Color histogram

It can readily be seen that color histogram measures are invariant to both $O(2, \Re)$ and $T(2)$, and hence, $E(2)$, because histogram measures are only dependent on summations of identical pixel values and do not incorporate orientation and position. The extraction of the normalized histogram ${\bf X_{\cal H}} \in \Re^{512}$ is used as a representation of an isotropic mapping.

A color space is perceptually uniform if a small perturbation to a component value is approximately equally perceptible across the range of that value. The $RGB$ color space does not exhibit perceptual uniformity. However, the CIE $LAB$ space [19], conceived in 1976, improves the perceptual uniformity of $RGB$ space considerably. $LAB$ color space is an approximately uniform color space that maps equally distinct color differences into approximately equal Euclidean distances in space. In this space, $L$ defines lightness, $A$ denotes red/green chrominance and $B$ the yellow/blue chrominance. Presently, it is one of the most popular color spaces for color measurement.

Given an image ${I}_{RGB}(x,y)$ in $RGB$ space we generate $I_{LAB}(x,y)$, where the pair $(x,y)$ denotes the coordinates in an image $I$. A 512-dimensional feature vector ${\bf X}_{\cal H}$, representing the 512-bin normalized histogram, is extracted from the image $I_{Lab}(x,y)$ by uniformly quantizing the $LAB$ space, i.e,

$${\bf X}_{\cal H} = (\tilde{\bf x}_{{\cal H}_0},\cdots,\tilde{\bf x}_{{\cal H}_{511}})^t$$ (11)

where $\tilde{\bf x}_{{\cal H}_j}$ (where the index integer $j \in [0,511]$) represents the normalized value of the $j^{th}$ bin of the histogram such that $\sum_{j=0}^{511} \tilde{\bf x}_{{\cal H}_j} = 1$. This feature space represents a unit hypercube.