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 (HVS) 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 modify the usual integral formulas used to define \(L^2\) distances between functions. The pointwise differences \(|u(x)-v(x)|\) which comprise the \(L^2\) (or \(L^p\)) integrands are replaced with measures of the appropriate greyscale intervals \(\nu _a ( \min \{ u(x),v(x) \}, \max \{ u(x),v(x) \} ]\). These measures \(\nu _a\) are defined in terms of density functions \(\rho _a(y)\) which decrease at rates that conform with Weber’s model of perception. The existence of such measures is proved in the paper. We also define the “best Weberized approximation” of a function in terms of these metrics and also prove the existence and uniqueness of such an approximation.

我们考虑修改基于\（L ^ 2 \）的近似值，以便它们以更好的方式“符合”韦伯感知模型的问题：给定灰度背景强度\（I> 0 \），强度的最小变化人类视觉系统（HVS ）感知到的\（\ varDelta I \）是\（\ varDelta I / I ^ a = C \），其中\（a> 0 \）和\（C> 0 \）是常量。两个图像函数u和v之间的“韦伯化距离”应以与上述公式一致的方式，在其假定较高（较低）强度值的区域上，承受更大（较小）的差异。在本文中，我们修改了用于定义的常用积分公式函数之间的\（L ^ 2 \）距离。组成\（L ^ 2 \）（或\（L ^ p \））被积的逐点差异\（| u（x）-v（x）| \）被适当灰度级间隔\（ \ nu _a（\ min \ {u（x），v（x）\}，\ max \ {u（x），v（x）\}] \）这些度量\（\ nu _a \）被定义在密度函数\（\ rho _a（y）\）方面以符合韦伯感知模型的速率递减，本文证明了此类度量的存在。我们还定义了函数的“最佳韦伯化近似”就这些度量而言，也证明了这种近似的存在性和唯一性。