Synthetic Holograms of Splines

In this paper we present an approach for the holographic imaging of objects composed of splines. It is based on an imaging method for lines and extended to fit the requirements for the visualization of 3D curves. The input object consisting of splines is decomposed into line segments. These segments are subsequently transferred into the holographic equivalent—a set of textured rectangles appropriately placed in 3D space. Proper rectangle orientation and texture distortion allow a narrow approximation of holographic patterns generated by cylindrical and conical waves. Our approach allows faster hologram generation compared to traditional methods. We employ computer graphics rendering, exploiting hardware support for the rendering of textured geometry. 1 Holographic Imaging An important goal in 3D visualization is to provide the cues known from natural vision which make the perception of depth possible. These phenomena include for instance binocular disparity, ocular accommodation, and motion parallax. A number of display techniques [11] exist to provide a 3D sensation, but holographic displays are the only ones known today to provide all the depth cues. Therefore we believe that holography will play an important role in the future of 3D imaging. The method of holographic imaging of objects was discovered by Dennis Gabor [7]. Holography is a two-step process. It consists of the recording and reconstruction stage [2]. The hologram is a diffraction screen reconstructing the wave field emitted by the object to be imaged. In the hologram recording step, the output from a laser is separated into two beams. One beam illuminates the object, the other (reference beam) is directed to the holographic plate. The hologram records the interference pattern caused by the light reflected by the object and the reference beam. During reconstruction, the hologram is again illuminated with the reference beam. The diffracted wave field contains a three-dimensional copy of the original wave reflected by the object. Thus, the viewer looking through the hologram sees the image of the object in depth with changing perspective. In synthetic holography the processes of optical holographic imaging are simulated. The transmittance of the holographic plate is calculated by a computer. The resulting hologram can be optically reconstructed as described above. The reconstruction also can be performed by means of computer simulation of diffraction [1]. Even the real time display of holographic images is possible [10]. A couple of methods dealing with hologram generation will be briefly described. In most cases the object to be imaged is assembled of a set of radiating points. Each point emits a spherical wave. In direct simulation the complex amplitude of the wave originating from each point is calculated for all hologram locations (pixels). This is computationally expensive. One method to reduce the effort is described in [9]. The contributions of each possible 3D point in the image volume are precomputed and stored in a look-up table. In the hologram generation step just the contributions of the actual points have to be accumulated. This method was extended to holographic stereograms [10]. The methods described so far are point-based. If the 3D input object consists of lines, the contribution of each line to the hologram can be calculated instead of decomposing the object into points. A line represents a large number of points. In computer graphics, line drawings are used to express the essential in the shape of an object [17] with less resources. In our research we developed a method for the efficient holographic imaging of lines [14] and curves in 3D. 2 Holograms of Lines 2.1 The Conventional Approach In holography, objects to be imaged are assembled of luminous geometric primitives. There are three categories of such primitives: points, lines and planes. These primitives emit waves which generate characteristic patterns on the hologram. A point source emits spherical waves, cylindrical waves originate from infinite luminous lines and plane waves are emitted by infinite planes [4]. In the hologram reconstruction step the waves are again generated and focus in their original objects. Thus the reconstruction of these primitives is accomplished. In our case the patterns generated by cylindrical waves are of special interest [6]. Their description is illustrated in Figures 1(a) and 1(b). The z axis of our