The use of intensity-dependent weight functions to “Weberize” $$L^2$$ L 2 -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 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.