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Fast calculation and correct depth cue are crucial issues in the calculation of computer-generated hologram (CGH) for high quality three-dimensional (3-D) display. An angular-spectrum based algorithm for layer-oriented CGH is proposed. Angular spectra from each layer are synthesized as a layer-corresponded sub-hologram based on the fast Fourier transform without paraxial approximation. The proposed method can avoid the huge computational cost of the point-oriented method and yield accurate predictions of the whole diffracted field compared with other layer-oriented methods. CGHs of versatile formats of 3-D digital scenes, including computed tomography and 3-D digital models, are demonstrated with precise depth performance and advanced image quality.
© 2015 Optical Society of America
1. Introduction
Holographic display is a promising technique for three-dimensional (3-D) data visualization since it can reconstruct the whole optical wave field of a 3-D scene and can provide all the depth cues that human eye can perceive [1]. With the developments of spatial light modulators (SLMs) and computing technology, electro-holography can use SLMs to present computer-generated holograms (CGHs) dynamically, which avoids photosensitive medium and complicated interference recording procedure [2–4 ]. Multiple formats of 3-D data can be encoded into the CGHs as long as their mathematical descriptions are provided. The algorithms for generating these CGHs are directly related with the computing efficiency and the image reconstruction quality [5–7 ].
Major current CGH calculation algorithms are physically based algorithms, including point-oriented [8–10 ] and polygon-oriented algorithms [11, 12 ], which simulate the optical transmission process from the 3-D scene to the hologram plane. The 3-D objects are divided into millions of point sources or polygons segments, which can provide precise geometrical information of the 3-D scene [13]. Continuous motion parallax and accurate depth information can be optically reconstructed. However, since the amount of the primitives would be extremely large for reconstructing a complicated 3-D scene, the computational load would also be aggravated drastically. In point-oriented methods, many optimization techniques can be implemented to accelerate the computing speed, including look-up Tables [14], GPU parallel computing platform [15, 16 ], and so on. Polygon-oriented methods often use rotation-based angular spectrum or analytic formulation to process the tilted 3-D meshes [17, 18 ]. The above algorithms focus on the wave propagation simulations of the individual primitives, and are difficult to integrate with computer graphics rendering techniques, which would affect the fidelity of the reconstructed 3-D scenes. Another method of CGHs is holographic stereogram based, which takes advantages of computer graphics rendering techniques for reconstructing versatile formats of 3-D scenes efficiently [19–21 ]. Since each holographic element corresponds to a two-dimensional parallax image with a fixed resolution, holographic stereograms would sacrifice the depth performance.
Recently, in order to accelerate the computing speed, several layer-based approaches using Fresnel diffraction algorithms were proposed [22–25 ]. Due to the paraxial approximation of the Fresnel diffraction based methods, the quality of the optical reconstruction would be reduced. The calculation error would be more serious in high numerical aperture (NA) systems, and these systems are commonly used in holographic 3-D display applications. Another disadvantage of the Fresnel diffraction calculation is that the sampling interval on the destination plane is proportionate to the propagation distance. Several algorithms were proposed to solve this problem, such as convolution based Fresnel method [24], shifted Fresnel method [26], and multi-step Fresnel method [25, 27 ], which would increase the computational loads.
In this work, an angular-spectrum based algorithm for layer-oriented CGH is proposed. The 3-D complex scene is divided into multiple layers according to their depth information. The angular-spectrum algorithm can be used in simulating the optical transmission processes from these parallel layers to the hologram without paraxial approximation with accurate depth information [28]. Angular spectra from each layer are synthesized as a layer-corresponded sub-hologram based on the fast Fourier transform (FFT). The superimposition of all the sub-holograms could yield accurate predictions of the whole diffracted field without paraxial approximation. Moreover, the amount of calculation has no relationship with the complexity of the 3-D scene, which can ease the computational load efficiently without degrading the image resolution. By using the angular-spectrum based algorithm without paraxial approximation, the quality of the optical reconstructions could be greatly improved compared with the conventional paraxial approximation methods. Experimental demonstrations are presented by using a computed tomography organ and two computer-generated 3-D objects. Numerical simulations and optical experimental results are in good agreement, which shows our proposed method can perform high-quality reconstructions of 3-D scenes with versatile formats.
2. Angular-spectrum layer-oriented method
The angular spectrum algorithm decomposes the optical wavefront into many plane waves with different spatial frequencies and superimposes them in the observation plane, which can be implemented with the help of fast Fourier transform algorithm. It is equivalent to the Reileigh-Sommerfeld diffraction formula, hence both non-paraxial fields and paraxial fields can be simulated with high accuracy [29]. The angular spectrum algorithm is based on the scalar diffraction theory, which yields identical predictions of diffracted fields. The diagram of angular-spectrum layer-oriented algorithm is shown in Fig. 1 . The 3-D model is sliced into S layers as parallel layers with depth cues, where i = 1, 2… S is the index. Li represents the ith layer. For each layer, amplitude information can be extracted from the rendered image. The random phase distribution is added to the amplituede information to simulate the diffusive effect of the object surface. The complex amplitude distribution of the hologram Hi is calculated by the angular spectrum method Hi = F −1 {HF ∙ F (Li)}. Hi is related to the i th layer Li. HF is the angular spectrum transfer function. F denotes the Fourier transform. F −1 denotes the inverse Fourier transform. By adding wavefront distributions of all the sliced layers, the complex distribution of the entire hologram is obtained. The phase distribution of the CGH is extracted and uploaded to the phase-only SLM. Coherent light illumination is applied to the SLM and the 3-D reconstruction is detected by a camera.
Based on the angular spectrum diffraction theory, HF, the transfer function of the angular spectrum can be expressed as
where u and v are spatial frequencies, k = 2π/λ is the wave number, z is the transmission distance. The calculation width of diffraction field is L, and the sampling number is N × N. The discrete form of the Eq. (1) can be expressed aswhere u 0 = v 0 = 1/L, m, n = -N/2, -N/2 + 1, … N/2 - 1. For simplicity, consider one-dimensional wave field of HF, the transfer function can be expressed asThe phase of H(u; z) is . The local frequency flu of the functionφ(u) is given as follows [29, 30 ],In order to avoid the aliasing error in the sampled transfer function, the sampling interval of the transfer function Δu = (2L)−1 should meet the Nyquist theorem,
From Eqs. (4) and (5) , the effective transmission distance z is determined as follows:When umax = N/2L,When L = ΔxN,As shown in Fig. 2(a) , when the sampling interval is fixed, the effective distance increases linearly with the increase of the calculation width. As shown in Fig. 2(b), when the calculation width is fixed, the effective distance decreases with the sampling number. According to the above equations, z, L, N, Δx can be selected. When L = 15.36 mm, λ = 532 nm, N = 1920, z < 461.7 mm; When L = 8.192 mm, λ = 532 nm, N = 1024, z < 246.2 mm; When L = 4.096 mm, λ = 532 nm, N = 512, z < 123.1 mm.
The angular-spectrum layer-oriented method has the advantage of low complexity. According to the point-oriented method, 3-D objects are divided into millions of points. The amount of the computational load is huge as the computation time including millions of exponent calculation together with evolution. The angular-spectrum layer-oriented method can avoid the enormous computing of the point-oriented method. There is only one exponent calculation and the on-line computing is one two-dimensional FFT together with one two-dimensional IFFT. The total computing time TN = TFFT + TIFFT. The complexity of a two-dimensional FFT is O(N2log2N) for N sampling points.
The phase distribution is essential for image formation. The bit depth of phase of the hologram could be reduced to accelerate the speed. We reduce the bit depth of phase values of the hologram, and then evaluate the quality of the related optical reconstruction by peak signal to noise ratio (PSNR). PSNR is the ratio between the maximum possible power of the original image and the power of reconstructed image, and the equation is
where m and n are the horizontal and vertical numbers of pixels, I0 and Ir are the original image and reconstructed image, respectively. A higher PSNR generally indicates that the reconstruction is of higher quality. The PSNRs of the optical reconstructed images related to the different bit depth of the phase are shown in Fig. 3 . The reconstruction exhibits almost the same when the bit depth larger than 2 bits. The results are essential for data compression, and are especially useful in dynamic 3-D display as the phase modulation time of the devices could be very fast when the bit depth is small.3. Demonstration of versatile 3-D digital models
To demonstrate the feasibility of the proposed method, numerical simulations and optical experiments are performed. The demonstrations in this work include 3-D digital models and computed tomography images. Three demonstrations of reconstructions are applied in the same holographic display system. At first, a train is modeled in the Autodesk 3ds Max. The length of the train is 20 mm, and the train is sliced into 50 layers. Then, we calculate the sub-hologram of every layer by the angular spectrum method and accumulate all of the layer holograms into an integral hologram. Then, the phase distribution of the hologram is extracted for the optical reconstruction. The second demonstration is the reconstruction of a medical image of the headangio. The original continuous headangio computed tomography (CT) images are acquired from a 64 slices volume CT scanner. A length of 10 mm headangio is divided into 100 cases. Each case is saved to DICOM image at the resolution of 512 × 512 pixels. The complex amplitude distribution of the sub-hologram related to a case is calculated. Then all of sub-holograms are accumulated into an integral hologram for the optical construction. The third demonstration is a holographic reconstruction of a 3-D movie of a rotating hall. The 3-D model is set up in the Autodesk 3ds Max. The movie has a refresh frequency of 50 Hz with total 360 frames.
The performed holographic display system is shown in Fig. 4 . In the experiment, we utilize the PLUTO Phase-only SLM (HOLOEYE Photonics AG), which is a reflective liquid crystal on silicon (LCOS) micro display with 1920 × 1080 pixels. The pixel pitch of the SLM is 8 μm, and the SLM is addressed with 256 gray-scale levels. The wavelength used in our experiment is 532 nm. The light beam passing through a pinhole is then collimated by the collimator lens to achieve a uniform plane wave illumination. After the uniform plane wave passes through the polarizer, the polarized light beam incidents into the phase-only LCOS SLM behind the beam splitter. The SLM is already loaded with the computer generated phase hologram. Each pixel modulates the phase depth and reflects the light back. The 3-D image is then magnified by a 4-f system, which is formed by the Fourier lenses. With the help of the modified 4-f system, the zero-order interruption is eliminated. Reconstructed images can be viewed directly along the optical axis. A Cannon 500D camera is used to capture the reconstructed images at different depths.
The numerical simulation results and optical reconstruction of the first demonstration are shown in Fig. 5 , respectively. The reconstruction results of the toy train are shown in Figs. 5(a) and 5(d) when the reconstruction plane focused on the front part of the train d = 210 mm. Figures 5(b) and 5(e) are the results when the reconstruction plane focused on the middle part d = 220 mm. Figures 5(c) and 5(f) are the results when reconstruction plane focused on the back part of the train d = 230 mm. In both the simulation and experimental images, the back part would become blur when the front part is focused, and vice versa. The 3-D train is reconstructed successfully, which verifies the method can be used in holographic display.
Fig. 5 Reconstructed images of the train. (a), (b) and (c) are numerical reconstructions of the train at distance d = 210mm, 220mm, 230mm, respectively. (d), (e) and (f) are optical reconstructions of the train at distance d = 210mm, 220mm, 230mm, respectively.
The numerical simulation results and optical reconstruction of the second demonstration are shown in Fig. 6 . The headangio is reconstructed at the distance of 210 mm, and the total length is 10 mm headangio. Figures 6(a) and 6(c) are the simulation and experimental results focused on the front of the headangio. Figures 6(b) and 6(d) are the simulation and experimental results focused on the backend of the headangio.
Fig. 6 Reconstructed images of the headangio. (a) and (b) are numerical reconstructions at distance d = 210 mm, 220 mm, respectively. (c) and (d) are optical reconstructions at distance d = 210 mm, 220 mm, respectively.
The 3-D model and optical reconstruction of the third demonstration are shown in Figs. 7(a) and 7(b) respectively. The third demonstration is a holographic reconstruction of a movie. We calculated 360 holograms of the hall’s different degree. The movie is a rotating hall with 360 frames. The frequency of the movie is 50 Hz, and the media is attached as Visualization 1.
Fig. 7 Reconstructed images of a hall. (a) is the 3-D model and (b) is the optical reconstruction (see Visualization 1).
4. Discussion
The optical performance of the proposed method is directly related to the accuracy of diffraction simulation between the parallel planes. To verify the accuracy of angular spectrum compared to FFT based Fresnel diffraction, one-dimensional diffraction by a chirp function shown in Fig. 8(a) is computed by the two algorithms. The chirp function is used to simulate the source plane with different spatial frequencies. Here, the parameters of the simulation are in accordance with the optical experiment. The sampling distance is 8 μm and the number of the samplings is 1920, hence the calculation width L is 15.36 mm.
Fig. 8 Numerical simulation of angular spectrum method and FFT based Fresnel diffraction. (a) Simulation diagram. (b) SNR of the algorithms.
The accuracies of two algorithm with different propagation distances are shown in Fig. 8(b). The accuracy is defined as the signal-to-noise ratio (SNR) of the wave field calculated by the algorithm compared with the Rayleigh-Sommerfeld diffraction integral. It is shown that the angular spectrum method remains better quality than FFT based Fresnel diffraction within 300 mm. The advantage becomes significant with a higher NA or a shorter propagation distance because the error caused by the paraxial approximation of the Fresnel diffraction is unneglectable in a higher NA system. High NA is preferred in holographic 3-D display system to produce better viewing parameters, and angular spectrum method is capable to support accurate calculations for these kinds of applications.
Computation of the CGHs of the 3-D objects is executed by using a PC with the CPU of the Intel Core i7 (34 GHz) processor and a shared memory of 8 Gbytes. The total computation time is shown in Table1 . The parameters for computing the train and headangio with the layer-oriented method are shown in Table2 . Compared with the point-oriented method, the angular- spectrum layer-oriented method dramatically reduces the computational load.
Table 1. Computation time of the train and headangio
Table 2. Parameters of computing the train and headangio with the layer-oriented method
5. Conclusion
We have presented an angular-spectrum layer-oriented method to generate CGHs of versatile formats of 3-D scenes. 3-D scenes of different digital models and computational tomography scanners can be easily encoded into CGHs with the proposed algorithm. Optical demonstrations have been performed using a phase-only SLM. Experimental results illustrate that the proposed method can perform quality optical reconstructions of the 3-D scenes with high spatial precision. And the reduction of bit depth of phases of the hologram can be also applied in fast dynamic holographic display. The proposed method can relieve the computational load efficiently without degrading the image resolution compared with the point-oriented method. Compared with other layer-oriented methods with paraxial approximation, the angular-spectrum algorithm can be used in a higher NA system which is preferred in holographic 3-D display to produce better viewing parameters and produce accurate calculations. This work could probably provide a new solution for the versatile 3-D visualization of the large-capacity data.
Acknowledgments
This work is supported by National Natural Science Foundation of China (No. 61205013), National Basic Research Program of China (No. 2013CB328801). The authors are very thankful to Prof. Yunlong Sheng for fruitful discussions about the optical reconstruction. The authors also thank the anonymous reviewers for their insightful comments and suggestions that significantly improve and help clarify this paper.
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