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.
More generally, ambiguities in surface reconstruction have been considered in a number of settings. @cite_6 consider the problem of reconstructing a smooth surface from local information that contains a discrete ambiguity. The ambiguities studied here are in the local surface orientation or gradient, a problem that occurs in photometric shape reconstruction. @cite_31 study the ambiguities that arise in monocular nonrigid structure from motion under perspective projection.
@cite_31 @cite_6
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We consider the problem of reconstructing a smooth surface under constraints that have discrete ambiguities. These problems arise in areas such as shape from texture, shape from shading, photometric stereo and shape from defocus. While the problem is computationally hard, heuristics based on semidefinite programming may reveal the shape of the surface. In this paper, we propose a non-local structured prior for volumetric multi-view 3D reconstruction. Towards this goal, we present a novel Markov random field model based on ray potentials in which assumptions about large 3D surface patches such as planarity or Manhattan world constraints can be efficiently encoded as probabilistic priors. We further derive an inference algorithm that reasons jointly about voxels, pixels and image segments, and estimates marginal distributions of appearance, occupancy, depth, normals and planarity. Key to tractable inference is a novel hybrid representation that spans both voxel and pixel space and that integrates non-local information from 2D image segmentations in a principled way. We compare our non-local prior to commonly employed local smoothness assumptions and a variety of state-of-the-art volumetric reconstruction baselines on challenging outdoor scenes with textureless and reflective surfaces. Our experiments indicate that regularizing over larger distances has the potential to resolve ambiguities where local regularizers fail. We study from a theoretical standpoint the ambiguities that occur when tracking a generic deformable surface under monocular perspective projection given 3D to 2D correspondences. We show that, additionally to the known scale ambiguity, a set of potential ambiguities can be clearly identified. From this, we deduce a minimal set of constraints required to disambiguate the problem and incorporate them into a working algorithm that runs on real noisy data. Resolving local ambiguities is an important issue for shape from shading (SFS). Pixel ambiguities of SFS can be eliminated by propagation approaches. However, patch ambiguities still exist. Therefore, we formulate the global disambiguation problem to resolve these ambiguities. Intuitively, it can be interpreted as flipping patches and adjusting heights such that the result surface has no kinks. The problem is intractable because exponentially many possible configurations need to be checked. Alternatively, we solve the integrability testing problems closely related to the original one. It can be viewed as finding a surface which satisfies the global integrability constraint. To encode the constraints, we introduce a graph formulation called configuration graph. Searching the solution on this graph can be reduced to a Max-cut problem and its solution is computuble using semidefinite programming (SDP) relaxation. Tests carried out on synthetic and real images show that the global disambiguation works well for complex shapes.