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Complex modulation computer-generated hologram by afast hybrid point-source/wave-field approach

Antonin Gilles, Patrick Gioia, Rémi Cozot, Luce Morin

To cite this version:Antonin Gilles, Patrick Gioia, Rémi Cozot, Luce Morin. Complex modulation computer-generatedhologram by a fast hybrid point-source/wave-field approach. IEEE International Conference on ImageProcessing (ICIP 2015), Sep 2015, Quebec, Canada. pp.4962 - 4966, �10.1109/ICIP.2015.7351751�.�hal-01229462v2�

Complex modulation computer-generated hologram by a fast

hybrid point-source/wave-field approach

Antonin Gilles1∗ Patrick Gioia1,2 Remi Cozot1,3 Luce Morin1,4

1 IRT b<>com 2 Orange Labs 3 University of Rennes 1 4 INSA RennesCesson-Sevigne Rennes Rennes Rennes

France France France France

Abstract

We propose a fast Computer-Generated Hologram(CGH) computation method based on a hybrid point-source/wave-field approach. Whereas previously pro-posed methods tried to reduce the computationalcomplexity of the point-source or the wave-field ap-proaches independently, our method uses the twoapproaches together and therefore takes advantagesfrom both of them. The algorithm consists of threesteps. First, the 3D scene is sliced into several depthlayers parallel to the hologram plane. Then, for eachlayer, we compute the complex wave scattered by thislayer either using a wave-field or a point-source ap-proach according to a threshold criterion on the num-ber of points within the layer. Finally, we sum up thecomplex waves scattered by all the depth layers inorder to obtain the final CGH. Experimental resultsreveal that this combination of approaches does notproduce any visible artifact and outperforms both thepoint-source and wave-field approaches.

Keywords : Computer-Generated Hologram,Color holography, Real-time holography, Three-dimensional imaging

∗This work has been achieved within the Institute of Re-search and Technology b<>com, dedicated to digital technolo-gies. It has been funded by the French government throughthe National Research Agency (ANR) Investment referencedANR-A0-AIRT-07. Authors can be reached at {antonin.gilles,patrick.gioia, remi.cozot, luce.morin}@b-com.com.

1 Introduction

Holography is often considered as the most promising3D visualization technology, since it can provide themost authentic and natural three-dimensional illusionto the naked eye. Indeed, it provides complete humandepth cues without the need for special viewing de-vices and without causing eye-strain [1]. Over thepast decades, several methods have been proposedto generate holograms by computer calculation. Us-ing these methods, it is possible to obtain Computer-Generated Holograms (CGH) of synthetic or existingscenes by simulating the propagation of light scat-tered by the scene towards the hologram plane. CGHcomputation techniques usually sample 3D scenes bya set of primitives and calculate light propagation asthe sum of complex light waves scattered by each ofthe primitives. Commonly used primitives includepoints (point-source approach) and planar segments(wave-field approach).

The point-source approach samples 3D scenes by acollection of self-luminous points, and calculates com-plex wave scattered by each of the points using themonochromatic spherical light wave equation. Thisapproach is very flexible and does not impose anyrestriction on the scene geometry. However, its com-plexity is very high since it requires one calculationper point of the scene per pixel of the hologram.Moreover, to produce shapes that appear solid andcontinuous, the scene needs to be sampled at veryhigh densities, making the CGH computation pro-

1

hibitively slow. In order to reduce the computationalcomplexity, several methods have been proposed, in-cluding geometric symmetry [2], look-up tables [3, 4],interframe and interline redundancy reduction [5, 6],difference and recurrence formulas [7, 8], image holo-grams [9, 10], wave-front recording planes [11, 12, 13],using GPU hardware [14, 15], and special purposehardware [16, 17].

The wave-field approach samples 3D scenes by acollection of self-luminous planar segments, and com-putes complex wave scattered by each of the seg-ments using the angular spectrum of plane waves [18].The computation of the angular spectrum of planewaves involves the use of the Fast Fourier Trans-form (FFT) algorithm twice, and is therefore moretime-consuming than the computation of the spher-ical light wave scattered by a single point. How-ever, complex waves scattered by scene points lo-cated within a single planar segment are calculated allat once using the angular spectrum of plane waves.Therefore, this approach is more efficient than thepoint-source approach when objects in a scene con-sist of large planar segments containing many points.However, when the scene geometry contains complexshapes, a large number of small planar segments con-taining only one or a few points are needed to sampleit, making the wave-field approach less efficient thanthe point-source approach. In order to reduce thecomputation burden, several methods have been pro-posed, including the use of analytic expression of theangular spectrum [19, 20, 21, 22, 23], and color spaceconversion [24, 25].

In this paper, we propose a fast CGH computa-tion method based on a hybrid point-source/wave-field approach. Whereas previously proposed meth-ods tried to reduce the computational complexity ofthe point-source or the wave-field approaches inde-pendently, our method uses the two approaches to-gether and therefore takes advantages from both ofthem. Section 2 gives a detailed description of ourmethod, Section 3 gives experimental results, andSection 4 concludes this paper.

���� ����

�� depth layers

�

�

�

Hologram

��

��

3D Scene

Figure 1: Scene geometry and coordinate system usedby the proposed method

2 Proposed method

2.1 Overview

Figure 1 shows the scene geometry and coordinatesystem used by the proposed method. The coordinatesystem is defined by (x, y, z) so that the hologramlies on the (x, y, 0) plane. The 3D scene is treatedas a set of Nz depth layers parallel to the hologramplane and located between zmin and zmax. The holo-gram is sampled on a regular 2D grid of resolutionNx × Ny with a sampling pitch p. Figure 2 showsthe overall block-diagram of the proposed method,which consists of three steps. First, the 3D scene issliced into Nz depth layers parallel to the hologramplane. Then, for each layer d, if the number of pointsNd within the layer exceeds a maximum value Nd,max

(selection criterion which will be determined in sec-tion 2.4), we compute the complex wave scattered bythis layer using a wave-field approach. Otherwise, ifNd is smaller than Nd,max, the method calculates thecomplex wave scattered by this layer using a point-source approach. Finally, the method sums up thecomplex waves scattered by all the depth layers inorder to obtain the final CGH. Afterwards, the sceneimage can be reconstructed from the computed CGHpattern.

2.2 Wave-field approach

When the number of points Nd within layer d ex-ceeds Nd,max, the complex wave Uw

d scattered by this

2

NO YES

Slice scene into ��

depth layers

Select first

depth layer �

�� ← Number of points

within the layer �

�� �� ���

� ← � ��

Compute complex wave ��

scattered by the layer � based on

a wave-field approach

� ← � ��

Compute complex wave ��

scattered by the layer � based on

a point-based approach

Select next

depth layer �

� ← � �

Figure 2: Block-diagram of the proposed method

layer towards the hologram plane is computed using awave-field approach. To this end, we use the angularspectrum of plane waves [18], which is given by

Uwd (x, y) = F−1

{

F{

Ad(x, y)ejφd(x,y)

}

× e−j2π√

λ−2−u2

−v2zd

}

, (1)

where Ad(x, y) and φd(x, y) are the amplitude andphase of the (x, y) point in layer d; λ is the wavelengthof light, u and v are the spatial frequencies, zd is thedepth of layer d, and F and F−1 are respectivelythe forward and inverse Fourier Transform. Thesetransforms can be computed using the Fast FourierTransform algorithm (FFT). In order to render a dif-fusive scene, the phase φd(x, y) is set to a randomvalue. Finally, the complex wave Uw scattered by allthe layers whose number of points Nd exceeds Nd,max

in the hologram plane is given by

Uw(x, y) =

Nz∑

d=0Nd>Nd,max

Uwd (x, y). (2)

In order to avoid one FFT per layer and thereforeto speed-up the computation, the algorithm sums upthe complex waves scattered by each layer directlyin the frequency domain, and then inverse Fouriertransforms the result to get Uw, as proposed in [19]:

Uwd (u, v) = F

{

Ad(x, y)ejφd(x,y)

}

e−j2π√

λ−2−u2

−v2zd ,

Uw(x, y) = F−1

Nz∑

d=0Nd>Nd,max

Uwd (u, v)

. (3)

2.3 Point-source approach

When the number of points within layer d ∈ {0..Nz}is smaller than Nd,max, the complex wave scatteredby this layer towards the hologram plane is computedusing a point-source approach. The complex wavescattered by a point source i located within layer d isgiven by the angular spectrum of plane waves [18] as

Upd,i(x, y) = Aie

jφiF−1{

e−j2π√

λ−2−u2

−v2zd

}

⊗ δ(x− xi, y − yi), (4)

where Ai and φi are the amplitude and phase of thepoint, xi and yi its coordinates within the layer, and⊗ is the convolution operator. In order to avoid in-terference between the points, the phase φi is set toa random value.

Convolving a function with a Dirac delta shifts itaround the delta impulse. Therefore, if we know theinverse Fourier transform term in Eq. (4) before-hand, Up

d,i can be computed simply by scaling thisterm with the point’s amplitude and phase factor,followed by a shifting operation. In order to speedup the computation, we use a pre-calculated LUT, asproposed in [4]. The LUT T (x, y, z) is pre-computedas

T (x, y, z) = F−1{

e−j2π√

λ−2−u2

−v2zd

}

h(x, y, z).

(5)

3

h being an envelope function used to restrict the re-gion of contribution of a given point source, equalto one within the region of contribution of the pointand zero elsewhere. This function limits the spatialfrequencies of the complex wave to avoid aliasing inthe CGH.

According to the Nyquist Sampling Theorem, themaximum spatial frequency fmax which can be rep-resented with a sampling pitch p is given by fmax =(2p)−1. The grating equation [18] gives the relationbetween the maximum spatial frequency fmax andthe maximum diffraction angle θ as sin(θ) = λfmax.Therefore, the region of contribution of a point sourceat depth z is given by its maximum radius Rmax by

Rmax = z tan(θ) = z tan

(

arcsin

(

λ

2p

))

, (6)

as shown in Figure 3. The envelope function h canthus be defined as

h(x, y, z) =

{

1 if√

x2 + y2 < Rmax

0 otherwise.(7)

In order to limit its number of pixels, the LUT ispre-computed only within the circumscribing squareof the region of contribution defined by the envelopefunction h. Therefore, the number of pixels NT,d ofthe LUT for depth zd is given by

NT,d =

(

2Rmax

p

)2

NT,d =

[

2zdp

tan

(

arcsin

(

λ

2p

))]2

. (8)

�

�����

CGH

Point

(� � �)

Figure 3: Region of contribution of a given pointsource

Then, the complex wave Upd scattered by layer d

in the hologram plane can be obtained by simply ad-dressing this pre-calculated LUT, as

Upd (x, y) =

Nd∑

i=1

AiejφiT (x− xi, y − yi, zd). (9)

Finally, the complex wave scattered by all the layerswhose number of points Nd is smaller than Nd,max inthe hologram plane is given by

Up =

Nz∑

d=0Nd<Nd,max

Upd (x, y). (10)

2.4 Determination of the selection cri-

terion

The first step to implement the proposed method isto determine the value of Nd,max. We call tp the timeneeded to compute the complex wave scattered by alayer at depth zd with Nd luminous points using thepoint-source approach presented in Section 2.3, andtw the time needed to compute it using the wave-field approach presented in Section 2.2. Since thewave-field approach involves one complex multiplica-tion per pixel and a Fourier transform, tw is linearlydependent on the number of pixels of the hologramNpix = Nx×Ny. The point-source approach involvesone complex multiplication per pixel of the LUT perpoint within the layer, so tp is dependent on the num-ber of pixels NT,d of the LUT for depth zd and on thenumber of points Nd within the layer. tw and tp areexperimentally found to be expressed by{

tw(Npix) = kNpix

tp(Nd, NT,d) =[

a (bNT,d + c)1

2 + dNT,d + e]

Nd.

(11)

We find the numerical values for the coefficients a,b, c, d, e and k in Eq. (11) using the Gnuplot imple-mentation of the nonlinear least-squares Levenberg-Marquardt algorithm [26]:{

a = 1, 11.10−5 b = 0, 59 c = 1, 0d = 2, 59.10−8 e = 4, 78.10−4 k = 5, 54.10−7

(12)

4

In order to maximize the efficiency of our method,Nd,max must be set such that

tp(Nd,max, NT,d) = tw(Npix) (13)

⇔Nd,max =kNpix

[

a (bNT,d + c)1

2 + dNT,d + e] . (14)

3 Experimental results and dis-

cussion

The proposed method was implemented inC++/CUDA on a PC system employing an In-tel Core i7-4930K CPU operating at 3.40 GHz, amain memory of 16 GB and an operating system ofMicrosoft Windows 8 as well as three GPUs NVIDIAGeForce GTX 780Ti.

For the experiments, we used the Middlebury’sviews and disparity maps datasets [27] as test scenes(Figures 4a and 4b). From each view and disparitymap pair, a 3D point cloud is extracted, where eachpoint is given an amplitude proportional to its corre-sponding pixel value in the view image and a randomphase. Since each disparity map is encoded as an 8-bits gray level image, the extracted 3D point cloudis naturally sliced as a set of Nz = 255 depth lay-ers parallel to the CGH plane1. The total number ofpoints Nscene within the point cloud is given by thenumber of pixels of the disparity map minus the num-ber of unknown disparity pixels. The 3D point cloudis considered to be located between zmin = −d andzmax = d in front of the CGH plane, where 2d = 2cmis the depth extent of the scene. Finally, the CGH tobe computed has a resolution of 4096 × 4096 with asampling pitch p = 8, 1µm.

We compare our method with GPU implementa-tions of two other methods: (1) the wave-field methodproposed in [28], which computes complex wave scat-tered by each layer using a wave-field approach, and(2) the point-source method proposed in [4], whichcomputes complex wave scattered by each layer using

1The 8-bits pixels in the disparity maps can have 256 dif-ferent values, but in this dataset, the 0 value is used to encodean unknown disparity. Points with unknown disparity are notextracted from the disparity maps.

(a) (b) (c)

(d) (e) (f)

Figure 4: (a) Intensity view and (b) disparity map ofthe test scene ”Moebius” from Middlebury’s dataset.Figure (c) shows in blue the scene points whose com-plex waves are computed by our method using thewave-field approach and in yellow the scene pointswhose complex waves are computed using the point-source approach. On the second line, scene imagesnumerically reconstructed from the CGH patternsgenerated by (d) the wave-field method, (e) the point-source method, and (f) our method.

a point-source approach. We adapted both methodsto produce colorful complex modulation CGH. Figure4 shows the scene images numerically reconstructedfrom the CGH patterns of the scene ”Moebius” gener-ated by the wave-field method (Figure 4d), the point-source method (Figure 4e), and our method (Figure4f). Figure 4c shows in blue the scene points whosecomplex wave is computed by our method using thewave-field approach and in yellow the scene pointswhose complex wave is computed using the point-source approach. As seen in Figure 4, our methoddoes not produce any visible artifact, even at theboundaries between these two categories of points.

In order to evaluate the objective quality of the re-constructed images compared to the original view im-age, we used the Peak Signal-to-Noise Ratio (PSNR).The PSNR of the reconstructed images of the scene”Moebius” were found to be 21, 20dB, 21, 19dB, and

5

0

0.5

1

1.5

2

2.5

3

3.5

1M Nt 3M 4M

Exe

cuti

on

tim

e t

(in

se

con

ds)

Number of scene points N

Wave-field method

Point-source method

Proposed method

(a)

20%

40%

60%

68%

80%

100%

1M Nt 3M 4M

Exe

cuti

on

tim

e r

ed

uct

ion

Number of scene points N

Proposed method

(b)

Figure 5: (a) CGH computation time for a syn-thetic 3D scene using the wave-field method (in blue),the point-source method (in red), and our method(in green) depending on the number of scene pointsN . (b) CGH computation time reduction using ourmethod depending on the number of scene points N .

21, 20dB for the wave-field method, the point-sourcemethod and our method, respectively. These resultsshow that our method does not reduce the quality thereconstructed scene images compared to the conven-tional point-source and wave-field methods. It mustbe noted that unlike the original view image, the nu-merically reconstructed images have a low depth offield due to the reconstruction technique used. As aconsequence, the PSNR of the reconstructed imagesare found to be below 30dB. Additionally, we com-pared the CGH pattern generated by our method tothose generated by the wave-field and point-sourcemethods using the PSNR. The PSNR of the CGHpattern generated by our method was found to be40, 59dB and 42, 88dB, compared to the wave-fieldand point-source methods, respectively.

In Figure 5a, we compared the CGH computationtime of the wave-field method (in blue), the point-source method (in red), and our method (in green)depending on the number of scene points N usingviews and depth maps pairs of a single synthetic 3Dscene with different resolutions. As shown on Fig-ure 5a, while the computation time of the point-source method increases linearly with the numberof scene points, the computation time of the wave-field method does not depend on it. Therefore, while

the point-source method is faster than the wave-fieldmethod for scenes with few points, the wave-fieldmethod is still more efficient than the point-sourcemethod for scenes with a large number of points. Bycombining these two approaches, our method takesadvantages from both of them and is therefore alwaysthe most efficient.

Figure 5b shows the reduction of the CGH com-putation time using our method depending on thenumber of scene points N . As seen in Figure 5b,our method allows the CGH computation time tobe reduced by a percentage that increases quicklyuntil N passes a threshold Nt, and then decreasesslowly. This threshold corresponds to the number ofscene points for which the computation time of thepoint-source method reaches the computation timeof the wave-field method. The value of Nt dependson the number of hologram pixels Npix = Nx × Ny,on the number of depth layers Nz, and on the dis-tance between the scene and the CGH plane. Asshown on Figure 5b, the CGH computation time isreduced by 68% using our method when the num-ber of scene points is equal to Nt. Moreover, ourmethod outperforms both the point-source and wave-field methods even when the number of scene pointsis higher than Nt. In addition to the results shownhere, we have conducted many tests on both real andsynthetic scenes with different number of hologrampixels and depth layers. A reduction of over 65%of the computation time has been reached for eachtest scene when the number of scene points is equalto Nt. These experimental results confirm the per-formance superiority of our method over the conven-tional point-source and wave-field methods in termsof computation time.

4 Conclusion

In this paper, we proposed a fast Computer-Generated Hologram (CGH) computation methodbased on a hybrid point-source/wave-field approach.The algorithm consists of three steps. First, the 3Dscene is sliced into several depth layers parallel to thehologram plane. Then, for each layer, if the numberof points within the layer exceeds a determined max-

6

imum value, we compute the complex wave scatteredby this layer using a wave-field approach. Otherwise,we compute the complex wave scattered by this layerusing a point-source approach. Finally, we sum upthe complex waves scattered by all the depth lay-ers in order to obtain the final CGH. Experimentalresults reveal that the CGH computation time hasbeen reduced up to 68% compared to the conven-tional point-source and wave-field methods withoutproducing any visible artifact. This confirms the per-formance superiority of our method over the conven-tional point-source and wave-field methods in termsof computation time.

Our method does not take into account occlusionsbetween objects in the scene, so in future study weplan to improve this method in order to handle sceneocclusions properly.

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