  A face image contains geometric cues in the form of configurational
information and contours that can be used to estimate 3D face shape. While it
is clear that 3D reconstruction from 2D points is highly ambiguous if no
further constraints are enforced, one might expect that the face-space
constraint solves this problem. We show that this is not the case and that
geometric information is an ambiguous cue. There are two sources for this
ambiguity. The first is that, within the space of 3D face shapes, there are
flexibility modes that remain when some parts of the face are fixed. The second
occurs only under perspective projection and is a result of perspective
transformation as camera distance varies. Two different faces, when viewed at
different distances, can give rise to the same 2D geometry. To demonstrate
these ambiguities, we develop new algorithms for fitting a 3D morphable model
to 2D landmarks or contours under either orthographic or perspective projection
and show how to compute flexibility modes for both cases. We show that both
fitting problems can be posed as a separable nonlinear least squares problem
and solved efficiently. We demonstrate both quantitatively and qualitatively
that the ambiguity is present in reconstructions from geometric information
alone but also in reconstructions from a state-of-the-art CNN-based method.
