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 invari-ance in greyscale space R_g is contained in R according to Weber's model leads to the unique (unnormalized) measure in R_g with density function ρ(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.
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