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# Optimization of two images mutual information estimation

Keywords:

G.L. Safina – Ph.D.(Eng.), Associate Professor, Department «Applied Mathematics», National Research Moscow State University of Civil Engineering
E-mail: minkinag@mail.ru
A.G. Tashlinsky – Dr.Sc.(Eng.), Professor, Head of Department «Radio Engineering», Ulyanovsk State Technical University
E-mail: tag@ulstu.ru
M.G. Tsarev – Post-graduate Student, Ulyanovsk State Technical University
E-mail: michael.tsaryov@gmail.com

Shannon's mutual information (MI) is one of the measures of two images similarity. It is used for their mismatch. The MI estimate can be determined from a local sample of image samples. The volume of useful information contained in brightness differences depends on the distance between the samples that are in the local sample. For a small distance, the noise suppresses useful information. If the distance is too long, the useful information is suppressed by the random component of the brightness, that increases as the correlation coefficient between the samples decreases. Thus, there is an optimal distance between the samples, providing with a given sample size a maximum of information on the mismatch of the images. The problem to determine this distance under conditions of additive noisy images is solved.
The solution of the problem is complicated by the fact that it is necessary to use all the differences in the brightness of the samples from the local sample to determine the MI of images. The solution is based on the fact that the maximum information about mismatch of images occurs with the maximum ratio of the mathematical expectation module of the MI gradient to its standard deviation. The analysis of the obtained expression for the gradient of mutual information estimation is analyzed. It is invariant to the form of the image mismatch model. It is shown that the minimum possible size of a local sample is equal to three samples. The relations allowing to determine the optimal distance are obtained for this local sampling size. They are generalized to the case of an arbitrary sampling size.
The studies have shown that increasing the sample size slightly influences on the optimal distance, which, as a rule, is much smaller than the size of the studied images. The images are discrete, this fact limits the local sampling size. On the other hand, increasing the sampling size leads to increasing MI volume. This contradiction can be resolved by using a set of optimized samples in the MI calculation. In this case the samples are chosen in different domains of the image, for example, regularly or randomly. For a given size each sample has a maximum of information about images mismatch. The required information volume can be obtained by specifying the desired number of samples.

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