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 redefiner Ɽ_t can be mimicked by a trail of infinitesimal unitary transforms. Each subsequent trail element has eigenvectors that differ from those of its predecessor. These eigenvectors are also eigenvectors of Ɽ_t. For a single vector, which is not an eigenvector of Ɽ_t, the action of Ɽ_t can be represented by the integrated activity of this trail on that vector. This can be interpreted as the activity of a genuine unitary transform U_t. When a redefiner Ɽ_t is applied to the eigenvector |q> of an operator Q with eigenvalue q, then the eigenvector is transferred into another vector |U_t q>. The expectation value for |QU_t q> is no longer q, but

Optics normally treats the behaviour of packages of light waves (photons). However, when passing through appropriately shaped fields, particles may behave similar as photons. A beam of electrons that is not too dense will under such conditions behave similar to light beams that pass comparable lenses. In a dense beam the electrons will influence each others path via their own Coulomb field.

The OTF