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 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 Weberize the \(L^2\) metric by inserting an intensity-dependent weight function into its integral. The weight function will depend on the exponent a so that Weber’s model is accommodated for all \(a> 0\). We also define the “best Weberized approximation” of a function 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” Weber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感知到的\（\ varDelta I \）是\（\ varDelta I / I ^ a = C \），其中\（a> 0 \）和\（C> 0 \）是常数。两个图像函数u和v之间的“ Weberized距离”应该以与上述公式一致的方式，在其假定较高（较低）强度值的区域上容许更大（较小）的差异。在本文中，我们对\（L ^ 2 \）进行Weberize通过将强度相关的权重函数插入其积分中来确定度量值。权重函数将取决于指数a，以便对所有\（a> 0 \）都适应Weber模型。我们还定义了函数的“最佳韦伯化近似”，并证明了这种近似的存在性和唯一性。