Method of quadratic approximation: A new approach to identification of analysers and channels in human vision

Abstract

Models comprising, in sequence, linear analysers, Ψ 1, …, Ψ n , non-linear transducer functions, and the Minkowski decision rule, are widely used to fit detection and discrimination data, especially when necessary to take into account the effect of probability summation. However, the analysers’ characteristics cannot be derived from detection/discrimination data because of an unavoidable trade-off between these characteristics and the decision rule. Here we show how to overcome this problem, i.e. how to identify the analysers Ψ i= ∑ m aijφ j=1 despite the probability summation between them. The observer's performance is assumed to be quantitatively defined in terms of an equi-detection (discrimination) surface. Each analyser Ψ i is expressed as a weighted sum of linear (coordinate) functionals φ j :Ψ 1,…,Ψ n , so that an identification of the analysers ϕ i is then reduced to evaluating the weight matrix A={ a ij }. It is proved that A can be uniquely recovered from a quadratic approximation of the equi-detection (discrimination) surface at the neighbourhood of at least two points. More specifically, the following equation holds true: A T D( A T ) #= H 1 H 2 #, where A # is the generalised inverse of the matrix A, D is an unknown diagonal matrix, H 1 and H 2 are the matrices of the quadratic forms determining the quadratic surfaces approximating the equi-detection (discrimination) surface at two different points. Thus, the matrix H 1 H 2 # known from experiment is a similarity transform of the diagonal matrix, the rows of A being the eigenvectors of H 1 H 2 #. Hence, any eigensystem routine can be used to derive A from H 1 H 2 #.