1. The contrast thresholds of a variety of grating patterns have been measured over a wide range of spatial frequencies.2. Contrast thresholds for the detection of gratings whose luminance profiles are sine, square, rectangular or saw-tooth waves can be simply related using Fourier theory.3. Over a wide range of spatial frequencies the contrast threshold of a grating is determined only by the amplitude of the fundamental Fourier component of its wave form.4. Gratings of complex wave form cannot be distinguished from sine-wave gratings until their contrast has been raised to a level at which the higher harmonic components reach their independent threshold.5. These findings can be explained by the existence within the nervous system of linearly operating independent mechanisms selectively sensitive to limited ranges of spatial frequencies.

Application of fourier analysis to the visibility of gratings

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

This article presents a review of coherence properties of electromagnetic fields and their measurements, with special emphasis on the optical region of the spectrum. Analyses based on both the classical and quantum theories are described. After a brief historical introduction, the elementary concepts which are frequently employed in the discussion of interference phenomena are summarized. The measure of second-order coherence is then introduced in connection with the analysis of a simple interference experiment and some of the more important second-order coherence effects are studied. Their uses in stellar interferometry and interference spectroscopy are described. Analysis of partial polarization from the standpoint of correlation theory is also outlined. The general statistical description of the field is discussed in some detail. The recently discovered universal "diagonal" representation of the density operator for free fields is also considered and it is shown how, with the help of the associated generalized phase-space distribution function, the quantum-mechanical correlation functions may be expressed in the same form as the classical ones. The sections which follow deal with the statistical properties of thermal and nonthermal light, and with the temporal and spatial coherence of blackbody radiation. Later sections, dealing with fourth- and higher-order coherence effects include a discussion of the photoelectric detection process. Among the fourth-order effects described in detail are bunching phenomena, the Hanbury Brown-Twiss effect and its application to astronomy. The article concludes with a discussion of various transient superposition effects, such as light beats and interference fringes produced by independent light beams.

Coherence Properties of Optical Fields

The Coherence Properties of Optical Fields

1. Optical quality of the eye was measured at eight pupil sizes between 1.5 and 6.6 mm diameter by recording the faint light emerging from the eye; this light was reflected from the bright image of a thin line on the fundus.2. The nature of the fundus reflexion was examined; it was found that the fundus acts very much like a perfect diffuser while retaining polarization.3. Using the result that the fundus acts like a diffuser, the recorded line images were Fourier analysed to provide modulation transfer functions. These functions indicate an optical quality considerably higher than that found in previous physical studies.4. Linespread profiles were then derived from the modulation transfer functions. These profiles are 40% narrower than those of previous physical studies for a 3.0 mm pupil. The narrowest profile occurred with a 2.4 mm pupil.5. Our results demonstrate that physical and psychophysical studies can yield similar estimates of optical quality. The influence of optical factors not common to both techniques is discussed. Evidence for the existence of neural ;image sharpening' mechanisms is reviewed.

Optical quality of the human eye.

The theory of image formation is formulated in terms of the coherence function in the object plane, the diffraction distribution function of the image-forming system and a function describing the structure of the object. There results a four-fold integral involving these functions, and the complex conjugate functions of the latter two. This integral is evaluated in terms of the Fourier transforms of the coherence function, the diffraction distribution function and its complex conjugate. In fact, these transforms are respectively the distribution of intensity in an 'effective source', and the complex transmission of the optical system-they are the data initially known and are generally of simple form. A generalized 'transmission factor' is found which reduces to the known results in the simple cases of perfect coherence and complete incoherence. The procedure may be varied in a manner more suited to non-periodic objects. The theory is applied to study inter alia the influence of the method of illumination on the images of simple periodic structures and of an isolated line.

On the diffraction theory of optical images

The theory of image formation is formulated in terms of the coherence function in the object plane, the diffraction distribution function of the image-forming system and a function describing the structure of the object. There results a four-fold integral involving these functions, and the complex conjugate functions of the latter two. This integral is evaluated in terms of the Fourier transforms of the coherence function, the diffraction distribution function and its complex conjugate. In fact, these transforms are respectively the distribution of intensity in an ‘effective source’, and the complex transmission of the optical system— they are the data initially known and are generally of simple form. A generalized ‘transmission factor’ is found which reduces to the known results in the simple cases of perfect coherence and complete incoherence. The procedure may be varied in a manner more suited to non-periodic objects. The theory is applied to study inter alia the influence of the method of illumination on the images of simple periodic structures and of an isolated line.

On the Diffraction Theory of Optical Images

The response of a defocused aberration-free optical system to line-frequencies in the object is studied analytically. Curves are given showing the response as a function of line-frequency for a range of values of defect of focus. A comparison is made with the results to be expected from geometrical optics. A tolerance for defect of focus is obtained, which accords well with published experimental results. Both circular and rectangular apertures are considered.

The Frequency Response of a Defocused Optical System

Wave theory of aberrations

A die for use in a food chopper downstream of the coarse-cut die is formed with an array of like small-diameter holes and on each of its faces with an array of grooves having a width substantially greater than the diameter of the through-going holes and extending at an angle to the blade which sweeps over the upstream face of this die. Particles which have passed through the upstream die and which are unable to pass through the small-diameter holes in the downstream die are caught in the grooves and cut between the downstream edge of the grooves and the cutter blade. These grooves are arrayed such that the cutter blade conducts the food along the grooves, and the grooves may be of decreasing depth in the direction of travel of the material therealong so as to allow the material to be reduced to very small size. At the inner ends the grooves may terminate in large-diameter throughgoing holes which are connected to an arrangement for carrying unchoppable particles out of the chopper.

https://iopscience.iop.org/article/10.1088/0370-1301/70/10/410/pdf

H. H. Hopkins

Papers

On the diffraction theory of optical images

On the Diffraction Theory of Optical Images

The Frequency Response of a Defocused Optical System

Wave theory of aberrations

The Numerical Evaluation of the Frequency Response of Optical Systems