Fitting Procedures for the Characteristic Curve

Abstract

The characteristic curve gives the relationship between the blackening or optical density D of a photographic emulsion and the intensity I the emulsion was exposed to. The theory of the photographic process deals with the explanation of the characteristic curve in the form

$$ \begin{array}{*{20}{c}} {D = D(y)} \\ {y = \log (I)} \\ \end{array} $$

(1)

and offers some mathematical approximations of this relationship (MEES 1954, FRIESER 1975,GERTH 1978). Unlike that for the reduction of photographic plates the astronomer is interested in the relationship

$$ y = y(D) $$

(2)

A simple method of obtaining an alytical approximation of equation (2) is a least squares fit by a polynomial. The disadvantage of this method is well known: The strong curvature in the region of underexposure requires a high degree of polynomial so that the method often fails in the case of a finite sample of measured points.