A. W. Lohmann

Binary fraunhofer holograms, generated by computer.

When a hologram is desired from an object which does not exist physically but is known in mathematical terms, one can compute the hologram. An automatic plotter will make a drawing at a large scale which is then reduced photographically. Since the drawing can contain only black and white areas, we have developed a theory for binary holograms. They are equivalent in terms of image reconstruction with ordinary holograms. This has been proven theoretically and verified experimentally.

Complex Spatial Filtering with Binary Masks

Usually a hologram is produced by means of an interference experiment. Here, however, we let a computer- guided plotter draw the hologram. The plot, which has to be minified and recorded on film, contains no grey, only binary transmittance values. Our binary holograms yield reconstructed images of a quality equal to that of images obtained from usual holograms of comparable dimensions. When a Fourier hologram is inserted into the Fraunhofer plane of a coherent image forming system, it acts as a special type of a spatial filter, a so-called optical matched filter. Our binary matched filter is suitable for optical character recognition, the same as the usual optical matched filter introduced by Vander Lugt.

Computer Generated Spatial Filters for Coherent Optical Data Processing

The setup used in many optical data processing schemes is a coherent optical image forming system. The most important lement in this setup is the complex spatial filter. It can perform a large variety of linear operations upon the object or input. In general, it is difficult to produce complex filters, since both amplitude transmission and phase delay may vary across the filter plane in a complicated manner. Our own filters which are very similar to binary holograms, consist of many little transparent rectangles on opaque background. They can easily be drawn on a large scale by a computer-guided plotter, and then photographically reduced in size. We show that our filters, despite containing only amplitude values zero and one, can perform any data processing operation which could be performed by any complex filter. After explaining the principle, we present three groups of applications. First, we describe new versions of some classical methods: schlieren observation and phase contrast. Next, we report on spatial which perform differential operations upon the object in order to enhance gradients or corners. Finally, we use our binary filters for signal detection.

Theta Modulation in Optics

The experiments reported in this paper are similar to the famous Abbe experiments. However, they were done for quite different reasons, namely, to perform certain information processing operations by optical means. Our technique, called theta modulation, allows production of a color image from a black and white film, on which the color object is recorded in encoded form. Furthermore, nonlinear characteristics (H & D curves) of any shape can be realized. A special application of theta modulation, called multiplex storage, will be described. By this technique, more than one image can be recorded in the same area on a piece of film. Subsequently, the individual images can be recovered with a minimum of crosstalk.

Single-Sideband Holography*

In making in-line holograms of amplitude objects with a strong background, the single-sideband technique can be used to improve the quality of the reconstruction. The main advantage of this method is the suppression of the twin image. Modifications of the same technique are also presented for making holograms of complex objects and objects with a weak background.

Space-Variant Image Formation*

The application of optical transfer theory to the process of image formation requires that the image-forming system be linear and space invariant. In a space-invariant system, the point image retains its shape while the point source explores the object plane. The purpose of this paper is to investigate image-forming systems which are linear but space variant. Such systems may exceed performance limitations which are inherent in linear space-invariant systems. A method for experimentally determining space variance is devised. The degree of space invariance is defined and evaluated for several examples of space-variant systems.

Variable Fresnel Zone Pattern

Among the conceivable uses of Fresnel Zone Plates (FZP) are image formation, synthesis of holograms, coherence measurements, spectrometry, optical analog computation, and optical testing. Sometimes it is desirable to change the scale of the FZP continuously, for example to give a zoom lens effect when the FZP is used for image formation. Here we describe four ways of creating a FZP pattern as a moiré effect by superposing pairs of suitable masks. The relative position of the two masks determines the FZP scale. The theory presented here is sufficiently general to allow the synthesis of patterns other than the FZP by means of a variable moiré effect.

Nonlinear Effects in Holography

The ideal photographic material in holography would have a linear relationship between amplitude transmittance and exposure. Here we study the case where this relationship can be described instead by a polynomial. This nonlinearity in reconstruction gives rise to some extra images, autocorrelations, autoconvolutions, and ambiguity functions of the object, which may be found superposed on normal images, or spatially separated both laterally and in depth.

Character Recognition by Incoherent Spatial Filtering

The character recognition method described here is based on the principle of incoherent spatial matched filtering. The input to this matched filter is not the unknown character itself, but its Fraunhofer diffraction pattern. The intensity distribution in this diffraction pattern is insensitive against shifting of the unknown character, avoiding the need for character registration. The incoherent matched filter is easier to implement than the coherent matched filter, since only binary rather than continuous-tone masks are required. The theory and some experiments will be discussed and compared with other optical character recognition methods.

Superresolution for Nonbirefringent Objects

The theory of a superresolution experiment is developed. To achieve superresolution one must know in advance some properties of the objects, e.g., nonbirefringence, time independence, or wavelength independence. Assuming that the objects are nonbirefringent, it would be wasteful to use the two possible states of independent linear polarization of the light for simultaneously carrying the same information twice through the image-forming system. One can avoid this waste by inserting polarizers and certain double-refracting components into the system, so that the two states of polarization instead carry different information through the conventional image-forming system. The transfer function of such a superresolution system is derived for coherent and incoherent object illumination. It confirms qualitatively the results of previously reported experiments. A modification of the system is then proposed so that the one-dimensional restriction of the original concept is eliminated. The transfer function for the modified system is derived and numerical examples are presented. The modification imposes a further constraint on the class of allowed objects: the objects must be time-independent or only slowly time-varying.