  We give a review of the most important results on optimal tomography as
mathematical wave-pattern recognition theory emerged in the 70's in connection
with the problems of optimal estimation and hypothesis testing in quantum
theory. In quantum theory such problems are sometimes referred as the problem
of optimal measurement of an unknown quantum state, and are the main subject of
the emerging mathematical theory of quantum statistics.
  We develop the results of quantum pattern recognition theory, most of which
belong to VPB, further into the direction of wave, rather than particle
statistical estimation and hypothesis testing theory, with the aim to include
not only quantum matter waves but also classical wave patterns like optical and
acoustic waves. We conclude that Hilbert space and operator methods developed
in quantum theory are equally useful in the classical wave theory, as soon as
the possible observations are restricted to only intensity distributions of
waves, i.e. when the wave states are not the allowed observables, as they are
not the observables of individual particles in the quantum theory. We show that
all characteristic attributes of quantum theory such as complementarity,
entanglement or Heisenberg uncertainty relations are also attributes of the
generalized wave pattern recognition theory.
