Histogram-similarity criteria

We calculate the histogram $H_{O'}$ from the sensed subsample of features analogously to Section 3. The first criterion for comparison with a database histogram $H_O$ is the intersection

$$\bigcap(H_O,H_{O'}) = \sum \limits_{i=1}^d \min(H_{O}(i), H_{O'}(i))\ ,$$ (11)

often used with fuzzy-set techniques and previously applied to color-histogram classification [10]. It is very fast to compute, because, apart from summation, no arithmetic operations are needed. Another straightforward criterion is the squared Euclidian distance

$${\cal{E}}(H_O,H_{O'}) = \sum \limits_{i=1}^d (H_{O}(i) - H_{O'}(i))^2\ ,$$ (12)

which is known to be sensitive to noise and does not generalize very well. Next, the statistical $\chi^2$-test is examined in its two forms

$$\chi_1^2(H_O,H_{O'}) = \sum \limits_{i=1}^d \frac{(H_{O}(i) - H_{O'}(i))^2} {H_{O}(i)}$$ (13)

and

$$\chi_2^2(H_O,H_{O'}) = \sum \limits_{i=1}^d \frac{(H_{O}(i) - H_{O'}(i))^2} {H_{O}(i) + H_{O'}(i)}\ .$$ (14)

Finally, we test the symmetric form of the Kullback-Leibler divergence

$${\cal{K}}(H_O,H_{O'}) = \sum \limits_{i=1}^d (H_{O'}(i) - H_{O}(i))\ln \frac{H_{O'}(i)}{H_{O}(i)}\ .$$ (15)

Because of the logarithmic operation, it is the computationally most expensive of all six criteria.