What Is a Weighted Local Histogram?
A weighted local histogram would mean filtering the image $I(x,y)$ with a localised filter $H(x,y)$ (gaussian in this example). The resultant image is the 2D convolution $Y(x,y) = I(x,y)*H(x,y)$. The normal histogram of $Y(x,y)$ is the localised histogram of $I(x,y)$. It is used to view certain specific characteristics of an image depending on the filter used