The use of intensity-dependent weight functions to "Weberize" L2-based methods of signal and image approximation

We consider the problem of modifying L2L2-based approximations so that they “conform” in a better way to Weber’s model of perception: Given a greyscale background intensity I>0I>0, the minimum change in intensity ?I?I perceived by the human visual system is ?I/Ia=C?I/Ia=C, where a>0a>0 and C>0C>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 L2L2 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>0a>0. We also define the “best Weberized approximation” of a function and also prove the existence and uniqueness of such an approximation.