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A face image contains geometric cues in the form of configurationalinformation and contours that can be used to estimate 3D face shape. While itis clear that 3D reconstruction from 2D points is highly ambiguous if nofurther constraints are enforced, one might expect that the face-spaceconstraint solves this problem. We show that this is not the case and thatgeometric information is an ambiguous cue. There are two sources for thisambiguity. The first is that, within the space of 3D face shapes, there areflexibility modes that remain when some parts of the face are fixed. The secondoccurs only under perspective projection and is a result of perspectivetransformation as camera distance varies. Two different faces, when viewed atdifferent distances, can give rise to the same 2D geometry. To demonstratethese ambiguities, we develop new algorithms for fitting a 3D morphable modelto 2D landmarks or contours under either orthographic or perspective projectionand show how to compute flexibility modes for both cases. We show that bothfitting problems can be posed as a separable nonlinear least squares problemand solved efficiently. We demonstrate both quantitatively and qualitativelythat the ambiguity is present in reconstructions from geometric informationalone but also in reconstructions from a state-of-the-art CNN-based method.

What does 2D geometric information really tell us about 3D face shape?