  The Paradigms introduced in philosophy of science one century ago are shown
to be quite more satisfactory of that introduced by Galileo. This is
particularly evident in the physics based on Hilbert Spaces and related
mathematical structures that we apply in this paper to Quantum Mechanics and to
Theory of Images. An exhaustive discussion, that include the algebraic analysis
of the operators acting on them, exhibits that the Hilbert Spaces -- that have
fixed dimension -- must be generalized to the Rigged Hilbert Spaces that
contains right inside spaces with continuous and discrete dimensions. This is
the property of Rigged Hilbert Spaces that allows a consistent formal
description of the physics we are considering. Theory of Quantum Mechanics and
of Images are similar and the fundamental difference between them come from the
definition of measure that is outside the theory of the spaces: while in
Quantum Mechanics the measure is a probabilistic action, in Images it is a
classical functional.
  KEYWORDS: Optics, Quantum Mechanics, Rigged Hilbert Spaces, Lie Algebras
