On the assessment of optical images

Abstract

In the formation of an optical image, each surface element of the object gives rise to a more or less blurred distribution in the image surface, of total brightness proportional to that of the object element. The image is the sum of these distributions in the appropriate sense: when the object is coherently lit, the image is built up by adding their complex amplitudes; when the object elements are regarded as incoherent it is the intensities which are added. In both cases the image can be expressed as the convolution of the object with a spread function which characterizes the optical system. In systems for which the spread function does not change appreciably from one part of the field to another, the Fourier transform of the image is obtained to a sufficient approximation on multiplying the Fourier transform of the object with that of the spread function. More generally, this holds for any part of the field of a non-isoplanatic system over which the changes in the form of the spread function are small enough to be disregarded; we call such an area an ‘isoplanatism-patch’. Working over such an area, an optical system can be regarded as a linear filter in which the Fourier components of the object reappear in the image multiplied by ‘transmission factors’. These factors, first considered by Duffieux, depend on the aperture and aberrations of the system, and in §2 they are evaluated in terms of an ikonal function. The qualities required of an optical image are so varied that an assessment valid over the whole range of practical applications seems out of the question. Two extreme cases are considered in the present paper. In the first of these it is assumed that the aim of an optical design is to produce an image which is directly similar to the object. This is appropriate when no process of image interpretation or reconstruction is envisaged. In the second case, the aim is to produce an image containing the greatest possible amount of information about the object, without regard to the complexity of the interpretation processes which may be needed to extract it. For the first case, a criterion of image fidelity is proposed in §2.4 which gives a numerical measure of the resemblance of image to object in terms of the transmission factors of the optical system. In the second case, assessment is based on the information content of the image in Shannon’s sense. This depends not only on the transmission factors of the system but also on the statistical properties of the presumed object set and of the unpredictable fluctuations which necessarily disturb observation; the analysis is carried through in §3. In §4 the assessment of optical images is discussed in terms of these two criteria.