Section: Scientific Foundations
Framework of reconstructible systems and motion analysis
The main strength of the MMF consists in its ability to compute accurately the value of a singularity exponent around any point x in the domain of a complex signal s. The following , which is one of the functionals used in implementing the MMF (precisely, in this case, a measure in the signal domain) is defined through the density of a generalized gradient:
and whose singularity exponents derived from the behaviour described in equation (4 ) can be computed by appropriate wavelet projections ( : ball of radius r centered at point x). A multiscale hierarchy directly related to information content (and, in the case of turbulence, to cascading properties ) is defined from the distribution of the singularity exponents . That distribution being bounded from below for physical signals, a specific geometric super-structure, the Most Singular Manifold is the geometrical set associated to the lowest value :
In the framework of reconstructible systems (  ,  ,  ,  ) the set is shown to correspond to the statistically most informative part in the signal, and, consequently, an operator can be defined to recover the whole signal from its restriction to the Most Singular Manifold :
and the operator can be completely specified (usually in Fourier space) from physical considerations about processes  ,  ,  . The framework of reconstructible systems opens the way to a whole area of research, for instance for the problem of motion analysis in oceanographic acquisition datasets  ,  ,  . In contrast to conservation methods in Image Processing (optical flow, correlation methods etc.) the reconstruction formula permits the determination of the dynamics from one single acquisition in a temporal image sequence. See figure 3 .