Optimization of Hologram Computation for Real-Time Display

Several methods of increasing the speed and simplicity of the computation of off-axis transmission holograms are presented, with applications to the real-time display of holographic images. A bipolar intensity approach enables a linear summation of interference fringes, a factor of two speed increase, and the elimination of image noise caused by object self-interference. An order of magnitude speed increase is obtained through the use of precomputed look-up tables containing a large array of elemental interference patterns corresponding to point source contributions from each of the possible locations in image space. Results achieved using a data-parallel supercomputer to compute horizontal-parallax-only holographic patterns containing 6 megasamples indicate that an image comprised of 10,000 points with arbitrary brightness (grayscale) can be computed in under one second. INTRODUCTION The real-time display of holographic images has recently become a reality. The MIT Spatial Imaging Group has reported the successful generation of small three-dimensional (3D) computer-generated holographic images reconstructed in real time using a display system based on acousto-optic modulation of light[1, 2, 3]. In any real-time display system, a computer-generated hologram (CGH) must be computed as quickly as possible in order to provide for dynamic and interactive images. However, numerical synthesis of a holographic interference pattern demands an enormous amount of computation, making rapid ( 1 second) generation of holograms of even limited size impossible with conventional computers. A holographic fringe pattern is computed by numerically simulating the physical phenomena of light diffraction and interference. In general, light diffracts from a three-dimensional object to the hologram plane. Since the analytical expressions that model diffractive propagation through free space resemble the Fourier transform integral, computation of holographic interference patterns often utilizes the Fast Fourier Transform (FFT) algorithm[4]. Though relatively fast, this approach is useful only for images possessing discrete depth surfaces[5, 6], and becomes slow when applied to images that extend throughout an image volume. A more general approach is a ray-tracing method in which the contribution from each object point source is computed at each point in the hologram plane. This method can produce arbitrary three-dimensional (3D) images, but is slow, since it requires one calculation per image point per hologram sample. As presented in this paper, the application of several methods of reducing computation complexity leads to computation times as short as one second on a data-parallel-processing supercomputer. First, a “bipolar intensity” representation of the holographic interference pattern is developed and shown to eliminate unwanted image artifacts and simplify calculations without loss of image quality or generality. Second, a look-up table approach is described and shown to provide further speed increase, though image resolution and quantization noise become issues. Finally, exemplary computation times are presented. HOLOGRAPHIC IMAGING SPECIFICS This paper focuses on the computation of off-axis transmission holograms possessing horizontal parallax only (HPO), a quality of the “rainbow” or Benton hologram. It is possible to represent an HPO hologram with a vertically stacked array of one-dimensional holographic lines[6, 7]. Consider an HPO hologram made optically using a reference beam with a horizontal angle of incidence. Spatial frequencies are large in the horizontal direction ( 1000 lp/mm) and increase with the reference beam angle. However, by limiting the view zone to only a single vertical view, vertical spatial frequencies are low ( 10 lp/mm). It is evident that elimination of vertical parallax provides a factor of 100 (or greater) reduction of CGH size. During reconstruction of this hologram, diffraction occurs predominantly in the horizontal direction. It is appropriate to represent this holographic pattern with a relatively low vertical sample spacing (or “pitch”), roughly that used in a two-dimensional (2D) imaging system. In the horizontal dimension, however, the sampling pitch must be very high in order to accurately represent the holographic information. For each horizontal plane (“scan-plane”), the associated horizontal line of the hologram diffracts light to form image points in that plane only. Therefore, the 2D holographic pattern representing an HPO 3D image can be thought of as a stack of one-dimensional (1D) holograms or “holo-lines”. The goal of this paper, then, is to compute these 1D holographic lines as quickly as possible. y x z holo −line E R