Three-dimensional shape reconstruction of 2D landmark points on a single image is a hallmark of human vision, but is a task that has been proven difficult for computer vision algorithms. We define a feed-forward deep neural network algorithm that can reconstruct 3D shapes from 2D landmark points almost perfectly (i.e., with extremely small reconstruction errors), even when these 2D landmarks are from a single image. Our experimental results show an improvement of up to two-fold over state-of-the-art computer vision algorithms; 3D shape reconstruction of human faces is given at a reconstruction error < .004, cars at .0022, human bodies at .022, and highly-deformable flags at an error of .0004. Our algorithm was also a top performer at the 2016 3D Face Alignment in the Wild Challenge competition (done in conjunction with the European Conference on Computer Vision, ECCV) that required the reconstruction of 3D face shape from a single image. The derived algorithm can be trained in a couple hours and testing runs at more than 1, 000 frames s on an i7 desktop. We also present an innovative data augmentation approach that allows us to train the system efficiently with small number of samples. And the system is robust to noise (e.g., imprecise landmark points) and missing data (e.g., occluded or undetected landmark points).
These works assume a low-dimensional embedding of the underlying shape space, and represent each 3D shape as a linear combination of shape bases. For example, Principal Component Analysis (PCA) and other linear methods are employed to obtain the low-dimensional shape bases @cite_32 @cite_0 . A major limitation of linear models is that when the true (but unknown) mapping function is nonlinear, the accuracy drops significantly @cite_38 . Therefore, these methods cannot efficiently handle highly deformable or articulated shapes.
@cite_0 @cite_38 @cite_32
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Non-rigid structure from motion (NRSFM) is a difficult, underconstrained problem in computer vision. The standard approach in NRSFM constrains 3D shape deformation using a linear combination of K basis shapes; the solution is then obtained as the low-rank factorization of an input observation matrix. An important but overlooked problem with this approach is that non-linear deformations are often observed; these deformations lead to a weakened low-rank constraint due to the need to use additional basis shapes to linearly model points that move along curves. Here, we demonstrate how the kernel trick can be applied in standard NRSFM. As a result, we model complex, deformable 3D shapes as the outputs of a non-linear mapping whose inputs are points within a low-dimensional shape space. This approach is flexible and can use different kernels to build different non-linear models. Using the kernel trick, our model complements the low-rank constraint by capturing non-linear relationships in the shape coefficients of the linear model. The net effect can be seen as using non-linear dimensionality reduction to further compress the (shape) space of possible solutions. We address the classical computer vision problems of rigid and nonrigid structure from motion (SFM) with occlusion. We assume that the columns of the input observation matrix W describe smooth 2D point trajectories over time. We then derive a family of efficient methods that estimate the column space of W using compact parameterizations in the Discrete Cosine Transform (DCT) domain. Our methods tolerate high percentages of missing data and incorporate new models for the smooth time trajectories of 2D-points, affine and weak-perspective cameras, and 3D deformable shape. We solve a rigid SFM problem by estimating the smooth time trajectory of a single camera moving around the structure of interest. By considering a weak-perspective camera model from the outset, we directly compute euclidean 3D shape reconstructions without requiring postprocessing steps such as euclidean upgrade and bundle adjustment. Our results on real SFM data sets with high percentages of missing data compared positively to those in the literature. In nonrigid SFM, we propose a novel 3D shape trajectory approach that solves for the deformable structure as the smooth time trajectory of a single point in a linear shape space. A key result shows that, compared to state-of-the-art algorithms, our nonrigid SFM method can better model complex articulated deformation with higher frequency DCT components while still maintaining the low-rank factorization constraint. Finally, we also offer an approach for nonrigid SFM when W is presented with missing data.