Some “Weberized” $L^2$-Based Methods of Signal/Image Approximation

Abstract

We examine two approaches of modifying $L^2$-based approximations so that they conform to Weber’s model of perception, i.e., higher/lower tolerance of deviation for higher/lower intensity levels. The first approach involves the idea of intensity-weighted $L^2$ distances. We arrive at a natural weighting function that is shown to conform to Weber’s model. The resulting “Weberized $L^2$ distance” involves a ratio of functions. The importance of ratios in such distance functions leads to a consideration of the well-known logarithmic $L^2$ distance which is also shown to conform to Weber’s model.

In fact, we show that the imposition of a condition of perceptual invariance in greyscale space ${\mathbb {R}}_g \subset {\mathbb {R}}$ according to Weber’s model leads to the unique (unnormalized) measure in ${\mathbb {R}}_g$ with density function $\rho (t)=1/t$. This result implies that the logarithmic $L^1$ distance is the most natural “Weberized” image metric. From this result, all other logarithmic $L^p$ distances may be viewed as generalizations.