Planar shape interpolation using relative velocity fields

Abstract

A novel approach is presented for interpolating two planar shapes using a shape sequence such that one shape is deformed into the other in an as-isometric-as-possible (AIAP) manner. The two shapes can have arbitrarily connected edge soup as long as they have the same connectivity. The interpolation is described by a nonlinear optimization problem that models the deformation energy, which penalizes the non-isometric component of the shape motion, based on the relation between two orthogonal vectors fields, namely, the edge vector field and the relative velocity field. Noticing that the nonlinear optimization includes two types of unknowns and admits a quadratic form with respect to each type, we address the optimization by iteratively solving two linear optimization procedures. One procedure admits a closed form, and the other procedure is associated with a quadratic energy that measures the deviation of the shape sequence from an AIAP motion. To speed up the processing, a local algorithm is devised to reduce the dimensionality of the linear optimization. This algorithm first addresses the AIAP interpolation of individual edges, and then reconstructs the shape sequence, frame by frame, using the generated edge vectors. Furthermore, an efficient initialization strategy is explored to greatly alleviate face-overlapping artifacts that are caused by using linear interpolation to make an initial guess for the shape sequence. Finally, relative velocity fields are employed to explore applications of the AIAP interpolation to shape transfer and sequence editing, in which the relative velocity fields of the given shape sequence are warped and copied onto the shape to be manipulated.