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.
A related problem is to describe the remaining flexibility in a statistical shape model that is partially fixed. If the position of some points, curves or subset of the surface is known, the goal is to characterise the space of shapes that approximately fit these observations. @cite_17 show how to compute the subspace of faces with the same profile. @cite_5 extended this approach into a probabilistic setting.
@cite_5 @cite_17
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Statistical shape models, and in particular morphable models, have gained widespread use in computer vision, computer graphics and medical imaging. Researchers have started to build models of almost any anatomical structure in the human body. While these models provide a useful prior for many image analysis task, relatively little information about the shape represented by the morphable model is exploited. We propose a method for computing and visualizing the remaining flexibility, when a part of the shape is fixed. Our method, which is based on Probabilistic PCA, not only leads to an approach for reconstructing the full shape from partial information, but also allows us to investigate and visualize the uncertainty of a reconstruction. To show the feasibility of our approach we performed experiments on a statistical model of the human face and the femur bone. The visualization of the remaining flexibility allows for greater insight into the statistical properties of the shape. The second edition of this reference provides an update on the best methods for the measurement of the surfaces of the head and neck. The newest techniques are explained and the latest normative data on age-related changes in measurements (i.e., population norms) are completely examined in nearly 20 chapters and six appendices. Topics covered include sources of error in anthropometry and anthroposcopy; age-related changes in selected measurements of the craniofacial complex; anthropometry of minor defects in the craniofacial complex; anthropometry in craniomaxillofacial surgery, genetics, and aesthetic surgery of the nose; and anthropometry of the attractive face. Other chapters discuss the reconstruction of a photograph of a missing child, medical photography in clinical practice, interpolation of growth curves, racial and ethnic morphometric differences in the craniofacial complex, and basic statistical methods in clinical research. Complementing the text throughout are detailed illustrations and highly informative tables.