On the reliable readout of random discrete-point structures

Abstract

An original approach is proposed to solve rather complex probabilistic problems of the readout of random discrete-point fields; these problems lack an exact analytical solution. Schemes are proposed for the direct, iterative, and combinatorially recursive analytical calculation of multidimensional integral expressions that describe the partial solutions of such problems; these solutions are further used to find general closed analytical dependencies. The huge volume of computations forced us to completely formalize the algorithms and to turn over the entire burden of routine analytical calculations to the computer. This helped us to find (and subsequently to prove) a number of new and previously unknown probability formulas that characterize the reliability of the readout of random discrete-point images when the readout is performed by multilevel integrators. Thus, we managed to carry out (which happens rather rarely in scientific practice) the idea expressed by J. von Neumann: when facing a complex problem that defies solution, the researcher resorts to computer calculations that "prompt" the correct answer to him and then strictly prove this answer. Another important point in our investigations is that we introduce the new concept of "three-dimensional generalized Catalan numbers" and find their explicit form, which is advantageously used for solving problems of registering and analyzing random discrete-point images.