There are plenty of distance measures between two histogram. You can read a good categorization of these measures in:
K. Meshgi, and S. Ishii, “Expanding Histogram of Colors with Gridding
to Improve Tracking Accuracy,” in Proc. of MVA’15, Tokyo, Japan, May
2015.
The most popular distance functions are listed here for your convenience:
- $L_0$ or Hellinger Distance
$D_{L0} = \sum\limits_{i} h_1(i) \neq h_2(i) $
- $L_1$, Manhattan, or City Block Distance
$D_{L1} = \sum_{i}\lvert h_1(i) - h_2(i) \rvert $
- $L=2$ or Euclidean Distance
$D_{L2} = \sqrt{\sum_{i}\left( h_1(i) - h_2(i) \right) ^2 }$
- L$_{\infty}$ or Chybyshev Distance
$D_{L\infty} = max_{i}\lvert h_1(i) - h_2(i) \rvert $
- L$_p$ or Fractional Distance (part of Minkowski distance family)
$D_{Lp} = \left(\sum\limits_{i}\lvert h_1(i) - h_2(i) \rvert ^p \right)^{1/p}$ and $0<p<1$
$D_{\cap} = 1 - \frac{\sum_{i} \left(min(h_1(i),h_2(i) \right)}{min\left(\vert h_1(i)\vert,\vert h_2(i) \vert \right)}$
$D_{CO} = 1 - \sum_i h_1(i)h2_(i)$
$D_{CB} = \sum_i \frac{\lvert h_1(i)-h_2(i) \rvert}{min\left( \lvert h_1(i)\rvert,\lvert h_2(i)\rvert \right)}$
- Pearson's Correlation Coefficient
$ D_{CR} = \frac{\sum_i \left(h_1(i)- \frac{1}{n} \right)\left(h_2(i)- \frac{1}{n} \right)}{\sqrt{\sum_i \left(h_1(i)- \frac{1}{n} \right)^2\sum_i \left(h_2(i)- \frac{1}{n} \right)^2}} $
- Kolmogorov-Smirnov Divergance
$ D_{KS} = max_{i}\lvert h_1(i) - h_2(i) \rvert $
$D_{MA} = \sum\limits_{i}\lvert h_1(i) - h_2(i) \rvert $
- Cramer-von Mises Distance
$D_{CM} = \sum\limits_{i}\left( h_1(i) - h_2(i) \right)^2$
$D_{\chi^2} = \sum_i \frac{\left(h_1(i) - h_2(i)\right)^2}{h_1(i) + h_2(i)}$
$ D_{BH} = \sqrt{1-\sum_i \sqrt{h_1(i)h_2(i)}} $ & hellinger
$ D_{SC} = \sum_i\left(\sqrt{h_1(i)}-\sqrt{h_2(i)}\right)^2 $
- Kullback-Liebler Divergance
$D_{KL} = \sum_i h_1(i)log\frac{h_1(i)}{m(i)}$
$D_{JD} = \sum_i \left(h_1(i)log\frac{h_1(i)}{m(i)}+h_2(i)log\frac{h_2(i)}{m(i)}\right)$
- Earth Mover's Distance (this is the first member of Transportation distances that embed binning information $A$ into the distance, for more information please refer to the abovementioned paper or Wikipedia entry.
$ D_{EM} = \frac{min_{f_{ij}}\sum_{i,j}f_{ij}A_{ij}}{sum_{i,j}f_{ij}}$
$ \sum_j f_{ij} \leq h_1(i) , \sum_j f_{ij} \leq h_2(j) , \sum_{i,j} f_{ij} = min\left( \sum_i h_1(i) \sum_j h_2(j) \right) $ and $f_{ij}$ represents the flow from
$i$ to $j$
$D_{QU} = \sqrt{\sum_{i,j} A_{ij}\left(h_1(i) - h_2(j)\right)^2}$
$D_{QC} = \sqrt{\sum_{i,j} A_{ij}\left(\frac{h_1(i) - h_2(i)}{\left(\sum_c A_{ci}\left(h_1(c)+h_2(c)\right)\right)^m}\right)\left(\frac{h_1(j) - h_2(j)}{\left(\sum_c A_{cj}\left(h_1(c)+h_2(c)\right)\right)^m}\right)}$ and $\frac{0}{0} \equiv 0$
A Matlab implementation of some of these distances is available from my GitHub repository:
https://github.com/meshgi/Histogram_of_Color_Advancements/tree/master/distance
Also you can search guys like Yossi Rubner, Ofir Pele, Marco Cuturi and Haibin Ling for more state-of-the-art distances.
Update: Alternative explaination for the distances appears here and there in the literature, so I list them here for sake of completeness.
- Canberra distance (another version)
$D_{CB}=\sum_i \frac{|h_1(i)-h_2(i)|}{|h_1(i)|+|h_2(i)|}$
- Bray-Curtis Dissimilarity, Sorensen Distance (since the sum of histograms are equal to one, it equals to $D_{L0}$)
$D_{BC} = 1 - \frac{2 \sum_i h_1(i) = h_2(i)}{\sum_i h_1(i) + \sum_i h_2(i)}$
- Jaccard Distance (i.e. intersection over union, another version)
$D_{IOU} = 1 - \frac{\sum_i min(h_1(i),h_2(i))}{\sum_i max(h_1(i),h_2(i))}$