Unsupervised Learning of Object Deformation Models Supplemental Material

Non-trivial deformations result in local contractions and expansions. These naturally capture object scalings but can have a negative side effect, namely making object features disappear or inflate. For this reason we want the deformation fields to have zero acceleration in the direction perpendicular to image features; this guarantees that the features are only ‘transported’. This requirement can be phrased as follows: consider a deformation field h = (hx, hy) = (x + fx, y + fx); fx and fy are the deformation increments calculated from the linear basis synthesis. Along orientation nx, ny this deformation field has ‘speed’ fxnx + fyny; a constant speed means that the motion of features in this orientation does not distort them, i.e. it is purely translating them. We can thus enforce our constraint by requiring that the directional derivative of this speed function equals zero, i.e.