A Method of Identification of Analyser Characteristics That is Free of Probability Summation Effect

Abstract

A detection model (originally proposed by Quick) comprising, in a sequence of linear analysers, varphi1, …, varphi n, nonlinear transducer functions, and the Minkowski decision rule, is widely used, especially when it is necessary to take into account the effect of probability summation. However, there is a general belief that the analyser characteristics cannot be determined in detection experiments since there is a trade-off between these characteristics and the decision rule. Here we show how to overcome this problem, ie how to identify the analysers varphi1, …, varphi n despite the probability summation between them. The observer's performance is assumed to be quantitatively defined in terms of an equidetection surface (EDS). Each analyser varphi i is expressed as a weighted sum of linear (coordinate) analysers {phi j}: varphi i=sum j=1 n a ijphi j, so that an identification of the analysers {phi i} is then reduced to evaluating the weight matrix A={ a ij}. It has been proven that A can be uniquely recovered from an ellipsoidal approximation of EDS in the neighbourhood of at least two points. More specifically, the following equation holds true: A−1 DA= H1−1 H2, where D is a diagonal matrix, H1 and H2 are the matrices of the quadratic forms determining the n-dimensional ellipsoids approximating EDS. Thus, the matrix H1−1 H2 known from experiment is a similarity transform of the diagonal matrix, the columns of A being the eigenvectors of H1−1 H2. Hence, any eigensystem routine can be used to derive A from H1−1 H2.