Abstract:
An improvement to the above described techniques for producing a holographic stereogram. The present invention uses a hologram lens in the process of creating the holographic stereogram, and also uses Fourier transforms of the images for projection onto the recording medium, not the images themselves. A lens between the generated images and the recording medium performs the inverse Fourier transform to convert the image back into a normal image.

Description:
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   BACKGROUND OF THE INVENTION 
   The present invention relates to an improved technique for forming a holographic stereo gram. 
   A hologram is a device that can produce a three dimensional image of an object. In order to produce a hologram, an object is illuminated by a laser in an optical set up. Light reflected off the object is combined with a reference beam on the surface of a photographic plate. The interference pattern between these two light beams is recorded to form the hologram. This process of making a hologram requires an actual object and precise optical arrangement between the object and the reference beam to produce the interference fringes. However, the human brain can combine views perceived by the right and left eyes to produce a perception of a three dimensional object. The two dimensional views can be photographs of an object or views of an object created by a computer. 
   A stereogram is a pair of pictures presenting two different views of an object. The three dimensional image produced by a stereogram typically has only horizontal parallax. There are many methods of presenting views of an object to the right and left eyes separately. The most common one is using a stereo viewer, which simply restricts the right eye to see one view and the left eye to see the other view of an object. Stereograms can also be printed in red and green color. By means of using color filter in front of the right and left eye, each eye will see different views of the object. Examples of stereograms can be found in U.S. Pat. Nos. 6,037,971 and 5,795,154. 
   One type of stereogram is called a lenticular stereogram. This is one of a number of methods for viewing stereo grams without the use of a viewing aid. The lenticular stereogram technique interlaces narrow strips of the two views and placed them behind a set of prisms so that alternate segments of the two views are separately projected to the right and left eyes. 
   A holographic stereogram is another method whereby two images are encoded with different spatial frequency so that when the hologram is illuminated by light, the two images will emerge from the hologram at different angles. The diffraction angle is determined by the separation of the eyes and the viewing distance. For a typical eye separation of 50 mm and a viewing distance of 400 mm, the angle is about 7.5 degrees. 
     FIG. 1  shows light beam  102  as bounded by ray  102   a  and ray  102   b  and beam  103  as bounded by ray  103   a  and ray  103   b  diffracted from the holographic stereogram  101  toward the right eye  104  and left eye  105 . Ray  102   a  and ray  103   a  are emitted from hologram element  106  in the holographic stereogram  101 . This suggests a technique for constructing such a holographic stereogram with only horizontal parallax by constructing the stereogram by having two beams such as  102   a  and  103   a  interfering within a narrow slit and composing the stereograms one narrow segment at a time. See Mark Holzbach, “Three dimensional image processing for synthetic holographic stereograms”, M.S. thesis. Massachusetts Institute of Technology, September 1986, pp. 1 86; C. K. Lee et al., “Optical configuration and color-representation of a variable-pitch dot matrix holographic printer” Appl. Opt., Vol. 39, No. 1, p. 40 (2000); U.S. Pat. Nos. 5,237,433, 5,475,511 and 5,793,503. 
     FIG. 2(   a ) shows how the hologram of  FIG. 1  is formed. A converging cone of laser light  201  illuminates a transparency  202  (which could be an LCD display). An image of the transparency is projected on a rotating diffuser  205 , which produces uniform illumination at the recording plane  207 . To record a stereogram the image segment  204  corresponds to the image for the left eye and the image segment  203  corresponds to the image for the right eye. The diffused light from these two image segments propagates to the recording plane  207 . A reference laser beam  206  is introduced to interfere with the light from the diffuser and produce interference fringes on a hologram recording area  209  of the recording plane  207 . Slit  208  confines the hologram recording to a narrow stripe. The width of the slit determines the image resolution of the hologram plane. After one hologram stripe has been recorded, the recording plane is moved to the next position and a new set of images is projected on the diffuser for the next recording. This process is repeated until the recording surface  207  as shown in  FIG. 2(   b ) is filled with a holographic stereogram. This is a simple process to record a pair of stereo images in the same hologram. When this hologram is illuminated by light, the eyes positioned at location near the diffuser will see a stereo image of the recorded object as shown in  FIG. 1 . The diffuser  203  in  FIG. 1  can also replaced by a cylindrical lens, which focuses the laser beam into a line with a width matching the width of slit  208  as shown in  FIG. 3(   a ). 
     FIG. 3  further extends the concept illustrated in  FIG. 2 . Instead of recording just a stereo pair, the film  302  contains many views of the object illuminated similarly by a converging cone of laser light. All these views on film are combined into the same hologram unit  309 . Pixels  303 ,  304  are two of these images corresponding to certain views of the object.  FIG. 3(   b ) shows the process of constructing such views. Layers  311 ,  312  represent two-dimensional images of certain views of the object. View images are stacked together to form a cube  310 . On the front side of this cube, stripes  313 ,  314  are image units in certain locations in a view such as  311  or any other view. To properly record the hologram unit j which corresponds to image location x=j, the view recorded on film is g(z=nδ, y, x=j) where j indicates location on the x-y plane and n indicates the view frame and δ is the width of stripes such as  310  or  312 . Mathematically, the light distribution on the focal plane of the cylindrical lens can be written as: 
                   G   ⁡     (       u   -     j   ⁢           ⁢   Δ       ,   y     )       =       ∑     n   =       -   M     /   2         M   /   2       ⁢           ⁢       g   ⁡     (     n   ,   y   ,   j     )       ⁢     ⅇ       j   ⁢           ⁢   2   ⁢           ⁢   π   ⁢           ⁢   n   ⁢           ⁢   δ   ⁢           ⁢   u       λ   ⁢           ⁢   F                     (   1   )               
where G(u−jΔ, y) is the light distribution on the recording plane. As can be seen, the image segment g(z=nδ, y, x=j) is incident on the hologram with an angle given by sin θ n =nδ/λF. When such hologram is recorded, the eye will see a gradual change of the views of the object as the eyes scan through the stereogram. See Mark Holzbach, “Three dimensional image processing for synthetic holographic stereograms”, M.S. thesis. Massachusetts Institute of Technology, September 1986, pp. 1 86. From a practical point of view, the width of a hologram unit Δ determines the resolution of the stereo image reproduced by this holographic stereogram. It is the objective of this present invention to describe a technique whereby each hologram unit contains more than one image pixel.
 
   BRIEF SUMMARY OF THE INVENTION 
   The present invention provides an improvement to the above described techniques for producing a holographic stereogram. The present invention uses computer generated holograms in the process of creating the holographic stereogram, and also uses Fourier transforms of the images for display on the LCD display panel, not the images themselves. A lens between the generated images and the recording medium performs the inverse Fourier transform to convert the image back into a normal image. 
   The use of a computer-generated Fourier transform hologram allows more than one pixel to be encoded and recorded at a time. This allows image resolution to be independent from the slit width in recording the hologram unit. Moreover, the encoding of the image pixels by a Fourier transformation allows the incorporation of random phase in the image pixels. Such a random phase encoding gives the effect of uniform illumination, eliminating the need for the diffuser of the prior art. This significantly simplifies the illumination of the film transparency or LCD display panel. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a diagram of a user viewing a holographic stereogram. 
       FIGS. 2(   a ) and  2 ( b ) are diagrams of a prior art method of constructing a holographic stereogram. 
       FIGS. 3(   a ) and  3 ( b ) are diagrams of a prior art method of constructing a holographic stereogram with more than two views. 
       FIG. 4  is a diagram showing the principle of recording holographic stereogram using the present invention. 
       FIGS. 5(   a ) and  5 ( b ) are top and side views of an optical system for constructing a holographic stereogram implementing the principle of  FIG. 4  according to the present invention. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
     FIG. 4  illustrates the principle of the present invention. A laser light beam  401  illuminates a transparency or a display panel  402 . This could be done by illumination from the back for a transparency, or illumination from the front using a beam splitter (see, e.g., U.S. Pat. No. 6,043,913). Instead of displaying the view pixels on the transparency or LCD screen as shown in  FIG. 3 , the Fourier transform of a group of image pixels are displayed. The function g(n,y,i) represents the nth view of an object at location x=i. The Fourier transform of a segment of g(n,y,i) is defined as: 
                     G   n     ⁡     (     k   ,   y     )       =       ∑     k   =   1     m     ⁢           ⁢       g   ⁡     (     n   ,   y   ,   i     )       ⁢     ⅇ     j   ⁢           ⁢   2   ⁢           ⁢   π   ⁢           ⁢   ⅈ   ⁢           ⁢   k                   (   2   )               
where k=1, . . . m, n=1, . . . N and i=0, . . . ,N/m. The view displayed on film or the display panel is equal to
 
   
     
       
         
           
             
               
                 
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   Segments  404  and  403  in  FIG. 4  are representatives of the function G n (k,y). Since G n (k,y) is a complex function which can not be displayed on film or a display panel, instead an equivalent function below is used:
 
 H   n ( k,y )= A+Re{G   n ( k,y )}  (4)
 
where A is a constant and Re{ } means the real part of the function within the bracket. The function H n  (k, y) is called the Fourier transform hologram of the function g(n, y, j).
 
   Lens  405  provides an inverse Fourier transform to convert segment  404  into image pixels  409 , 410 , 411 , 412 . The number of image pixels at the recording plane is determined by the construction of the Fourier segment  404 . Four image pixels are illustrated, although other numbers could be used. A mask with a slit  408  confines the hologram unit to just recording the image pixels reconstructed from the Fourier segment. Similar Fourier segments which correspond to the different views are then displayed on  402  to record the hologram unit in the next position until the complete holographic stereogram is recorded. A major distinction between the present invention and the prior art is that more than one image pixel may be recorded inside a hologram unit. In a computer, the Fourier transform G n (k,y) can incorporate a random phase into the image as shown by the following equation: 
   
     
       
         
           
             
               
                 
                   
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   The addition of a random phase to each pixel renders the pixels incoherent from each other so that there will be no interference fringes produced in the reconstructed stereo image. In the prior art system a rotating diffuser is used in front of the film or display panel. The present invention removes this added complexity from the recording system. 
     FIGS. 5(   a ) and  5 ( b ) are top and side views of an optical system which implements the concept of the present invention described in  FIG. 4 . A collimated laser beam illuminates a liquid crystal display (LCD) panel  502  which takes the place of the film transparency in  FIG. 2 . Too avoid confusing the illustration of the invention, the optics for producing the collimated laser beam and directing it through or at a transparency or LCD panel are not shown.  FIG. 5(   a ) shows the top view of this anamorphic optical system. Segment  503  is the Fourier transform hologram Hn(k,y). The beam, after passing through LCD panel  502 , is focused by a spherical lens  504 . Lens  504  performs an inverse Fourier transform function to produce a diffracted, reconstructed image. The reconstructed image from Hn(k,y) is diffracted off the optical axis due to the properties of Hn(k,y). A slit  505  is used to pass only one of the diffracted orders from the function Hn(k,y). The beam from lens  504  is collimated by a spherical lens  506  and focused again by cylindrical lens  507  to the recording plane  508 . Lenses  506  and  507  form a telecentric optical system which basically images the reconstructed image on plane  505  to the recording plane  508 . Since Hn(k,y) is the Fourier transform hologram of g(n,y,j), the inverse Fourier transform of Hn(k,y) performed by lens  504 , and imaged by lenses  506  and  507 , reproduces the function g(n,y,j) on the recording plane  508 . A slit  509  is used to restrict the recording width of hologram unit  510 . The hologram on plane  508  is recorded with the aid of a reference beam  511 . 
   The orthogonal, side view of the optical system in  FIG. 5(   b ) shows the focused beam at plane  505  is collimated by a spherical lens  506 . In this orientation, the LCD panel is imaged to the recording plane  508 . The image on recording plane  508  is a magnified version of the image on LCD panel  502 . The magnification is given by the ratio of the focal lengths of lens  506  and lens  504 . The light beam from the LCD panel is combined with a reference beam  511  to form the hologram unit  510 . A unique property of this recording system is that each hologram unit at u=jΔ corresponds to a group of image pixels as given in Equation (2). The resolution of the image is independent of the width of a hologram. 
   As will be understood by those of skill in the art, the present invention could be embodied in other specific forms without departing from the essential characteristics thereof. For example, a different lens could be used to perform the inverse Fourier transformation. Accordingly, the foregoing description is illustrative, but not limiting, of the scope of the invention which is set forth in the following claims.