(x) can be obtained by HT of the real signal. Thus ‘hilbert’ MATLAB function can generate an analytical signal from a real signal. The real part is the original signal and imaginary part is the HT of the original signal which is phase shifted by /2. Hilbert transform is equivalent to the filtering, in which the amplitudes of the spectral components are left unchanged, except that their phases are altered by /2, positively or negatively according to the sign of x. The HT of the signal f(x) = cos(ϕ(x)) gives sin(ϕ(x)). The success of the use of the HT method in this application depends on (a) preservation of the signal amplitude factor (Io V) during the phase shift process, (b) accuracy of the numerical phase shift, and (c) elimination of the background intensity, Io . The bias Io (uniform or non-uniform) must be eliminated prior to the application of Hilbert transformation. The uniform bias can be removed by subtracting the mean value of the signal from the original signal. The bias-free signal is [15] P = I − I = Io V cos()
(2)
The non-uniform bias can be removed by local average filtering [13,15]. It is a simple and effective digital filter which operates by averaging a number of points in a space domain. The HT of bias-free signal P is Q = HT {P} = −Io V sin()
Fig. 3. Bias-free signal and corresponding HT signal.
with bias. Fig. 3 shows the bias free signal and corresponding HT signal. The phase shift can be seen clearly between the bias-free signal and HT signal due to the application of HT. Fig. 4 shows the wrapped phase map obtained using Eq. (4) which is subsequently unwrapped. However, the value of calculated from Eq. (4) differs from the correct argument of the cosine function due the finite number of fringe cycles. This error can be expressed as
=+ε
(5)
where ε is the error. The error ε can be calculated for any value of to create a look-up table. Fig. 5 shows a look-up table (graph) obtained by simulation [15]. The look-up table is used to correct the calculated phase ( ). The phase of the reference plane (Fig. 1) can be extracted in the same way. The reference phase (r ) is then subtracted from to get (= − r ) which is directly related to the surface height or depth of the object.
(3)
2.2. Phase extraction The phase has the information about the surface height/depth variations. In the phase shifting techniques the phase information can be obtained with 3 or more phase shifted frames. Hilbert transformation is used to generate a phase shifted frame from a single fringe pattern. The two frames represented by Eqs. (2) and (3) can be used to determine phase [13–15] = arctan
Q P
(4)
Fig. 2(a) shows the original interferogram (Eq. (1)) captured on a curved surface by projecting the linear fringe pattern and Fig. 2(b) shows line scan from the fringe pattern representing the signal
Fig. 4. Wrapped phase map obtained from bias-free fringe pattern. Traces on the right side are central line scan of the phase maps on the left.
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U. Paul Kumar et al. / Optik 124 (2013) 166–169
Fig. 5. (a) The look-up table for the error ε between and and (b) phase error due to the Hilbert transformation.
2.3. Relationship between surface height and phase In an off axis optical setup, a simple triangulation principle establishes the relation between the phase distribution and the height, H of the 3-D object. The relation that governs is given by [1–5] H=
L + 2fD
(6)
where L and D are the parameters of the optical setup. D represents the distance between the CCD camera and the projector. f is the spatial frequency of the projected fringes and (= − r ) is the object shape-related phase obtained after subtracting from the reference phase. Camera and projector are assumed to be in a parallel plane at a distance L from the reference plane. If the distance between the CCD sensor and the reference plane is large compared to the pitch of projected fringes and under normal viewing conditions, the phase and height relationship from method of triangulation can be rewritten as H=
L 2fD
= K
(7)
K = L/(2fD) is the calibration coefficient, which is related to the configuration of the optical measuring system. The resolution of such a triangulation system is dH =
1 tan
dx
(8)
where is the angle between the CCD camera and projector. 3. Digital fringe projection profilometry (DFPP) system Fig. 1 shows the schematic of the experimental setup for fringe projection profilometry system. A portable digital light processing (DLP) BENQ MP523 interfaced with a computer is used to focus the sinusoidal fringes generated onto the object surface. It can provide a 1024 × 768 resolution. It uses a Digital Micromirror Device (DMD) to project seamless images with very high brightness and contrast, high image quality and spatial repeatability. In the present case a gray scale linear interference fringe pattern is generated with MATLAB at a fixed frequency. The pattern is projected on to the object with the projector interface with the PC. A Jai CV-A1 CCD camera with 1384(H) × 1035(V) pixels spatial resolution and the size of each pixel is 4.65 m is used to capture the fringe patterns on the object under investigation. The CCD is interfaced with the PC with a NI1409 frame grabber card. The sensitivity of the system depends on the angle between the axes of the projector and the CCD camera. The sensitivity increases with the angle (Eq. (8)). However it also creates shadow areas on the object and the height measurement range is reduced. Therefore, a suitable angle has to be set based on the height of the object to be investigated. The contour fringe
Fig. 6. The single-frame digital fringe projection analysis on a stratum of Buddha statue: (a) fringe pattern generated in MATLAB, (b) fringe pattern on a reference plane, (c) fringe pattern on the stratum of a Buddha statue, (d) wrapped phase, (e) 2-D plot, (f) 3-D plot, and (g) central line scan profile.
patterns from the CCD camera are received and are displayed on the monitor in real time and these frames are stored for the fringe analysis. 4. Experimental results A linear fringe pattern generated at fixed frequency (shown in Fig. 6(a)) is projected on to the reference plane using the projector. The reference plane is situated at a distance (L = 100 cm) from the CCD sensor. The projector is at a distance of (D = 50 cm) from the CCD camera. The CCD camera, projector, and object are situated on a right angled triangle as shown in Fig. 1. When the linear fringes are projected on an object, the fringes will be straight and regularly spaced for a planar object (z = 0) as shown in Fig. 6(b), and distorted otherwise as shown in Fig. 6(c). The divergence is related to the shape of the object under investigation. The distorted fringe pattern on a stratum of Buddha statue is shown in Fig. 6(c). The phase corresponding to the fringe patterns shown in Fig. 6(b) and (c) is extracted using the HT method as explained in Section 2. The shape phase obtained by subtracting the reference phase and the object phase is shown in Fig. 6(d). The wrapped phase shown in Fig. 6(d) is then unwrapped to get the continuous phase which is a function of surface variations. Fig. 6(e) and (f) shows the 2-D and 3-D plots scaled using Eq. (8). The central line scan profile of the measured shape along x-axis is shown in Fig. 6(g). Experiment is also carried out on (i) a combined convex–concave surface and (ii) on object consisting of IITM letters in front of reference plane. The systematic analysis on these objects is shown in Figs. 7 and 8 respectively. Fig. 8(d) and (e) represents the uncorrected (A) and corrected (B) profiles along central x-axis and the corresponding difference plot along central x-axis respectively. The radius of curvature of the convex and concave surfaces measured is 16.575 cm and 21.395 cm respectively. Similarly the average thickness of the IITM letters is 0.515 cm as shown in Fig. 8(d). The interference between a spherical wavefront and a reference generally results in closed fringes. In such cases, the application of 1-D HT method for phase calculation is not straight forward [15].
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HT method on a closed fringe pattern results in an ambiguous phase. 2-D Hilbert transformation methods have been developed to solve such problems [16–18]. However in the digital fringe projection technique, one can project linear fringes on to the curved surface and calculate the surface shape phase using the 1-D HT method as explained in Section 2. Thus the digital fringe projection technique makes the application of HT method straight forward even in case of curved surfaces. 5. Conclusions We have demonstrated the usefulness of the Hilbert transformation with the single-frequency digital fringe projection technique for 3-D shape measurement. This method requires only one frame for evaluation of the surface shape. The working of the method has been demonstrated by experimental results. Acknowledgement This work is supported by Defense Research and Development Organization (DRDO). References
Fig. 7. 3-D shape measurement on a convex and concave combined surface: (a) fringe pattern on the sample, (b) wrapped phase, (c) 2-D plot, (d) uncorrected (A) and corrected (B) central line scan profiles, and (e) difference plot, and (f) 3-D plot.
Fig. 8. Measurement of 3-D shape on IITM: (a) fringe pattern, (b) phase map, (c) 3-D plot, and (d) central line scan profile.
[1] W. Osten (Ed.), Optical Inspection of Microsystems, CRC Press, Boca Raton, FL, 2007. [2] S.S. Gorthi, P. Rastogi, Fringe projection techniques: whither we are? Opt. Lasers Eng. 48 (2010) 133–140. [3] S.S Gorthi, G. Rajshekhar, P. Rastogi, Three dimensional shape measurement using high-order instantaneous moments based fringe projection method, Opt. Laser Technol. 43 (2011) 40–44. [4] T. Yoshizawa (Ed.), Handbook of Optical Metrology: Principles and Applications, CRC Press, Boca Raton, FL, 2009. [5] F. Chen, G.M. Brown, M. Song, Overview of three-dimensional shape measurement using optical methods, Opt. Eng. 39 (2000) 10–22. [6] K. Leonhardt, U. Droste, H.J. Tiziani, Micro shape and rough-surface analysis by fringe projection, Appl. Opt. 33 (1994) 7477–7488. [7] C. Quan, X.Y. He, C.F. Wang, C.J. Tay, H.M. Shang, Shape measurement of small objects using LCD fringe projection with phase shifting, Opt. Commun. 189 (2001) 21–29. [8] C. Quan, C.J. Tay, X.Y. He, X. Kang, H.M. Shang, Microscopic surface contouring by fringe projection method, Opt. Laser Technol. 34 (2002) 547–552. [9] K.G. Larkin, B.E. Oreb, Design and assessment of symmetrical phase-shifting algorithms, J. Opt. Soc. Am. 9 (1992) 1741–1748. [10] K. Hibino, B.E. Oreb, D.I. Farrant, Phase shifting for non-sinusoidal waveforms with phase-shifting errors, J. Opt. Soc. Am. 12 (1995) 761–768. [11] M. Takeda, H. Ika, S. Kobayasha, Fourier transform method of fringe pattern analysis for computer based topography and interferometry, J. Opt. Soc. Am. 72 (1982) 156–160. [12] M.A. Sutton, W. Zhao, S.R. McNeill, H.W. Schreier, Y.J. Chao, Development and assessment of a single-image fringe projection method for dynamic applications, Exp. Mech. 41 (2001) 205–217. [13] V.D. Madjarova, H. Kadono, Dynamic electronic speckle pattern interferometry (DESPI) phase analyses with temporal Hilbert transform, Opt. Express 11 (2003) 617–623. [14] V.D. Madjarova, H. Kadono, S. Toyooka, Use of dynamic electronic speckle pattern interferometry with the Hilbert transform method to investigate thermal expansion of a joint material, Appl. Opt. 45 (2006) 7590–7596. [15] U. Paul Kumar, N. Krishna Mohan, M.P. Kothiyal, Time average vibration fringe analysis using Hilbert transformation, Appl. Opt. 49 (2010) 5777–5786. [16] K.G. Larkin, D.J. Bone, M.A. Oldfield, Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform, J. Opt. Soc. Am. A 18 (2001) 1862–1870. [17] K.G. Larkin, Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform, J. Opt. Soc. Am. A 18 (2001) 1871–1881. [18] K.G. Larkin, A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns, Opt. Exp. 9 (2001) 236–253.
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Single frame digital fringe projection profilometry for 3-D surface shape measurement U. Paul Kumar, U. Somasundaram, M.P. Kothiyal, N. Krishna Mohan ∗ Applied Optics Laboratory, Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
a r t i c l e
i n f o
Article history: Received 21 June 2011 Accepted 15 November 2011
Keywords: Fringe projection Hilbert transformation 3-D shape Fringe analysis
a b s t r a c t Multiple frame digital fringe projection technique is widely used for measuring the 3-D surface shape. In dynamic situations single frame analysis techniques are desirable. In this paper we discuss a Hilbert transform based single-frame analysis. Hilbert transformation method requires only one fringe pattern for the extraction of phase reducing the calculation time. The method is easy to implement, and it is capable of conducting automated measurements at video frame rate. The application of the proposed method for curved surfaces is emphasized. A few experimental results are presented. © 2011 Elsevier GmbH. All rights reserved.
1. Introduction Digital fringe projection profilometry (DFPP) to retrieve the surface topography of the 3-D objects is one of the active research areas in optical metrology [1–5]. Its applications range from measuring the 3-D surface shape of small systems such as Micro Electro Mechanical Systems (MEMS) [6–8] to large engineering structures such as aircraft honeycomb panels [1–5]. A recent review article by Gorthi and Rastogi [2] covers the recent trends in fringe projection techniques. The technique is used in material testing, quality control, reverse engineering, biomedical engineering, on-line inspection, surface roughness measurements, MEMS characterization, etc. It is whole-field, non-contact, non-invasive, flexible, inexpensive, and capable of providing high resolution. Further, it is non-scanning and provides 3-D reconstruction at video frame rates. Multiple frame methods such as phase shifting techniques which require at least three images with known or unknown phase shifts between the subsequent images are used for extracting phase information [9,10]. Single fringe methods such as Fast Fourier Transform (FFT) [11] and Hilbert transform (HT) [12] have also been employed. In FFT method the image is processed in the frequency domain whereas HT method can be implemented in spatial domain. The HT method has several advantages: (a) it has simplicity in the calculation algorithms, (b) it can be used in wide range of interferometric applications, (c) it has relatively shorter calculation time compared to FFT method, and (d) the method can be made fully
∗ Corresponding author. E-mail address: [email protected] (N. Krishna Mohan). 0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.11.030
automatic [13–15]. However, not much work has been reported on fringe projection techniques using Hilbert transformation. We have used the inbuilt 1-D Hilbert transform function in MATLAB for our application. This function results in proper phase if the fringes are linear. The direct application of 1-D HT method on a closed fringe system (indicate a curved surface) results in an ambiguous phase which requires further rigorous image processing to remove the ambiguity [15–18]. So the phase evaluation procedure in case of curved surfaces using the 1-D HT method is not straight forward. But the digital fringe projection technique allows the projection of linear fringes on to the object (plane or curved) surface which makes it easier to implement HT method even on curved surfaces as well in a straight forward manner. In this paper we demonstrate the use of the Hilbert transformation method in fringe projection technique and address issues such as bias removal and phase error correction.
2. Theory: single frame fringe projection method Fig. 1 shows a typical experimental arrangement of the fringe projection method. If a sinusoidal fringe pattern with a spatial frequency f which is parallel to x-axis is projected on to the surface of a 3-D object, the intensity distribution of the distorted fringe pattern captured by the camera can be expressed as [1–5] I(x, y) = Io (x, y)(1 + V (x, y) cos(2fx + (x, y)))
(1)
where Io (x, y) is the bias intensity, V(x, y) is the fringe visibility, and (x, y) is the phase introduced due to height variations, H(x, y). To profile the height variations, (x, y) needs to be determined.
U. Paul Kumar et al. / Optik 124 (2013) 166–169
167
Fig. 2. (a) Original interferogram (Eq. (1)) and (b) central line scan profiles of the fringe pattern.
Fig. 1. Schematic of opto-electronic arrangement for fringe projection profilometry system.
2.1. Numerical phase shifting According to the signal theory, with each real wave function u(x), we may associate a complex wave function (x) = u(x) + iv(x) [13]. The imaginary part v(x) can be obtained by HT of the real signal. Thus ‘hilbert’ MATLAB function can generate an analytical signal from a real signal. The real part is the original signal and imaginary part is the HT of the original signal which is phase shifted by /2. Hilbert transform is equivalent to the filtering, in which the amplitudes of the spectral components are left unchanged, except that their phases are altered by /2, positively or negatively according to the sign of x. The HT of the signal f(x) = cos(ϕ(x)) gives sin(ϕ(x)). The success of the use of the HT method in this application depends on (a) preservation of the signal amplitude factor (Io V) during the phase shift process, (b) accuracy of the numerical phase shift, and (c) elimination of the background intensity, Io . The bias Io (uniform or non-uniform) must be eliminated prior to the application of Hilbert transformation. The uniform bias can be removed by subtracting the mean value of the signal from the original signal. The bias-free signal is [15] P = I − I = Io V cos()
(2)
The non-uniform bias can be removed by local average filtering [13,15]. It is a simple and effective digital filter which operates by averaging a number of points in a space domain. The HT of bias-free signal P is Q = HT {P} = −Io V sin()
Fig. 3. Bias-free signal and corresponding HT signal.
with bias. Fig. 3 shows the bias free signal and corresponding HT signal. The phase shift can be seen clearly between the bias-free signal and HT signal due to the application of HT. Fig. 4 shows the wrapped phase map obtained using Eq. (4) which is subsequently unwrapped. However, the value of calculated from Eq. (4) differs from the correct argument of the cosine function due the finite number of fringe cycles. This error can be expressed as
=+ε
(5)
where ε is the error. The error ε can be calculated for any value of to create a look-up table. Fig. 5 shows a look-up table (graph) obtained by simulation [15]. The look-up table is used to correct the calculated phase ( ). The phase of the reference plane (Fig. 1) can be extracted in the same way. The reference phase (r ) is then subtracted from to get (= − r ) which is directly related to the surface height or depth of the object.
(3)
2.2. Phase extraction The phase has the information about the surface height/depth variations. In the phase shifting techniques the phase information can be obtained with 3 or more phase shifted frames. Hilbert transformation is used to generate a phase shifted frame from a single fringe pattern. The two frames represented by Eqs. (2) and (3) can be used to determine phase [13–15] = arctan
Q P
(4)
Fig. 2(a) shows the original interferogram (Eq. (1)) captured on a curved surface by projecting the linear fringe pattern and Fig. 2(b) shows line scan from the fringe pattern representing the signal
Fig. 4. Wrapped phase map obtained from bias-free fringe pattern. Traces on the right side are central line scan of the phase maps on the left.
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U. Paul Kumar et al. / Optik 124 (2013) 166–169
Fig. 5. (a) The look-up table for the error ε between and and (b) phase error due to the Hilbert transformation.
2.3. Relationship between surface height and phase In an off axis optical setup, a simple triangulation principle establishes the relation between the phase distribution and the height, H of the 3-D object. The relation that governs is given by [1–5] H=
L + 2fD
(6)
where L and D are the parameters of the optical setup. D represents the distance between the CCD camera and the projector. f is the spatial frequency of the projected fringes and (= − r ) is the object shape-related phase obtained after subtracting from the reference phase. Camera and projector are assumed to be in a parallel plane at a distance L from the reference plane. If the distance between the CCD sensor and the reference plane is large compared to the pitch of projected fringes and under normal viewing conditions, the phase and height relationship from method of triangulation can be rewritten as H=
L 2fD
= K
(7)
K = L/(2fD) is the calibration coefficient, which is related to the configuration of the optical measuring system. The resolution of such a triangulation system is dH =
1 tan
dx
(8)
where is the angle between the CCD camera and projector. 3. Digital fringe projection profilometry (DFPP) system Fig. 1 shows the schematic of the experimental setup for fringe projection profilometry system. A portable digital light processing (DLP) BENQ MP523 interfaced with a computer is used to focus the sinusoidal fringes generated onto the object surface. It can provide a 1024 × 768 resolution. It uses a Digital Micromirror Device (DMD) to project seamless images with very high brightness and contrast, high image quality and spatial repeatability. In the present case a gray scale linear interference fringe pattern is generated with MATLAB at a fixed frequency. The pattern is projected on to the object with the projector interface with the PC. A Jai CV-A1 CCD camera with 1384(H) × 1035(V) pixels spatial resolution and the size of each pixel is 4.65 m is used to capture the fringe patterns on the object under investigation. The CCD is interfaced with the PC with a NI1409 frame grabber card. The sensitivity of the system depends on the angle between the axes of the projector and the CCD camera. The sensitivity increases with the angle (Eq. (8)). However it also creates shadow areas on the object and the height measurement range is reduced. Therefore, a suitable angle has to be set based on the height of the object to be investigated. The contour fringe
Fig. 6. The single-frame digital fringe projection analysis on a stratum of Buddha statue: (a) fringe pattern generated in MATLAB, (b) fringe pattern on a reference plane, (c) fringe pattern on the stratum of a Buddha statue, (d) wrapped phase, (e) 2-D plot, (f) 3-D plot, and (g) central line scan profile.
patterns from the CCD camera are received and are displayed on the monitor in real time and these frames are stored for the fringe analysis. 4. Experimental results A linear fringe pattern generated at fixed frequency (shown in Fig. 6(a)) is projected on to the reference plane using the projector. The reference plane is situated at a distance (L = 100 cm) from the CCD sensor. The projector is at a distance of (D = 50 cm) from the CCD camera. The CCD camera, projector, and object are situated on a right angled triangle as shown in Fig. 1. When the linear fringes are projected on an object, the fringes will be straight and regularly spaced for a planar object (z = 0) as shown in Fig. 6(b), and distorted otherwise as shown in Fig. 6(c). The divergence is related to the shape of the object under investigation. The distorted fringe pattern on a stratum of Buddha statue is shown in Fig. 6(c). The phase corresponding to the fringe patterns shown in Fig. 6(b) and (c) is extracted using the HT method as explained in Section 2. The shape phase obtained by subtracting the reference phase and the object phase is shown in Fig. 6(d). The wrapped phase shown in Fig. 6(d) is then unwrapped to get the continuous phase which is a function of surface variations. Fig. 6(e) and (f) shows the 2-D and 3-D plots scaled using Eq. (8). The central line scan profile of the measured shape along x-axis is shown in Fig. 6(g). Experiment is also carried out on (i) a combined convex–concave surface and (ii) on object consisting of IITM letters in front of reference plane. The systematic analysis on these objects is shown in Figs. 7 and 8 respectively. Fig. 8(d) and (e) represents the uncorrected (A) and corrected (B) profiles along central x-axis and the corresponding difference plot along central x-axis respectively. The radius of curvature of the convex and concave surfaces measured is 16.575 cm and 21.395 cm respectively. Similarly the average thickness of the IITM letters is 0.515 cm as shown in Fig. 8(d). The interference between a spherical wavefront and a reference generally results in closed fringes. In such cases, the application of 1-D HT method for phase calculation is not straight forward [15].
U. Paul Kumar et al. / Optik 124 (2013) 166–169
169
HT method on a closed fringe pattern results in an ambiguous phase. 2-D Hilbert transformation methods have been developed to solve such problems [16–18]. However in the digital fringe projection technique, one can project linear fringes on to the curved surface and calculate the surface shape phase using the 1-D HT method as explained in Section 2. Thus the digital fringe projection technique makes the application of HT method straight forward even in case of curved surfaces. 5. Conclusions We have demonstrated the usefulness of the Hilbert transformation with the single-frequency digital fringe projection technique for 3-D shape measurement. This method requires only one frame for evaluation of the surface shape. The working of the method has been demonstrated by experimental results. Acknowledgement This work is supported by Defense Research and Development Organization (DRDO). References
Fig. 7. 3-D shape measurement on a convex and concave combined surface: (a) fringe pattern on the sample, (b) wrapped phase, (c) 2-D plot, (d) uncorrected (A) and corrected (B) central line scan profiles, and (e) difference plot, and (f) 3-D plot.
Fig. 8. Measurement of 3-D shape on IITM: (a) fringe pattern, (b) phase map, (c) 3-D plot, and (d) central line scan profile.
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