3D display by binary computer-generated holograms with localized random down-samplingand adaptive intensity accumulation

LIU, Jung-Ping; WU, Ming-Hsuan; TSANG, Peter W. M.

Published in:Optics Express

Published: 17/08/2020

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Publication details:LIU, J-P., WU, M-H., & TSANG, P. W. M. (2020). 3D display by binary computer-generated holograms withlocalized random down-sampling and adaptive intensity accumulation. Optics Express, 28(17), 24526-24537.https://doi.org/10.1364/OE.399011

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Research Article Vol. 28, No. 17 / 17 August 2020 / Optics Express 24526

3D display by binary computer-generatedholograms with localized randomdown-sampling and adaptive intensityaccumulationJUNG-PING LIU,1,* MING-HSUAN WU,1 AND PETER W. M. TSANG2

1Department of Photonics, Feng Chia University, 100 Wenhwa Rd., Seatwen, Taichung 40724, Taiwan2Department of Electrical Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon,Hong Kong*jpliu@fcu.edu.tw

Abstract: In this paper, we proposed a new technique to realize a high-quality three-dimensional(3D) display by using binary holograms. First, we applied a localized random down-sampling(LRDS) mask to down-sample the object function and generated a binary CGH by direct sign-thresholding. Subsequently, we devised the display by adaptive intensity accumulation (AIA). InAIA, multiple CGHs of the same object are generated. However, selective sampling points of thesame scene are removed according to the reconstructed image of previous binary CGHs as thesecond and more binary CGHs are generated. Finally, these holograms are sequentially displayedon a fast spatial light modulator, a digital micromirror device (DMD). Thus, a high-quality 3Dimage is reconstructed without artifacts and speckle noise.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Computer-generated hologram (CGH) is a promising technique for three-dimensional (3D)display [1]. Color holography [2], dynamic holography [3,4], and near-eye holographic display[5,6] have been demonstrated to prove the abilities of CGH. In the applications of holographicdisplay, usually a phase-only spatial light modulator (SLM), such as a liquid-crystal on silicon(LCoS) device, is applied. Ideally, high-quality 3D image can be reconstructed by a phase-onlyhologram. Nevertheless, there is usually zeroth-order light and noise among the reconstructedimage of the LCoS SLM, which is caused by the Fresnel reflection, modulation error, or phaseflicker of the LCoS. An alternative display device is a binary-modulation SLM, such as theFerroelectric liquid crystal SLM (binary phase) or digital micromirror device (DMD) (binaryintensity). Because of the less degrees of freedom, the quality of image reconstructed by a binarySLM is worse than that by a phase-only SLM. On the upside, the modulation speed of binarySLM can be as fast as 104 binary frames per second. The high frame rate provides additionaldegrees of freedom, including the manipulation of color and viewing directions [7–11].

In using a binary SLM, the complex hologram must be converted to a binary hologram. Typicalcomplex-to-binary conversion methods include detour-phase method [12], down-sampling[13–17], error diffusion [18], direct binary search (DBS) [19–24], and Gerchberg–Saxton (GS)algorithm [25–27]. Particularly, DBS and GS are iteration-based methods and thus the computingspeed is very slow. As a feedback, they can generate binary holograms with better quality. Inusing either of the above methods, the reconstructed image always contains significant artifact andspeckle due to the binarization. There have been many studies on the speckle reduction methodsfor laser display. One kind of the methods are based on the decoherence of the light source[28–31], and thus it will also blur the reconstructed image in holographic display [32]. The otherkind of speckle reduction methods is based on intensity accumulation. Intensity accumulation, inwhich a scene is reconstructed using multiple holograms, has been applied in the 3D display by

#399011 https://doi.org/10.1364/OE.399011Journal © 2020 Received 8 Jun 2020; revised 18 Jul 2020; accepted 24 Jul 2020; published 4 Aug 2020

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phase-only CGHs [33–35]. Recently, intensity accumulation was also applied to the display bybinary CGHs [27,36,37]. Although intensity accumulation can partially suppress the speckle,there is still residual speckle in the reconstructed image. It is time-consuming to further suppressthe residual speckle because fifty GS binary holograms are demanded for the display of a singlescene [27].In this paper, we proposed a new method for high-quality 3D display by using a binary SLM

with low computing load. First, the binary holograms are generated by random down-sampling,which is much faster than DBS or GS, and is compatible to both point-based and layer-based CGHalgorithms. The other merit of random down-sampling is that it will not produce artifact dueto regular-grid or regular-point down sampling, which is a significant problem in conventionaldown-sampling binary hologram. To suppress the residual artifact and speckle in the reconstructedimage, we propose adaptive intensity accumulation (AIA). The concept of AIA is that multipleholograms of the scene are not independently generated. As the first binary hologram is generated,the artifact in the reconstructed image is measured and compensated in the second binaryhologram, and so on. By this way, we can produce a high-quality reconstructed image withrelatively low computing load. The remaining of this paper is organized as follows. In section 2,the principle of down-sampling-based binary CGH and display by intensity accumulation areexplained. In section 3, simulations of various binary CGHs are provided and discussed. Someexperimental results are shown and discussed in section 4. Finally, the concluding remarks areprovided in section 5.

2. Principle

2.1. Localized random down-sampling

A computer-generated complex Fresnel hologram is calculated by

Hc(m, n) =X∑

x=1

Y∑y=1

O(x, y; z)r(m − x, n − y; z)

exp[ik0r(m − x, n − y; z) − iφr(m, n)], (1)

where O(x, y; z) is the object function; (x, y) and (m, n) are the indices of sampling gridat the object plane and the hologram plane, respectively; k0 is the wave number of light.

r(m − x, n − y; z) =√(m − x)2∆2 + (n − y)2∆2 + z2 is the distance between object point and

hologram point, where ∆ is pixel pitch and z the axial distance. X and Y are the pixel numberalong x and y directions, respectively. φr(m, n) is the phase of the reference light. For a plane-wavereference light with an offset angle θy between the wave direction and the z-axis, the phasefunction is expressed as φr(m, n) = k0 sin(θy)n∆. Subsequently, a binary hologram is obtained byapplying sign-thresholding binarization on the complex hologram, given by

Hb(m, n) =

1, Re{Hc(m, n)}>0

0, otherwise.(2)

It should be noted that sign-thresholding binarization [Eq. (2)] will produce serious error on thebinary hologram. The error can be suppressed by down sampling the object [13]. That is, theobject function must be replaced by a down-sampled version of the object,

Od(x, y; z) = O(x, y; z)S(x, y), (3)

where S(x, y) is the function of down-sampling mask. One of the simplest masks is a regularpoint down sampling (RPDS) mask, which is defined as

Srp(x, y) =

1, x = kp ∩ y = kq

0, otherwise,(4)

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where p, q ∈ 1, 2, 3 . . ., and k the down-sampling factor (an integer larger than one). The operator∩ denotes the intersection of the two sets of data. Because only 1/k2 over the total XY pixelsare selected by Srp(x, y), the down sampling rate of Srp(x, y) is R = 1/k2. Equation (4) onlydefines a single down-sampling mask. We can just shift the down-sampling points to generateother down-sampling masks. By this way, total k2 various masks can be generated, as shown inFig. 1(a). Nevertheless, the number of RPDS masks is still limited. This issue can be solved bylocalized random down-sampling (LRDS).

Fig. 1. Sampling grid (black lines) and the selected points (circles) for k = 2 down-sampling.The different colors of circles represent samples from different tiles. j denotes the index ofgenerated mask. (a) RPDS, and (b) LRDS.

In LRDS, the object space is divided to many tiles, and each tile contains k × k pixels.Subsequently, one over the k2 points in the tile is randomly selected as the down-sampling point.Therefore, the down-sampling rate is also R = 1/k2. The LRDS mask can be modeled as

Slr(x, y) =

1, x = kp − τa ∩ y = kq − τb

0, otherwise,(5)

where both τa and τb are random integer and 0 ≤ τa, τb ≤ k − 1. Therefore, for every (p, q), i.e.within a tile, a combination of (τa, τb) is randomly selected, as shown in Fig. 1(b). In this example,every point of a tile is selected once. However, there is no limitation on the selection rule and thustotal k2XY/k2 various masks can be generated. The LRDS mask also have other advantages. Theperiodic structure due to regular-point down-sampling is significantly suppressed. In addition,the random process is localized in a tile. Thus, the sample points distribute uniformly on thewhole object plane, which is unachievable for conventional random down sampling. An exampleof RPDS mask and LRDS mask is shown in Fig. 2.

2.2. Display by adaptive intensity accumulation

In the display by intensity accumulation, multiple holograms of the same scene are fast displayedsequentially. As the refresh rate of the SLM is fast enough, the human vision only detects theintensity accumulation of all reconstructed images. Explicitly, the resulting image reconstructed

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Fig. 2. Example of down-sampling masks for k = 4, X = Y = 128. (a) RPDS, and (b)LRDS.

from J holograms and detected by human eye can be modeled as

I(u, v; zr) =

J∑j=1

Ij(u, v; zr) =

J∑j=1|Hj

b(u, v) ⊗ PSF(u, v; zr)|2, (6)

where (u, v) are the indices of the sampling points at the reconstruction plane. PSF(u, v; zr) isthe free-space point spread function; zr denotes the propagation distance, and the superscriptj is corresponding to the j-th hologram generated by the j-th down-sampling mask. It hasbeen proved that the display by direct intensity accumulation (DIA) [Eq. (6)] can significantlysuppress the speckle noise of binary CGHs [27]. However, the CGHs must be optimized bythe time-consuming GS algorithm. In addition, we will show in the next section that it cannotcompensate the artifacts and the speckles on the reconstructed images of binary holograms bysolely down-sampling. For this reason, we propose adaptive intensity accumulation (AIA) forthe compensation of artifacts and speckles.The flow chart of generating the binary holograms for AIA is shown in Fig. 3. First a binary

hologram is generated by LRDS. It should be noted that for display by AIA, the down-sampledobject function is obtained by

Ojd(x, y; z) = O(x, y; z)Sj

lr(x, y)P(x, y), (7)

where P(x, y) is an adaptive binary mask and initially P(x, y) = 1. As the first binary hologramis generated, it is reconstructed by applying J = 1 in Eq. (6). Subsequently, the normalizeddifference function is evaluated as

D(x, y) =d(x, y) − dmindmax − dmin

, (8)

where d(x, y) = |O(x, y; z)|2 − I(x, y); dmax and dmin are the maximum and minimum of d(x, y),respectively. A small value in D(x, y) means that the reconstructed light in this location isstronger. Hence the region with smaller D(x, y) can be ignored in generating the next hologram.Accordingly, a new adaptive mask for the next binary hologram is produced by

P(x, y) =

1, D(x, y) ≥ T

0, otherwise,(9)

where T is a thresholding value (0<T<1). Therefore, the adaptive mask will remove theobject points at the region with stronger reconstructed light. Other object points are still down

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sampled by the localized random down-sampling mask in the next round of hologram generation[Eq. (7)]. This process repeats in generating each hologram. Because the accumulated intensityof reconstructed image varies in each round, the adaptive mask is changed accordingly. As aresult, the artifact in the reconstructed image can be significantly compensated in the display byAIA.

Fig. 3. Flow chart of binary hologram generation for display by AIA.

3. Simulation

A series of simulations are performed to verify our method. The simulation parameters areas follows. The wavelength is 650 nm. The size of both the object and the hologram areX × Y = 512 × 512 pixels, and the pixel pitch is 7.56 µm. The object distance is 190mm. Thetarget object is shown in Fig. 4(a), and a binary hologram generated by LRDS is shown inFig. 4(b). The intensity of the full reconstructed field calculated by angular spectrum method[38] is shown in Fig. 4(c). Because there is always zeroth-order light and the conjugate term, onlythe first order in the lower right corner is selected as the region of interest for quality evaluation.The peak signal-to-noise ratio (PSNR) is adopted as the quality metric in this paper.

In the first set of simulation, various down-sampling factor k is applied for generating theLRDS holograms. First, the display by DIA is evaluated. The dependence of image quality andthe number of holograms is shown in Fig. 5(a). The curves of k = 3 ∼ 5 are similar, and the PSNRof reconstructed image is significantly improved from 7.80 dB (J = 1) to 21.8 dB (J = 100). As acomparison, by using RPDS together with DIA, the best PSNR is 15.7 dB, at k = 9 (J = k2 = 81).The dependence of the quality and the number of LRDS holograms for AIA is shown in Fig. 5(b).In comparison with Fig. 5(a), the image quality can be improved to PSNR = 29.6 dB (J = 100).We also performed simulations in various thresholding value T, as shown in Fig. 6. It is notedthat the influence of T is larger than k and thus must be carefully selected. We have appliedvarious target images to the simulations and concluded that good image quality can be alwaysachieved by using T = 0.6 ∼ 0.7. Finally, it should be mentioned that the same set of LRDSmasks are applied in all the above simulations. However, the resulting display quality possiblydepends on the applied LRDS masks. Therefore, we generated ten different sets of LRDS masks

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Fig. 4. (a) The original object (Image by 192635 from Pixabay). (b) Central 128 × 128pixels of a binary hologram generated by LRDS. (c) The intensity of full reconstructed fieldof the LRDS binary hologram.

for evaluating both the DIA and the AIA display. For the LRDS+DIA, the difference of theresulting PSNR is little; the standard deviation of the PSNR is always smaller than 0.1 dB. Forthe LRDS+AIA, the PSNR curve is slightly affected by the applied LRDS masks. This is shownin Fig. 7, in which the standard deviation bars are marked on the PSNR curve. The maximumstandard deviation (0.35 dB) occurs at J = 37. Nevertheless, the standard deviation at J = 100 isonly 0.18 dB, implying that the influence of LRDS masks can be mostly omitted, provided thenumber of accumulation is large enough.

Fig. 5. PSNR of the reconstructed image versus the number of holograms generated byvarious down-sampling factor (k). (a) Direct intensity accumulation; (b) Adaptive intensityaccumulation (T = 0.6).

Selected reconstructed images of simulations are shown in Fig. 8. In Fig. 8(a), the reconstructedimages of RPDS holograms with DIA display show that the gridding structure and speckle can besuppressed, but serious artifacts cannot. These artifacts can be completely suppressed by usingthe LRDS holograms with DIA display, as shown in Fig. 8(b). However, speckle is detectedeven though in the case that 100 intensities are accumulated. By using the LRDS hologramswith AIA display, both the artifacts and the speckle can be significantly suppressed, as shownin Fig. 8(c). Finally, we also generated binary holograms by using modified iterative Fresnelping-pong algorithm (MIFA) [27], and display by DIA. In MIFA, every binary hologram isgenerated by five iterations. The quality of the reconstructed image is a little better than thatof LRDS+DIA [Fig. 8(b)], but still worse than that of LRDS+AIA. We have also measured

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Fig. 6. PSNR of the reconstructed image versus the number of holograms generated byvarious thresholding value (T). The down-sampling factor is k = 3.

Fig. 7. PSNR of the reconstructed image versus the number of holograms generated byLRDS+AIA (k = 3, T = 0.6). The mean of ten simulations is plotted as the curve, on whichthe red bars mark ±1 standard deviations.

the average diffraction efficiency, which is defined as the ratio of power of first-order image andtotal power. The diffraction efficiencies of LRDS+DIA, LRDS+AIA, and MIFA+DIA are17.3%, 18.0%, and 19.7%, respectively. The computing times per hologram are 0.28 seconds(LRDS+DIA), 0.55 seconds (LRDS+AIA), and 3.86 seconds (MIFA), respectively. Thecomputing is at the platform of Intel Core i7-870 @ 2.93 GHz and MATLAB 2018b.

4. Experiment and discussion

The experimental setup is depicted in Fig. 9. We applied a fiber-coupled diode laser withwavelength of 650 nm as the light source. The SLM is a DMD (DLP6500FYE) by TexasInstruments with pixel pitch 7.56 µm [39]. The DMD was illuminated by a collimated laserbeam with a diagonal tilt angle (θ = 24◦) to locate the strongest diffraction order at the center ofoptical axis [40–42]. A lensless monochromatic CCD (FLIR FL3-GE-50S5M-C, 2448 × 2048pixels with pitch 3.45 µm) is set at the reconstruction plane (190mm measured from the DMD)to acquire the reconstructed image. The exposure time of a single image is 10.5 ms. The DMD issynchronized with the CCD, and thus all holograms of the same object are displayed sequentiallywithin the time of a single shot. For example, in the AIA of 100 holograms, the display timeof each hologram is 10.5ms/100 = 105 µs, which is the shortest display time of the DMD. Theacquired reconstructed images of different holograms are shown in Fig. 10. The experimental

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Fig. 8. Reconstructed images. (a) Display by DIA of RPDS holograms. (b) Display by DIAof LRDS holograms. (c) Display by AIA of LRDS holograms. (d) Display by DIA of MIFAholograms. The red number under images denotes (J, PSNR).

images are very similar to the images in simulation (Fig. 8). However, the images are a littleblurred due to the difficult alignment of blazed-grating condition of DMD. It is noted thatFig. 10(d) is slightly brighter than other images. This is the reason that the diffraction efficiencyof MIFA+DIA is a little higher than that of other methods. It is also noted that in the best resultsof LRDS+AIA, there is a little speckle noise, which is different from simulation. The residualspeckle noise may be due to the scattering light from the optical elements (e.g. the lens). Thiskind of unpredicted noise source cannot be compensated in the calculation of AIA.We have also conducted a simplified demonstration of 3D display by LRDS+AIA method.

The object consists of two layers. The first layer is the letters “FCU” at z = 190mm, whilethe other layer is the letters “OIP” at z = 210mm. In generating the holograms for AIA, everyhologram are reconstructed at both z = 190mm and z = 210mm, respectively. Thus, thecompensation thresholding is applied on both planes to generate the next hologram. By this way,100 holograms are generated and displayed by AIA. The simulation results together with theexperimental results are shown in Fig. 11. Both layers are successfully reconstructed with goodquality.

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Fig. 9. Schematic of experimental setup.

Fig. 10. Experimentally reconstructed images of (a) RPDS holograms, (b) LRDS holograms,(c) LRDS holograms, (d) MIFA holograms, respectively. The red number under imagedenotes the accumulation number (J).

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Fig. 11. Reconstructed images of a hologram of a 2-layer object. (a) and (b) are simulationresults, and (c) and (d) are experimental results. The reconstruction distances of the twolayers are 190mm [(a) and (c)] and 210mm [(b) and (d)].

5. Conclusion

We have discussed two methods for 3D display by binary holograms. In the first method, multiplebinary holograms based on localized random down-sampling (LRDS) are generated, and thedisplay is accomplished by direct intensity accumulation (DIA) of reconstructed images of allholograms. LRDS+DIA is efficient to suppress the artifacts in the reconstructed image, and itscomputing time is the shortest. However, it still contains speckle noise. In the second method,in addition to LRDS, we propose the adaptive intensity accumulation (AIA) method. That is,selective object points are removed by an adaptive mask, which is generated according to thereconstructed image from previous holograms. The LRDS+AIA method suppresses not onlythe artifacts, but also the speckle noise. The PSNR of the reconstructed image can reach nearly30 dB, which is high enough for most visual applications. The holographic display by either DIAor AIA demands high-frame rate SLM and high transfer-rate interface. The time for displaying afull-color 3D scene is 105 µs × 100 × 3 = 31.5ms, which is short enough for dynamic display.The transfer rate is 1920 × 1080 bits/105 µs = 19.7Gbits/s, which is also achievable by currenthardware framework. Therefore, the proposed LRDS+AIA method can be applied for dynamiccolor 3D display by using three lasers and time-multiplexing theoretically. In our demonstration,the proposed LRDS+AIA method is about seven times faster than the MIFA+DIA. However,the computing time for a scene is still too long to realize real-time holographic display. Thehardware acceleration methods or algorithm optimization for realizing real-time calculation isworth further studying in the future.

Funding

Ministry of Science and Technology, Taiwan (106-2628-E-035-002-MY3, 109-2221-E-035-076-MY3); Research Grants Council, University Grants Committee (11200319).

Disclosures

The authors declare no conflicts of interest.

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