The 22nd International Display Workshops (IDW '15), Otsu (Shiga), 3D1-1, (2015.12.10, Invited).

Occlusion Processing in Computer Holography—With a Focus on Switch-Back Technique—
Kyoji Matsushima
Kansai University, Yamate-cho 3-3-35, Suita, Osaka 564-8680, Japan
Keywords: 3D image, Computer-generated hologram, Holographic display, Hidden surface removal
ABSTRACT
A powerful technique called the switch-back technique
is presented for occlusion processing. The techniqueallows us to process polygon-by-polygon light-shielding
very fast. The computation times for various sizes ofcomputer-generated holograms as well as opticalreconstruction are presented for verifying the performanceand validity of the technique.
1. INTRODUCTION
Occlusion is definitely the most important depth cue incomputer holography as well as other types of 3Dtechniques such as stereoscopic and multi-viewpoint 3Dimages. Occlusion processing is, however, also mostdifficult process in computer holography, because highquality computer-generated holograms (CGH) require verylarge space-band product [1].
In the point cloud techniques chosen by many
researchers because of its simplicity, occlusion processingcommonly results in the problem of ‘the visibility test.’ Thisis the test whether something obstructs the ray from apoint source in the object surface to a sampling point in thehologram, as shown in Fig.1. However, the visibility test
inevitably makes computation by the point cloud, which is
known as a very time-consuming technique, even slower.
In the polygon-based method [2-4], mask-based
techniques called the silhouette method is used forocclusion processing [5]. The silhouette method isclassified into two groups; one is object-by-object (O-O)
light-shielding and another is polygon-by-polygon (P-P)
light shielding. Since the former is very simple and fast, OO light-shielding is commonly used for creating high-Hologram
Point source

Obstacle

Point cloudRaysFig. 1 The visibility test in point cloud
definition CGHs which have several billions andsometimes several ten billons pixels. However, the O-O
light-shielding has several drawbacks. One of them isthat it is difficult and sometimes impossible toautomatically determine the position of the masks. Someheuristic decision is always required for fixing thepositions. This means that for the complex object model
it is very difficult to simply convert the model into theCGH.
In contrast, by using the P-P light-shielding, the CGH
fringe pattern can be generated from any complicatedmodel. However, P-P light-shielding was very timeconsuming because it requires the same number of field
propagation as the number of polygons of the object.
Recently, a novel technique called the switch-backtechnique is proposed for acceleration of the P-P lightshielding [6].
In this article, we describe the principle and procedure

Background wave
fieldBackgroundwaveOpaqueobject
Object wave

Masking plane
(a)

(b)

Wave field of object

Binary mask

Fig. 2 The principle of silhouette-masking method [6](a) Light-shielding by opaque object in the real world. (b) Light-shielding simulated by silhouettemasking.

(a)Hologram(b)
Hologram(c)

Backgroundfield

BinaryMask

ObjectfieldNext
Propagation plane
zn+1

Propagationun ( x, y )M n ( x, y )On ( x, y )u n 1 ( x , y )Fig. 3 The procedure for object-by-object (O-O) light-shielding [1](a) The 3D scene of high-definition CGH ‘Brothers.’ (b) Field propagation. (c) The operation in a masking plane.
of the switch-back technique. The measured performanceand optical reconstruction of high-definition CGH createdby using the technique is also presented in this paper.

respectively, as shown in Fig. 3 (c). The notation
Pn  1, n  ( x , y ) stands for numerical propagation of the

2. TWO TYPES OF THE SILHOUETTE METHOD
2.1 Principle of silhouette-masking
Figure 2 shows the principle of the silhouette masking.
Any opaque object shields the background light, as shownin Fig. 2 (a). This can be simulated by silhouette masking
as in (b). In this technique, the wave field is calculated in
the masking plane by using some numerical field
propagation technique. The field is simply masked by abinary mask whose shape is the same as the silhouette of
the obstacles.
2.2 Formulation of the procedure of O-O lightshielding for multiple objects
Figure 3 (a) shows the 3D scene of a high-definition
CGH named ‘Brothers’ that was on display at
Massachusetts Institute of Technology (MIT) Museum [7].
In this CGH, two objects are arranged in front of thebackground object. Thus, the field of the backgroundobject is sequentially propagated to the masking plane ofthe obstacles in order of depth as in (b). The procedure of
the silhouette masking is given by recurrence formula asfollows:
u n  1 ( x , y )  Pn  1, n  M n ( x , y ) u n ( x , y )  O n ( x , y ) ,(1)2.3 Drawbacks of the O-O and P-P light-shielding
If the object has complicated shape such as selfocclusion as shown in Fig. 4 (a), the object isreconstructed as a partially see-through phantom image.
The viewers may also see black shadows in some cases.
In addition, we cannot automatically determine theposition of the masking plane.
These problems can be avoided by the polygon-bypolygon (P-P) light-shielding as in (b). The formulationand procedure of the P-P light-shielding is exactly thesame as that of the O-O. The difference is that theobstacle is not an object but all polygons. However, weneed many number of numerical propagation in this case.
The number of propagation is actually the same as thatof the polygons. This is very time-consuming especiallyin high-definition CGHs such as Brothers.

where M n ( x , y ) , u n ( x , y ) and O n ( x , y ) are the binary mask,
background field and object field of the object n,wave field  ( x , y ) from the plane at zn to zn+1.

See-through portion

Self-occluded
objectObject mask

Black shadow(a)Self-occluded
objectPolygon masks(b)Fig. 4 Comparison between the object-by-object and polygon-by-polygon light-shielding [6](a) Object-by-object (O-O) light-shielding. (b) Polygon-by-polygon (P-P) light-shielding.
3. THE SWITCH-BACK TECHNIQUE
3.1 Formulation and procedure
The following simultaneous recurrence formulas can bederived from the recurrence formula of Eq. (1) after somesteps [6]:obj(2)u n ( x , y )  Pn , obj  u n ( x , y )u n  1 ( x , y )  u n ( x , y )  Pobj, n  O n ( x , y )  An ( x , y ) u n ( x , y ) (3)objobjobj

where u n ( x , y ) is the temporally accumulated wave field
in a plane called the object plane. The operation
Pobj, n {u n ( x , y )} propagates the field u n ( x , y ) in zn plane tothe object plane. The operation P n ,obj { ( x , y )} is theopposite. An ( x , y ) is called silhouette aperture function

An ( x , y )  1  M n ( x , y ).

Figure 5 shows the interpretation of the simultaneousrecurrence formulas of Eq. (2) and (3). Suppose that theaccumulated field u no b j ( x , y ) of polygon 0 to n-1 is alreadycalculated with P-P light-shielding in the object plane.
When we add light-shielding of polygon n, (i) the fieldu n ( x , y ) in the masking plane is calculated by the
backward propagation of the accumulated field u no b j ( x , y ) ,as in Eq. (2). And then, (ii) the field is masked by thesilhouette aperture and the polygon field is added to themasked field. Finally, (ii) the field is backwardly
propagate to the object plane and subtracted from theaccumulated field u no b j ( x , y ) , as in Eq. (3). Here,Babinet’s low insures validity of the procedure.
This procedure, however, requires a double field
propagation, while the original
Object plane
procedure
needs
only
singleWave field ofpropagation. The new procedure
polygon 0 to n –1
seems to be inefficient. However, let
us emphasize a point; the field isPolygon nrequired only around the silhouette
Subtraction
aperture in the propagation in thiscase, as shown in Fig. 6 (b). In
Polygons 0 to n –1
contrast, the original procedure
always need whole field propagation(i) Backward
as in (a). This means that the number
propagationof samplings of the field and thus thecomputation time can be drasticallyreduced in the new procedure.
3.2 Measurements of performance
The 3D scene and optical(ii) Masking
reconstruction of the test CGH are(iii) Forward
shown in Fig 7. Here, the pixel pitches
propagationof the CGH are 0.8 m in bothhorizontal and vertical direction andFig. 5 The recursively defined procedure for P-P light-shielding
the number of polygon of the object isusing silhouette apertures, based on Babinet's low
5000. The lengths indicated in (a) andThe roman number indicates the order of process.

given by simple inversion of the silhouette mask;

Object planes

Objectmodel

Hologram

52.4(a)(b)
Fig. 6 Comparison between (a) silhouette mask and(b) silhouette aperture [6]40optical reconstruction is of the CGH composed of 64K 64K pixels, where 1K =1024.
Computation times measured for various sizes ofholograms are shown in Fig. 8. Here, pixel pitches are notchanged in any cases, whereas the lengths indicated inFig. 7 (a) are changed proportionally to the hologram size.
The computation time increases linearly in the log scalewith increasing the number of pixels. The measured
maximum computation time is 1.7 hour of 64K  64K CGH.
For the 2K  2K CGH intended to be reconstructed byelectro-holography, the computation time is only 4.5
seconds. These are 20 to 60 times shorter than that ofconventional P-P light-shielding [6].

ACKNOWLEDGEMENT
This work was supported by JSPS. KAKENHI(15K00512) and the MEXT strategic research foundation
at private universities (2013-2017).
REFERENCES[1] K. Matsushima, “Digital signal processing of light inholographic 3D imaging,” SPIE Proc. vol.9599,(2015) in press.
[2] K. Matsushima, “Computer-generated holograms forthree-dimensional surface objects with shade andtexture,” Appl. Opt., vol. 44, pp. 4607-4614 (2005).
[3] K. Matsushima, S. Nakahara, “Extremely highdefinition full-parallax computer-generated hologram
created by the polygon-based method,” Appl. Opt.,
vol. 48, pp. H54-H63 (2009).
[4] K. Matsushima, H. Nishi, S. Nakahara, “Simple wavefield rendering for photorealistic reconstruction inpolygon-based high-definition computer holography,”
J. Electron. Imaging, vol. 21, 023002 (2012).
[5] K. Matsushima, A. Kondoh, “A wave optical algorithm
for hidden-surface removal in digitally synthetic fullparallax holograms for three-dimensional objects,”

90

Hologram

52.4

(a)

(b)
Fig. 7 The test CGH [6](a) The 3D scene. (b) Optical reconstruction of
64K×64K CGH.
10000

Computation time [s]4. CONCLUSION
A new technique, called the switch-back technique, ispresented for occlusion processing. This technique usessilhouette apertures instead of silhouette masks to reduce
the number of samplings required in field propagation andaccelerate computation of polygon-by-polygon lightshielding. In fact, the computation is speeded-up 20 to 60
times in comparison with the conventional technique.

Units: mm

1.7h
0.44h

1000

0.12h

100

60.8s

15.5s
104.5s12Kx2K

4Kx4K

8Kx8K 16Kx16K 32Kx32K 64Kx64K

Number of pixels
Fig. 8 Computation times of the switch-backtechnique for various sizes of the test CGH [6]

[6]

[7]

Proc. SPIE vol. 5290, pp. 90-97 (2004).
K. Matsushima, M, Nakamura, S. Nakahara,“Silhouette method for hidden surface removal incomputer holography and its acceleration using theswitch-back technique,” Opt. Express, vol. 22, pp.
24450-24465 (2014).
K. Matsushima, S. Nakahara, “Stepping closer tothe perfect 3D digital image,” SPIE Newsroom, (6Nov. 2012).
http://dx.doi.org/10.1117/2.1201210.004526