Image processing techniques find applications in many areas, chief among which are image enhancement, pattern recognition, and efficient picture coding. Some aspects of image processing are discussed--specifically: the mathematical operations one is likely to encounter, and ways of implementing them by optics and on digital computers; image description; and image quality evaluation. Many old results are reviewed, some new ones presented, and several open questions are posed.

Image processing

The functional Fourier transform (FRFT), which is a generalization of the classical Fourier transform, was introduced a number of years ago in the mathematics literature but appears to have remained largely unknown to the signal processing community, to which it may, however, be potentially useful. The FRFT depends on a parameter /spl alpha/ and can be interpreted as a rotation by an angle /spl alpha/ in the time-frequency plane. An FRFT with /spl alpha/=/spl pi//2 corresponds to the classical Fourier transform, and an FRFT with /spl alpha/=0 corresponds to the identity operator. On the other hand, the angles of successively performed FRFTs simply add up, as do the angles of successive rotations. The FRFT of a signal can also be interpreted as a decomposition of the signal in terms of chirps. The authors briefly introduce the FRFT and a number of its properties and then present some new results: the interpretation as a rotation in the time-frequency plane, and the FRFT's relationships with time-frequency representations such as the Wigner distribution, the ambiguity function, the short-time Fourier transform and the spectrogram. These relationships have a very simple and natural form and support the FRFT's interpretation as a rotation operator. Examples of FRFTs of some simple signals are given. An example of the application of the FRFT is also given. >

The fractional Fourier transform and time-frequency representations

Imaging with visible light today uses numerous contrast mechanisms, including bright- and dark-field contrast, phase-contrast schemes and confocal and fluorescence-based methods. X-ray imaging, on the other hand, has only recently seen the development of an analogous variety of contrast modalities. Although X-ray phase-contrast imaging could successfully be implemented at a relatively early stage with several techniques, dark-field imaging, or more generally scattering-based imaging, with hard X-rays and good signal-to-noise ratio, in practice still remains a challenging task even at highly brilliant synchrotron sources. In this letter, we report a new approach on the basis of a grating interferometer that can efficiently yield dark-field scatter images of high quality, even with conventional X-ray tube sources. Because the image contrast is formed through the mechanism of small-angle scattering, it provides complementary and otherwise inaccessible structural information about the specimen at the micrometre and submicrometre length scale. Our approach is fully compatible with conventional transmission radiography and a recently developed hard-X-ray phase-contrast imaging scheme. Applications to X-ray medical imaging, industrial non-destructive testing and security screening are discussed.

Hard-X-ray dark-field imaging using a grating interferometer

An algorithm for efficient and accurate computation of the fractional Fourier transform is given. For signals with time-bandwidth product N, the presented algorithm computes the fractional transform in O(NlogN) time. A definition for the discrete fractional Fourier transform that emerges from our analysis is also discussed.

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Digital computation of the fractional Fourier transform

In high precision holographic imagery of weak objects of small angular subtense, electronic detection and digital image formation have distinct advantages. Experiments with a vidicon detector and a PDP‐6 computer have yielded reconstructed images of good quality with computation times of five minutes.

Digital Image Formation From Electronically Detected Holograms

In this study the degree p = 1 is assigned to the ordinary Fourier transform The fractional Fourier transform, for example with degree P = 1/2, performs an ordinary Fourier transform if applied twice in a row Ozaktas and Mendlovic [ “ Fourier transforms of fractional order and their optical implementation,” Opt Commun (to be published)] introduced the fractional Fourier transform into optics on the basis of the fact that a piece of graded-index (GRIN) fiber of proper length will perform a Fourier transform Cutting that piece of GRIN fiber into shorter pieces corresponds to splitting the ordinary Fourier transform into fractional transforms I approach the subject of fractional Fourier transforms in two other ways First, I point out the algorithmic isomorphism among image rotation, rotation of the Wigner distribution function, and fractional Fourier transforming Second, I propose two optical setups that are able to perform a fractional Fourier transform

Image rotation, Wigner rotation, and the fractional Fourier transform

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.

Binary Fraunhofer Holograms, Generated by Computer

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.

Complex Spatial Filtering with Binary Masks

Holograms synthesized by computer are used for constructing optical wavefronts from numerically specified objects. Elimination of the need for a physical object has made new applications possible, for example, three-dimensional computer output displays, synthetic prototypes for interferometric testing, and filters for various optical data processing operations. Our computer holograms differ from a normal hologram in that the transmittance is binary, yet they are able to construct general wavefronts and images efficiently and have several practical advantages over holograms with a continuous range of transmittance. Recent improvements that simplify the production of binary holograms and improve their performance are described and experimental work showing reconstruction of two- and three-dimensional images is presented.

Adolf W. Lohmann

Papers

Image rotation, Wigner rotation, and the fractional Fourier transform

Binary Fraunhofer Holograms, Generated by Computer

Complex Spatial Filtering with Binary Masks

Computer-generated binary holograms

An interferometer based on the Talbot effect