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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 16 — Aug. 1, 2011
  • pp: 15205–15211
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Holographic video at 40 frames per second for 4-million object points

Peter Tsang, W.-K. Cheung, T.-C. Poon, and C. Zhou  »View Author Affiliations


Optics Express, Vol. 19, Issue 16, pp. 15205-15211 (2011)
http://dx.doi.org/10.1364/OE.19.015205


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Abstract

We propose a fast method for generating digital Fresnel holograms based on an interpolated wavefront-recording plane (IWRP) approach. Our method can be divided into two stages. First, a small, virtual IWRP is derived in a computational-free manner. Second, the IWRP is expanded into a Fresnel hologram with a pair of fast Fourier transform processes, which are realized with the graphic processing unit (GPU). We demonstrate state-of-the-art experimental results, capable of generating a 2048x2048 Fresnel hologram of around 4 × 10 6 object points at a rate of over 40 frames per second.

© 2011 OSA

1. Introduction

Past research has demonstrated that the Fresnel hologram of a three-dimensional scene can be generated numerically by computing the fringe patterns emerged from each object point to the hologram plane. In brief, given a scene is of self-illuminating object pointsO=[o0(x0,y0,z0),o1(x1,y1,z1),.....,oN1(xN1,yN1,zN1)],the diffraction pattern D(x,y) on the hologram plane can be derived as
D(x,y)=j=0N1ajrjexp(ikrj)=j=0N1[ajrjcos(krj)+iajrjsin(krj)],
(1)
where ajand rj represents the intensity of the ‘jth’ point in O and its distance to the position (x,y) on the diffraction plane, k=2π/λ is the wavenumber and λ is the wavelength of the light. Although the method is effective, the computation involved in generating a hologram is extremely high. In the past lots of research attempts have been conducted to overcome the above problems, such as the works developed in [2

2. S. C. Kim and E. S. Kim, “Fast computation of hologram patterns of a 3D object using run-length encoding and novel look-up table methods,” Appl. Opt. 48(6), 1030–1041 (2009). [CrossRef]

- 12

12. P. W. M. Tsang, J.-P. Liu, W. K. Cheung, and T.-C. Poon, “Fast generation of Fresnel holograms based on multirate filtering,” Appl. Opt. 48(34), H23–H30 (2009). [CrossRef] [PubMed]

]. Recently, a fast method has been reported by Shimobaba et al. in [13

13. T. Shimobaba, H. Nakayama, N. Masuda, and T. Ito, “Rapid calculation algorithm of Fresnel computer-generated-hologram using look-up table and wavefront-recording plane methods for three-dimensional display,” Opt. Express 18(19), 19504–19509 (2010). [CrossRef] [PubMed]

]. In their approach, Eq. (1) is first applied to compute the fringe pattern of each object point within a small window on a virtual wavefront recording plane (WRP) which is placed very close to the scene. Subsequently, the hologram is generated from the WRP with Fresnel diffraction. However, as the number of object points increases, the time taken to derive the WRP will be lengthened in a linear manner and real-time generation of holographic video sequence is not possible. In this paper, a method to overcome the limitation in [13

13. T. Shimobaba, H. Nakayama, N. Masuda, and T. Ito, “Rapid calculation algorithm of Fresnel computer-generated-hologram using look-up table and wavefront-recording plane methods for three-dimensional display,” Opt. Express 18(19), 19504–19509 (2010). [CrossRef] [PubMed]

], is proposed. Essentially, we have formulated a novel, computation-free algorithm for generating what we call an interpolated WRP (IWRP). We then expand the IWRP into a Fresnel hologram. Experimental evaluation demonstrates that our proposed method is capable of generating a 2048x2048 hologram for an object scene with around 4×106 object points in less than 25ms.

2. Background of the wavefront-recording plane (WRP) method

For clarity of explanation, a brief outline of the method in [13

13. T. Shimobaba, H. Nakayama, N. Masuda, and T. Ito, “Rapid calculation algorithm of Fresnel computer-generated-hologram using look-up table and wavefront-recording plane methods for three-dimensional display,” Opt. Express 18(19), 19504–19509 (2010). [CrossRef] [PubMed]

] is summarized in this section. To begin with, the following terminology is adopted. The hologram u(x,y) is a vertical, 2D image that is positioned at the origin. A virtual WRP, uw(x,y), is placed at a depth zw from u(x,y). The object scene is composed of a set of self-illuminating pixels, each having an intensity value of aj and located at a perpendicular distance of djfrom the WRP. Without loss of generality we assume that both the hologram, the WRP, and the object scene have the same horizontal and vertical extents of X and Y pixels, respectively, as well as identical sampling pitch p. The hologram generation process can be divided into two stages. In the first stage, the complex wavefront contributed by the object points is computed as
uw(x,y)=j=0N1(Aj/Rwj(xxj,yyj))exp(i2πλRwj(xxj,yyj)),
(2)
where 0xj<X and 0yj<Y are the horizontal and vertical positions of the jth object point, and Rwj(x,y)=(xxj)2+(yyj)2+dj2 is the distance of the point from the WRP. As the object scene is very close to the WRP, the diffracted beam of each object point is assumed to cover a small square window of size W×W (hereafter refer as the virtual window). As such, Eq. (2) can be rewritten as
uw(x,y)=j=0N1fj,
(3)
where fj={AjRwj(x,y)exp(i2πλRwj(x,y))if|xxj|and|yyj|<12W0otherwise

In Eq. (3), the computation of the WRP for each object point is only confined to the region of the virtual window on the WRP. As W is much smaller than X and Y, the computation load is significantly reduced as compared with Eq. (2). In [13

13. T. Shimobaba, H. Nakayama, N. Masuda, and T. Ito, “Rapid calculation algorithm of Fresnel computer-generated-hologram using look-up table and wavefront-recording plane methods for three-dimensional display,” Opt. Express 18(19), 19504–19509 (2010). [CrossRef] [PubMed]

], the calculation is further simplified by pre-computing the exponential terms for all combinations of (xj,yj,dj), and the estimated computational amount is 2αNL¯2. L¯ is the mean perpendicular distance of the object points to the WRP, and α is the arithmetic operations involved in computing the wavefront contributed by each object point. In the second stage, the WRP is expanded to the hologram as
u(x,y)=KF1[F[uw(x,y)]F[h(x,y)]],
(4)
where F[] and F1[] denote the forward and inverse Fourier transform, respectively. K=i/(λzw)exp(i2πzw/λ) is a constant and h(x,y)=exp(iπ(x2+y2)/(λzw)) is an impulse function which is fixed for a given separation zw between the WRP and the hologram. In Eq. (4), the term F[h(x,y)]can be pre-computed in advance, and hence it is only necessary to compute the forward and an inverse Fourier transform operations. As reported in the article, these two processes can be conducted swiftly with GPU.

3. Proposed computational-free interpolated wavefront-recording plane (IWRP) method

Our proposed method is described as follows. First, we note that the resolution of the scene image is generally smaller than that of the hologram. Hence, it is unnecessary to convert every object point of the scene to its wavefront on the WRP. On this basis, we propose to sub-sample the scene image evenly by M times (where M>0) along the horizontal and the vertical directions. Let I(m,n) and d(m,n) represent the intensity and distance from the WRP of the sample object point located at the nth row and the mth column of the object scene. With the sample pitch set to Mp, the physical horizontal and vertical positions of the sample point are located at xm=mMp+Mp/2 and yn=nMp+Mp/2, respectively, where m and n are integer values. A square support, as shown in Fig. 1a
Fig. 1 a. Sampling lattice and square support Fig. 1b. A pair of sample points, each associate with a square support and a virtual window on the scene image and the WRP, respectively. Note that the depth (i.e., z position) of the object points are not shown in the diagram.
, is defined for each sample point, with the left side lm, and the right side rm given bylm=xmMp/2, and rm=xm+(M1)p/2. Similarly, the bottom side tn and top side bn of the square support are given by bn=ynMp/2, and tn=yn+(M1)p/2. We point out that the square supports of adjacent sample points are non-overlapping and just touching each other at their boundaries. Next, we assume the contribution of each sample point is contributing to a square virtual window in the WRP with side length equals to Mp as shown in Fig. 1b.

The virtual window is aligned with the square support of the object point, and the wavefront within the virtual window is only contributed by the object point in the square support. Under this approximation, Eq. (2) therefore can be re-written as
uw(x,y)|lmx<rm,bny<tn=I(m,n)exp(i2πRd(m,n)(xxm,yyn)/λ),
(5)
where Rd(m,n)(x,y)=x2+y2+d(m,n)2 is the distance of the sample object point to the location (x,y) on the WRP. As such, the right hand side of Eq. (4) can be further encapsulated into a single function of two independent variables (d(m,n) and I(m,n)) as given by

uw(x,y)=I(m,n)exp(i2πRd(m,n)(xxm,yyn)/λ)=G(xxm,yyn,I(m,n),d(m,n))
(6)

It can be inferred from Eq. (6) that the function G(xxm,yxn,I(m,n),d(m,n)) represents a Fresnel zone plate G(x,y,I(m,n),d(m,n)), within the virtual window, which is shifted to the position (xm,yn). The Fresnel zone plate is contributed by an object point of intensity I(m,n) and at distance d(m,n) from the wavefront plane. Hence for a finite variations of d(m,n) andI(m,n), all the possible combinations of G(x,y,I(m,n),d(m,n)) can be pre-computed in advance, and store in a look up table (LUT). For example, if the depth d(m,n) and I(m,n) are quantized into Nd and NI levels, respectively, there will be a total of Nd×NI combinations. As a result, in the generation of uw(x,y), each of its constituting virtual window can be retrieved from the corresponding entries in the LUT. In another words, the process is computational-free.

Although the decimated has effectively reduced the computation time, as will be shown later, the reconstructed images obtained with the WRP derived from Eq. (6) are weak, noisy, and difficult to observe. This is caused by the sparse distribution of the object points caused by the sub-sampling of the scene image. To overcome this problem, we propose the interpolated WRP (IWRP) to interpolate the associated support of each object point with padding, i.e., the object point is duplicated to all the pixels within each square support. After the interpolation, the wavefront of a virtual window will be contributed by all the object points (which are identical in intensity and depth) within the support as given by

uw(x,y)|lmx<rm,tny<bn=I(m,n)τx=M2M21τy=M2M21exp(i2πRd(m,n)(xxm+τxp,yyn+τyp)/λ)
(7)
=GA(xxm,yyn,I(m,n),d(m,n))
(8)

Similar to Eq. (6), the wavefront function =GA(xxm,yyn,I(m,n),d(m,n)) in Eq. (8) is simply a shifted version of GA(x,y,I(m,n),d(m,n)) which can be pre-computed and stored in a LUT for different combinations of I(m,n) andd(m,n). Consequently, each virtual window in the IWRP can be generated in a computation-free manner by retrieving, from the LUT, the wavefront corresponding to the intensity and depth of the corresponding object point. Comparing Eq. (6) and Eq. (8), it can also be inferred that the number of combination on the values of G(x,y,I(m,n),d(m,n)) and GA(x,y,I(m,n),d(m,n)), and hence the size of the corresponding LUTs, are identical. After uw(x,y)|lmx<rm,tny<bn is generated within the IWRP, Eq. (4) is applied to generate the hologram u(x,y).

4. Experimental results

5. Conclusion

In this paper, we propose a method for real-time generation of Fresnel holograms. A proposed interpolated wavefront recording plane (IWRP) is first constructed with a computation-free process. Subsequently, the IWRP is expanded into a Fresnel hologram via a pair of fast Fourier transform operations that are realized with the GPU. Based on our method a hologram size of 2048x2048, representing an image scene comprising of over 4×106 points, can be generated in less than 25ms, equivalent to 40 frames per second. These results correspond to state-of-the-art speed in the calculation of CGH.

Acknowledgments

The work is partly supported by the Chinese Academy of Sciences Visiting Professorships for Senior International scientists. Grant Number: 2010T2G17

References and links

1.

T.-C. Poon, ed., “Digital holography and three-dimensional display: Principles and Applications,” Springer (2006).

2.

S. C. Kim and E. S. Kim, “Fast computation of hologram patterns of a 3D object using run-length encoding and novel look-up table methods,” Appl. Opt. 48(6), 1030–1041 (2009). [CrossRef]

3.

S.-C. Kim and E.-S. Kim, “Effective generation of digital holograms of three-dimensional objects using a novel look-up table method,” Appl. Opt. 47(19), D55–D62 (2008). [CrossRef] [PubMed]

4.

S.-C. Kim, J.-H. Yoon, and E.-S. Kim, “Fast generation of three-dimensional video holograms by combined use of data compression and lookup table techniques,” Appl. Opt. 47(32), 5986–5995 (2008). [CrossRef] [PubMed]

5.

H. Sakata and Y. Sakamoto, “Fast computation method for a Fresnel hologram using three-dimensional affine transformations in real space,” Appl. Opt. 48(34Issue 34), H212–H221 (2009). [CrossRef] [PubMed]

6.

T. Yamaguchi, G. Okabe, and H. Yoshikawa, “Real-time image plane full-color and full-parallax holographic video display system,” Opt. Eng. 46(12), 125801 (2007). [CrossRef]

7.

H. Yoshikawa, “Fast computation of Fresnel holograms employing difference,” Opt. Rev. 8(5), 331–335 (2001). [CrossRef]

8.

T. Ito, N. Masuda, K. Yoshimura, A. Shiraki, T. Shimobaba, and T. Sugie, “Special-purpose computer HORN-5 for a real-time electroholography,” Opt. Express 13(6), 1923–1932 (2005). [CrossRef] [PubMed]

9.

L. Ahrenberg, P. Benzie, M. Magnor, and J. Watson, “Computer generated holography using parallel commodity graphics hardware,” Opt. Express 14(17), 7636–7641 (2006). [CrossRef] [PubMed]

10.

H. Kang, F. Yaraş, and L. Onural, “Graphics processing unit accelerated computation of digital holograms,” Appl. Opt. 48(34), H137–H143 (2009). [CrossRef] [PubMed]

11.

Y. Seo, H. Cho, and D. Kim, “High-performance CGH processor for real-time digital holography,” Laser App. Chem., Sec. and Env. Ana., OSA Tech. Digest (CD) (OSA, 2008), paper JMA9.

12.

P. W. M. Tsang, J.-P. Liu, W. K. Cheung, and T.-C. Poon, “Fast generation of Fresnel holograms based on multirate filtering,” Appl. Opt. 48(34), H23–H30 (2009). [CrossRef] [PubMed]

13.

T. Shimobaba, H. Nakayama, N. Masuda, and T. Ito, “Rapid calculation algorithm of Fresnel computer-generated-hologram using look-up table and wavefront-recording plane methods for three-dimensional display,” Opt. Express 18(19), 19504–19509 (2010). [CrossRef] [PubMed]

OCIS Codes
(090.0090) Holography : Holography
(090.1760) Holography : Computer holography
(090.1995) Holography : Digital holography

ToC Category:
Holography

History
Original Manuscript: June 29, 2011
Manuscript Accepted: July 16, 2011
Published: July 22, 2011

Virtual Issues
August 12, 2011 Spotlight on Optics

Citation
Peter Tsang, W.-K. Cheung, T.-C. Poon, and C. Zhou, "Holographic video at 40 frames per second for 4-million object points," Opt. Express 19, 15205-15211 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-16-15205


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References

  1. T.-C. Poon, ed., “Digital holography and three-dimensional display: Principles and Applications,” Springer (2006).
  2. S. C. Kim and E. S. Kim, “Fast computation of hologram patterns of a 3D object using run-length encoding and novel look-up table methods,” Appl. Opt. 48(6), 1030–1041 (2009). [CrossRef]
  3. S.-C. Kim and E.-S. Kim, “Effective generation of digital holograms of three-dimensional objects using a novel look-up table method,” Appl. Opt. 47(19), D55–D62 (2008). [CrossRef] [PubMed]
  4. S.-C. Kim, J.-H. Yoon, and E.-S. Kim, “Fast generation of three-dimensional video holograms by combined use of data compression and lookup table techniques,” Appl. Opt. 47(32), 5986–5995 (2008). [CrossRef] [PubMed]
  5. H. Sakata and Y. Sakamoto, “Fast computation method for a Fresnel hologram using three-dimensional affine transformations in real space,” Appl. Opt. 48(34Issue 34), H212–H221 (2009). [CrossRef] [PubMed]
  6. T. Yamaguchi, G. Okabe, and H. Yoshikawa, “Real-time image plane full-color and full-parallax holographic video display system,” Opt. Eng. 46(12), 125801 (2007). [CrossRef]
  7. H. Yoshikawa, “Fast computation of Fresnel holograms employing difference,” Opt. Rev. 8(5), 331–335 (2001). [CrossRef]
  8. T. Ito, N. Masuda, K. Yoshimura, A. Shiraki, T. Shimobaba, and T. Sugie, “Special-purpose computer HORN-5 for a real-time electroholography,” Opt. Express 13(6), 1923–1932 (2005). [CrossRef] [PubMed]
  9. L. Ahrenberg, P. Benzie, M. Magnor, and J. Watson, “Computer generated holography using parallel commodity graphics hardware,” Opt. Express 14(17), 7636–7641 (2006). [CrossRef] [PubMed]
  10. H. Kang, F. Yaraş, and L. Onural, “Graphics processing unit accelerated computation of digital holograms,” Appl. Opt. 48(34), H137–H143 (2009). [CrossRef] [PubMed]
  11. Y. Seo, H. Cho, and D. Kim, “High-performance CGH processor for real-time digital holography,” Laser App. Chem., Sec. and Env. Ana., OSA Tech. Digest (CD) (OSA, 2008), paper JMA9.
  12. P. W. M. Tsang, J.-P. Liu, W. K. Cheung, and T.-C. Poon, “Fast generation of Fresnel holograms based on multirate filtering,” Appl. Opt. 48(34), H23–H30 (2009). [CrossRef] [PubMed]
  13. T. Shimobaba, H. Nakayama, N. Masuda, and T. Ito, “Rapid calculation algorithm of Fresnel computer-generated-hologram using look-up table and wavefront-recording plane methods for three-dimensional display,” Opt. Express 18(19), 19504–19509 (2010). [CrossRef] [PubMed]

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