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

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 _ij phi _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⁻¹DA=H₁⁻¹H₂, 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 H₁⁻¹H₂ known from experiment is a similarity transform of the diagonal matrix, the columns of A being the eigenvectors of H₁⁻¹H₂. Hence, any eigensystem routine can be used to derive A from H₁⁻¹H₂.