We address the classical computer vision problems of rigid and non-rigid 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 post-processing steps such as Euclidean upgrade and bundle adjustment. Our results on real SFM datasets with high percentages of missing data were positively compared to those in the literature. In non-rigid 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 non-rigid 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 non-rigid SFM when W is presented with missing data.

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.

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Non-rigid structure from motion (NRSFM) is a classical underconstrained problem in computer vision. A common approach to make NRSFM more tractable is to constrain 3D shape deformation to be smooth over time. This constraint has been used to compress the deformation model and reduce the number of unknowns that are estimated. However, temporal smoothness cannot be enforced when the data lacks temporal ordering and its benefits are less evident when objects undergo abrupt deformations. This paper proposes a new NRSFM method that addresses these problems by considering deformations as spatial variations in shape space and then enforcing spatial, rather than temporal, smoothness. This is done by modeling each 3D shape coefficient as a function of its input 2D shape. This mapping is learned in the feature space of a rotation invariant kernel, where spatial smoothness is intrinsically defined by the mapping function. As a result, our model represents shape variations compactly using custom-built coefficient bases learned from the input data, rather than a pre-specified set such as the Discrete Cosine Transform. The resulting kernel-based mapping is a by-product of the NRSFM solution and leads to another fundamental advantage of our approach: for a newly observed 2D shape, its 3D shape is recovered by simply evaluating the learned function.

Image‐based lighting has allowed the creation of photo‐realistic computer‐generated content. However, it requires the accurate capture of the illumination conditions, a task neither easy nor intuitive, especially to the average digital photography enthusiast. This paper presents an approach to directly estimate an HDR light probe from a single LDR photograph, shot outdoors with a consumer camera, without specialized calibration targets or equipment. Our insight is to use a person's face as an outdoor light probe. To estimate HDR light probes from LDR faces we use an inverse rendering approach which employs data‐driven priors to guide the estimation of realistic, HDR lighting. We build compact, realistic representations of outdoor lighting both parametrically and in a data‐driven way, by training a deep convolutional autoencoder on a large dataset of HDR sky environment maps. Our approach can recover high‐frequency, extremely high dynamic range lighting environments. For quantitative evaluation of lighting estimation accuracy and relighting accuracy, we also contribute a new database of face photographs with corresponding HDR light probes. We show that relighting objects with HDR light probes estimated by our method yields realistic results in a wide variety of settings.