The intensity-based measure approach 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 (HVS) 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 modify the usual integral formulas used to define L2L2 distances between functions. The pointwise differences |u(x)-v(x)||u(x)-v(x)| which comprise the L2L2 (or LpLp) integrands are replaced with measures of the appropriate greyscale intervals ?a(min{u(x),v(x)},max{u(x),v(x)}]?a(min{u(x),v(x)},max{u(x),v(x)}]. These measures ?a?a are defined in terms of density functions ?a(y)?a(y) which decrease at rates that conform with Weber’s model of perception. The existence of such measures is proved in the paper. We also define the “best Weberized approximation” of a function in terms of these metrics and also prove the existence and uniqueness of such an approximation.