(Non-)rigid motion interpretation : a regularized approach

Determining 3D motion from a time-varying 2D image is an ill-posed problem; unless we impose additional constraints, an infinite number of solutions is possible. The usual constraint is rigidity, but many naturally occurring motions are not rigid and not even piecewise rigid. A more general assumption is that the parameters (or some of the parameters) characterizing the motion are approximately (but not exactly) constant in any sufficiently small region of the image. If we know the shape of a surface we can uniquely recover the smoothest motion consistent with image data and the known structure of the object, through regularization. This paper develops a general paradigm for the analysis of nonrigid motion. The variational condition we obtain includes many previously studied constraints as ‘special cases’. Among them are isometry, rigidity and planarity. If the variational condition is applied at multiple scales of resolution, it can be applied to turbulent motion. Finally, it is worth noting that our theory does not require the computation of correspondence (optic flow or discrete displacements), and it is effective in the presence of motion discontinuities.