A.V. Sorokin, E.N. Khokhlov

Abundant design expertise and wide field condition trials experience of the tertiary information processing (TIP) algorithms indicate the algorithms to be efficiency in tracking of observable objects (OO) on basis of the data from differently accurate and differently discrete sources. This efficiency corresponds with a certain environment, which involves most of all complexity of OO motion paths. Under complication of an environment the realized algorithms cannot guarantee a sufficient quality of OO tracking without their considerable enhancement. In the paper the track-to-track association (TTA) task decision to use the mathematical tool of complex numbers and quaternions and many-dimensional vectors is proposed. The appropriateness of describing procedure data TTA in terms of space, which is differ from the real three-dimensional space, is substantiated. The representation current tracking information about OO by the way of a sheaf of vectors (SV), interpreted as many-dimensional vectors, is posed. The introduction of a generalized similarity measure allows withdrawing more information about positional relationship of the OO paths in that case than using known TIP algorithms. A problem of SV track-to-track association in a two- and a three-space is set. Balancing a maximal dimension of a SV and a pole of it by means of an analytical estimation of effect this choice on SV discriminating properties under occurrence of telemetry errors is carried out. To place a pole of SV in its centroid is the most effective. Vectors on a plane are representable as complex numbers, and vectors in a three-space are representable as quaternions. A change from real to complex numbers and quaternions yield more informative SV similarity measure. Therefore to specify three-space vectors as vector quaternions is proposed on basis of the known mathematical tool of quaternion algebra. Innerproduct of quaternions are used as a primitive operation to utilize in computation of a similarity measure. A few variants of similarity measure constructed because of innerproduct of quaternions are offered, their analytical treatment is carried out. The appropriateness of specification of any OO individualizing additional information of any physical meaning as a real component of quaternion is justified. The way of three-space SV specification as a many-dimensional quaternion signal is described. An operation of innerproduct of such signals is obtained, basic properties of the operation are selected, and based on it two signals similarity measure in the form of a distance is noted. Estimation of potential attainable efficiency of the TTA task decision by using minimum-distance criterion is conducted. A TTA potential efficiency attainment condition for a SV is developed. The main result of carried out research concludes in the application of the quaternion signals mathematical tool for TTA task decision to enable introducing a generalized similarity measure, that use tracks prehistory information, and is not concerned with prior information about statistical performance of telemetry errors, and is not founded on a definite model of OO motion. In order to obtain typical characterization for proposed approach it is necessary to make computing experiments, which allow to compare the acquired results with characters of realized algorithms and to estimate degree of TTA effectiveness increase.