The Use of Intensity-Based Measures to Produce Image Function Metrics Which Accommodate Weber’s Models of Perception

Abstract

We are concerned with the construction of “Weberized” metrics for image functions – distance functions which allow greater deviations at higher intensity values and lower deviations at lower intensity values in accordance with Weber’s models of perception. In this paper, we show how the use of appropriate nonuniform measures over the image function range space can be used to produce “Weberized” metrics. In the case of Weber’s standard model, the resulting metric is an $L^1$ distance between logarithms of the image functions. For generalized Weber’s law, the metrics are $L^1$ distances between appropriate powers of the image functions. We then define the corresponding $L^2$ analogues of these metrics which are easier to work with because of their differentiability properties. Finally, we extend the definition of these “Weberized” metrics to vector-valued functions.

Appendix: Sketch of Proof of Theorem 2

We shall consider the “reverse” problem: Given the density function $\rho (y)=1/y^a$, find the leading-order behaviour of f(x) such that

$$\begin{aligned} F(x) = \int _x^{x + Cf(x)} \frac{1}{y^a} \, dy = K , ~~ \text {constant} . \end{aligned}$$

(21)

Differentiation with respect to x and a little manipulation yields the following differential equation (DE) for f(x),

$$\begin{aligned} 1 + Cf'(x) = \left[ 1 + \frac{Cf(x)}{x} \right] ^a . \end{aligned}$$

(22)

In the case $a=1$, we easily find that $f(x)=x = x^1$. In the case $a=0$, $f'(x)=0$ which implies that $f(x)=x^0$. For $0< a < 1$, we shall assume that for x sufficiently large, the RHS of Eq. (22) may be approximated using the binomial theorem, which yields the following DE for f(x),

$$\begin{aligned} f'(x) = a \, \frac{f(x)}{x} . \end{aligned}$$

(23)

The solution of this DE is, up to a constant, $f(x) = x^a$.