At one time the only criterion for the performance of an optical system was its resolving power. In an ideal system (i.e. one without any aberrations) this bears a simple relation to its
The difficulty is that this tells us only the point at which the optical system fails to produce an image. It gives no information as to the quality of what it does resolve. The nature of this defect became clear during the Second World War, when photo interpreters were examining reconnaissance photographs under high magnification, and saw that some types of camera lens produced more readily interpretable images than others of higher resolving power. Others performed less well than would be expected, and even produced misleading distortions in fine detail.
By careful ray tracing (nowadays done by computer programs) lens designers could predict the shape and size of the image of a point of light. However, this point spread function is somewhat difficult to relate to image quality.
The modern approach
In the early 1950s it was noticed that in the image of a Sayce chart, as the spatial frequency increased the image contrast fell steadily, and the intensity profile became markedly sinusoidal (see fourier optics, Fig. 5b). Further trials using a test chart with a sinusoidal profile instead of black and white bars showed that there was no change in the profile as the spatial frequency increased apart from a progressive decrease in contrast. Taking a leaf from the electronic engineering book, it became clear that any image (analogously to complex sound waveforms) could be analysed into a spectrum of sinusoidal gratings of various amplitudes (i.e. contrasts) and (spatial) frequencies, and, for two-dimensional images, orientations. If an optical system were to be tested using sinusoidal gratings it would be possible to plot the ratio of image contrast to subject contrast for the whole range of spatial frequencies, and thus obtain data on the performance of the system for every type of subject and at all levels of detail.
Modulation
Contrast is a subjective perception, and it is difficult to express as a measurable quantity. In photography it is traditionally defined as the logarithm of the highlight-to-shadow luminance ratio. This matches visual sensations fairly closely until the contrast becomes very high, and as it increases further the logarithmic figure continues to rise while the eye perceives little or no change. A closer fit to perceived contrast—again familiar to electronics engineers—is modulation. The modulation M of a scene is given by the expression
M = (Lmax -Lmin)/ (Lmax + Lmin)where Lmax and Lmin are the respective luminances of the brightest and darkest parts of the scene. Modulation cannot go higher than 1, corresponding to a black that is completely devoid of shadow detail (Lmin = 0).
The optical transfer function
The optical transfer function (OTF) provides a full description of the imaging quality of an optical system. It is a combination of the modulation transfer function (MTF) and the phase transfer function (PTF). The MTF is a graph of image modulation ÷ object modulation plotted against image spatial frequency in cycles per millimetre (mm-1). (With a sinusoidal test object the spatial frequency is counted in cycles per mm rather than line pairs.) The PTF is a graph of the displacement of the image compared with the geometrically correct position in radians (rad), plotted against image spatial frequency. A displacement of one whole cycle is ±2π radians or 360 degrees. Fig. 1 shows a typical OTF for a wide-aperture camera lens. The MTF shows how the system behaves at all levels of detail, and is an accurate predictor of image quality. The PTF is in general less useful, but it does predict whether image artefacts (such as doubled edges to details) are likely to occur at high spatial frequencies.
OTFs can be predicted by computer programs that design optical systems, and can be tested on existing lenses by a device that projects an image of a continuously variable grating on to the test system via a high-quality collimator. A collimator is an optical device that produces an accurately parallel (collimated) beam of light. Two sets of data are produced, one with the grating bars orientated radially from the optical centre, and the other tangentially to it. Readings are taken at intervals from the centre of the field to its periphery. The PTF for the radial OTF will be zero throughout unless the lens has become decentred; but it will depart from zero for the tangential readings if the lens suffers from coma or distortion (see lens aberrations). A lens that is defocused will show an MTF that dips below zero, while the PTF switches between +π, 0, and -π. Examination of the image of the test object reveals spurious reversals of the image, with white bars appearing where dark bars belong and vice versa (Fig. 2). This is a consequence of the overlapping of spread functions distorted by the defocusing. It can also be caused by certain lens aberrations, and it was one of the more serious problems for photo interpreters involved in wartime reconnaissance photography.
The OTF of an ‘ideal’ lens can be calculated precisely using diffraction theory. Fig. 3 shows MTFs for several different f-numbers. In practice, any lens will have an MTF lower than this ideal, but once it reaches ‘aperture limitation’ its MTF will follow the ‘ideal’ curve for that aperture.
Cascading of OTFs
Most optical imaging systems have more than one component. A simple camera system has separate OTFs for the lens and the film. Multiplying all the MTFs and adding all the PTFs obtains the aggregate OTF. Some optical systems can have five or more components, each with its own OTF; they can all be combined in the same way to give a final OTF.
Historical note
The photographic division of the former Royal Aircraft Establishment at Farnborough, Hampshire, was responsible for the development of aerial cameras and processing systems during the Second World War, and treated the problem of image artefacts seriously. For reasons of secrecy, little was said in public then or later. But the concept of the OTF had already been evolved from the analogy with electronic audio amplification, and it had been realized that it was simply the Fourier transform of the point spread function, a concept familiar to audio engineers. The team built an analogue computer to perform Fourier transforms, to obtain the OTF from the PSF. The first OTF measuring system was designed and built at Farnborough in the early 1950s, and a company was set up to exploit the new insights, but without commercial success. In the meantime, Japanese manufactures had eagerly accepted the principles of OTF theory, and had discovered that a slight defocusing of an existing camera lens from the optimum resolution point gave an OTF that had a more satisfactory shape for pictorial purposes. As a result, Japanese lenses acquired a reputation for image quality that resulted in their domination of the lens market for nearly 50 years.
Fig. 1
Fig. 2a
Fig. 2b
Fig. 3
— Graham Saxby
Bibliography
- Williams, T., The Optical Transfer Function of Imaging Systems (1999).
- Jacobson, R. E. (ed.), The Manual of Photography (9th edn. 2000).
- Ray, S., Applied Photographic Optics (3rd edn. 2003)
