Non-rigid structure from motion using quadratic deformation models

In this paper we present a new approach to the modelling of non-rigid 3D surfaces from the observation of 2D motion in images captured by an orthographic camera. Our aim is to characterize strong variations of the shape due, for instance, to bending motions. Such motions are hard to describe with previously used deformation models, such as the linear basis shapes model, which would tend to overestimate the dimensionality of the deformable data. Our approach uses a quadratic deformation model which is able to represent non-linear non-rigid motions such as bending, stretching, shearing and twisting. The model is bilinear and thus fits easily into previous schemes for Non-Rigid Structure from Motion (NRSfM). We formulate the NRSfM problem using a non-linear optimization scheme to minimize image reprojection error and recover the camera parameters, the 3D shape at rest and the quadratic deformation transformations. Our experiments with synthetic and real data show examples in which methods based on the linear basis shape model perform poorly or do not converge and instead the quadratic model is able to achieve accurate 3D reconstructions.