This paper revisits the problem of continuous-time structure from motion, and introduces a number of extensions that improve convergence and efficiency. The formulation with a @math -continuous spline for the trajectory naturally incorporates inertial measurements, as derivatives of the sought trajectory. We analyse the behaviour of split interpolation on @math and on @math , and a joint interpolation on @math , and show that the latter implicitly couples the direction of translation and rotation. Such an assumption can make good sense for a camera mounted on a robot arm, but not for hand-held or body-mounted cameras. Our experiments show that split interpolation on @math and on @math is preferable over @math interpolation in all tested cases. Finally, we investigate the problem of landmark reprojection on rolling shutter cameras, and show that the tested reprojection methods give similar quality, while their computational load varies by a factor of 2.
When re-projecting a landmark in a frame there is an additional complication in the rolling shutter case. As one image coordinate (typically the image row) corresponds to observation time, the reprojection of a landmark at time @math will not necessarily end up at the row corresponding to that time. Early methods handled this by setting the reprojection time to the landmark observation time . This was improved upon by @cite_4 who also linearize the reprojection time error and convert it to a spatial error covariance. @cite_7 instead use the Newton method to iteratively find a reprojection time with a consistent row coordinate, and this approach is also followed by @cite_2 . Yet another approach is to add a projection time parameter for each landmark observation, as well as a cost term for the projection time deviation . In the experiments, we refer to this approach as lifting , which is the common term for elimination of alternating optimization by adding variables and constraints . No previous publication has compared these choices, instead each paper makes a hard commitment to one of the methods. In @cite_0 some of the choices are discussed, but a comparison is left for future work.
@cite_0 @cite_4 @cite_7 @cite_2
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We address the problem of estimating the pose of a cam- era relative to a known 3D scene from a single RGB-D frame. We formulate this problem as inversion of the generative rendering procedure, i.e., we want to find the camera pose corresponding to a rendering of the 3D scene model that is most similar with the observed input. This is a non-convex optimization problem with many local optima. We propose a hybrid discriminative-generative learning architecture that consists of: (i) a set of M predictors which generate M camera pose hypotheses, and (ii) a 'selector' or 'aggregator' that infers the best pose from the multiple pose hypotheses based on a similarity function. We are interested in predictors that not only produce good hypotheses but also hypotheses that are different from each other. Thus, we propose and study methods for learning 'marginally relevant' predictors, and compare their performance when used with different selection procedures. We evaluate our method on a recently released 3D reconstruction dataset with challenging camera poses, and scene variability. Experiments show that our method learns to make multiple predictions that are marginally relevant and can effectively select an accurate prediction. Furthermore, our method outperforms the state-of-the-art discriminative approach for camera relocalization. Many traditional challenges in reconstructing 3D motion, such as matching across wide baselines and handling occlusion, reduce in significance as the number of unique viewpoints increases. However, to obtain this benefit, a new challenge arises: estimating precisely which cameras observe which points at each instant in time. We present a maximum a posteriori (MAP) estimate of the time-varying visibility of the target points to reconstruct the 3D motion of an event from a large number of cameras. Our algorithm takes, as input, camera poses and image sequences, and outputs the time-varying set of the cameras in which a target patch is visibile and its reconstructed trajectory. We model visibility estimation as a MAP estimate by incorporating various cues including photometric consistency, motion consistency, and geometric consistency, in conjunction with a prior that rewards consistent visibilities in proximal cameras. An optimal estimate of visibility is obtained by finding the minimum cut of a capacitated graph over cameras. We demonstrate that our method estimates visibility with greater accuracy, and increases tracking performance producing longer trajectories, at more locations, and at higher accuracies than methods that ignore visibility or use photometric consistency alone. We propose a dense continuous-time tracking and mapping method for RGB-D cameras. We parametrize the camera trajectory using continuous B-splines and optimize the trajectory through dense, direct image alignment. Our method also directly models rolling shutter in both RGB and depth images within the optimization, which improves tracking and reconstruction quality for low-cost CMOS sensors. Using a continuous trajectory representation has a number of advantages over a discrete-time representation (e.g. camera poses at the frame interval). With splines, less variables need to be optimized than with a discrete representation, since the trajectory can be represented with fewer control points than frames. Splines also naturally include smoothness constraints on derivatives of the trajectory estimate. Finally, the continuous trajectory representation allows to compensate for rolling shutter effects, since a pose estimate is available at any exposure time of an image. Our approach demonstrates superior quality in tracking and reconstruction compared to approaches with discrete-time or global shutter assumptions. Low cost CMOS cameras can have an acquisition mode called rolling shutter which sequentially exposes the scan-lines. When a single object moves with respect to the camera, this creates image distortions. Assuming 2D-3D correspondences known, previous work showed that the object pose and kinematics can be estimated from a single rolling shutter image. This was achieved using a suboptimal initialization followed by local iterative optimization. We propose a polynomial projection model for rolling shutter cameras and a constrained global optimization of its parameters. This is done by means of a semidefinite programming problem obtained from the generalized problem of moments method. Contrarily to previous work, our optimization does not require an initialization and ensures that the global minimum is achieved. This allows us to build automatically robust 2D-3D correspondences using a template to provide an initial set of correspondences. Experiments show that our method slightly improves previous work on both simulated and real data. This is due to local minima into which previous methods get trapped. We also successfully experimented building 2D-3D correspondences automatically with both simulated and real data.