The main features of this chapter are original methods developed that help in solving complex fundamental problems arising in the study of random discrete fields and digital images. These methods are based on not numerical calculations but on analytical computer transformations.
The second distinctive feature of the methods described in this chapter is that with their help it is possible to implement the idea in pure form expressed by John von Neumann: the researcher faces a problem that he can not solve, so he uses a computer to do the labor-intensive calculations, which can lead him to “right” answer, and if he’s lucky (with the computer-obtained solution), he finds a rigorous mathematical proof.
Another feature of this chapter is that mathematical proof of the calculated probability formulas and dependencies forced us to determine the explicit analytical form of the generalized Catalan numbers (that naturally extend the classical Catalan sequence known since Leonhard Euler’s time and arising in many applications of probability theory and mathematical statistics).
The developed algorithms were successfully applied to solving problems related to digital error-free discrete-point image recording, the construction of asymptotically optimal decorrelating transformations of signals with different smoothness, and the development of time-optimal localization algorithms for pulsed objects (dots) generating extremely short delta pulses in random time.
