What Does 2D Geometric Information Really Tell Us About 3D Face Shape?

A face image contains geometric cues in the form of configurational information (semantically meaningful landmark points and contours). In this thesis, we explore to what degree such 2D geometric information allows us to estimate 3D face shape. First, we focus on the problem of fitting a 3D morphable model to single face images using only sparse geometric features. We propose a novel approach that explicitly computes hard correspondences which allow us to treat the model edge vertices as known 2D positions, for which optimal pose or shape estimates can be linearly computed. Moreover, we show how to formulate this shape-from-landmarks problem as a separable nonlinear least squares optimisation. Second, we show how a statistical model can be used to spatially transform input data as a module within a convolutional neural network. This is an extension of the original spatial transformer network in that we are able to interpret and normalise 3D pose changes and self-occlusions. We show that the localiser can be trained using only simple geometric loss functions on a relatively small dataset yet is able to perform robust normalisation on highly uncontrolled images. We consider another extension in which the model itself is also learnt. The final contribution of this thesis lies in exploring the limits of 2D geometric features and characterising the resulting ambiguities. 2D geometric information only provides a partial constraint on 3D face shape. In other words, face landmarks or occluding contours are an ambiguous shape cue. Two faces with different 3D shape can give rise to the same 2D geometry, particularly as a result of perspective transformation when camera distance varies. We derive methods to compute these ambiguity subspaces, demonstrate that they contain significant shape variability and show that these ambiguities occur in real-world datasets.