Abstract
Recent research on the logarithm of matrices is likely to have a significant impact on optom-etry. The purpose of this paper is to draw atten-tion to this research and show application to the calculation of an average centred Gaussian eye in particular. The paper makes use of Cardoso’s method to obtain an explicit expression for the logarithm of a transference. The arithmetic aver-age of log-transferences leads to a transference which can be regarded as the transference of the average eye for the set. A numerical example is presented.
Highlights
Most quantitative studies require, at some point, the calculation of an average
The problem of calculating an average refractive error was solved as recently as 1983 by Keating; he showed that the average could be obtained as the arithmetic average of dioptric power matrices.[3]
While some unresolved issues remain it appears that the exponential-mean-logarithm of the transference may be the solution to the problem of the average eye.[4]
Summary
The first-order optical nature of a centred Gaussian optical system is completely characterised by the transference[12, 13]. If one attempts to calculate an average in this naïve way of averaging the transferences one typically obtains an ‘average eye’ with a ‘transference’ whose determinant is generally not 1. (despite being fundamentally unsound, such naïve averages can be adequate This is an issue that is discussed further below.). A transference is necessarily symplectic, which means, in this case, that the transference has a unit determinant or, in other words, the fundamental properties are related by is called a logarithm of M. Given the transferences of N eyes application of equation 5 to obtain an average eye is routine. Even with MATLAB the calculation is purely mechanical and gives little insight
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