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OPTICS LETTERS / Vol. 39, No. 7 / April 1, 2014
Voxel model for evaluation of a three-dimensional display and reconstruction in integral imaging Liqiu Zhou,1 Xing Zhao,1,* Yong Yang,1 and Xiaocong Yuan2 1
2
Institute of Modern Optics, Key Laboratory of Optical Information Science and Technology, Nankai University, Ministry of Education of China, Tianjin 300071, China
Institute of Micro & Nano Optics, College of Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China *Corresponding author: [email protected] Received December 23, 2013; revised February 22, 2014; accepted March 3, 2014; posted March 4, 2014 (Doc. ID 203451); published March 26, 2014 An approximate voxel model for integral imaging is proposed by ray tracing. By analyzing the case of corresponding pixels overlapping completely and partially in the image space, the voxel is defined with an appropriate approximation, and the voxel size and its distribution feature in imaging space are derived. The model is verified in a reconstruction experiment of a resolution target and compared with the calculation result of an integral imaging display or reconstruction system. The proposed model is simple and easy to calculate and thus useful for the evaluation and optimization of integral imaging systems. © 2014 Optical Society of America OCIS codes: (110.6880) Three-dimensional image acquisition; (110.3000) Image quality assessment. http://dx.doi.org/10.1364/OL.39.002032
Integral imaging has been an important branch of threedimensional (3D) display technology owing to its unique advantages compared with holographic display and binocular parallax display [1,2]. Employing digital reconstruction methods, it can also provide digitized 3D information for medical diagnosis, architectural design, animation design, and many other applications. The realization of 3D display and reconstruction with high performance requires high display and reconstruction resolutions [3,4]. There have thus been many reports on the resolution of integral imaging [5–7]. However, most of these results focus on the improvement of viewing effects instead of detailed analysis of the characters of the display resolution. Recently, Cho et al. and Wu et al. optimized an integral imaging system according to the relations between the resolutions and system parameters [8,9]. However, they did not discuss the distribution features of the spatial resolution. Kavehvash et al. proposed the concepts of the depth-resolution plane and lateral-resolution plane to demonstrate the variation in lateral and longitudinal resolutions along the axial direction [10]. However, their analysis and experimental verification were qualitative, and the limits of the lateral and depth resolutions cannot be achieved exactly. Navarro et al. calculated the lateral resolution limit and studied its nonhomogeneity in integral imaging [11]. Nonetheless, the depth resolution limit cannot be derived employing their method. Therefore, for the purpose of evaluating the display and reconstruction of integral imaging, the current research results mentioned above are not complete enough to derive lateral and depth resolutions quantitatively and demonstrate their distribution features in image space. In this Letter, for the first time to our knowledge, an approximate voxel model for integral imaging is proposed by ray tracing. The size and distribution of the 3D voxel in imaging space are derived with this model and confirmed by reconstruction experiments. The model is simple and needs no complicated calculations. It could be used to evaluate the resolution characteristics of integral imaging systems. In the capturing stage of integral imaging, an object point located within the common field of view of the lens 0146-9592/14/072032-04$15.00/0
array is recorded as corresponding image points through each elemental lens. In integral imaging display and reconstruction, 3D images are achieved in the image space by the integration of the elemental images (EIs) through the elemental lens. Since the evaluation of the display and reconstruction system should be objective and independent of EIs, the capturing stage is treated as an ideal recording process in our analysis. Then owing to the limited size of the pixel in the display device, the pixels containing the corresponding image points in each EI, referred to as the corresponding pixels, overlap in the image space and lead to a spatial energy distribution of the light field with 3D information of the object. In fact, this overlapping region cannot be resolved by the viewer unless the signal-to-noise ratio (SNR) is high enough. Therefore, when the overlapped light comes from the corresponding pixel, the energy-overlap region with the highest SNR, called the voxel, will be the minimum unit to be resolved in the space. When all the corresponding pixels overlap in the image space through the elemental lens completely, such as in Fig. 1, all the corresponding image points locate in the corresponding pixel center, and the dashed lines in the figure connecting the center of the corresponding pixel with the elemental lens center will intersect at point A. In Fig. 1, A is the integral image point in image space, A0n is the corresponding pixel in the nth EI (n 1; …; N), N is the number of the elemental lenses, k is the index of the elemental lens, δ is the pixel size, p is the aperture of the elemental lens, and g is the distance between the lens array and the EIs. The shadow in the figure is the region where the light beams from A0n pixels overlap. Therefore, this energy-overlap region with the highest SNR is a voxel in the image space. For convenience and simplification, the voxel is approximated as a cuboid, whose lateral size is decided by the projection of the corresponding pixel on the image plane where the integral image point A is located. The longitudinal size of the voxel is decided by the projection of the corresponding pixel in the marginal EI on the z axis. The following equations apply: © 2014 Optical Society of America
April 1, 2014 / Vol. 39, No. 7 / OPTICS LETTERS
Fig. 1. Voxel defined by complete overlapping of the corresponding pixels. (a) 2D view and (b) 3D view.
ZC
pg LP − p
HC
ZC δ pδ ; g LP − p
DC N − 1pg
LP p mδ
m 1; 2; …
p ; (1) 2δ
(2)
2δ : N − 1 LP − p2 − δ2 2
(3)
Here Z C is the coordinate of point A; H C and DC are the lateral size and longitudinal size of the voxel, respectively; and LP is the disparity of the corresponding image points. In Fig. 2, A and B are two neighboring integral image points in the image space, and pixels A0n and B0n are the corresponding pixels of points A and B in the nth EI, respectively, and they are adjacent in the marginal Nth EI. Since A is integrated by complete overlapping of its corresponding pixels, as shown in Fig. 1, the difference between the disparities of the corresponding image points of A and B is less than half a pixel due to different recording angles of the elemental lens, which can be derived according to the principles of integral imaging: LPA − LPB
δ : N −k
2033
Fig. 2. Voxel defined by partial overlap of the corresponding pixels. (a) 2D view and (b) 3D view.
shadow in Fig. 2 is the region where the light beams from B0n corresponding pixels overlap partially, and it has a local maximum of the SNR. This partial overlapping phenomenon will last until the distance between points A and B is large enough that all corresponding pixels of point B in each EI are distinguishable from the corresponding pixels of point A. At that time, the corresponding pixels of point B overlap completely in the image space, just like the case shown in Fig. 1. Hence, the partial overlapping region between two adjacent complete overlapping regions can be considered a voxel. For convenience and simplification, the voxel is approximated as a cuboid whose longitudinal size is the length of the partial overlapping region. The lateral size is decided by the projection of half the corresponding pixel on the middle plane of the partial overlapping region. The following equations apply:
HU
ZU δ 2g N −12 pδ 2LP −2p−δ ; 4 N −1LP −p−δδN −1LP −p−δ (5)
(4)
This means that corresponding pixels of points B and A are separated in the marginal EI, while they are overlapped in other EIs. Then the corresponding pixels of point B will overlap partially in the image space through the elemental lens, and the dashed lines connecting the center of the corresponding pixel with the elemental lens center do not intersect at point B, as shown in Fig. 2. The
DU N −1pg
N −3δ : N −1LP −p−δδN −1LP −p−δ (6)
Here H U and DU are the lateral size and longitudinal size of the voxel, and Z U is the coordinate of the middle plane of the partial overlapping region and is expressed as follows:
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ZU
OPTICS LETTERS / Vol. 39, No. 7 / April 1, 2014
N −12 pg 2LP −2p−δ : 2 N −1LP −p−δδN −1LP −p−δ (7)
To illustrate this model, the voxel size and its distribution along the axis of an integral imaging display or reconstruction system are calculated using Eqs. (2), (3), (5), and (6). The lens array in the system has 11 × 11 elemental lenses, with aperture width of 10.24 mm. Each EI has 512 × 512 pixels, and the pixel size is δ 0.02 mm. The gap g between EIs and the lens array is 40 mm. The voxel size and its distribution along the axis can then be derived and are shown as Fig. 3. Since the voxel is the minimum unit that can be resolved in the space, the lateral and depth resolution can be defined as the reciprocal of the voxel size; then their variation can also be derived and is shown in Fig. 3. As described previously, the voxel size is decided by the projection of the corresponding pixel in image space. So the projection area will increase with axis distance z, which makes the voxel size increase also. But it is opposite for lateral and depth resolutions. At the same time, the range with resolution invariable will increase with voxel size as well. From the derivation of voxel size, it is known that the lateral size of the voxel derived from the partial overlapping region is smaller and the longitudinal size is larger due to pixelation, as compared with that of the voxel derived from the complete overlapping region. So these voxels will distribute nonhomogeneously because of the alternative distributions of complete and partial overlapping regions, which can be shown by the step-shape curves of voxel size in Fig. 3. The upconvex part in the curve of the lateral size of voxels
Fig. 3. Voxel size and resolution along the z axis: (a) lateral size of the voxel and (b) lateral resolution of the voxel and the depth resolution.
shown in Fig. 3(a) corresponds to the voxel determined by the complete overlapping region; the down-convex part in this curve corresponds to the voxel determined by the partial overlapping region. This pattern is reversed in the curve of longitudinal size of voxel shown in Fig. 3(b). For the resolution curves, it is indicated that the lateral and depth resolution also have nonhomogeneous distributions but with reverse variations to the curves of voxel size. By comparing with the results derived by Kavehvash et al. [10], the quantitative results in Fig. 3 are consistent with the qualitative analysis,
Fig. 4. Result of the digital reconstruction experiment. (a) Reconstruction image at z 696 mm, marked with a dashed line in (c) and (d); (b) extracted gray level of the transverse line in (a), with the limit decided by the Rayleigh energy criterion marked by an ellipse; (c) and (d) experimental and calculated voxel sizes, respectively, and their distribution along the z axis from 680 to 720 mm.
April 1, 2014 / Vol. 39, No. 7 / OPTICS LETTERS
which can be treated as a verification of the concept proposed in [10]. To verify the model and the above calculation results, a digital reconstruction experiment is implemented. The object used in the experiment is a resolution target, and the line widths in the target are discrete values from 0.155 to 0.08 mm with steps of 0.005 mm in Fig. 4(a). The recording distance varies from 298.424 to 315.968 mm with intervals of 0.878 mm, and the EIs corresponding to each recording distance can be generated through computer-generated integral imaging [12]. Using the system parameters mentioned above, the reconstructed images located within the range from z 680 to 720 mm can be obtained by digital reconstruction. After extracting the gray level of the line pairs in each reconstructed image and considering the magnification factor M z∕g, the transverse minimum line width that can be resolved in each reconstruction plane is determined by the Rayleigh energy criterion, while the longitudinal minimum length that can be resolved is determined by the range having constant transverse minimum line width in Fig. 4(b). According to the definition of the voxel, this transverse minimum line width is the lateral size of the voxel, and the longitudinal minimum length is the longitudinal size of the voxel. The variation of voxel size along the z axis is shown in Figs. 4(c) and 4(d), which also represent the distribution features of the lateral and depth resolutions. Because of the limited precision of the reconstruction method and the approximation used in the definition of the voxel, there are differences in voxel size between the experiment and the calculation results, as shown in Fig. 4. However, the trends of the variation in voxel size in the experiment are well consistent with the calculation result, thus verifying the voxel model. In conclusion, an approximate voxel model for integral imaging was proposed for the first time by ray tracing. By analyzing the case of corresponding pixels overlapping completely and partially in the image space, the voxel was defined with an appropriate approximation. Employing this model, the voxel size and its distribution features
2035
in the imaging space were derived, and the calculation of an integral imaging display or reconstruction system was carried out. Finally, the model was verified in a reconstruction experiment using a resolution target. Since the voxel is defined by ray tracing, the proposed model is simple and needs no complicated calculations. It can be used to derive lateral and depth resolutions and demonstrate their distribution features in image space, which is significant to the evaluation and optimization of integral imaging systems. This work was supported by the Ministry of Science and Technology of China under grant 2010CB327702, the National Natural Science Foundation of China (NSFC) under grant 61108046, and the Research Fund for the Doctoral Program of Higher Education of China under grant 20100031120033. References 1. A. Stern and B. Javidi, Proc. IEEE 94, 591 (2006). 2. X. Xiao, B. Javidi, M. Martinez-Corral, and A. Stern, Appl. Opt. 52, 546 (2013). 3. B. Javidi, R. Ponce-Díaz, and S. H. Hong, Opt. Lett. 31, 1106 (2006). 4. D. C. Hwang, D. H. Shin, S. C. Kim, and E. S. Kim, Appl. Opt. 47, D128 (2008). 5. S. Kishk and B. Javidi, Opt. Express 11, 3528 (2003). 6. D. Shin and B. Javidi, Opt. Lett. 37, 2130 (2012). 7. H. Liao, T. Dohi, and M. Iwahara, Opt. Express 15, 4814 (2007). 8. M. Cho and B. Javidi, J. Display Technol. 8, 357 (2012). 9. C. H. Wu, Q. Q. Wang, H. X. Wang, and J. H. Lan, J. Opt. Soc. Am. A 30, 2328 (2013). 10. Z. Kavehvash, M. Martinez-Corral, K. Mehrany, S. Bagheri, G. Saavedra, and H. Navarro, J. Opt. Soc. Am. A 29, 525 (2012). 11. H. Navarro, E. Sánchez-Ortiga, G. Saavedra, A. Llavador, A. Dorado, M. Martínez-Corral, and B. Javidi, J. Display Technol. 9, 37 (2013). 12. S. H. Hong, J. S. Jang, and B. Javidi, Opt. Express 12, 483 (2004).
OPTICS LETTERS / Vol. 39, No. 7 / April 1, 2014
Voxel model for evaluation of a three-dimensional display and reconstruction in integral imaging Liqiu Zhou,1 Xing Zhao,1,* Yong Yang,1 and Xiaocong Yuan2 1
2
Institute of Modern Optics, Key Laboratory of Optical Information Science and Technology, Nankai University, Ministry of Education of China, Tianjin 300071, China
Institute of Micro & Nano Optics, College of Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China *Corresponding author: [email protected] Received December 23, 2013; revised February 22, 2014; accepted March 3, 2014; posted March 4, 2014 (Doc. ID 203451); published March 26, 2014 An approximate voxel model for integral imaging is proposed by ray tracing. By analyzing the case of corresponding pixels overlapping completely and partially in the image space, the voxel is defined with an appropriate approximation, and the voxel size and its distribution feature in imaging space are derived. The model is verified in a reconstruction experiment of a resolution target and compared with the calculation result of an integral imaging display or reconstruction system. The proposed model is simple and easy to calculate and thus useful for the evaluation and optimization of integral imaging systems. © 2014 Optical Society of America OCIS codes: (110.6880) Three-dimensional image acquisition; (110.3000) Image quality assessment. http://dx.doi.org/10.1364/OL.39.002032
Integral imaging has been an important branch of threedimensional (3D) display technology owing to its unique advantages compared with holographic display and binocular parallax display [1,2]. Employing digital reconstruction methods, it can also provide digitized 3D information for medical diagnosis, architectural design, animation design, and many other applications. The realization of 3D display and reconstruction with high performance requires high display and reconstruction resolutions [3,4]. There have thus been many reports on the resolution of integral imaging [5–7]. However, most of these results focus on the improvement of viewing effects instead of detailed analysis of the characters of the display resolution. Recently, Cho et al. and Wu et al. optimized an integral imaging system according to the relations between the resolutions and system parameters [8,9]. However, they did not discuss the distribution features of the spatial resolution. Kavehvash et al. proposed the concepts of the depth-resolution plane and lateral-resolution plane to demonstrate the variation in lateral and longitudinal resolutions along the axial direction [10]. However, their analysis and experimental verification were qualitative, and the limits of the lateral and depth resolutions cannot be achieved exactly. Navarro et al. calculated the lateral resolution limit and studied its nonhomogeneity in integral imaging [11]. Nonetheless, the depth resolution limit cannot be derived employing their method. Therefore, for the purpose of evaluating the display and reconstruction of integral imaging, the current research results mentioned above are not complete enough to derive lateral and depth resolutions quantitatively and demonstrate their distribution features in image space. In this Letter, for the first time to our knowledge, an approximate voxel model for integral imaging is proposed by ray tracing. The size and distribution of the 3D voxel in imaging space are derived with this model and confirmed by reconstruction experiments. The model is simple and needs no complicated calculations. It could be used to evaluate the resolution characteristics of integral imaging systems. In the capturing stage of integral imaging, an object point located within the common field of view of the lens 0146-9592/14/072032-04$15.00/0
array is recorded as corresponding image points through each elemental lens. In integral imaging display and reconstruction, 3D images are achieved in the image space by the integration of the elemental images (EIs) through the elemental lens. Since the evaluation of the display and reconstruction system should be objective and independent of EIs, the capturing stage is treated as an ideal recording process in our analysis. Then owing to the limited size of the pixel in the display device, the pixels containing the corresponding image points in each EI, referred to as the corresponding pixels, overlap in the image space and lead to a spatial energy distribution of the light field with 3D information of the object. In fact, this overlapping region cannot be resolved by the viewer unless the signal-to-noise ratio (SNR) is high enough. Therefore, when the overlapped light comes from the corresponding pixel, the energy-overlap region with the highest SNR, called the voxel, will be the minimum unit to be resolved in the space. When all the corresponding pixels overlap in the image space through the elemental lens completely, such as in Fig. 1, all the corresponding image points locate in the corresponding pixel center, and the dashed lines in the figure connecting the center of the corresponding pixel with the elemental lens center will intersect at point A. In Fig. 1, A is the integral image point in image space, A0n is the corresponding pixel in the nth EI (n 1; …; N), N is the number of the elemental lenses, k is the index of the elemental lens, δ is the pixel size, p is the aperture of the elemental lens, and g is the distance between the lens array and the EIs. The shadow in the figure is the region where the light beams from A0n pixels overlap. Therefore, this energy-overlap region with the highest SNR is a voxel in the image space. For convenience and simplification, the voxel is approximated as a cuboid, whose lateral size is decided by the projection of the corresponding pixel on the image plane where the integral image point A is located. The longitudinal size of the voxel is decided by the projection of the corresponding pixel in the marginal EI on the z axis. The following equations apply: © 2014 Optical Society of America
April 1, 2014 / Vol. 39, No. 7 / OPTICS LETTERS
Fig. 1. Voxel defined by complete overlapping of the corresponding pixels. (a) 2D view and (b) 3D view.
ZC
pg LP − p
HC
ZC δ pδ ; g LP − p
DC N − 1pg
LP p mδ
m 1; 2; …
p ; (1) 2δ
(2)
2δ : N − 1 LP − p2 − δ2 2
(3)
Here Z C is the coordinate of point A; H C and DC are the lateral size and longitudinal size of the voxel, respectively; and LP is the disparity of the corresponding image points. In Fig. 2, A and B are two neighboring integral image points in the image space, and pixels A0n and B0n are the corresponding pixels of points A and B in the nth EI, respectively, and they are adjacent in the marginal Nth EI. Since A is integrated by complete overlapping of its corresponding pixels, as shown in Fig. 1, the difference between the disparities of the corresponding image points of A and B is less than half a pixel due to different recording angles of the elemental lens, which can be derived according to the principles of integral imaging: LPA − LPB
δ : N −k
2033
Fig. 2. Voxel defined by partial overlap of the corresponding pixels. (a) 2D view and (b) 3D view.
shadow in Fig. 2 is the region where the light beams from B0n corresponding pixels overlap partially, and it has a local maximum of the SNR. This partial overlapping phenomenon will last until the distance between points A and B is large enough that all corresponding pixels of point B in each EI are distinguishable from the corresponding pixels of point A. At that time, the corresponding pixels of point B overlap completely in the image space, just like the case shown in Fig. 1. Hence, the partial overlapping region between two adjacent complete overlapping regions can be considered a voxel. For convenience and simplification, the voxel is approximated as a cuboid whose longitudinal size is the length of the partial overlapping region. The lateral size is decided by the projection of half the corresponding pixel on the middle plane of the partial overlapping region. The following equations apply:
HU
ZU δ 2g N −12 pδ 2LP −2p−δ ; 4 N −1LP −p−δδN −1LP −p−δ (5)
(4)
This means that corresponding pixels of points B and A are separated in the marginal EI, while they are overlapped in other EIs. Then the corresponding pixels of point B will overlap partially in the image space through the elemental lens, and the dashed lines connecting the center of the corresponding pixel with the elemental lens center do not intersect at point B, as shown in Fig. 2. The
DU N −1pg
N −3δ : N −1LP −p−δδN −1LP −p−δ (6)
Here H U and DU are the lateral size and longitudinal size of the voxel, and Z U is the coordinate of the middle plane of the partial overlapping region and is expressed as follows:
2034
ZU
OPTICS LETTERS / Vol. 39, No. 7 / April 1, 2014
N −12 pg 2LP −2p−δ : 2 N −1LP −p−δδN −1LP −p−δ (7)
To illustrate this model, the voxel size and its distribution along the axis of an integral imaging display or reconstruction system are calculated using Eqs. (2), (3), (5), and (6). The lens array in the system has 11 × 11 elemental lenses, with aperture width of 10.24 mm. Each EI has 512 × 512 pixels, and the pixel size is δ 0.02 mm. The gap g between EIs and the lens array is 40 mm. The voxel size and its distribution along the axis can then be derived and are shown as Fig. 3. Since the voxel is the minimum unit that can be resolved in the space, the lateral and depth resolution can be defined as the reciprocal of the voxel size; then their variation can also be derived and is shown in Fig. 3. As described previously, the voxel size is decided by the projection of the corresponding pixel in image space. So the projection area will increase with axis distance z, which makes the voxel size increase also. But it is opposite for lateral and depth resolutions. At the same time, the range with resolution invariable will increase with voxel size as well. From the derivation of voxel size, it is known that the lateral size of the voxel derived from the partial overlapping region is smaller and the longitudinal size is larger due to pixelation, as compared with that of the voxel derived from the complete overlapping region. So these voxels will distribute nonhomogeneously because of the alternative distributions of complete and partial overlapping regions, which can be shown by the step-shape curves of voxel size in Fig. 3. The upconvex part in the curve of the lateral size of voxels
Fig. 3. Voxel size and resolution along the z axis: (a) lateral size of the voxel and (b) lateral resolution of the voxel and the depth resolution.
shown in Fig. 3(a) corresponds to the voxel determined by the complete overlapping region; the down-convex part in this curve corresponds to the voxel determined by the partial overlapping region. This pattern is reversed in the curve of longitudinal size of voxel shown in Fig. 3(b). For the resolution curves, it is indicated that the lateral and depth resolution also have nonhomogeneous distributions but with reverse variations to the curves of voxel size. By comparing with the results derived by Kavehvash et al. [10], the quantitative results in Fig. 3 are consistent with the qualitative analysis,
Fig. 4. Result of the digital reconstruction experiment. (a) Reconstruction image at z 696 mm, marked with a dashed line in (c) and (d); (b) extracted gray level of the transverse line in (a), with the limit decided by the Rayleigh energy criterion marked by an ellipse; (c) and (d) experimental and calculated voxel sizes, respectively, and their distribution along the z axis from 680 to 720 mm.
April 1, 2014 / Vol. 39, No. 7 / OPTICS LETTERS
which can be treated as a verification of the concept proposed in [10]. To verify the model and the above calculation results, a digital reconstruction experiment is implemented. The object used in the experiment is a resolution target, and the line widths in the target are discrete values from 0.155 to 0.08 mm with steps of 0.005 mm in Fig. 4(a). The recording distance varies from 298.424 to 315.968 mm with intervals of 0.878 mm, and the EIs corresponding to each recording distance can be generated through computer-generated integral imaging [12]. Using the system parameters mentioned above, the reconstructed images located within the range from z 680 to 720 mm can be obtained by digital reconstruction. After extracting the gray level of the line pairs in each reconstructed image and considering the magnification factor M z∕g, the transverse minimum line width that can be resolved in each reconstruction plane is determined by the Rayleigh energy criterion, while the longitudinal minimum length that can be resolved is determined by the range having constant transverse minimum line width in Fig. 4(b). According to the definition of the voxel, this transverse minimum line width is the lateral size of the voxel, and the longitudinal minimum length is the longitudinal size of the voxel. The variation of voxel size along the z axis is shown in Figs. 4(c) and 4(d), which also represent the distribution features of the lateral and depth resolutions. Because of the limited precision of the reconstruction method and the approximation used in the definition of the voxel, there are differences in voxel size between the experiment and the calculation results, as shown in Fig. 4. However, the trends of the variation in voxel size in the experiment are well consistent with the calculation result, thus verifying the voxel model. In conclusion, an approximate voxel model for integral imaging was proposed for the first time by ray tracing. By analyzing the case of corresponding pixels overlapping completely and partially in the image space, the voxel was defined with an appropriate approximation. Employing this model, the voxel size and its distribution features
2035
in the imaging space were derived, and the calculation of an integral imaging display or reconstruction system was carried out. Finally, the model was verified in a reconstruction experiment using a resolution target. Since the voxel is defined by ray tracing, the proposed model is simple and needs no complicated calculations. It can be used to derive lateral and depth resolutions and demonstrate their distribution features in image space, which is significant to the evaluation and optimization of integral imaging systems. This work was supported by the Ministry of Science and Technology of China under grant 2010CB327702, the National Natural Science Foundation of China (NSFC) under grant 61108046, and the Research Fund for the Doctoral Program of Higher Education of China under grant 20100031120033. References 1. A. Stern and B. Javidi, Proc. IEEE 94, 591 (2006). 2. X. Xiao, B. Javidi, M. Martinez-Corral, and A. Stern, Appl. Opt. 52, 546 (2013). 3. B. Javidi, R. Ponce-Díaz, and S. H. Hong, Opt. Lett. 31, 1106 (2006). 4. D. C. Hwang, D. H. Shin, S. C. Kim, and E. S. Kim, Appl. Opt. 47, D128 (2008). 5. S. Kishk and B. Javidi, Opt. Express 11, 3528 (2003). 6. D. Shin and B. Javidi, Opt. Lett. 37, 2130 (2012). 7. H. Liao, T. Dohi, and M. Iwahara, Opt. Express 15, 4814 (2007). 8. M. Cho and B. Javidi, J. Display Technol. 8, 357 (2012). 9. C. H. Wu, Q. Q. Wang, H. X. Wang, and J. H. Lan, J. Opt. Soc. Am. A 30, 2328 (2013). 10. Z. Kavehvash, M. Martinez-Corral, K. Mehrany, S. Bagheri, G. Saavedra, and H. Navarro, J. Opt. Soc. Am. A 29, 525 (2012). 11. H. Navarro, E. Sánchez-Ortiga, G. Saavedra, A. Llavador, A. Dorado, M. Martínez-Corral, and B. Javidi, J. Display Technol. 9, 37 (2013). 12. S. H. Hong, J. S. Jang, and B. Javidi, Opt. Express 12, 483 (2004).
