In this paper we tackle the problem of learning Structure-from-Motion (SfM) through the use of graph attention networks. SfM is a classic computer vision problem that is solved though iterative minimization of reprojection errors, referred to as Bundle Adjustment (BA), starting from a good initialization. In order to obtain a good enough initial-ization to BA, conventional methods rely on a sequence of sub-problems (such as pairwise pose estimation, pose averaging or triangulation) which provide an initial solution that can then be refined using BA. In this work we re-place these sub-problems by learning a model that takes as input the 2D keypoints detected across multiple views, and outputs the corresponding camera poses and 3D key-point coordinates. Our model takes advantage of graph neural networks to learn SfM-specijic primitives, and we show that it can be used for fast inference of the reconstruction for new and unseen sequences. The experimental results show that the proposed model outperforms com-peting learning-based methods, and challenges COLMAP while having lower runtime. Our code is available at: https://github.com/lucasbrynte/gasfm/.

Bundle adjustment is a key component in practically all available Structure from Motion systems. While it is crucial for achieving accurate reconstruction, convergence to the right solution hinges on good initialization. The recently introduced factorization-based pOSE methods formulate a surrogate for the bundle adjustment error without reliance on good initialization. In this paper, we show that pOSE has an undesirable penalization of large depths. To address this we propose expOSE which has an exponential regularization that is negligible for positive depths. To achieve efficient inference we use a quadratic approximation that allows an iterative solution with VarPro. Furthermore, we extend the method with radial distortion robustness by decomposing the Object Space Error into radial and tangential components. Experimental results confirm that the proposed method is robust to initialization and improves reconstruction quality compared to state-of-the-art methods even without bundle adjustment refinement.1

Lines are among the most used computer vision features, in applications such as camera calibration to object detection. Catadioptric cameras with rotationally symmetric mirrors are omnidirectional imaging devices, capturing up to a 360 degrees field of view. These are used in many applications ranging from robotics to panoramic vision. Although known for some specific configurations, the modeling of line projection was never fully solved for general central and non-central catadioptric cameras. We start by taking some general point reflection assumptions and derive a line reflection constraint. This constraint is then used to define a line projection into the image. Next, we compare our model with previous methods, showing that our general approach outputs the same polynomial degrees as previous configuration-specific systems. We run several experiments using synthetic and real-world data, validating our line projection model. Lastly, we show an application of our methods to an absolute camera pose problem.

Low rank inducing penalties have been proven to successfully uncover fundamental structures considered in computer vision and machine learning; however, such methods generally lead to non-convex optimization problems. Since the resulting objective is non-convex one often resorts to using standard splitting schemes such as Alternating Direction Methods of Multipliers (ADMM), or other subgradient methods, which exhibit slow convergence in the neighbourhood of a local minimum. We propose a method using second order methods, in particular the variable projection method (VarPro), by replacing the nonconvex penalties with a surrogate capable of converting the original objectives to differentiable equivalents. In this way we benefit from faster convergence.The bilinear framework is compatible with a large family of regularizers, and we demonstrate the benefits of our approach on real datasets for rigid and non-rigid structure from motion. The qualitative difference in reconstructions show that many popular non-convex objectives enjoy an advantage in transitioning to the proposed framework.1

Factorization methods are frequently used for structure from motion problems (SfM). In the presence of noise they are able to jointly estimate camera matrices and scene points in overdetermined settings, without the need for accurate initial solutions. While the early formulations were restricted to affine models, recent approaches have been show to work with pinhole cameras by minimizing object space errors.In this paper we propose a factorization approach using the so called radial camera, which is invariant to radial distortion and changes in focal length. Assuming a known principal point our approach can reconstruct the 3D scene in settings with unknown and varying radial distortion and focal length. We show on both real and synthetic data that our approach outperforms state-of-the-art factorization methods under these conditions.1

In this paper we present a study of global optimality conditions for Point Set Registration (PSR) with missing data. PSR is the problem of aligning multiple point clouds with an unknown target point cloud. Since non-linear rotation constraints are present the problem is inherently non-convex and typically relaxed by computing the Lagrange dual, which is a Semidefinite Program (SDP). In this work we show that given a local minimizer the dual variables of the SDP can be computed in closed form. This opens up the possibility of verifying the optimally, using the SDP formulation without explicitly solving it. In addition it allows us to study under what conditions the relaxation is tight, through spectral analysis. We show that if the errors in the (unknown) optimal solution are bounded the SDP formulation will be able to recover it.

In this paper we address the 3D reconstruction of points, on a non-central catadioptric system, composed by a mirror, a projector, and a perspective camera. The goal of the paper is to propose a framework to build an omnidirectional depth camera, towards an omnidirectional RGB-D camera system. The main contributions are: an efficient technique to project 3D points from the world to an image of a general non-central catadioptric camera; the definition of the template pattern (for both the projector and camera's images); and the matching between the projection of these features to the world and its respective images. The 3D depth is directly recovered using the template matching approach. In conclusion, we apply some filtering techniques to improve the results. To evaluate the proposed framework, we test the method using synthetic data, under different levels and types of noises, proving that the framework is robust to noise and, thus, can be put into practice.