A long time ago (≈1975) I was involved in establishing a world standard for the measurement of the Optical Transfer Function (OTF). It is better known as its modulus, the Modulation Transfer Function (MTF). The OTF combines the MTF with the Phase Transfer Function (PTF). The OTF is a two dimensional Fourier transform of the Point Spread Function (PTF). Thus, it is a two dimensional frequency characteristic used for qualifying imaging devices and chains of imaging devices.

Since the image of a point contains very little energy, the OTF is measured by analyzing the Line Spread Function (LSF). The measurement using the LSF will provide a convolution of the OTF with the Fourier transform of the line, which results in a one dimensional Fourier transform that corresponds to a cut through the center of the two dimensional OTF oriented perpendicular to the direction of the line. Every cut through the center of the MTF is symmetric. Thus, only one half of it is specified. Off-axis the lateral and the radial cuts usually differ.

And again in order to improve the available energy, the image of a thin slit rather than a line is analyzed. One way to analyze the slit image is to filter it by a pattern formed by two bar patterns that are crossing with a varying angle. When properly oriented the slit image meets a triangle wave. What is left then is to measure the modulus of the intensity of the light that falls through the optical filter. For high spatial frequencies the blurring already diminishes the high frequency components of the triangle wave such that they can be ignored. With sufficient electronic filtering the triangle wave converts also at low spatial frequencies to its sinusoidal ground frequency. This procedure delivers the MTF. For inhomogeneous light imaging the MTF is an appropriate qualifier. In inhomogeneous light all phases are scrambled, so the PTF does not make sense. In contrast for holographic imaging the MTF is hardly of importance. There the PTF is crucial as an objective imaging qualifier.

(Remark: In inhomogeneous light imaging ray tracing is a useful technique for designing imaging devices. In contrast holographic imaging is mainly determined by interferences of waves. It is interesting to interpret wave mechanics as the three dimensional equivalent of optics. In this respect the particle view corresponds with inhomogeneous imaging and the wave view corresponds with holographic imaging. The hole in the wall of the camera obscura corresponds with the holographic screen that surrounds a black hole.)

The theory behind the OTF considers the imaging characteristics as invariant with respect to the position in the image field. This fact does NOT correspond to common practice. Cylindrical lenses feature Seidel aberrations and chromatic aberrations that increase with the distance from the axis. Electronic imaging devices show similar characteristics and besides of that screens and fiber plates also offer fine structured irregularities and vignetting. Imaging characteristics depend on the angular distribution and on the homogeneity of the light.

In order to reach a well-defined measuring result, all these influences must be reckoned. Most importantly the measuring area must be restricted. This comes down to restricting the length of the measuring slit and restricting the validity of the measuring result. The criterion here will be supplied by the accepted measuring accuracy. A small variation in the slit length or a small move in lateral direction of the slit must not cause a variation in the course measuring result that varies beyond the accepted measuring accuracy.

Internal reflections inside imaging device often induce an effect that is named veiling glare. It reduces the brilliance of the produced image. Our visual system can filter this effect partly away but when more that 4% of the light is contributing to this effect, the veiling glare becomes noticeable in an annoying way. The veiling glare appears in the MTF as a fast drop near zero frequency. It is affected by the aperture of the imaging apparatus.

The conclusion is that we must adapt our theoretical concepts to these deviations of ideal circumstances. Just like nature, general technology does not behave in an ideal sense. The same thing that happens to our tools in imaging qualification also happens to all of our linear tools that we use in physics. Nature does not tend to behave invariant neither in space nor in time. Further its behavior depends on all kinds of circumstances.

In most cases the linear operators that we apply have only validity in a small region of space and time. In many cases we are forced to work with infinitesimal versions of operators that are combined in trails. This brings strong restrictions, but it also brings opportunities. These opportunities are offered by a special kind of hyper complex numbers: the 2ⁿ-ons. The high dimensional 2ⁿ-ons behave like 2^m-ons in their lower 2^m dimensions. This turns out to be favorable at very small scales. At very small scales hyper complex numbers behave as complex numbers. However, the imaginary part of the complex number appears to have inherited a direction from its higher dimensional parent.

There exists another opportunity. Locally smoothly curved manifolds show tangent spaces that can be interpreted as parts of a hyper complex number field. This possibility is especially suited to tackle curved spaces that tend to appear in nature. Where our regular linear tools fail, trails of infinitesimal forms of these tools may take over. This fact may be exploited by applying higher dimensional hyper complex numbers as eigenvalues. The sets of eigenvectors may differ per trail element. In this way a trail of infinitesimal unitary transforms can do what a single unitary transform can never do: transport a multidimensional subspace of Hilbert space over a significant distance. Thus when dynamics must be implemented, the trails come in.

If you think about it, the tools that we learned about in university, hardly ever fit practical purposes. Even the most sophisticated tools usually only work in a short range and under special, mostly quite abstract circumstances. When we mature we must learn to adapt these tools to reality, then reality does not adapt to us.