The Topography of Vision in Mammals of Contrasting Life Style: Comparative Optics and Retinal Organisation

The Wigner distribution function applied to optical signals and systems

Utilizing a cosine transform in image compression has several recognized performance benefits, resulting in the ability to attain large compression ratios with small quality loss. Also, incorporation of a model of the human visual system into an image compression or quality assessment technique intuitively should (and has often proven to) improve performance. Clearly, then, it should prove highly beneficial to combine the image cosine transform with a visual model. In the past, combining these two has been hindered by a fundamental problem resulting from the scene alteration that is necessary for proper cosine transform utilization. A new analytical solution to this problem, taking the form of a straightforward multiplicative weighting function, is developed in this paper. This solution is readily applicable to image compression and quality assessment in conjunction with a visual model and the image cosine transform. In the development, relevant aspects of a human visual system model are discussed, and a refined version of the mean square error quality assessment measure is given which should increase this measure's utility.

A Visual Model Weighted Cosine Transform for Image Compression and Quality Assessment

The concept of a quasihomogeneous source is introduced. Unlike a source that is strictly homogeneous in its statistical properties, a quasihomogeneous source may be finite. Many physical sources, both primary and secondary ones, are adequately approximated by this model. Coherence and radiometric properties of light generated by such sources (assumed, for simplicity, to be planar) are discussed and an important reciprocity relation is shown to exist between light in the far zone and in the source plane. This relation implies that the degree of coherence in the far zone is given by the classic form of the van Cittert-Zernike theorem, even though the source may have a high degree of spatial coherence over arbitrarily large areas. The reciprocity relation also provides a generalization of a recently derived result that expresses the angular dependence of the radiant intensity in terms of the degree of spatial coherence of light in the source plane. The dependence of all the basic radiometric quantities on the distribution of the optical intensity across the source and on the degree of spatial coherence of the light emerging from the source is discussed and is illustrated, for some typical sources, by computed curves.

Coherence and radiometry with quasihomogeneous planar sources

A new method for tracing rays through graded-index media is presented. The method essentially consists of transforming the ray equation into a convenient form and solving the resulting equation using a standard numerical technique. A detailed comparison of this method with existing methods has also been made, and it is shown that for obtaining a desired accuracy this method requires much less computational effort.

Tracing rays through graded-index media: a new method

The basic laws of radiometry are generalized to fields generated by a two-dimensional stationary source of any state of coherence. Important in this analysis is the concept of the generalized radiance function, introduced by Walther in 1968. The concepts of generalized radiant emittance and of generalized radiant intensity are introduced and it is shown how all these quantities may be expressed in terms of coherence functions of the source. Both the generalized radiance of Walther and the generalized radiant emittance may take on negative values, indicating that these quantities have, in general, a less-direct physical meaning than have the corresponding quantities of traditional radiometry (which presumably represents the incoherent limit of the present theory). The generalized radiant intensity is, however, always found to be non-negative and, just as in the incoherent limit, represents the angular distribution of the energy flux in the far zone.

Radiometry with sources of any state of coherence

In the first part of this paper, the concept of the angular correlation function of a stationary optical field is introduced. This function characterizes the correlation that exists between the complex amplitudes of any two plane waves in the angular spectrum description of the statistical ensemble that represents the field. Relations between this function and the more commonly known correlation functions are derived. In particular, it is shown that the angular correlation function is essentially the four-dimensional spatial Fourier transform of the cross-spectral density function of the source. The angular correlation function is shown to characterize completely the second-order coherence properties of the far field. An expression for the intensity distribution in the far zone of a field generated by a source of any state of coherence is deduced. Some generalizations of the far-zone form of the Van Cittert–Zernike theorem are also obtained.

Angular Correlation and the Far-Zone Behavior of Partially Coherent Fields*

With a view to elucidating the effect of a well-known mathematical inconsistency in Kirchhoff’s diffraction theory, a comparison is made of the predictions relating to the field diffracted at an aperture, based on Kirchhoff’s theory (UK) and on formulas due to Rayleigh and Sommerfeld (UR). It is shown that, when the incident wave is plane or spherical, the difference δ = UK−UR represents a boundary wave, i.e., a wave which may be thought of as originating at each point of the edge of the aperture. It is shown further that, when the linear dimensions of the aperture are large compared with the wavelength, the boundary values of δ in the plane of the aperture change very rapidly and almost periodically from point to point, with the mean period close to the wavelength of the incident radiation. This result is shown to imply that if the linear dimensions of the aperture are large compared with the wavelength, the two theories predict essentially the same behavior for the diffracted field in the far zone, at moderate angles of diffraction.

Comparison of the Kirchhoff and the Rayleigh–Sommerfeld Theories of Diffraction at an Aperture

In dealing with optical imaging systems, it is more feasible experimentally to measure line spread functions than point spread functions. When the intensity distribution is known to possess rotational symmetry, the point spread function can be obtained mathematically from the corresponding line spread function by solving an integral equation. A direct solution of this equation is given which represents a procedure that is simpler for practical use than the usual one involving Fourier transforms.

Derivation of the Point Spread Function from the Line Spread Function

A gradient-index element is one in which the refractive index varies appreciably in a layer just below the surface of the glass. The present paper deals with practical methods of tracing rays through such a medium. One method, due to L. Montagnino, lends itself to rapid computer ray tracing. A second method, through restricted to the case of spherical symmetry in the index function, lends itself both to computer tracing and to analytical investigations.

Erich W. Marchand

Papers

Radiometry with sources of any state of coherence

Angular Correlation and the Far-Zone Behavior of Partially Coherent Fields*

Comparison of the Kirchhoff and the Rayleigh–Sommerfeld Theories of Diffraction at an Aperture

Derivation of the Point Spread Function from the Line Spread Function

Ray Tracing in Gradient-Index Media