ChiSquareDistance

Introduction

The Chi-Square Distance is a statistical measure often used to compare two histograms or distributions. It quantifies the difference between two distributions by calculating the normalized squared difference between corresponding bin values.

Sense of the Distance

The Chi-Square Distance computes how two histograms differ by comparing the bin values. This distance is sensitive to both the magnitude of the difference between the bins and their overall sizes, making it useful in many contexts such as image processing and histogram comparison.

Formal Definition

The Chi-Square Distance between two histograms ( H_1 ) and ( H_2 ) is given by:

\[\chi^2(H_1, H_2) = \sum_{i=1}^{n} \frac{(H_1[i] - H_2[i])^2}{H_1[i] + H_2[i]}\]

Where: - ( H_1[i] ) and ( H_2[i] ) represent the values in the (i)-th bin of histograms ( H_1 ) and ( H_2 ), - ( n ) is the number of bins in the histograms.

Academic Reference

Pele and Werman[1]:

[1]

Ofir Pele and Michael Werman. A linear time histogram metric for improved sift matching. In David Forsyth, Philip Torr, and Andrew Zisserman, editors, Computer Vision – ECCV 2008, 495–508. Berlin, Heidelberg, 2008. Springer Berlin Heidelberg.

Conclusion

The Chi-Square Distance provides an effective method for comparing distributions, particularly histograms of pixel intensities or color distributions. Its sensitivity to the relative size of bin values makes it well-suited for tasks involving statistical comparison.