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.
Other ambiguities There are other known ambiguities in the monocular estimation of 3D shape. The bas relief ambiguity arises in photometric stereo with unknown light source directions. A continuous class of surfaces (differing by a linear transformation) can produce the same set of images when an appropriate transformation is applied to the illumination and albedo. For the particular case of faces, @cite_4 resolve this ambiguity by exploiting the symmetries and similarities in faces. Specifically they assume: bilateral symmetry; that the forehead and chin should be at approximately the same depth; and that the range of facial depths is about twice the distance between the eyes.
@cite_4
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To deal with image variations due to illumination problem, recently Ramamoorthi and Basri have independently derived a spherical harmonic analysis for the Lambertian reflectance and linear subspace. Their theoretical work provided a new approach for face representation, however both of them had the assumption that the 3D surface normal and albedo are known. This assumption limits this algorithm's application. In this paper, we present a novel method for modeling 3D face shape and albedo from only three images with unknown light directions and this work well fills the blank, which Ramamoorthi and Basri left. By taking the advantage of similar 3D shape of all human faces, the highlight of the new method is that it circumambulates the linear ambiguity by 3D alignment. The experiment results show that our estimated model can be perfectly employed to face recognition and 3D reconstruction. Photometric stereo with uncalibrated lights determines surface orientations ambiguously up to any regular transformation. If the surface reflectance model is separable with respect to the illumination and viewing directions, its inherent symmetries enable to design two previously unrecognized constraints on normals that reduce this ambiguity. The two constraints represent projections of normals onto planes perpendicular to the viewing and illumination directions, respectively. We identify the classes of transformations that leave each constraint invariant. We construct the constraints using polarization measurement under the assumption of separable reflectance model for smooth dielectrics. We verify that applying the first constraint together with the integrability constraint results in bas-relief ambiguity, while application of the second constraint on integrable normals reduces the ambiguity to convex concave ambiguity. Importantly, the latter result is also obtained when the first and second constraints alone are combined. We present a novel process for acquiring detailed facial geometry with high resolution diffuse and specular photometric information from multiple viewpoints using polarized spherical gradient illumination. Key to our method is a new pair of linearly polarized lighting patterns which enables multiview diffuse-specular separation under a given spherical illumination condition from just two photographs. The patterns -- one following lines of latitude and one following lines of longitude -- allow the use of fixed linear polarizers in front of the cameras, enabling more efficient acquisition of diffuse and specular albedo and normal maps from multiple viewpoints. In a second step, we employ these albedo and normal maps as input to a novel multi-resolution adaptive domain message passing stereo reconstruction algorithm to create high resolution facial geometry. To do this, we formulate the stereo reconstruction from multiple cameras in a commonly parameterized domain for multiview reconstruction. We show competitive results consisting of high-resolution facial geometry with relightable reflectance maps using five DSLR cameras. Our technique scales well for multiview acquisition without requiring specialized camera systems for sensing multiple polarization states. We present a novel model based stereo system, which accurately extracts the 3D shape and pose of faces from multiple images taken simultaneously. Extracting the 3D shape from images is important in areas such as pose-invariant face recognition and image manipulation. The method is based on a 3D morphable face model learned from a database of facial scans. The use of a strong face prior allows us to extract high precision surfaces from stereo data of faces, where traditional correlation based stereo methods fail because of the mostly textureless input images. The method uses two or more uncalibrated images of arbitrary baseline, estimating calibration and shape simultaneously. Results using two and three input images are presented. We replace the lighting and albedo estimation of a monocular method with the use of stereo information, making the system more accurate and robust. We evaluate the method using ground truth data and the standard PIE image dataset. A comparison with the state of the art monocular system shows that the new method has a significantly higher accuracy.