Fast Hologram Pattern Generation by Removing Concentric Redundancy

Seok Lee, Hocheon Wey, Dongkyung Nam, Dusik Park and Changyeoung Kim Samsung Electronics, Gyeonggi-do, Korea

Abstract We propose a fast hologram pattern generation framework by removing concentric redundancy of point source hologram. We minimized repeated or redundant calculations in point hologram generation and accumulation process, and results shows that proposed method is over 1400 times faster than analytic method.

Author Keywords hologram generation; fringe pattern generation; holographic display.

1. Introduction Holographic display is considered as ideal 3D display because it can generate whole 3D information of light distribution which can be seen in any direction without visual discomfort. Computer generated hologram (CGH) which is proposed by Brown and Lohmann [1][2] can be made from discrete input signal such as color plus depth or graphics data. In CGH, the 3D object is considered as a set of point cloud and each point hologram is accumulated to make holographic fringe pattern. Because the computational burden is proportional to the number of object point, the complexity problem is one of the critical bottlenecks for the holographic display. Plesniak [3] proposed an incremental update method for point source based hologram. In their work, hologram pattern of previous time is updated only for the changed part of object point set. Thus the fringe data for other stationary points can be reused in generating next hologram frame. This approach can be applied especially in hologram video to reduce temporal redundancy. In this paper, we present a fast hologram pattern generation method by removing concentric redundancy of point source hologram.

2. Methods To speed up hologram pattern computation, we minimized repeated or redundant calculations in point hologram generation and accumulation process. In point hologram generation, the fringe pattern size is scaled adaptively and concentric redundancy of each pattern is removed by substituting the calculation of wave

propagation with interpolation and duplication. Also the depth of object volume is quantized into predefined levels according to the input depth data. Followings are detailed explanation for each process of proposed algorithm.

(a) Scalable fringe pattern size

We limited the pattern size of point hologram considering Nyquist sampling theory. The sampling rate of SLM is determined by the pixel pitch and high frequency region where phase difference of propagated wave is over half of (wave length) is excluded in calculation of point hologram pattern.

d(x)d(x+ p)

xHologram Plane p

z

Figure 1. Hologram 2D coordinate for point source.

In figure 1, p is the pixel pitch in x direction, x is pixel

position, z is the depth and )(xd is distance between point

source and pixel position. Then sampling condition is 2/)()( xdpxd , and this will be

p

zx

2

For each point source, the effective pattern region is limited in x, y direction by this equation.

(b) Remove concentric redundancy

After effective pattern region is determined as above, point source hologram is generated by removing concentric redundancy.

r(i, j)

(i, j)

d1d2

d3d4

d5d6

d7

Fast interpolate & Fillthe half-quarter

Duplicate thepattern

r(i, j)

(i, j)

Pre-calculate radiusdata & make LUT

Pre-calculate 1DFringe w.r.t. Depth

z = 0.5

z = 2.0

z = 10

M ake FringeLUT

Generate 1D information Symmetr ical pattern

Figure 2. Hologram 2D coordinate for point source

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Discrete 3D object data

1D elemental fringe LUT

Radius LUT

Intensity

depth

Hologram Fringe Pattern

Point extractionPattern

accumulationDepth layerextraction

1D elementalfringe extraction

First quadrantpattern extraction

Full fringepattern extraction

Pattern shifting

Point pattern generation

Input data

Figure 3. Block diagram for proposed hologram pattern generation framework.

Figure 2 shows the process to get fringe pattern of point hologram. In our algorithm, two look up tables are used. For each depth level, the 1D fringe data in diagonal direction is calculated and saved as elemental fringe LUT. For each pixel in half of first quadrant, the distance to center position is also calculated and saved as radius LUT. These two LUT is obtained in offline process, and for each point source the half of quadrant hologram pattern is first calculated. In figure 2, ),( jir is the

distance of current pixel and d is distance of each diagonal

fringe data, then the amplitude and phase at pixel ),( ji is

obtained by linear interpolation of wave information of corresponding diagonal fringe data. The other half of first quadrant is symmetrically copied, and other three quadrants are duplicated from first quadrant. Consequently we reduced almost wave computations because the wave propagation is only calculated for diagonal fringe line in offline process, and all the pattern data is obtained using interpolation and duplication.

(c) Depth quantization

In our algorithm, we assumed that the 3D object volume is limited and each depth level is quantized properly. For example, if input data is 8bit color plus depth format, the depth data is quantized into 256 levels with the corresponding physical distances. Thus the elemental fringe LUT described above section can be made for predefined object volume range in offline process. For each depth level, point hologram pattern is made only once, because the shape of pattern is exactly same except the position. The point hologram for object points in same depth level is generated by shifting the pattern according to each point location.

Figure 3 shows overall process of proposed hologram pattern generation framework. 1D fringe LUT for quantized depth levels and radius LUT for SLM resolution and pixel pitch size are saved in memory. Object layer extracted for each depth level from input data, and point hologram pattern is generated by proposed algorithms. First 1D elemental fringe copied from 1D fringe LUT, and using this and radius LUT first quadrant pattern is obtained. Then by duplication we get full fringe pattern for effective pattern region. Point hologram pattern is shifted by point location and finally accumulated to full fringe pattern. In proposed algorithm, there is no online calculation for wave propagation of 3D object.

(d) Depth layered PSNR

To evaluate the quality of reconstructed 3D object quantitatively, depth layered PSNR (dlPSNR) is proposed. dlPSNR is average of PSNRs for reconstructed 2D object layers for all depth levels. Using hologram pattern, we reconstructed all the 2D images of depth layer and compared the intensity of pixels which is present in original 3D object.

Figure 4 is an example of ‘Coca-can’ image data which has 5 depth layers. The five image shows point set which is present at each depth layer.

3. Results We compared our pattern generation method with analytic Rayleigh-Sommerfeld method which calculate all the wave propagation of 3D object point and accumulate it to obtain hologram pattern. Input data format is 8bit intensity and depth image, and ‘House’, ‘Bunny’, ‘Coca-can’ and ‘Dice’ data is used. The resolution, number of object point and number of depth layer are described in table 1.

Figure 4. Example for object layers of 5 depth level

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Table 1. Data ratio and dlPSNRs for proposed method

Test vector

House Bunny Coca-can Dice

Object resolution

320x240 280x280 320x240 200x200

Object point

40190 34220 14966 7170

Depth layer 26-levels 28-levels 5-levels 124-levels

Data amount (analytic)

1.04e+12 1.80e+12 1.06e+11 1.12e+11

Data amount (proposed)

154329600 131404800 57469440 27532800

Data ratio 0.015% 0.007% 0.054% 0.024%

dlPSNR difference

-0.12 dB -0.04 dB -0.05 dB -0.03 dB

In our simulation, we assumed that the resolution of SLM is 1920x1200, pixel pitch is 8.1um, and wavelength is 633 nm. To assess the computational performance, we compared the total SLM pixel number which we need to calculate the wave equation. We reconstructed the texture image at each depth level by computer simulation and measured dlPSNR with original data. As shown in table 1, the average data ratio of proposed method is 0.025%, which means 4000 times less pixel calculations are needed. The average difference of dlPSNR is -0.06 dB, which means the object quality is not so degraded by proposed method. Figure 5 is reconstruction result of proposed method for Bunny and House data by computer simulation. We can see that the visual quality of our method is almost same with the result of analytic calculation.

Figure 5. Reconstruction result of computer simulation: (a) original, (b) analytic method and (c) proposed method

To prove the computational efficiency of proposed method, we also compared the operation cycle number of each method. We made randomly distributed synthetic 3D data which consists of 90,000 points with 256 depth levels. SLM size was same with above experiment, and we estimated cycle number for each operation by reconfigurable processor case. Table 2 shows the cycle number of operations for proposed method, and table 3 shows the total cycle number ratio. The operation ratio is 0.069% and this means that our algorithm is 1449 times faster than analytic method.

Table 2. Cycle number for each operation in proposed method

Analytic Proposed

Addition 34,560,000 5,292,074

Multiplication 57,600,000 115,281

Division 55,296,000 12,700,974

Square root 27,648,000 69,168

Exponential 27,648,000 69,168

Power 110,592,000 276,672

Copy 0 57,103,228

Iteration 90,000 256

Total cycles 2.82e+13 1.94e+10

Table 3. Operation ratio of proposed method

Analytic method (direct calculation)

Proposed method

# of SLM pixels: 1920x1200

# of 3D points: 90,000

Avg. cycles /point

Total cycles

Avg. cycles/point

Total cycles

Operation ratio

1.10 e+7

2.82 e+13

7563 1.94 e+10

0.069%

Figure 6 is the reconstruction result of hologram by proposed method using the optical table. We used SLM of Holoeye LC-R 1080. The resolution is 1920x1200, and pixel pitch is 8.1um. The wavelength of laser is 633nm. The image is captured by digital video camera of Point Grey Flea2.

Figure 6. Reconstruction result for proposed method using optical table

4. Conclusion In this paper, we established holographic contents creation framework based on fast pattern generation algorithms exploiting the concentric redundancy of point source hologram. The effectiveness of proposed algorithm is proved by simulation and experiment, and the results shows that proposed method is over 1400 times faster than analytic method while the quality

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degradation is less than 0.06 dB in dlPSNR. The study on applying our framework to hardware of parallel and pipeline architecture using GPU/FPGA is ongoing.

In our framework the computational complexity of hologram generation is much reduced, and consequently the resolution of SLM or frame rate of hologram video can be raised accordingly. This means that the algorithm can be used in high quality or high speed holographic display. Though the state of art of hologram technology is not able to give sufficient impressions as commercial 2D or stereo display, software and hardware

technology is rapidly developing, and we can expect to see realistic holograms in TV and mobile devices in near future.

5. References [1] B. Brown and A. Lohmann, "Complex spatial filtering with

binary masks", Applied Optics, 5(5), pp.967-968, 1996.

[2] A. Lohmann,"Binary fraunhofer holograms, generated by computer", Applied Optics 6(10), pp.1739-1748, 1967.

[3] W. Plesniak "Incremental update of computer-generated holograms", Opt. Eng. 42(6), pp.1560-1571, 2003.

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