ГРАНИЦЫ КРАМЕРА-РАО И БХАТТАЧАРИЯ ДЛЯ ОЦЕНКИ ПОТЕНЦИАЛЬНОЙ ТОЧНОСТИ СУБПИКСЕЛЬНОГО СОВМЕЩЕНИЯ ИЗОБРАЖЕНИЙ

The subject matter of the article is theoretical lower bounds of parameter estimates applied to the problem of image co-registration. The goal is to study and compare the Cramer-Rao and Bhattacharyya bounds. The tasks to be solved are: to formulate algorithms for calculating the Cramer-Rao and Bhattacharyya bounds for estimating the subpixel shifts of two images; using the Monte Carlo methods to compare the calculated bounds with the results of the real registration algorithm. The methods used are computer simulation; Monte Carlo methods. Monte Carlo experiments were used both for calculating theoretical bounds (partial derivatives were estimated using numerical differentiation formulas) and for calculating the accuracy of the real algorithm. The subpixel accuracy of the registration algorithm was achieved by the intensity interpolation method, in this case, the problem of image coordinates determination was considered as an optimization problem solved by a numerical method. The following results were obtained. It is experimentally confirmed that, when calculating the lower bounds of the registration accuracy in the traditional formulation (when we do not take into account the errors of interpolation used to achieve subpixel accuracy), the Bhattacharyya bound always passes above the Cramer-Rao bound, that corresponds to the theory. However, although the Bhattacharyya bound provides more accurate estimates than the Cramer-Rao bound, its values at high signal-to-noise ratios are still too optimistic for registration accuracy in real situations. Both artificially modeled images (based on the fractal Brownian motion model) and fragments of real satellite images were used in computational experiments. Conclusions. The scientific novelty of the results obtained is that this work, in many respects following the research carried out by other authors, in contrast to them considers the maximum likelihood function taking into account the possibility of linear brightness transformation between two images, which is the most frequently used model in practice. However, the possibility of rotating two images in this article is not taken into account

A note on Hammersley's inequality for estimating the normal integer mean

LetX1,X2,…,Xnbe a random sample from a normalN(θ,σ2)distribution with an unknown meanθ=0,±1,±2,…. Hammersley (1950) proposed the maximum likelihood estimator (MLE)d=[X¯n], nearest integer to the sample mean, as an unbiased estimator ofθand extended the Cramér-Rao inequality. The Hammersley lower bound for the variance of any unbiased estimator ofθis significantly improved, and the asymptotic (asn→∞) limit of Fraser-Guttman-Bhattacharyya bounds is also determined. A limiting property of a suitable distance is used to give some plausible explanations why such bounds cannot be attained. An almost uniformly minimum variance unbiased (UMVU) like property ofdis exhibited.