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Anamorphic imagery in holographic stereograms
Volume
7, number
4
OPTICS
ANAMORPHIC
IMAGERY
COMMUNICATIONS
IN HOLOGRAPHIC
April
1973
STEREOGRAMS
I. GLASER* Department of Electronics, The Weizmann Institute of Science, Rehovot, Israel
Received
19 January
1973
Horizontal parallax only (HPO) holographic stereograms may exhibit anamorphic imagery in the reconstructed image. We calculated the anamorphic ratio, using geometrical optics, as a function of recording and viewing geometry. An experimental example verifying the calculated results is presented. Criteria for minimizing the distortion are suggested.
The holographic stereogram is a hologram synthesized from a set of two-dimensional, non-holographic pictures, so as to exhibit a three-dimensional image t . Holographic stereograms may be synthesized to exhibit images with both vertical and horizontal parallax. However, the synthesis of such holograms is complex because a large number of individual pictures are required. Furthermore, the usefulness of the vertical parallax as a stereoscopic vision cue is rather limited. For these reasons it is advantageous to use simpler, horizontal parallax only (HPO) holographic stereogram. In this note we deal with the special properties of such holograms. The preparation of a holographic stereogram is a two-step process. First we make a set of two-dimensional pictures of the object, each representing a view of the object from a specific point. Then a holographic stereogram is synthesized from these pictures. The synthesis may be done by multiple exposure of the entire holographic plate [ 11, or, with the aid of a slit, resulting in a spatial division of the area of the plate into small holograms [2-51. The first method is difficult to implement in practice. Furthermore, holograms made by large numbers of multiple exposures are usually relatively inefficient.
For that reason, the second method is preferred by most workers on application of the holographic stereogram. Thus we concentrate on the second method. In the holographic recording arrangement, illustrated in fig. 1, a section of the holographic plate H is illuminated by both reference and object beams through a slit S. The object beam passes through a diffuser D and a transparency T; the transparency is a two-dimensional picture from the set prepared in the first step. After one component hologram is recorded, the transparency of the next picture is indexed into position, and the slit is translated horizontally by a distance of the slit’s own width. Using such a step and record procedure, a set of narrow holograms is recorded on a common plate, each reconstructing a diffused image of the picture recorded on it. The entire set of holograms reconstructs a three-dimensional image, provided a proper choice of points of view for the original set of pictures is made.
* This work has been submitted
in partial fulfilment of the Degree of Ph.D. Sponsored in part by the National Council for Research and Development, Israel. t Note that a three-dimensional geometric image is reconstructed here rather than the original three-dimensional wave front, as in direct holography.
Fig. 1. Arrangement grams.
for the synthesis
of holographic
stereo-
323
Volume
7, number
4
April
OPTICS COMMUNICATIONS
1973
From fig. 2a, we get: -ax,
= x, (a-b)/(b-c).
(2)
However, from fig. 2b we note that in order to view object point projected at Xi we must look through the partial hologram recorded at slit position X, so that: Xc = XL c/a
(3)
By simple substitution [(a-b)/@ Fig. 2. Top view of the viewing geometry text for explanation.
of stercogram.
See
-c)]X,
t x:, = x,,
x;=x,+Axa. * Some authors prefer not to use the term “distortion” regard to anamorphic imagery. See for example ref.
324
(1) with
[ 61.
x, -(c/a)
x:, = 0. (4)
Solving this for Xi we obtain. x:, = [a(&c)/b(a-c)]
With the aid of geometrical optics we analyze the distortions* of the holographic stereogram as a function of the recording geometry and the viewing position. We assume that other possible causes for distortions have been removed; the original pictures used for the synthesis are undistorted and taken from the correct points of view, and the wavelength and radius of curvature of the reconstructing beam are identical with those of the reference beam. Fig. 2a represents a top view of the reconstructing geometry. The “input” denotes the location of the transparency during recording (T in fig. l), “slit” represents the slit plane (A in fig. I), and “object” represents a plane where an image of a certain point of the object is expected due to the synthesized parallax. The distances from the viewing plane to the input plane, to the object plane and to the slit plane are a. b, and c respectively. All x’s denote lateral displacements. Consider now a point of the object viewed from the central component hologram; its pr-ejection on the input plane is seen at a distance X, from the axis. The projection of this point would be displaced by AX, when viewed through another component hologram. that was recorded at the slit position X,. The new position will be:
of (1) into (2) and (3) we get:
The horizontal is thus:
x,
magnification
(5) (MH) at the input plane
MH = XL/X, = a(&c)/b(a-c)
(6)
The anamorphic distortion is defined as the ratio of the vertical to horizontal magnification. Since no parallax synthesis took place in the vertical dimension. the vertical magnification (MV) at input plane is unity. The distortion (D) is thus, D =M,/M,
= l/M,
= b(u-c)/a(b-c).
(7)
We calculated the distortion as a function of the object location. b, for some fixed values of a and c. The results arc shown in fig. 3. As shown in the figure, the distortion has several interesting special cases. For c = 0, the distortion is unity (no distortion) regardless of the value of a and b. At c = 0, however, the eye is at the slit plane. This may seem to be unpractical. However, by making a duplicate hologram [3 1from the one made as described above, we may have an aerial real image of the slit plane, where the eye can be placed. Another interesting case is that of b = a. This also leads to D 2 1. However, b = a implies that the object is flat and there is little sense in making a holographic stereogram of a flat object. A third case is the asymptotic one: b --f m. In that case D has a limiting value, where,
Volume 7, number 4 Distortion
in
HPO-Holograms.
a _.
Is
zn, c.
Yi;.::; a=42, c=32 I 25
I 50
Fig. 3. The anamorphic distance b.
April 1973
OPTICS COMMUNICATIONS
I 75 distortion
I loo
as a function
b (cm)
of the object
In the experimental example, presented in fig. 4, we note that the object, a circle, was distorted into an elipse whose horizontal axis is shorter. In our recording arrangement a was larger than b. From eq. (7) we note that for a > b, D > 1, in agreement with our observation. It is possible to minimize the distortion by a proper choice of the geometry of the recording system. Usually, a and c are fixed by practical considerations such as the viewing conditions and the size of the reconstructed image, so that only b can be varied. For a three-dimensional object, b will traverse from a fixed point at b, to b, + I, 1 being the length of the object in the z (axial) direction. A possible minimization criteria may be: bo+l
dlogDi)=l-’
1logDl db ,
s
(9)
ho which, for b, t I> a > b, gives: D(bo) D(b,
Fig. 4. Experimental example of anamorphic distortion in holographic stereogram. Left: undistorted reconstruction from conventional hologram. Right: distorted reconstruction from HP0 holographic stereogram when a > b (D > 1). The values used for this experiment are: u = 96.3 cm, b = 89.6 cm,
c = 76.3 cm. ;rT+lf = (a-c)/a
.
(8)
Thus, for b S a, the distortion is virtually a constant and can be compensated for by introducing an opposite distortion into the original pictures. Another interesting effect is that due to the dependence of D on b; the anamorphic distortion may cause straight lines to bend (or curve). While the anamorphic distortion is a linear distortion for each object plane, it is not linear for a line that goes from one value of b to another. Indeed, we have noticed such “curving” phenomena in some extreme cases.
+ 1) = 1,
(10)
whereby b, must satisfy the equation for minimum average distortion. Other criteria may be chosen for different specific applications. It is important to note that all horizontal parallax only holograms would exhibit anamorphic distortion and that eq. (7) is applicable to all such holograms provided the parameters a, b and c are properly defined?. In summary, we have shown that distortion will usually exist for horizontal parallax only holographic stereograms, and that this distortion may be minimized by a proper choice of the recording an viewing geometry. The help of Professor D. Treves (of this Department) and Dr. A.A. Friesem of Radiation Inc. is greatly appreciated. Thanks are also due to Dr. J. Bulabois of Laboratoire d’optique, Besancon, France, for his helpful correspondence. in the case of Redman’s method [ 11, for synthesizing the holographic stereogram by multiple exposure of is the plane of the holographic plate the entire plate, “input” itself, and the “slit” plane is the plane of the exit pupil of the projection lens.
t As an example,
325
Volume
7, number
4
OPTICS
References
]l] J.D. Redman, J. Phys. E, Ser. 2, l(1968) 831. [2] N. George and J.T. McCrickerd, Phot. Sci. Eng. 13 (1969) 342.
326
COMMUNICATIONS
April
1973
(31 M.C. King, A.M. Noll and D.H. Berry, Appl. Opt. 9 (1970) 471. [4] G. Groh and M. Kock, Appl. Opt. 9 (1970) 775. (51 B.D. Sollish and I. Glaser, Acoustical holography, Vol. 4 (Plenum Press, New York, 1972) p. 157. ]6] J.F. Miles, Opt. Acta 19 (1972) 165.
7, number
4
OPTICS
ANAMORPHIC
IMAGERY
COMMUNICATIONS
IN HOLOGRAPHIC
April
1973
STEREOGRAMS
I. GLASER* Department of Electronics, The Weizmann Institute of Science, Rehovot, Israel
Received
19 January
1973
Horizontal parallax only (HPO) holographic stereograms may exhibit anamorphic imagery in the reconstructed image. We calculated the anamorphic ratio, using geometrical optics, as a function of recording and viewing geometry. An experimental example verifying the calculated results is presented. Criteria for minimizing the distortion are suggested.
The holographic stereogram is a hologram synthesized from a set of two-dimensional, non-holographic pictures, so as to exhibit a three-dimensional image t . Holographic stereograms may be synthesized to exhibit images with both vertical and horizontal parallax. However, the synthesis of such holograms is complex because a large number of individual pictures are required. Furthermore, the usefulness of the vertical parallax as a stereoscopic vision cue is rather limited. For these reasons it is advantageous to use simpler, horizontal parallax only (HPO) holographic stereogram. In this note we deal with the special properties of such holograms. The preparation of a holographic stereogram is a two-step process. First we make a set of two-dimensional pictures of the object, each representing a view of the object from a specific point. Then a holographic stereogram is synthesized from these pictures. The synthesis may be done by multiple exposure of the entire holographic plate [ 11, or, with the aid of a slit, resulting in a spatial division of the area of the plate into small holograms [2-51. The first method is difficult to implement in practice. Furthermore, holograms made by large numbers of multiple exposures are usually relatively inefficient.
For that reason, the second method is preferred by most workers on application of the holographic stereogram. Thus we concentrate on the second method. In the holographic recording arrangement, illustrated in fig. 1, a section of the holographic plate H is illuminated by both reference and object beams through a slit S. The object beam passes through a diffuser D and a transparency T; the transparency is a two-dimensional picture from the set prepared in the first step. After one component hologram is recorded, the transparency of the next picture is indexed into position, and the slit is translated horizontally by a distance of the slit’s own width. Using such a step and record procedure, a set of narrow holograms is recorded on a common plate, each reconstructing a diffused image of the picture recorded on it. The entire set of holograms reconstructs a three-dimensional image, provided a proper choice of points of view for the original set of pictures is made.
* This work has been submitted
in partial fulfilment of the Degree of Ph.D. Sponsored in part by the National Council for Research and Development, Israel. t Note that a three-dimensional geometric image is reconstructed here rather than the original three-dimensional wave front, as in direct holography.
Fig. 1. Arrangement grams.
for the synthesis
of holographic
stereo-
323
Volume
7, number
4
April
OPTICS COMMUNICATIONS
1973
From fig. 2a, we get: -ax,
= x, (a-b)/(b-c).
(2)
However, from fig. 2b we note that in order to view object point projected at Xi we must look through the partial hologram recorded at slit position X, so that: Xc = XL c/a
(3)
By simple substitution [(a-b)/@ Fig. 2. Top view of the viewing geometry text for explanation.
of stercogram.
See
-c)]X,
t x:, = x,,
x;=x,+Axa. * Some authors prefer not to use the term “distortion” regard to anamorphic imagery. See for example ref.
324
(1) with
[ 61.
x, -(c/a)
x:, = 0. (4)
Solving this for Xi we obtain. x:, = [a(&c)/b(a-c)]
With the aid of geometrical optics we analyze the distortions* of the holographic stereogram as a function of the recording geometry and the viewing position. We assume that other possible causes for distortions have been removed; the original pictures used for the synthesis are undistorted and taken from the correct points of view, and the wavelength and radius of curvature of the reconstructing beam are identical with those of the reference beam. Fig. 2a represents a top view of the reconstructing geometry. The “input” denotes the location of the transparency during recording (T in fig. l), “slit” represents the slit plane (A in fig. I), and “object” represents a plane where an image of a certain point of the object is expected due to the synthesized parallax. The distances from the viewing plane to the input plane, to the object plane and to the slit plane are a. b, and c respectively. All x’s denote lateral displacements. Consider now a point of the object viewed from the central component hologram; its pr-ejection on the input plane is seen at a distance X, from the axis. The projection of this point would be displaced by AX, when viewed through another component hologram. that was recorded at the slit position X,. The new position will be:
of (1) into (2) and (3) we get:
The horizontal is thus:
x,
magnification
(5) (MH) at the input plane
MH = XL/X, = a(&c)/b(a-c)
(6)
The anamorphic distortion is defined as the ratio of the vertical to horizontal magnification. Since no parallax synthesis took place in the vertical dimension. the vertical magnification (MV) at input plane is unity. The distortion (D) is thus, D =M,/M,
= l/M,
= b(u-c)/a(b-c).
(7)
We calculated the distortion as a function of the object location. b, for some fixed values of a and c. The results arc shown in fig. 3. As shown in the figure, the distortion has several interesting special cases. For c = 0, the distortion is unity (no distortion) regardless of the value of a and b. At c = 0, however, the eye is at the slit plane. This may seem to be unpractical. However, by making a duplicate hologram [3 1from the one made as described above, we may have an aerial real image of the slit plane, where the eye can be placed. Another interesting case is that of b = a. This also leads to D 2 1. However, b = a implies that the object is flat and there is little sense in making a holographic stereogram of a flat object. A third case is the asymptotic one: b --f m. In that case D has a limiting value, where,
Volume 7, number 4 Distortion
in
HPO-Holograms.
a _.
Is
zn, c.
Yi;.::; a=42, c=32 I 25
I 50
Fig. 3. The anamorphic distance b.
April 1973
OPTICS COMMUNICATIONS
I 75 distortion
I loo
as a function
b (cm)
of the object
In the experimental example, presented in fig. 4, we note that the object, a circle, was distorted into an elipse whose horizontal axis is shorter. In our recording arrangement a was larger than b. From eq. (7) we note that for a > b, D > 1, in agreement with our observation. It is possible to minimize the distortion by a proper choice of the geometry of the recording system. Usually, a and c are fixed by practical considerations such as the viewing conditions and the size of the reconstructed image, so that only b can be varied. For a three-dimensional object, b will traverse from a fixed point at b, to b, + I, 1 being the length of the object in the z (axial) direction. A possible minimization criteria may be: bo+l
dlogDi)=l-’
1logDl db ,
s
(9)
ho which, for b, t I> a > b, gives: D(bo) D(b,
Fig. 4. Experimental example of anamorphic distortion in holographic stereogram. Left: undistorted reconstruction from conventional hologram. Right: distorted reconstruction from HP0 holographic stereogram when a > b (D > 1). The values used for this experiment are: u = 96.3 cm, b = 89.6 cm,
c = 76.3 cm. ;rT+lf = (a-c)/a
.
(8)
Thus, for b S a, the distortion is virtually a constant and can be compensated for by introducing an opposite distortion into the original pictures. Another interesting effect is that due to the dependence of D on b; the anamorphic distortion may cause straight lines to bend (or curve). While the anamorphic distortion is a linear distortion for each object plane, it is not linear for a line that goes from one value of b to another. Indeed, we have noticed such “curving” phenomena in some extreme cases.
+ 1) = 1,
(10)
whereby b, must satisfy the equation for minimum average distortion. Other criteria may be chosen for different specific applications. It is important to note that all horizontal parallax only holograms would exhibit anamorphic distortion and that eq. (7) is applicable to all such holograms provided the parameters a, b and c are properly defined?. In summary, we have shown that distortion will usually exist for horizontal parallax only holographic stereograms, and that this distortion may be minimized by a proper choice of the recording an viewing geometry. The help of Professor D. Treves (of this Department) and Dr. A.A. Friesem of Radiation Inc. is greatly appreciated. Thanks are also due to Dr. J. Bulabois of Laboratoire d’optique, Besancon, France, for his helpful correspondence. in the case of Redman’s method [ 11, for synthesizing the holographic stereogram by multiple exposure of is the plane of the holographic plate the entire plate, “input” itself, and the “slit” plane is the plane of the exit pupil of the projection lens.
t As an example,
325
Volume
7, number
4
OPTICS
References
]l] J.D. Redman, J. Phys. E, Ser. 2, l(1968) 831. [2] N. George and J.T. McCrickerd, Phot. Sci. Eng. 13 (1969) 342.
326
COMMUNICATIONS
April
1973
(31 M.C. King, A.M. Noll and D.H. Berry, Appl. Opt. 9 (1970) 471. [4] G. Groh and M. Kock, Appl. Opt. 9 (1970) 775. (51 B.D. Sollish and I. Glaser, Acoustical holography, Vol. 4 (Plenum Press, New York, 1972) p. 157. ]6] J.F. Miles, Opt. Acta 19 (1972) 165.