Distortion Correction for Conical Multiplex Holography Using Direct Object-Image Relationship

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1 Proc. Natl. Sci. Counc. ROC(A) Vol. 25, No. 5, pp Distortion Correction for Conical Multiplex Holography Using Direct Object-Image Relationship YIH-SHYANG CHENG, RAY-CHENG CHANG, AND SHIH-YU CHEN Institute of Optical Sciences National Central University Chungli, Taiwan, R.O.C. (Received April 15, 2000; Accepted November 27, 2000) ABSTRACT A direct mathematical theory which relates the final image point, as seen by the observer, to a point on the original 3D object is built. This theory utilizes the imaging property of lenses and coordinate transformation. From these equations, the whole image of the object as seen by one eye of the observer can be simulated. The 3D position of an image point is determined by the line of sight of both eyes of the observer. This offers the possibility of distortion correction for the image. If one starts from a desired ideal image and uses the mathematical theory to calculate the 2D distorted objects at all angles, the hologram fabricated by these 2D photographs can greatly correct distortion of the image. Computer simulation and experimental results are also given to demonstrate this method. Key Words: holography, multiplex holography, rainbow holography I. Introduction The multiplex holographic process, which combines the photographic process and the rainbow holographic process, was introduced by Cross in 1977 (Saxby, 1994). Many photographs are taken around the object along a horizontal arc. Each photograph is then enlarged in a coherent optical system onto a cylindrical lens, which compresses the object information into a vertical strip on the holographic film. A diverging spherical wave from a point source, either above or below the cylindrical lens, serves as the reference wave. Successive frames of photographs are converted into successive long thin holograms, resulting in a composite hologram. This hologram, when bent into a cylinder (or part of a cylinder) and viewed with a white light source, exhibits the characteristics of a rainbow hologram. To eliminate the rainbow effect in the vertical direction, anamorphic optical systems have been proposed (Huff and Fusek, 1980; Leith and Voulgaris, 1985; Cheng and Chang, 1998). In these systems, the horizontal and the vertical structures of each photograph are imaged independently. The horizontal structure is imaged onto the final cylindrical lens, while the vertical counterpart is imaged onto the holographic film, resulting in an image-plane hologram (Rosen, 1966) in the vertical direction. More recently, a hologram called the conical multiplex hologram was introduced (Okada et al., 1989; Murillo-Mora et al., 1995). This type of hologram is fabricated as part of a disk on the holographic film. The hologram disk is then curved as a certain cone for viewing. In a conical hologram, there exists the possibility of displaying a real image of the object at the center of the cone, which can be seen from all angles around the image. Due to mismatch among the parameters, the image observed inherently possesses some degree of distortion, which is difficult to completely remove by adjusting all the parameters. In this paper, we propose a method which enables us to not only evaluate the distortion of the final image under arbitrary fabrication parameters, but also correct the distortion of the observed image. For simplicity of analysis, we consider that the hologram is lying between the final image and the observer. This method is based on the direct mathematical relationship relating the original object point to the observed image point. The relationship is found through the imaging property of lenses and coordinate transformation. In this paper, we first present mathematical development of the theory. Then, computer simulation and experimental results are given. Finally, by reversing the process and specifying the desired image, we obtain, using the derived mathematical formulas, a series of 2D distorted objects which belongs to the 3D object at all angles. By using these distorted 2D photographs to fabricate the multiplex hologram, the distortion of the final observed image can be greatly reduced. II. Theory Referring to Fig. 1, an object is placed in the X j Y j Z j coordinate system (object coordinate system), where the Y j -axis is the axis of rotation for the object. The camera (in our case, a CCD camera), which is taking 2D photographs of the 3D object, lies parallel to and with its axis on the Z D - axis. The X D Y D Z D coordinate system (CCD coordinate 300

2 Distortion Correction for Conical Holography optical system in which the object is imaged onto the recording holographic film vertically while the horizontal image is imaged onto the horizontal image plane, which is located at a distance (Rtanθ)/2 in front of the plane of the holographic film. This optical system consists of two spherical lenses, L 1 and L 2, together with a cylindrical lens, L 3. Besides the anamorphic property on the image of the object, it images the illuminating light point onto the plane, at a distance q behind lens L 2, and converts it into a horizontal line, which later acts as the viewing slit for the rainbow holographic process. Using the simple imaging property of lenses, we find the distance between the plane of the recording holographic film and the lens L 2 as follows: Fig. 1. The CCD coordinate system X D Y D Z D is the version of the object coordinate system X j Y j Z j rotated by an angle φ with respect to the minus X j -axis. system) is a rotated version of the X j Y j Z j coordinate system with respect to the minus X j -axis by an angle φ. A generalized object point at (x j, y j, z j ) in the object coordinate system can then be transformed into CCD coordinates (x D, y D, z D ) through the following equation : f 2 (d f 1p ) p f q = 1 d f, (4) 1p f p f 2 1 where f 1 and f 2 are the focal lengths of and d is the separation between the two spherical lenses, and p is the distance between the object and the first lens. Hence, the vertical magnification ratio from the object to the hologram plane is x D y D z D = cosφ sinφ 0 sinφ cosφ x j y j. (1) z j Since the detector plane in the CCD camera is parallel to the X D Y D plane, we can find the position of the image for this generalized object point with the help of Fig. 2. Through simple geometry, the coordinates of the image point at the CCD detector can be found as follows: and x a = x D d OL z D d LF, (2a) Fig. 2. (x D, y D, z D ) represent the coordinates of the generalized object point in the CCD coordinate system. (x a, y a ) are the coordinates of its image on the CCD detector plane. y a = y D d OL z D d LF, (2b) where d OL (d LF ) is the distance between the CCD camera lens and the origin of the x D y D z D coordinate system (the detector plane). Suppose that the magnification ratio from the CCD detector to the input plane (in our case, a liquid crystal television LCTV) of the hologram forming optical system (Fig. 3) is M L ; the coordinates of this object point at the LCTV will be at x L = M L x a, y L = M L y a. (3a) (3b) The optical system shown in Fig. 3 is an anamorphic Fig. 3. Anamorphic optical system for fabrication of a multiplex hologram, which consists of two spherical lenses, L 1 and L 2, and one cylindrical lens, L 3. The illuminating source point is imaged onto a plane at a distance q from the lens L 2 and becomes a horizontal line, which, later, is utilized as the viewing slit for holographic image reconstruction. The object at the input plane is imaged onto the holographic film vertically while it is imaged horizontally onto a plane at a distance (Rtanθ)/2 in front of the holographic film. 301

3 Y.S. Cheng et al. Figure 4 shows the configuration of all the light rays. The hologram disk is situated in the plane X p Y p with its center of rotation at (0, R/2). We now use T ch to denote (Rtanθ)/2. For the object point considered previously, the object ray goes through the point (0, y 1 ) on the fan-shaped slit hologram and then reaches the horizontal line image of the illuminating source at the point ( x o T hp /T ch, 0, T hp ). The distance y 1 is related to y a by Fig. 4. Configuration showing the relationships among all the light rays. The object ray goes from the horizontal image plane of the object, at (x o, y o, T ch ), through the vertical image of the object, at (0, y 1, 0), and reaches the horizontal line-image of the illuminating source at ( x o. T hp /T ch, 0, T hp ). The reference source point for hologram recording is at (0, y c, z c ), while that for hologram reading is at (0, y r, z r ). The slit in front of the recording hologram disk is used to limit each individual hologram within a fan-shaped region. m v = f 1 (p f 1 ) f 2 (d ( f. (5) 1p + f p f 2 )) 1 The distance between the image of the illuminating source point and the lens L 2 is q = f 2(d f 1 ) d (f 1 + f 2 ). (6) Due to the presence of the cylindrical lens, L 3, the magnification ratio of the horizontal image m h is different from that of the vertical one by a factor. Suppose that the distance between lenses L 2 and L 3 is d ; then, the magnification ratio from the LCTV to the horizontal image plane will be m h = m v f 3 q d + f 3. (7) The horizontal image plane is located at a distance q = f 3(q d ) q d + f 3 (8) from the cylindrical lens, which is equal to q d (Rtanθ)/ 2, where R is the radius of the recording holographic disk and the center of the coordinates for each individual hologram is located at a distance R/2 from the center of the disk. In the recording process, when a 2D image is transmitted to the input plane of the optical system, a fan-shaped hologram (limited by a slit of the corresponding shape in front of the recording film) is recorded. Next, a fan-shaped hologram is produced by using the 2D image of the original 3D object, which is rotated by a suitable unit angle about the Y j -axis while the recording holographic film is rotated by a different unit angle, reduced by a factor sinθ. This process continues until the 3D object is rotated for a full round. y 1 = M L M v x a. (9) If we trace this ray back to the horizontal image plane, we find that y o = T hp + T ch T hp y 1, (10) where T hp is the distance between the hologram plane and the plane of the image for the illuminating source. On the other hand, the position of the horizontal image in the horizontal direction is at x o = M L M h x a. (11) From the three points (x o, y o, T ch ), (0, y 1, 0), and ( x o T hp / T ch, 0, T hp ), the line equation for the object ray going through the hologram plane can be determined as follows: x 0 x o T ch T hp 0 = y y 1 0 y 1 = z 0 T hp 0. (12) The angles between this line and the X p, Y p, and Z p axes can be obtained from the above equation and are assumed to be α o, β o, and γ o. Since the reference source point for hologram recording is at (0, y c, z c ), the angles α c, β c, and γ c for the reference ray can be similarly obtained. Hence, the object wave and the reference wave at the hologram plane can be expressed as and u o = exp [i 2π λ (x cosα o + y cosβ o )], u c = exp [i 2π λ (x cosα c + y cosβ c )], (13a) (13b) where λ is the wavelength of the light. These two waves interfere with one another at the hologram plane and produce interference fringes in which the useful information for later reconstruction of the object wave is the term u o u c *. Suppose that a reference source point with wavelength λ is placed at the position (0, y r, z r ) to read the hologram. This wave can be expressed as u r = exp [i λ 2π (x cosα r + y cosβ r )], (13c) 302

4 Distortion Correction for Conical Holography and the reconstructed image ray is u i = exp [i λ 2π (x cosα i + y cosβ i )], (13d) where, again, α r, β r, γ r, α i, β i, and γ i are the corresponding angles with respect to the coordinate axes. Since the reconstructed image wave is related to the other waves by u i = u r u o u c *, (14) we can, from Eq. (13), obtain the following relations: cosα i =[ cosα r λ + cosα o λ cosα c ] λ, (15a) λ and cosβ i =[ cosβ r λ + cosβ o λ cosβ c ] λ (15b) λ cosγ i = 1 cos 2 α i cos 2 β i. (15c) Fig. 6. Representation of the point of diffraction, which was at (0, y 1, 0) in the X p Y p Z p coordinate system shown in Fig. 5, in the observation coordinate system X u Y u Z u. From Fig. 4, we note that the line equation for the reference ray in the recording process can be expressed as x =0 y y c y 1 y c = z z c 0 z c. (16) Since, in the reconstruction process, the reference source point is situated on the axis of the hologram cone and the relative orientations with respect to all the individual holograms are the same, the diffracted image ray can then be calculated in the coordinates of the recording geometry. The appearance of the position of the reconstruction reference source point in the recording coordinates can be calculated based on its relative orientation (Fig. 5) with respect to the slit hologram in the recording, which is at (x r, y r, z r ) = (0, (t + Rcosθ)cosθ R 2, (t + Rcosθ)sinθ). (17) Similarly, in the coordinates of the recording geometry, the line equation for the reconstruction reference ray can be expressed as x r =0 y y r y 1 y r = z z r 0 z r. (18) From the above equations, we obtain the equation for the reconstructed image-ray in the recording coordinates as x 0 cosα i = y y 1 cosβ i = z 0 cosγ i. (19) This line equation should be transformed back to the observation coordinate system. In order to obtain the equation for the image-ray in the observation coordinate system, we need to know the point of diffraction as well as the direction cosines of the image ray. From Fig. 6, we know that the point of diffraction at (0, y 1, 0) in the recording coordinates can be transformed into the point (x d, y d, z d ) = [(y 1 + R/2)sinθsinφ, y 1 cosθ, (y 1 + R/2)sinθcosφ] Fig. 5. The coordinates of the reconstruction reference source point in the X p Y p Z p coordinate system. in the observation coordinate system. The direction cosines of the image ray can be obtained by means of double coordinate 303

5 Y.S. Cheng et al. transformation. First, we use the X p -axis as the axis of rotation to rotate the Y p -axis until it is parallel to the Y u -axis. Then, we use this new Y-axis as the axis of rotation to rotate the X p and Z p axes till they are parallel to the X u and Z u axes of the observation coordinate system. This two-step rotation results in the following equation of direction cosines for the image ray in the observation coordinate system: cosα i cosβ i cosγ i = cosφ 0 sinφ sinφ 0 cosφ cosθ sinθ 0 sinθ cosθ cosα i cosβ i. (20) cosγ i Next, we simulate the whole image as seen by the eyes of the observer during holographic reconstruction. First, from the image ray of the particular object point, we determine from which hologram slit this image point is seen through the eyes of the observer and specify its 3D orientation in space. If the image ray intersects one eye of the observer, it is said to be seen by that eye. For all the object points, this process can be repeated using the above developed equations. Hence, the whole image of the object as seen by one eye of the observer can be plotted. Computer simulation and experiment result for variation of various holographic parameters will be presented in the next section. As one can see from the results, no matter how the parameters are varied the image will possess some degree of distortion. In the following, we propose a method to correct this distortion. Since, for each eye of the observer, one image point can be seen through a certain individual hologram slit, the 3D position of that image point can be determined by two lines of sight which belong to the two eyes of the observer. The intersection point of these two lines is where this image point appears to be. Through this process, the whole 3D image, with some distortion, can be generated. If we reverse this process by specifying a desired 3D image, then, tracing back from each image point to each eye of the observer, we can obtain the intersection point on the hologram cone. This intersection point is the point (0, y 1, 0) shown in Fig. 4, and the line from the intersection point to the eye specifies the direction of the image ray. From this intersection point, the number of hologram slit through which this image point is observed is also specified. Using Eq. (15), the direction cosines (or direction) for the object ray can be calculated. Then, by tracing back from the point (0, y 1, 0) along the inverse direction of the object ray to the horizontalimage plane, one can obtain the location of the horizontal image (x o, y o, T ch ). By using the vertical magnification ratio m v and the horizontal magnification ratio m h, the position of this object at the input plane of the optical system (Fig. 3) can be found. This process is repeated for all the desired image points and all the angles of rotation for the specified 3D image. Finally, a series of distorted 2D objects belonging to different viewing angles of a 3D distorted object is generated. After sending these distorted 2D images sequentially into the input plane of the hologram-forming optical system (Fig. 3), the resulted conical multiplex hologram will be capable of generating a distortion-free 3D image. III. Computer Simulation and Experimental Results Numerical simulation of some typical cases of conical multiplex holography (primarily limited by the size of holographic film) was performed in order to observe the effect of varying various parameters on the final reconstructed image. For a conical multiplex hologram with a half-tip angle equal to 30 (i.e., θ = 30 ), the correct CCD angle φ should be 30. We considered the case with a reconstruction reference source point situated on the axis and at a distance t = 10.5 cm from the open end of the hologram cone (or a distance t + Rcosθ from the tip of the hologram cone), although, in reality, we used a white-light line-source with an approximate length of 2 cm, lying on the axis of the hologram cone and symmetric with respect to the position t = 10.5 cm. The distance between the center of the hologram disk and the origin of the coordinate system for each individual hologram was taken to be 9 cm (i.e., R/2 = 9 cm). This hologram was designed to be viewed at a distance of about 70 cm from the hologram surface. The object used for simulation was a cube of size 3 cm 3 cm 3 cm. In the formation of a conical holographic stereogram, one constraint which should be satisfied is that, when the hologram is rotated for a full round, the object (or the final image) should be rotated a full circle. If this condition is violated (i.e., the rotation speeds of the image and the hologram cone are different.), the observed image may be quite different from the correct one. We define the ratio of the speed of rotation of the image to that of the hologram cone as the r-ratio. Even when r-ratio is equal to one, the reconstructed image does not appear as an ideal cube. Figure 7 shows the result of computer simulation and the optical reconstructed image (Fig. 7(d)) for this case. Figure 7(a) shows the front view of the reconstructed image from computer simulation, which exhibits the characteristic where the top of the cube is narrower than the bottom, and the front side is also narrower than the rear side. Figure 7(b) shows the top view, which shows that the cube bulges out at the back. The side view is shown in Fig. 7(c), in which the rear side of the cube is higher than the front side. As the r-ratio gets smaller (larger), the rear side of the cube becomes narrower (wider) as compared with the front side. At the same time, the image is compressed (expanded) in the z-direction with less (more significant) bulging out in the back and is tilted 304

6 Distortion Correction for Conical Holography Fig. 7. Computer simulation and experimental result for a cube object under the conditions described in Section III. (a) Numerically simulated front view of the reconstructed image using the theory given in Section II. (b) Top view of the image. (c) Side view of the image. (d) Reconstructed image from a conical multiplex hologram under white-light line-source illumination. (d) Fig. 8. Reconstructed image for the case with (R/2 = 7 cm) using computer simulation. (a) Front view. (b) Top view. (c) Side view. Fig. 9. Reconstructed image for the case with (R/2 = 11 cm) using computer simulation. (a) Front view. (b) Top view. (c) Side view. more (less) in the side view. When the origin of each individual fan-shaped hologram from the center of the hologram disk deviates from its proper distance (the case we consider is 9 cm), the reconstructed image will have additional distortion. Figures 8 and 9 show the front view, top view and side view of the reconstructed images from computer simulation for the cases in which R/2 equals 7 cm and 11 cm. We can see from these results that, as the radius of the hologram disk becomes smaller (larger), the image is more compressed (stretched) in the horizontal direction relative to the vertical direction, and the tilt of the cube in the z-direction is more (less) significant. The experimentally re- Fig. 10. Holographically reconstructed image under white-light line-source illumination. (a) The case with R/2 = 7 cm. (b) The case with R/2 = 11 cm. 305

7 Y.S. Cheng et al. Fig. 11. Reconstructed image for a CCD angle equal to 20 degrees using computer simulation. (a) Front view. (b) Top view. (c) Side view. Fig. 12. Reconstructed image for a CCD angle equal to 40 degrees using computer simulation. (a) Front view. (b) Top view. (c) Side view. Fig. 13. Corresponding images from an optical experiment. (a) For the case with a CCD angle equal to 20 degrees. (b) For the case with a CCD angle equal to 40 degrees. constructed images, taken by a CCD camera from in front and above, are shown in Fig. 10, which show some tendency as the results from computer simulation although they are not easy to compare. If the CCD camera is not aimed at the original 3D object at the right angle φ (the case we consider is φ = 30 ) in the hologram forming process, the reconstructed image will be different from the correct one (Fig. 7). Figures 11 and 12 show the front view, top view, and side view from computer simulation for the cases in which φ = 20 and φ = 40, respectively. We note that the horizontal curves on the front surface are flatter for the case with a smaller CCD angle. The tilt of the cube in the z-direction is steeper (flatter) for a larger (smaller) CCD angle as compared with that for the correct Fig. 14. Distorted 2D objects for generating a distortion-corrected 3D image. These objects were generated by specifying an ideal cube image and reversing the process described in Section II. N a denotes the number of distorted 2D objects which belong to a 3D distorted object at different angles. reconstruction. These characteristics can easily be seen in Fig. 13, which shows the result of optical experiments with CCD angles equal to 20 degrees and 40 degrees. From the above computer simulation results, we find that the method we have developed can be used to simulate holographic reconstruction under arbitrarily specified param- 306

8 Distortion Correction for Conical Holography Fig. 15. Optically reconstructed image from the conical multiplex hologram. The hologram was fabricated with the 2D distorted objects, shown in Fig. 14, as the input objects for the hologram forming optical system (Fig. 3). eters and can help one find the holographic parameters under which the reconstructed image will possess less distortion. However, from the wide range of variation of holographic parameters, we conclude that the reconstructed image for this type of hologram inherently possesses the characteristic that the front side is narrower than the rear side. Similarly, the top is narrower than the bottom. This characteristic suggests use of the method described at the end of the last section to generate distortion-corrected images. By specifying the desired 3D image and reversing the process based on the theory given in Section II, a series of 2D distorted images belonging to a 3D distorted object at various angles can be obtained. Each image has the characteristic that its top and front sides are wider than their counterparts. Figure 14 shows some of these images, where the notation Na denotes the number of images. When these images are used as the original 2D objects in the hologram forming optical system (Fig. 3), the resulting multiplex hologram can indeed produce distortion-compensated image (Fig. 15). In Fig. 15, we notice that the top and the front of the cube are no longer narrower than the bottom and the rear as compared with Fig. 7. However, due to some experimental error (primarily from incorrect bending of the hologram since it was attached to a piece of thick paper which was cut into a partial disk and then bent into a cone), the top corners of the reconstructed cube appear to be slanted down. IV. Conclusion By using the imaging property of lenses and coordinate transformation, a mathematical theory which relates the final image point to the original object point has been derived. From these equations, the whole image of the object as seen by one eye of the observer can be generated by means of numerical simulation. Since each image point is seen by the eye of the observer through a particular slit hologram, the 3D position of each image point is determined by the intersection point of the lines of sight of both eyes of the observer. The 3D image is generally distorted as compared with the original 3D object no matter how one varies all the parameters. The characteristic of the image reconstructed from a conical multiplex hologram is that its top is narrower than its bottom, and its front side is narrower than its rear side. Since each image point is related to each object point through the equations derived in Section II, this suggests us a way to correct the distortion of the image. If one reverses the process by specifying the desired image and uses the mathematical theory given in Section II to generate a series of distorted 2D objects which belong to a 3D object at different angles, the fabricated hologram will be capable of generating an image which is free from distortion. Computer simulation together with experimental results have been provided to verify the above theory. Results showing the characteristics of the images under variation of various parameters have been given. A preliminary experimental result showing a distortion-corrected image has also been provided to demonstrate the potential capability of the process proposed in this paper. Acknowledgment This work was supported by the National Science Council, R.O.C., under grant NSC E References Cheng, Y. S. and R. C. Chang (1998) Characteristics of a prism-pair anamorphic optical system for multiplex holography. Opt. Eng., 37, Huff, L. and R. L. Fusek (1980) Color holographic stereograms. Opt. Eng., 19, Leith, E. N. and P. Voulgaris (1985) Multiplex holography: some new methods. Opt. Eng., 24, Murillo-Mora, L. M., K. Okada, T. Honda, and J. Tsujiuchi (1995) Color conical holographic stereogram. Opt. Eng., 34, Okada, K., S. Yoshi, Y. Yamaji, and J. Tsujiuchi (1989) Conical holographic stereograms. Opt. Commun., 73, Rosen, L. (1966) Focused-image holography with extended sources. Appl. Phys. Lett., 9, Saxby, G. (1994) Practical Holography, 2nd Ed., pp Prentice- Hall, New York, NY, U.S.A. 307

9 Y.S. Cheng et al. 308