The use of intensity-dependent weight functions to “Weberize” $L^2$-based methods of signal and image approximation

Abstract

We consider the problem of modifying $L^2$-based approximations so that they “conform” in a better way to Weber’s model of perception: Given a greyscale background intensity $I > 0$, the minimum change in intensity $\varDelta I$ perceived by the human visual system is $\varDelta I / I^a = C$, where $a > 0$ and $C > 0$ are constants. A “Weberized distance” between two image functions u and v should tolerate greater (lesser) differences over regions in which they assume higher (lower) intensity values in a manner consistent with the above formula. In this paper, we Weberize the $L^2$ metric by inserting an intensity-dependent weight function into its integral. The weight function will depend on the exponent a so that Weber’s model is accommodated for all $a> 0$. We also define the “best Weberized approximation” of a function and also prove the existence and uniqueness of such an approximation.