The use of optical flow for the analysis of non-rigid motions

This paper analyses the 2D motion field on the image plane produced by the 3D motion of a plane undergoing simple deformations. When the deformation can be represented by a planar linear vector field, the projected vector field, i.e., the 2D motion field of the deformation, is at most quadratic. This 2D motion field has one singular point, with eigenvalues identical to those of the singular point describing the deformation. As a consequence, the nature of the singular point of the deformation is a projective invariant. When the plane moves and experiences a linear deformation at the same time, the associated 2D motion field is at most quadratic with at most 3 singular points. In the case of a normal rototranslation, i.e., when the angular velocity is normal to the plane, and of a linear deformation, the 2D motion field has one singular point and substantial information on the rigid motion and on the deformation can be recovered from it. Experiments with image sequences of planes moving and undergoing linear deformations show that the proposed analysis can provide accurate results. In addition, experiments with deformable objects, such as water, oil, textiles and rubber show that the proposed approach can provide information on more general 3D deformations.