The intensity-based measure approach to “Weberize” L2-based methods of signal and image approximation

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 (HVS) 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 modify the usual integral formulas used to define $$L^2$$ distances between functions. The pointwise differences $$|u(x)-v(x)|$$ which comprise the $$L^2$$ (or $$L^p$$ ) integrands are replaced with measures of the appropriate greyscale intervals $$\nu _a ( \min \{ u(x),v(x) \}, \max \{ u(x),v(x) \} ]$$ . These measures $$\nu _a$$ are defined in terms of density functions $$\rho _a(y)$$ 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.