Nicholas Dowson at the University of Surrey
Mutual Information
Introduction
Mutual Information was originally devised by Shannon, as a method to measure the information shared between two signals with discretised amplitudes over a period of time. It is a simple extension to consider discretised intensities over a 2D space.
Mutual information is measured by obtaining a joint histogram of intensities of the overlapping regions between a template image and a reference image, before applying the following mathematical operation:
I = \sum_{r,t} p_{rt} log( \frac{p_{rt}}{p_r p_t} )
Objective
To use mutual information to track small image patches.
Difficulties
The problem with small image patches is that the are often too small to obtain a fully populated histogram, the estimate of MI information is often incorrect.
As the position of the template relative to the reference image changes, the various Mutual Information values describe a surface. To obtain the position where the Mutual Information is highest it is important that the surface is smooth. In general this is not the case.
Many methods exist for measuring Mutual Information. Some of these cause the position of the maximum to shift, making the final solution inaccurate. How is this prevented, and which method should be used?
Most histogramming methods also ignore the structure of the image. This throws away important information, inherent to the data. Using the structure would give more accurate results.
Our contribution
We have placed all the methods of Mutual Information into a single mathematical framework and critically compared them. For more information read this.
We have developed a method, called NP windowing (Not Parzen windowing) to use the structure inherent to images, which is equivalent to sampling the images at infinite resolution. Comparison in terms of convergence (how often is the position of maximum Mutual Information found) and bias (how accurate is the solution relative to ground truth), shows that NP windowing outperforms all existing methods. For more information read this.