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.
Like us, @cite_30 also explore ambiguities in shape-from-landmarks in the context of objects represented by a linear basis (in their case, nonrigid deformations of an object rather than the space of faces). However, unlike in this paper, they assume that the intrinsic camera parameters are known. Hence, they do not model the perspective ambiguity that we describe (in which a change in distance is compensated by a change in focal length). Different to our flexibility modes, instead of analytically deriving a subspace, they use stochastic sampling to explore the set of possible solutions. They attempt to select from within this space using additional information provided by motion or shading.
@cite_30
2102813107 1968381457 2052094314 2138816964
Existing approaches to nonrigid structure from motion assume that the instantaneous 3D shape of a deforming object is a linear combination of basis shapes. These bases are object dependent and therefore have to be estimated anew for each video sequence. In contrast, we propose a dual approach to describe the evolving 3D structure in trajectory space by a linear combination of basis trajectories. We describe the dual relationship between the two approaches, showing that they both have equal power for representing 3D structure. We further show that the temporal smoothness in 3D trajectories alone can be used for recovering nonrigid structure from a moving camera. The principal advantage of expressing deforming 3D structure in trajectory space is that we can define an object independent basis. This results in a significant reduction in unknowns and corresponding stability in estimation. We propose the use of the Discrete Cosine Transform (DCT) as the object independent basis and empirically demonstrate that it approaches Principal Component Analysis (PCA) for natural motions. We report the performance of the proposed method, quantitatively using motion capture data, and qualitatively on several video sequences exhibiting nonrigid motions, including piecewise rigid motion, partially nonrigid motion (such as a facial expressions), and highly nonrigid motion (such as a person walking or dancing). Tracking across cameras with non-overlapping views is a challenging problem. Firstly, the observations of an object are often widely separated in time and space when viewed from non-overlapping cameras. Secondly, the appearance of an object in one camera view might be very different from its appearance in another camera view due to the differences in illumination, pose and camera properties. To deal with the first problem, we observe that people or vehicles tend to follow the same paths in most cases, i.e., roads, walkways, corridors etc. The proposed algorithm uses this conformity in the traversed paths to establish correspondence. The algorithm learns this conformity and hence the inter-camera relationships in the form of multivariate probability density of space-time variables (entry and exit locations, velocities, and transition times) using kernel density estimation. To handle the appearance change of an object as it moves from one camera to another, we show that all brightness transfer functions from a given camera to another camera lie in a low dimensional subspace. This subspace is learned by using probabilistic principal component analysis and used for appearance matching. The proposed approach does not require explicit inter-camera calibration, rather the system learns the camera topology and subspace of inter-camera brightness transfer functions during a training phase. Once the training is complete, correspondences are assigned using the maximum likelihood (ML) estimation framework using both location and appearance cues. Experiments with real world videos are reported which validate the proposed approach. If a physical object has a smooth or piecewise smooth boundary, its images obtained by cameras in varying positions undergo smooth apparent deformations. These deformations are locally well approximated by affine transforms of the image plane. In consequence the solid object recognition problem has often been led back to the computation of affine invariant image local features. Such invariant features could be obtained by normalization methods, but no fully affine normalization method exists for the time being. Even scale invariance is dealt with rigorously only by the scale-invariant feature transform (SIFT) method. By simulating zooms out and normalizing translation and rotation, SIFT is invariant to four out of the six parameters of an affine transform. The method proposed in this paper, affine-SIFT (ASIFT), simulates all image views obtainable by varying the two camera axis orientation parameters, namely, the latitude and the longitude angles, left over by the SIFT method. Then it covers the other four parameters by using the SIFT method itself. The resulting method will be mathematically proved to be fully affine invariant. Against any prognosis, simulating all views depending on the two camera orientation parameters is feasible with no dramatic computational load. A two-resolution scheme further reduces the ASIFT complexity to about twice that of SIFT. A new notion, the transition tilt, measuring the amount of distortion from one view to another, is introduced. While an absolute tilt from a frontal to a slanted view exceeding 6 is rare, much higher transition tilts are common when two slanted views of an object are compared (see Figure hightransitiontiltsillustration). The attainable transition tilt is measured for each affine image comparison method. The new method permits one to reliably identify features that have undergone transition tilts of large magnitude, up to 36 and higher. This fact is substantiated by many experiments which show that ASIFT significantly outperforms the state-of-the-art methods SIFT, maximally stable extremal region (MSER), Harris-affine, and Hessian-affine. Existing approaches to nonrigid structure from motion assume that the instantaneous 3D shape of a deforming object is a linear combination of basis shapes, which have to be estimated anew for each video sequence. In contrast, we propose that the evolving 3D structure be described by a linear combination of basis trajectories. The principal advantage of this approach is that we do not need to estimate any basis vectors during computation. We show that generic bases over trajectories, such as the Discrete Cosine Transform (DCT) basis, can be used to compactly describe most real motions. This results in a significant reduction in unknowns, and corresponding stability in estimation. We report empirical performance, quantitatively using motion capture data, and qualitatively on several video sequences exhibiting nonrigid motions including piece-wise rigid motion, partially nonrigid motion (such as a facial expression), and highly nonrigid motion (such as a person dancing).