Some ‘‘Weberized’’ L2-Based Methods of Signal/Image Approximation

We examine two approaches of modifying L2-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 L2 distances. We arrive at a natural weighting function that is shown to conform to Weber’s model. The resulting “Weberized L2 distance” involves a ratio of functions. The importance of ratios in such distance functions leads to a consideration of the well-known logarithmic L2 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 Rg ⊂ R according to Weber’s model leads to the unique (unnormalized) measure in Rg with density function ρ(t)=1/t. This result implies that the logarithmic L1 distance is the most natural “Weberized” image metric. From this result, all other logarithmic Lp distances may be viewed as generalizations