A.G. Tashlinskii - Dr. Sc. (Eng.), Professor, Head of Department «Radio Engineering», Ulyanovsk State Technical University E-mail: tag@ulstu.ru I.V. Voronov - Post-graduate Student, Department «Radio Engineering», Ulyanovsk State Technical University E-mail: i.voronov@ulstu.ru

Image registration is one of the typical problem in digital image processing. Recurren stochastic gradient descent-based registration procedure is one of the approaches to finding large images registration parameters wich combins small computational cost, sustainability of results in conditions of a priori uncertainty and the lack of a preliminary assessment of image parameters. The asymptotic properties of the estimates are well-studied, in particular the asymptotic rate of convergence and the conditions for the asymptotic normality. However, the capabilities of probabilistic analysis of the properties of estimates for a finite number of iterations are poorly studied, because of the large number of factors, the influence of which can not be neglected. It is the nature of probability density distribution of the correlation function, the image and of the interfering noise, the form of the objective function, gradient estimation function and learning rate matrix, the number of iterations and the initial approximation of the parameters. In the modelling process of estimation process these factors can be divided into two groups. The first group includes factors that do not depend on the kind of gradient estimation procedures, and the second - factors which characteristics can be changed by the procedure implementation. There is a method of probabilistic modeling of stochastic gradient-based estimation process based on an analysis of the probabilities of estimate change (probability with which an estimate will change in true or false directions). When finding the probability of estimate change the key step is estimation of the probability distribution function of the objective function gradient estimates which is usually assumed Gaussian. However, studies show that the probability density function of gradient estimates are close to Gaussian with the size of the local sample on each iteration more than 10−15. When the sample size is low it differs significantly from a Gaussian which results in significant errors in the calculation of the probability and reduce the model adequacy. In order to increase the accuracy of modeling an approximation of the distribution of the objective function gradient estimates via the family of Pearson distributions is proposed. To determine the distribution parameters the expressions for the third and fourth central moments of the gradient estimates are found under the assumption that the images have a Gaussian distribution of intensities with zero mean and image model is an additive mixture of the useful signal and noise. The analytical and experimental results show that the use of Pearson distributions can significantly increase the accuracy of modelling results in comparison with the use of Gaussian distribution. Simulated images formed by using the wave model are used in the experiments.

 

1. Cypkin JA.Z. Informacionnaja teorija identifikacii. M.: Nauka. Fizmatlit. 1995. 336 s.
2. Tashlinskii A.G. Computational expenditure reduction in pseudo-gradient image parameter estimation // Lecture Notes in Computer Science. 2003. V. 2658. P. 456−462.
3. Tashlinskijj A.G. Psevdogradientnoe ocenivanie prostranstvennykh deformacijj posledovatelnosti izobrazhenijj // Naukoemkie tekhnologii. 2002. T. 3. № 3. S. 32−43.
4. Sacks J. Asymptotic distribution of stochastic approximation // The Annals of Mathematical Statistics. 1958. V. 29. № 2. P. 373−405.
5. Repin V.G., Tarkovskijj G.P. Statisticheskijj analiz pri apriornojj neopredelennosti i adaptacija informacionnykh sistem. M.: Sov. radio. 1977. 432 s.
6. Tashlinskijj A.G., Tikhonov V.O. Metodika analiza pogreshnosti psevdogradientnogo izmerenija parametrov mnogomernykh processov // Izvestija VUZov. Ser. Radioehlektronika. 2001. T. 44. № 9. S. 75−80.
7. Tashlinskijj A.G., Minkina G.L., Sinicyn V.I. Metodika analiza tochnosti psevdogradientnogo ocenivanija geometricheskikh deformacijj posledovatelnosti izobrazhenijj // Naukoemkie tekhnologii. 2007. T. 8. № 9. S. 14−23.
8. Parzen E. On Estimation of a Probability Density Function and Mode // Annals of Math. Statistics. 1962. V. 33. P. 1065−1076.
9. Krasheninnikov V.R. Osnovy teorii obrabotki izobrazhenijj: Uchebnoe posobie. Uljanovsk: UlGTU. 2003. 152 s.
10. Tashlinskijj A.G., Voronov I.V. Verojatnost snosa ocenok parametrov mezhkadrovykh geometricheskikh deformacijj izobrazhenijj pri psevdogradientnom izmerenii // Izvestija Samarskogo nauchnogo centra Rossijjskojj akademii nauk. 2014. T. 16. № 6(2). S. 612−615.
11. Pearson K. Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous // Philosophical Transactions of the Royal Society of London. 1895. V. 186. P. 343− 414.
12. Karpov I.G., Gribkov A.N. Modernizacija raspredelenijj Pirsona dlja approksimacii dvukhstoronnikh zakonov raspredelenija ehksperimentalnykh dannykh // Izvestija Tomskogo politekhnicheskogo universiteta. 2014. T. 324. № 2. S. 5−10.