Information-theoretic solutions to early visual information processing: Analytic results

The time-dependent evolution equation for the connection strength of a three-layer feed-forward neural network derived from optimizing the information rate is solved accurately at the long-time limit. It is shown that (1) the receptive field is given by the eigenvector of the maximum eigenvalue of the evolution equation, (2) the morphology of the receptive field is divided by a parameter $k_2$ into distinct regions, (3) the receptive field obeys parity invariance for high values of $k_2$ but breaks the parity invariance for values of $k_2$ less than -0.891$C_Q$, and (4) the analytic expressions for the receptive fields are derived. Differences of our results from the theory of Marr and Hildreth [Proc. R. Soc. London Ser. B 207, 187 (1980)] of edge-detection based on the center-surround ${\nabla }^2$G filters are presented.