Parallel phase-shifting digital holography using spectral estimation technique Peng Xia,1 Yasuhiro Awatsuji,1,* Kenzo Nishio,2 Shogo Ura,1 and Osamu Matoba3 1
Graduate School of Science and Technology, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan
2
Advanced Technology Center, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan 3
Graduate School of System Informatics, Kobe University, Rokkodai 1-1, Nada-ku, Kobe 657-8501, Japan *Corresponding author: [email protected] Received 23 April 2014; revised 5 July 2014; accepted 6 July 2014; posted 8 July 2014 (Doc. ID 210606); published 6 August 2014
We propose a parallel phase-shifting digital holography using a spectral estimation technique, which enables the instantaneous acquisition of spectral information and three-dimensional (3D) information of a moving object. In this technique, an interference fringe image that contains six holograms with two phase shifts for three laser lines, such as red, green, and blue, is recorded by a space-division multiplexing method with single-shot exposure. The 3D monochrome images of these three laser lines are numerically reconstructed by a computer and used to estimate the spectral reflectance distribution of object using a spectral estimation technique. Preliminary experiments demonstrate the validity of the proposed technique. © 2014 Optical Society of America OCIS codes: (090.1995) Digital holography; (090.1760) Computer holography; (090.1705) Color holography. http://dx.doi.org/10.1364/AO.53.00G123
1. Introduction
Digital holography [1–4] shows a strong potential for providing three-dimensional (3D) information from various objects in many fields such as microscopy [5–8], particle measurement [9,10], object recognition [11,12], and phase measurement [13]. In general, the light source used in digital holography is a coherent light source whose optical bandwidth is very narrow so that it is impossible to record the spectral information of object. However, many studies on spectral measurement based on digital holography have been reported: a multiwavelength digital holography for 3D object visualization and recognition using two wavelengths based on the principal component analysis technique and the mixture discriminant analysis technique, which can provide 1559-128X/14/27G123-07$15.00/0 © 2014 Optical Society of America
spatial and spectral information of 3D objects, was proposed [14]. This technique provides just two images of two wavelengths. Another example, a fully passive interferometric technique capable of obtaining both a 3D image and spectral information of spatially incoherent and polychromatic objects, was investigated [15]. But the technique is useless for a moving object due to the scanning of the object. In the spectral imaging field, various spectroscopic measurement methods have been studied; especially Raman spectroscopy has been widely investigated in many fields, which can provide spectral information from various samples in a noninvasive way [16–19]. However, to record 3D information of an object, the scanning of the object is required, such as in the technique combining Raman spectroscopy with confocal microscopy [20,21], which is useless for measuring a moving object. In multispectral imaging, Haneishi et al. proposed a spectral estimation technique, which can estimate 20 September 2014 / Vol. 53, No. 27 / APPLIED OPTICS
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the reflectance of an object from a multiband image [22]. We introduced the spectral estimation technique to digital holography to achieve a technique capable of obtaining spectral information and 3D information of an object with only several exposures [23,24]. Because the pixel size of an image sensor is too large to record fine interference fringes in general, in-line digital holography has been actively researched. The phase-shifting digital holography is a widely used method because the undesired images such as the zeroth-order diffraction image and the conjugate image can be removed from the object image by a phase-shifting calculation [25,26]. Therefore, we demonstrated the feasibility of the technique in which the spectral estimation technique is used in digital holography by means of phase-shifting digital holography. Although the technique greatly reduced the recording time, it can only be used to measure some slowly varying objects such as the living cell. However, in some applications, any techniques that can obtain spectral reflectance information and 3D profiles of high-speed moving objects are urgently required. Parallel phase-shifting digital holography [27–45] is one of the fastest digital holographies. The technique can attain both high precision and instantaneous 3D measurement of a moving object with single-shot exposure and has been increasingly investigated in recent years. In this paper, we propose a parallel phase-shifting digital holography using a spectral estimation technique, which enables the instantaneous acquisition of spectral information and 3D information of an object with single-shot exposure. Preliminary experiments demonstrate the validity of the proposed technique. 2. Parallel Phase-Shifting Digital Holography
In this section, we will exposit the principle of parallel phase-shifting digital holography (in the case of two-step phase-shifting interferometry) [30,35]. In this technique, the information of multiple phaseshifted holograms is recorded by using a single image sensor with single-shot exposure as shown in Fig 1. A 3D image of the object is numerically reconstructed
Fig. 2. Flow of the image reconstruction algorithm.
by using the image reconstruction algorithm of parallel phase-shifting digital holography. Figure 2 shows the processing procedure for the image reconstruction algorithm. First, pixels containing the information of the same phase-shifted hologram are extracted. After that, the values of the vacant pixels are interpolated. Then phase-shifted holograms are numerically generated. By applying the two-step phase-shifting method [46–48], we can obtain a reconstructed image by using ux; y
1 π I0 − ax; y − i I − − ax; y : 2Ar 2 (1)
Here, i is the imaginary unit. I0 and I−π∕2 are the two phase-shifted holograms. Ar is the complex amplitude of the reference wave, and ax; y is defined as follows: p v2 − 2w ; ax; y 2 π v I0 I − 2A2r ; 2 v−
2 π w I02 I − 4A2r : 2
(2) (3)
(4)
3. Parallel Phase-Shifting Digital Holography Using Spectral Estimation Technique
Fig. 1. Principle of parallel phase-shifting digital holography (in case of two-step phase-shifting interferometry). G124
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There are many methods to implement the parallel phase-shifting digital holography using a spectral estimation technique in which six holograms are recorded simultaneously. Here we give an example of the optical implementation of the proposed technique as shown in Fig. 3(a). This is a classical parallel two-step phase-shifting digital holography system
Fig. 4. Configurations of the polarizer and wavelength-filter arrays attached to the image sensor. (a) Micropolarizer array; (b) wavelength-filter array; (c) interference fringe image that contains six holograms.
Fig. 3. Schematics of the proposed technique. (a) Example of the optical implementation; (b) schematic of the image sensor with the micropolarizer array and wavelength-filter array.
using three laser lines. The perpendicular linearly polarized wave from each laser line passes through a quarter-wave plate whose slow (or fast) axis is set so as to incline at an angle of 45 deg against the polarization direction of the perpendicular linearly polarized wave. Then the phase retardation of the laser beam in the slow axis direction comes to π∕2 in comparison with that in the fast axis direction. The laser beam is divided into an object wave and a reference wave. The object wave illuminates an object, and the scattered wave from the object is incident to an image sensor to which a micropolarizer array and a wavelength-filter array are attached, as shown the schematics of the camera in Fig. 3(b). Figures 4(a) and 4(b) show the configurations of the polarizer array and wavelength-filter array attached to the image sensor, respectively. Note that the extinction ratio of micropolarizers influences the phase-shifting steps greatly [36]. Therefore, the micropolarizers must be available to the corresponding light sources. Figure 4(c) shows the configuration of an interference fringe image that contains six holograms: I R 0, I R −π∕2, I G 0, I G −π∕2, I B 0, and I B −π∕2. The subscripts R, G, B mean the R, G, B laser lines. The interference
fringes of the blue component are the finest among the three wavelengths so that the number of pixels of the blue component is twice that of the green or red component. The wave with each phase retardation is selected by the polarizer array, and the wave of each wavelength is selected by the wavelengthfilter array. Then the interference fringe image that contains six holograms can be recorded by the camera. Figure 4 shows the processing procedure for the image reconstruction of parallel phase-shifting digital holography using the spectral estimation technique. Through the extraction and interpolation, six complete holograms can be generated. After that, the monochrome images of these three laser lines are numerically reconstructed and used to estimate the spectral reflectance of the object by using the spectral estimation technique. The processing procedure for the spectral estimation is described below. First, to easily understand the principle of the spectral estimation, we define the wavelength, the spectral reflectance of the object, the spectral power distribution of the illuminant for recording of the hologram, and the spectral sensitivity of the image sensor as λ, rλ, Iλ, and Sλ, respectively. The following process is performed on a pixel-by-pixel basis. Then a pixel value of the image is calculated by using the following equation: Z gj Sλj Iλj rλdλ Z j B; G; R etc: (5) Hλj rλdλ Here, j is the laser line, and Hλj is equal to Sλj Iλj . Then Eq. (5) can be written in vector and matrix notation as follows: 20 September 2014 / Vol. 53, No. 27 / APPLIED OPTICS
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g Hr:
(6)
If the spectral characteristics and the number of laser lines used for recording a hologram are set to m and n, H is the n × m matrix comprising the spectral power distribution of the illuminant and the spectral sensitivity of the image sensor. g is the n × 1 vector of the intensity value, and r is the m × 1 vector of the spectral reflectance of the object. n is smaller than m in spectral analyses of the object because the number of the laser lines used for recording holograms is finite in general. Then it is impossible to determine the unique solution of Eq. (6) for r when m > n. So an estimating constant matrix A is introduced to estimate the spectral reflectance distribution of the object from the three reconstructed intensity images. The estimated spectral reflectance distribution rest can be calculated by using rˆ est Ag:
(7)
Here, ˆ means the estimator. The estimating constant matrix A is determined such that it minimizes the ensemble average of the square error between the original and the estimated spectral reflectance [49]: h‖r − rˆ est ‖2 i h‖r − Ag‖2 i → min :
(8)
Here, hi and r denote ensemble average and spectral reflectances of samples, respectively. A is calculated by using the following equation: A Cr Ht HCr Ht −1 ;
(9)
where t denotes transposition. Cr is an m × m covariance matrix of the spectral reflectances of many samples and is calculated by using Cr hr − r¯r − r¯ t i:
(10)
Here, r¯ denotes the average spectral reflectance of many samples. Therefore, to estimate the spectral reflectance of the object, various samples should be prepared in advance. 4. Preliminary Experiment
We conducted a preliminary experiment to confirm the validity of the proposed technique. In this preliminary experiment, six holograms are recorded by the sequential phase-shifting digital holography, because the image sensor with both the micropolarizer array and wavelength-filter array has not been developed. Figure 5 shows the optical setup of the preliminary experiment. We used a mirror driven by a piezoelectric actuator to shift the phase of a reference wave. Six holograms, I R 0, I R −π∕2, I G 0, I G −π∕2, I B 0, and I B −π∕2, were sequentially recorded by driving the piezoelectric actuator mirror. Then the interference fringe image that contained six holograms obtained with the proposed technique was numerically generated. Three laser G126
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Fig. 5. Optical setup of the preliminary experiment.
lines, a He–Ne laser operated at 633 nm, a Nd: YVO4 laser operated at 532 nm, and a Nd:YAG laser operated at 473 nm, were used as the light sources. A monochrome CMOS camera with pixel size 1.67 μm × 1.67 μm was used to record the hologram. The object is a Mini Color Checker Chart with 24 colors, the size is 57.0 mm × 82.5 mm as shown in Fig. 6, and the size of each lattice is 9.5 mm × 9.5 mm. The object was located 390 mm apart from the image sensor plane. The experimental process is shown step by step as follows. 1. Measure the spectral power distribution of the illuminant to obtain the recording conditions H. 2. The reconstruction process as shown in Fig. 7 was applied to the interference fringe image, which contained six holograms, to reconstruct the images. 3. Calculate the covariance matrix Cr . When we estimated the spectral reflectance distribution of one color which was arbitrarily chosen as an object whose spectral reflectance distribution was unknown, the other 23 colors were used as the samples. 4. Calculate the estimating constant matrix A using the covariance matrix Cr and the recording conditions H.
Fig. 6. Photograph of the object.
Fig. 9. Reconstructed images in the preliminary experiment. Fig. 7. Processing procedure for image reconstruction of proposed technique.
5. Estimate the spectral reflectance distribution of the object. We estimated the spectral reflectance distribution of the object and arbitrarily showed two colors’ results in Fig. 8. The bands of the estimated spectral reflectance distribution of the object are set to 400–700 nm with a 5 nm step, and then the spectral characteristic m is 61. The two colors of the object shown in Figs. 8(a) and 8(b) were No. 16 and No. 20, respectively. The reconstructed images of these two colors in different wavelengths are shown in Fig. 9. It is shown that the spectral reflectance distribution of the object is obtained successfully. 5. Discussion
We compared the spectral reflectance distributions of the two colors with the true values, which are given
Fig. 8. Estimated spectral reflectance distribution of object in the preliminary experiment. (a) No. 16 color; (b) No. 20 color.
by the manufacturer, to verify the correctness of the results of the preliminary experiment. Figure 10 shows the results of the comparisons. We can see that the curve generated by the estimated spectral reflectance distributions in the preliminary experiment greatly agreed with that generated by the truth values given by the manufacturer, which confirmed the effectiveness of the technique. However, there exists some dissatisfaction between true color values and the experimental spectral measurement. Under the same conditions, the shorter the wavelength is, the finer the interference fringes are. In the proposed method, the interpolation is required. The finer the interference fringes are, the larger the interpolation error is. Therefore, the dissatisfaction between true color values and the experimental spectral measurement increased with decrease in wavelength. In Fig. 10(a), the estimated error is small in the short band because the reflectance is low, which decreased the influence of the interpolation error. Moreover, the number of laser lines used to estimate the spectral
Fig. 10. Comparisons of preliminary experiment and the true values. (a) No. 16 color; (b) No. 20 color. 20 September 2014 / Vol. 53, No. 27 / APPLIED OPTICS
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reflectance distributions are not sufficient, and the adjustment of the optical setup also influence the estimated results. If the laser lines operating at other wavelengths were added to record holograms, the results of the estimated spectral distributions could be improved possibly. In addition, the spectral reflectance distribution of the object in the infrared band or ultraviolet band can be estimated if the infrared laser or ultraviolet laser is applied to this technique. 6. Conclusion
We proposed a parallel phase-shifting digital holography using a spectral estimation technique capable of obtaining spectral information and 3D images of a moving object with single-shot exposure. The effectiveness of the proposed method has been confirmed by a preliminary experiment. The spectral reflectance distribution of the object can be obtained by the proposed technique, which will contribute to 3D spectral measurement of high-speed moving objects. This study was partially supported by the Grantin-Aid for JSPS Fellows from the Japan Society for the Promotion of Science (JSPS) and by the Funding Program for Next Generation World-Leading Researchers GR064 of JSPS. References 1. J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967). 2. U. Schnars and W. P. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994). 3. G. Pedrini, P. Froning, H. Fessler, and H. J. Tiziani, “In-line digital holographic interferometry,” Appl. Opt. 37, 6262–6269 (1998). 4. T.-C. Poon, Digital Holography and Three-Dimensional Display: Principles and Applications (Springer, 2006). 5. T.-C. Poon, K. Doh, B. Schilling, M. Wu, K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1344 (1995). 6. L. Yu, S. Mohanty, J. Zhang, S. Genc, M. K. Kim, M. W. Berns, and Z. Chen, “Digital holographic microscopy for quantitative cell dynamic evaluation during laser microsurgery,” Opt. Express 17, 12031–12038 (2009). 7. T. Zhang and I. Yamaguchi, “Three-dimensional microscopy with phase-shifting digital holography,” Opt. Lett. 23, 1221–1223 (1998). 8. P. Ferraro, G. Coppola, S. De Nicola, A. Finizio, and G. Pierattini, “Digital holographic microscope with automatic focus tracking by detecting sample displacement in real time,” Opt. Lett. 28, 1257–1259 (2003). 9. D. Lebrun, A. M. Benkouider, S. Coëtmellec, and M. Malek, “Particle field digital holographic reconstruction in arbitrary tilted planes,” Opt. Express 11, 224–229 (2003). 10. T. Yamaguchi, S. Murata, and T. Morihara, “Three-dimensional flow measurement by digital holographic particle image velocimetry with spatiotemporal derivative method,” JSME Int. J. 49, 1133–1140 (2005). 11. B. Javidi and E. Tajahuerce, “Three-dimensional object recognition by use of digital holography,” Opt. Lett. 25, 610–612 (2000). 12. T. J. Naughton, Y. Frauel, B. Javidi, and E. Tajahuerce, “Compression of digital holograms for three-dimensional object reconstruction and recognition,” Appl. Opt. 41, 4124– 4132 (2002). G128
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Graduate School of Science and Technology, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan
2
Advanced Technology Center, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan 3
Graduate School of System Informatics, Kobe University, Rokkodai 1-1, Nada-ku, Kobe 657-8501, Japan *Corresponding author: [email protected] Received 23 April 2014; revised 5 July 2014; accepted 6 July 2014; posted 8 July 2014 (Doc. ID 210606); published 6 August 2014
We propose a parallel phase-shifting digital holography using a spectral estimation technique, which enables the instantaneous acquisition of spectral information and three-dimensional (3D) information of a moving object. In this technique, an interference fringe image that contains six holograms with two phase shifts for three laser lines, such as red, green, and blue, is recorded by a space-division multiplexing method with single-shot exposure. The 3D monochrome images of these three laser lines are numerically reconstructed by a computer and used to estimate the spectral reflectance distribution of object using a spectral estimation technique. Preliminary experiments demonstrate the validity of the proposed technique. © 2014 Optical Society of America OCIS codes: (090.1995) Digital holography; (090.1760) Computer holography; (090.1705) Color holography. http://dx.doi.org/10.1364/AO.53.00G123
1. Introduction
Digital holography [1–4] shows a strong potential for providing three-dimensional (3D) information from various objects in many fields such as microscopy [5–8], particle measurement [9,10], object recognition [11,12], and phase measurement [13]. In general, the light source used in digital holography is a coherent light source whose optical bandwidth is very narrow so that it is impossible to record the spectral information of object. However, many studies on spectral measurement based on digital holography have been reported: a multiwavelength digital holography for 3D object visualization and recognition using two wavelengths based on the principal component analysis technique and the mixture discriminant analysis technique, which can provide 1559-128X/14/27G123-07$15.00/0 © 2014 Optical Society of America
spatial and spectral information of 3D objects, was proposed [14]. This technique provides just two images of two wavelengths. Another example, a fully passive interferometric technique capable of obtaining both a 3D image and spectral information of spatially incoherent and polychromatic objects, was investigated [15]. But the technique is useless for a moving object due to the scanning of the object. In the spectral imaging field, various spectroscopic measurement methods have been studied; especially Raman spectroscopy has been widely investigated in many fields, which can provide spectral information from various samples in a noninvasive way [16–19]. However, to record 3D information of an object, the scanning of the object is required, such as in the technique combining Raman spectroscopy with confocal microscopy [20,21], which is useless for measuring a moving object. In multispectral imaging, Haneishi et al. proposed a spectral estimation technique, which can estimate 20 September 2014 / Vol. 53, No. 27 / APPLIED OPTICS
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the reflectance of an object from a multiband image [22]. We introduced the spectral estimation technique to digital holography to achieve a technique capable of obtaining spectral information and 3D information of an object with only several exposures [23,24]. Because the pixel size of an image sensor is too large to record fine interference fringes in general, in-line digital holography has been actively researched. The phase-shifting digital holography is a widely used method because the undesired images such as the zeroth-order diffraction image and the conjugate image can be removed from the object image by a phase-shifting calculation [25,26]. Therefore, we demonstrated the feasibility of the technique in which the spectral estimation technique is used in digital holography by means of phase-shifting digital holography. Although the technique greatly reduced the recording time, it can only be used to measure some slowly varying objects such as the living cell. However, in some applications, any techniques that can obtain spectral reflectance information and 3D profiles of high-speed moving objects are urgently required. Parallel phase-shifting digital holography [27–45] is one of the fastest digital holographies. The technique can attain both high precision and instantaneous 3D measurement of a moving object with single-shot exposure and has been increasingly investigated in recent years. In this paper, we propose a parallel phase-shifting digital holography using a spectral estimation technique, which enables the instantaneous acquisition of spectral information and 3D information of an object with single-shot exposure. Preliminary experiments demonstrate the validity of the proposed technique. 2. Parallel Phase-Shifting Digital Holography
In this section, we will exposit the principle of parallel phase-shifting digital holography (in the case of two-step phase-shifting interferometry) [30,35]. In this technique, the information of multiple phaseshifted holograms is recorded by using a single image sensor with single-shot exposure as shown in Fig 1. A 3D image of the object is numerically reconstructed
Fig. 2. Flow of the image reconstruction algorithm.
by using the image reconstruction algorithm of parallel phase-shifting digital holography. Figure 2 shows the processing procedure for the image reconstruction algorithm. First, pixels containing the information of the same phase-shifted hologram are extracted. After that, the values of the vacant pixels are interpolated. Then phase-shifted holograms are numerically generated. By applying the two-step phase-shifting method [46–48], we can obtain a reconstructed image by using ux; y
1 π I0 − ax; y − i I − − ax; y : 2Ar 2 (1)
Here, i is the imaginary unit. I0 and I−π∕2 are the two phase-shifted holograms. Ar is the complex amplitude of the reference wave, and ax; y is defined as follows: p v2 − 2w ; ax; y 2 π v I0 I − 2A2r ; 2 v−
2 π w I02 I − 4A2r : 2
(2) (3)
(4)
3. Parallel Phase-Shifting Digital Holography Using Spectral Estimation Technique
Fig. 1. Principle of parallel phase-shifting digital holography (in case of two-step phase-shifting interferometry). G124
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There are many methods to implement the parallel phase-shifting digital holography using a spectral estimation technique in which six holograms are recorded simultaneously. Here we give an example of the optical implementation of the proposed technique as shown in Fig. 3(a). This is a classical parallel two-step phase-shifting digital holography system
Fig. 4. Configurations of the polarizer and wavelength-filter arrays attached to the image sensor. (a) Micropolarizer array; (b) wavelength-filter array; (c) interference fringe image that contains six holograms.
Fig. 3. Schematics of the proposed technique. (a) Example of the optical implementation; (b) schematic of the image sensor with the micropolarizer array and wavelength-filter array.
using three laser lines. The perpendicular linearly polarized wave from each laser line passes through a quarter-wave plate whose slow (or fast) axis is set so as to incline at an angle of 45 deg against the polarization direction of the perpendicular linearly polarized wave. Then the phase retardation of the laser beam in the slow axis direction comes to π∕2 in comparison with that in the fast axis direction. The laser beam is divided into an object wave and a reference wave. The object wave illuminates an object, and the scattered wave from the object is incident to an image sensor to which a micropolarizer array and a wavelength-filter array are attached, as shown the schematics of the camera in Fig. 3(b). Figures 4(a) and 4(b) show the configurations of the polarizer array and wavelength-filter array attached to the image sensor, respectively. Note that the extinction ratio of micropolarizers influences the phase-shifting steps greatly [36]. Therefore, the micropolarizers must be available to the corresponding light sources. Figure 4(c) shows the configuration of an interference fringe image that contains six holograms: I R 0, I R −π∕2, I G 0, I G −π∕2, I B 0, and I B −π∕2. The subscripts R, G, B mean the R, G, B laser lines. The interference
fringes of the blue component are the finest among the three wavelengths so that the number of pixels of the blue component is twice that of the green or red component. The wave with each phase retardation is selected by the polarizer array, and the wave of each wavelength is selected by the wavelengthfilter array. Then the interference fringe image that contains six holograms can be recorded by the camera. Figure 4 shows the processing procedure for the image reconstruction of parallel phase-shifting digital holography using the spectral estimation technique. Through the extraction and interpolation, six complete holograms can be generated. After that, the monochrome images of these three laser lines are numerically reconstructed and used to estimate the spectral reflectance of the object by using the spectral estimation technique. The processing procedure for the spectral estimation is described below. First, to easily understand the principle of the spectral estimation, we define the wavelength, the spectral reflectance of the object, the spectral power distribution of the illuminant for recording of the hologram, and the spectral sensitivity of the image sensor as λ, rλ, Iλ, and Sλ, respectively. The following process is performed on a pixel-by-pixel basis. Then a pixel value of the image is calculated by using the following equation: Z gj Sλj Iλj rλdλ Z j B; G; R etc: (5) Hλj rλdλ Here, j is the laser line, and Hλj is equal to Sλj Iλj . Then Eq. (5) can be written in vector and matrix notation as follows: 20 September 2014 / Vol. 53, No. 27 / APPLIED OPTICS
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g Hr:
(6)
If the spectral characteristics and the number of laser lines used for recording a hologram are set to m and n, H is the n × m matrix comprising the spectral power distribution of the illuminant and the spectral sensitivity of the image sensor. g is the n × 1 vector of the intensity value, and r is the m × 1 vector of the spectral reflectance of the object. n is smaller than m in spectral analyses of the object because the number of the laser lines used for recording holograms is finite in general. Then it is impossible to determine the unique solution of Eq. (6) for r when m > n. So an estimating constant matrix A is introduced to estimate the spectral reflectance distribution of the object from the three reconstructed intensity images. The estimated spectral reflectance distribution rest can be calculated by using rˆ est Ag:
(7)
Here, ˆ means the estimator. The estimating constant matrix A is determined such that it minimizes the ensemble average of the square error between the original and the estimated spectral reflectance [49]: h‖r − rˆ est ‖2 i h‖r − Ag‖2 i → min :
(8)
Here, hi and r denote ensemble average and spectral reflectances of samples, respectively. A is calculated by using the following equation: A Cr Ht HCr Ht −1 ;
(9)
where t denotes transposition. Cr is an m × m covariance matrix of the spectral reflectances of many samples and is calculated by using Cr hr − r¯r − r¯ t i:
(10)
Here, r¯ denotes the average spectral reflectance of many samples. Therefore, to estimate the spectral reflectance of the object, various samples should be prepared in advance. 4. Preliminary Experiment
We conducted a preliminary experiment to confirm the validity of the proposed technique. In this preliminary experiment, six holograms are recorded by the sequential phase-shifting digital holography, because the image sensor with both the micropolarizer array and wavelength-filter array has not been developed. Figure 5 shows the optical setup of the preliminary experiment. We used a mirror driven by a piezoelectric actuator to shift the phase of a reference wave. Six holograms, I R 0, I R −π∕2, I G 0, I G −π∕2, I B 0, and I B −π∕2, were sequentially recorded by driving the piezoelectric actuator mirror. Then the interference fringe image that contained six holograms obtained with the proposed technique was numerically generated. Three laser G126
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Fig. 5. Optical setup of the preliminary experiment.
lines, a He–Ne laser operated at 633 nm, a Nd: YVO4 laser operated at 532 nm, and a Nd:YAG laser operated at 473 nm, were used as the light sources. A monochrome CMOS camera with pixel size 1.67 μm × 1.67 μm was used to record the hologram. The object is a Mini Color Checker Chart with 24 colors, the size is 57.0 mm × 82.5 mm as shown in Fig. 6, and the size of each lattice is 9.5 mm × 9.5 mm. The object was located 390 mm apart from the image sensor plane. The experimental process is shown step by step as follows. 1. Measure the spectral power distribution of the illuminant to obtain the recording conditions H. 2. The reconstruction process as shown in Fig. 7 was applied to the interference fringe image, which contained six holograms, to reconstruct the images. 3. Calculate the covariance matrix Cr . When we estimated the spectral reflectance distribution of one color which was arbitrarily chosen as an object whose spectral reflectance distribution was unknown, the other 23 colors were used as the samples. 4. Calculate the estimating constant matrix A using the covariance matrix Cr and the recording conditions H.
Fig. 6. Photograph of the object.
Fig. 9. Reconstructed images in the preliminary experiment. Fig. 7. Processing procedure for image reconstruction of proposed technique.
5. Estimate the spectral reflectance distribution of the object. We estimated the spectral reflectance distribution of the object and arbitrarily showed two colors’ results in Fig. 8. The bands of the estimated spectral reflectance distribution of the object are set to 400–700 nm with a 5 nm step, and then the spectral characteristic m is 61. The two colors of the object shown in Figs. 8(a) and 8(b) were No. 16 and No. 20, respectively. The reconstructed images of these two colors in different wavelengths are shown in Fig. 9. It is shown that the spectral reflectance distribution of the object is obtained successfully. 5. Discussion
We compared the spectral reflectance distributions of the two colors with the true values, which are given
Fig. 8. Estimated spectral reflectance distribution of object in the preliminary experiment. (a) No. 16 color; (b) No. 20 color.
by the manufacturer, to verify the correctness of the results of the preliminary experiment. Figure 10 shows the results of the comparisons. We can see that the curve generated by the estimated spectral reflectance distributions in the preliminary experiment greatly agreed with that generated by the truth values given by the manufacturer, which confirmed the effectiveness of the technique. However, there exists some dissatisfaction between true color values and the experimental spectral measurement. Under the same conditions, the shorter the wavelength is, the finer the interference fringes are. In the proposed method, the interpolation is required. The finer the interference fringes are, the larger the interpolation error is. Therefore, the dissatisfaction between true color values and the experimental spectral measurement increased with decrease in wavelength. In Fig. 10(a), the estimated error is small in the short band because the reflectance is low, which decreased the influence of the interpolation error. Moreover, the number of laser lines used to estimate the spectral
Fig. 10. Comparisons of preliminary experiment and the true values. (a) No. 16 color; (b) No. 20 color. 20 September 2014 / Vol. 53, No. 27 / APPLIED OPTICS
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reflectance distributions are not sufficient, and the adjustment of the optical setup also influence the estimated results. If the laser lines operating at other wavelengths were added to record holograms, the results of the estimated spectral distributions could be improved possibly. In addition, the spectral reflectance distribution of the object in the infrared band or ultraviolet band can be estimated if the infrared laser or ultraviolet laser is applied to this technique. 6. Conclusion
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