Bundle adjustment is the gold standard for refining solutions to geometric computer vision problems. This paper develops an uncertainty visualization technique for bundle adjustment solutions to Structure from Motion problems. Propagating uncertainty through an optimization- from measurement uncertainties to uncertainties in the resulting parameter estimates- is well understood. However, the calculations involved fail numerically for real problems. Often we cope by considering only individual variances, but this ignores the important mutual dependencies between parameters. The dominant modes of uncertainty in most models are large motions involving nearly all parameters at once. These frequently look like flexions, stretchings, and bendings in the overall scene structure. In this paper we present a numerically tractable method for computing dominant eigenvectors of the covariance of a Bundle Adjustment solution. We pay careful attention to the mismatched scales of rotational and translational parameters. Finally, we animate this spectral information. The resulting interactive visualizations (included in the supplemental) give insight into the quality and failure modes of a model. We hope that this work is a step towards broader uncertainty-aware computation for Structure from Motion.

Rotation Averaging is a non-convex optimization problem that determines orientations of a collection of cameras from their images of a 3D scene. The problem has been studied using a variety of distances and robustifiers. The intrinsic (or geodesic) distance on SO(3) is geometrically meaningful; but while some extrinsic distance-based solvers admit (conditional) guarantees of correctness, no comparable results have been found under the intrinsic metric. In this paper, we study the spatial distribution of local minima. First, we do a novel empirical study to demonstrate sharp transitions in qualitative behavior: as problems become noisier, they transition from a single (easy-to-find) dominant minimum to a cost surface filled with minima. In the second part of this paper we derive a theoretical bound for when this transition occurs. This is an extension of the results of [24], which used local convexity as a proxy to study the difficulty of problem. By recognizing the underly- ing quotient manifold geometry of the problem we achieve an n-fold improvement over prior work. Incidentally, our analysis also extends the prior l2 work to general lp costs. Our results suggest using algebraic connectivity as an indicator of problem difficulty.

The Internet contains a wealth of rich geographic information about our world, including 3D models, street maps, and many other data sources. This information is potentially useful for computer vision applications, such as scene understanding for outdoor Internet photos. However, leveraging this data for vision applications requires precisely aligning input photographs, taken from the wild, within a geographic coordinate frame, by estimating the position, orientation, and focal length. To address this problem, we propose a system for aligning 3D structure-from-motion point clouds, produced from Internet imagery, to existing geographic information sources, including Google Street View photos and Google Earth 3D models. We show that our method can produce accurate alignments between these data sources, resulting in the ability to accurately project geographic data into images gathered from the Internet, by ``Googling'' a depth map for an image using sources such as Google Earth.

Repeated features are common in urban scenes. Many objects, such as clock towers with nearly identical sides, or domes with strong radial symmetries, pose challenges for structure from motion. When similar but distinct features are mistakenly equated, the resulting 3D reconstructions can have errors ranging from phantom walls and superimposed structures to a complete failure to reconstruct. We present a new approach to solving such problems by considering the local visibility structure of such repeated features. Drawing upon network theory, we present a new way of scoring features using a measure of local clustering. Our model leads to a simple, fast, and highly scalable technique for disambiguating repeated features based on an analysis of an underlying visibility graph, without relying on explicit geometric reasoning. We demonstrate our method on several very large datasets drawn from Internet photo collections, and compare it to a more traditional geometry-based disambiguation technique.