Monocular depth estimation (MDE) models have undergone significant advancements over recent years. Many MDE models aim to predict affine-invariant relative depth from monocular images, while recent developments in large-scale training and vision foundation models enable reasonable estimation of metric (absolute) depth. However, effectively leveraging these predictions for geometric vision tasks, in particular relative pose estimation, remains relatively under explored. While depths provide rich constraints for cross-view image alignment, the intrinsic noise and ambiguity from the monocular depth priors present practical challenges to improving upon classic keypoint-based solutions. In this paper, we develop three solvers for relative pose estimation that explicitly account for independent affine (scale and shift) ambiguities, covering both calibrated and uncalibrated conditions. We further propose a hybrid estimation pipeline that combines our proposed solvers with classic point-based solvers and epipolar constraints. We find that the affine correction modeling is beneficial to not only the relative depth priors but also, surprisingly, the "metric" ones. Results across multiple datasets demonstrate large improvements of our approach over classic keypoint-based baselines and PnP-based solutions, under both calibrated and uncalibrated setups. We also show that our method improves consistently with different feature matchers and MDE models, and can further benefit from very recent advances on both modules. Code is available at https://github.com/MarkYu98/madpose.

In this paper, we speed up robust two-view relative pose from dense correspondences. Previous work has shown that dense matchers can significantly improve both accuracy and robustness in the resulting pose. However, the large number of matches comes with a significantly increased runtime during robust estimation in RANSAC. To avoid this, we propose an efficient match summarization scheme which provides comparable accuracy to using the full set of dense matches, while having 10-100x faster runtime. We validate our approach on standard benchmark datasets together with multiple state-of-the-art dense matchers.

In this paper, we present a new generic Structure-From-Motion pipeline, GenSfM, that uses a non-parametric camera projection model. The model is self-calibrated during the reconstruction process and can fit a wide variety of cameras, ranging from simple low-distortion pinhole cameras to more extreme optical systems such as fisheye or catadioptric cameras. The key component in our framework is an adaptive calibration procedure that can estimate partial calibrations, only modeling regions of the image where sufficient constraints are available. In experiments, we show that our method achieves comparable accuracy to traditional Structure-From-Motion pipelines in easy scenarios, and outperforms them in cases where they are unable to self-calibrate their parametric models. Code is at https://github.com/Ivonne320/GenSfM.git

Finding local features that are repeatable across multiple views is a cornerstone of sparse 3D reconstruction. The classical image matching paradigm detects keypoints per-image once and for all, which can yield poorly-localized features and propagate large errors to the final geometry. In this article, we refine two key steps of structure-from-motion by a direct alignment of low-level image information from multiple views: we first adjust the initial keypoint locations prior to any geometric estimation, and subsequently refine points and camera poses as a post-processing. This refinement is robust to large detection noise and appearance changes, as it optimizes a featuremetric error based on dense features predicted by a neural network. This significantly improves the accuracy of camera poses and scene geometry for a wide range of keypoint detectors, challenging viewing conditions, and off-the-shelf deep features. Our system easily scales to large image collections, enabling pixel-perfect crowd-sourced localization at scale.

In this paper, we revisit the problem of estimating the relative pose from a sparse set of point-correspondences. For each point-correspondence we also estimate the relative depth, i.e. the relative distance to the scene point in the two images. This yields an additional constraint, allowing us to use fewer matches in RANSAC to generate the pose candidates. In the paper we propose two novel minimal solvers: one for general motion and one for the case of known vertical direction. To obtain the relative depth estimates, we explore using scale estimates obtained from a keypoint detector as well as a neural network that directly predicts the relative depth for a pair of patches. We show in experiments that while our estimates are more noisy compared to the purely point-based solvers, the smaller sample size leads to a significantly reduced runtime in settings with high outlier ratios.

Neural implicit representations have recently become popular in simultaneous localization and mapping (SLAM), especially in dense visual SLAM. However, existing works either rely on RGB-D sensors or require a separate monocular SLAM approach for camera tracking, and fail to produce high-fidelity 3D dense reconstructions. To address these shortcomings, we present NICER-SLAM, a dense RGB SLAM system that simultaneously optimizes for camera poses and a hierarchical neural implicit map representation, which also allows for high-quality novel view synthesis. To facilitate the optimization process for mapping, we integrate additional supervision signals including easy-to-obtain monocular geometric cues and optical flow, and also introduce a simple warping loss to further enforce geometric consistency. Moreover, to further boost performance in complex large-scale scenes, we also propose a local adaptive transformation from signed distance functions (SDFs) to density in the volume rendering equation. On multiple challenging indoor and outdoor datasets, NICER-SLAM demonstrates strong performance in dense mapping, novel view synthesis, and tracking, even competitive with recent RGB-D SLAM systems. Project page: https://nicer-slam.github.io/.

Two-view homography estimation is a classic and fundamental problem in computer vision. While conceptually simple, the problem quickly becomes challenging when multiple planes are visible in the image pair. Even with correct matches, each individual plane (homography) might have a very low number of inliers when comparing to the set of all correspondences. In practice, this requires a large number of RANSAC iterations to generate a good model hypothesis. The current state-of-the-art methods therefore seek to reduce the sample size, from four point correspondences originally, by including additional information such as keypoint orientation/angles or local affine information. In this work, we continue in this direction and propose a novel one-point solver that leverages different approximate constraints derived from the same auxiliary information. In experiments we obtain state-of-the-art results, with execution time speed-ups, on large benchmark datasets and show that it is more beneficial for the solver to be sample efficient compared to generating more accurate homographies.

Many problems in computer vision can be formulated as geometric estimation problems, i.e. given a collection of measurements (e.g. point correspondences) we wish to fit a model (e.g. an essential matrix) that agrees with our observations. This necessitates some measure of how much an observation “agrees” with a given model. A natural choice is to consider the smallest perturbation that makes the observation exactly satisfy the constraints. However, for many problems, this metric is expensive or otherwise intractable to compute. The so-called Sampson error approximates this geometric error through a linearization scheme. For epipolar geometry, the Sampson error is a popular choice and in practice known to yield very tight approximations of the corresponding geometric residual (the reprojection error). In this paper we revisit the Sampson approximation and provide new theoretical insights as to why and when this approximation works, as well as provide explicit bounds on the tightness under some mild assumptions. Our theoretical results are validated in several experiments on real data and in the context of different geometric estimation tasks.

One of the classical multi-view geometry problems is the so called P3P problem, where the absolute pose of a calibrated camera is determined from three 2D-to-3D correspondences. Since these solvers form a critical component of many vision systems (e.g. in localization and Structure-from-Motion), there have been significant effort in developing faster and more stable algorithms. While the current state-of-the-art solvers are both extremely fast and stable, there still exist configurations where they break down. In this paper we algebraically formulate the problem as finding the intersection of two conics. With this formulation we are able to analytically characterize the real roots of the polynomial system and employ a tailored solution strategy for each problem instance. The result is a fast and stable solver, that is able to correctly solve cases where competing methods might fail. Our experimental evaluation shows that we outperform the current state-of-the-art methods both in terms of speed and success rate.

We present novel solutions to previously unsolved prob-lems of relative pose estimation from images whose calibration parameters, namely focal lengths and radial distortion, are unknown. Our approach enables metric reconstruction without modeling these parameters. The minimal case for reconstruction requires 13 points in 4 views for both the calibrated and uncalibrated cameras. We describe and implement the first solution to these minimal problems. In the calibrated case, this may be modeled as a polynomial sys-tem of equations with 3584 solutions. Despite the apparent intractability, the problem decomposes spectacularly. Each solution falls into a Euclidean symmetry class of size 16, and we can estimate 224 class representatives by solving a sequence of three subproblems with 28, 2, and 4 solutions. We highlight the relationship between internal constraints on the radial quadrifocal tensor and the relations among the principal minors of a$4\times 4$matrix. We also address the case of 4 upright cameras, where 7 points are minimal. Finally, we evaluate our approach on simulated and real data and benchmark against previous calibration-free solutions, and show that our method provides an efficient startup for an SfM pipeline with radial cameras.

In contrast to sparse keypoints, a handful of line segments can concisely encode the high-level scene layout, as they often delineate the main structural elements. In addition to offering strong geometric cues, they are also omnipresent in urban landscapes and indoor scenes. Despite their apparent advantages, current line-based reconstruction methods are far behind their point-based counterparts. In this paper we aim to close the gap by introducing LIMAP, a library for 3D line mapping that robustly and efficiently creates 3D line maps from multi-view imagery. This is achieved through revisiting the degeneracy problem of line triangulation, carefully crafted scoring and track building, and exploiting structural priors such as line coincidence, parallelism, and orthogonality. Our code integrates seamlessly with existing point-based Structure-from-Motion methods and can leverage their 3D points to further improve the line reconstruction. Furthermore, as a byproduct, the method is able to recover 3D association graphs between lines and points / vanishing points (VPs). In thorough experiments, we show that LIMAP significantly outperforms existing approaches for 3D line mapping. Our robust 3D line maps also open up new research directions. We show two example applications: visual localization and bundle adjustment, where integrating lines alongside points yields the best results. Code is available at https://github.com/cvg/limap.

Line segments are ubiquitous in our human-made world and are increasingly used in vision tasks. They are complementary to feature points thanks to their spatial extent and the structural information they provide. Traditional line detectors based on the image gradient are extremely fast and accurate, but lack robustness in noisy images and challenging conditions. Their learned counterparts are more repeatable and can handle challenging images, but at the cost of a lower accuracy and a bias towards wireframe lines. We propose to combine traditional and learned approaches to get the best of both worlds: an accurate and robust line detector that can be trained in the wild without ground truth lines. Our new line segment detector, DeepLSD, processes images with a deep network to generate a line attraction field, before converting it to a surrogate image gradient magnitude and angle, which is then fed to any existing handcrafted line detector. Additionally, we propose a new optimization tool to refine line segments based on the attraction field and vanishing points. This refinement improves the accuracy of current deep detectors by a large margin. We demonstrate the performance of our method on low-level line detection metrics, as well as on several downstream tasks using multiple challenging datasets. The source code and models are available at https://github.com/cvg/DeepLSD.

In this paper, we revisit the rotation averaging problem applied in global Structure-from-Motion pipelines. We argue that the main problem of current methods is the minimized cost function that is only weakly connected with the input data via the estimated epipolar geometries. We propose to better model the underlying noise distributions by directly propagating the uncertainty from the point correspondences into the rotation averaging. Such uncertainties are obtained for free by considering the Jacobians of two-view refinements. Moreover, we explore integrating a variant of the MAGSAC loss into the rotation averaging problem, instead of using classical robust losses employed in current frameworks. The proposed method leads to results superior to baselines, in terms of accuracy, on large-scale public benchmarks. The code is public. https://github.com/zhangganlin/GlobalSfMpy

For a generalized (or non-central) camera model, the minimal problem for two views of six points has efficient solvers. However, minimal problems of three views with four points and three views of six lines have not yet been explored and solved, despite the efforts from the computer vision community. This paper develops the formulations of these two minimal problems and shows how state-of-the-art GPU implementations of Homotopy Continuation solver can be used effectively. The proposed methods are evaluated on both synthetic and real datasets, demonstrating that they are fast, accurate and that they improve on structure from motion estimations, when employed in an hypothesis and test setting.

Recent methods on privacy-preserving image-based localization use a random line parameterization to protect the privacy of query images and database maps. The lifting of points to lines effectively drops one of the two geometric constraints traditionally used with point-to-point correspondences in structure-based localization. This leads to a significant loss of accuracy for the privacy-preserving methods. In this paper, we overcome this limitation by devising a coordinate permutation scheme that allows for recovering the original point positions during pose estimation. The recovered points provide the full 2D geometric constraints and enable us to close the gap between privacy-preserving and traditional methods in terms of accuracy. Another limitation of random line methods is their vulnerability to density based 3D line cloud inversion attacks. Our method not only provides better accuracy than the original random line based approach but also provides stronger privacy guarantees against these recently proposed attacks. Extensive experiments on standard benchmark datasets demonstrate these improvements consistently across both scenarios of protecting the privacy of query images as well as the database map.

Line segments are powerful features complementary to points. They offer structural cues, robust to drastic viewpoint and illumination changes, and can be present even in texture-less areas. However, describing and matching them is more challenging compared to points due to partial occlusions, lack of texture, or repetitiveness. This paper introduces a new matching paradigm, where points, lines, and their descriptors are unified into a single wireframe structure. We propose GlueStick, a deep matching Graph Neural Network (GNN) that takes two wireframes from different images and leverages the connectivity information between nodes to better glue them together. In addition to the increased efficiency brought by the joint matching, we also demonstrate a large boost of performance when leveraging the complementary nature of these two features in a single architecture. We show that our matching strategy outperforms the state-of-the-art approaches independently matching line segments and points for a wide variety of datasets and tasks. The code is available at https://github.com/cvg/GlueStick.

In this paper we introduce the Tangent Sampson error, which is a generalization of the classical Sampson error in two-view geometry that allows for arbitrary central camera models. It only requires local gradients of the distortion map at the original correspondences (allowing for pre-computation) resulting in a negligible increase in computational cost when used in RANSAC or local refinement. The error effectively approximates the true-reprojection error for a large variety of cameras, including extremely wide field-of-view lenses that cannot be undistorted to a single pinhole image. We show experimentally that the new error outperforms competing approaches both when used for model scoring in RANSAC and for non-linear refinement of the relative camera pose.

A distinctive representation of image patches in form of features is a key component of many computer vision and robotics tasks, such as image matching, image retrieval, and visual localization. State-of-the-art descriptors, from hand-crafted descriptors such as SIFT to learned ones such as HardNet, are usually high-dimensional; 128 dimensions or even more. The higher the dimensionality, the larger the memory consumption and computational time for approaches using such descriptors. In this paper, we investigate multi-layer perceptrons (MLPs) to extract low-dimensional but high-quality descriptors. We thoroughly analyze our method in unsuper-vised, self-supervised, and supervised settings, and evaluate the dimensionality reduction results on four representative descriptors. We consider different applications, including visual localization, patch verification, image matching and retrieval. The experiments show that our lightweight MLPs trained using supervised method achieve better dimensionality reduction than PCA. The lower-dimensional descriptors generated by our approach outperform the original higher-dimensional descriptors in downstream tasks, especially for the hand-crafted ones. The code is available at https://github.com/PRBonn/descriptor-dr.

We study the problem of estimating the semi-generalized pose of a partially calibrated camera, i.e., the pose of a perspective camera with unknown focal length w.r.t. a generalized camera, from a hybrid set of 2D-2D and 2D-3D point correspondences. We study all possible camera configurations within the generalized camera system. To derive practical solvers to previously unsolved challenging configurations, we test different parameterizations as well as different solving strategies based on state-of-the-art methods for generating efficient polynomial solvers. We evaluate the three most promising solvers, i.e., the H51f solver with five 2D-2D correspondences and one 2D-3D match viewed by the same camera inside the generalized camera, the H32f solver with three 2D-2D and two 2D-3D correspondences, and the H13f solver with one 2D-2D and three 2D-3D matches, on synthetic and real data. We show that in the presence of noise in the 3D points these solvers provide better estimates than the corresponding absolute pose solvers.

Recently proposed privacy preserving solutions for cloud-based localization rely on lifting traditional point-based maps to randomized 3D line clouds. While the lifted representation is effective in concealing private information, there are two fundamental limitations. First, without careful construction of the line clouds, the representation is vulnerable to density-based inversion attacks. Secondly, after successful localization, the precise camera orientation and position is revealed to the server. However, in many scenarios, the pose itself might be sensitive information. We propose a principled approach overcoming these limitations, based on two observations. First, a full 6 DoF pose is not always necessary, and in combination with egomotion tracking even a one dimensional localization can reduce uncertainty and correct drift. Secondly, by lifting to parallel planes instead of lines, the map only provides partial constraints on the query pose, preventing the server from knowing the exact query location. If the client requires a full 6 DoF pose, it can be obtained by fusing the result from multiple queries, which can be temporally and spatially disjoint. We demonstrate the practical feasibility of this approach and show a small performance drop compared to both the conventional and privacy preserving approaches.

Low-dimensional parametric models are the de-facto standard in computer vision for intrinsic camera calibration. These models explicitly describe the mapping between incoming viewing rays and image pixels. In this paper, we explore an alternative approach which implicitly models the lens distortion. The main idea is to replace the parametric model with a regularization term that ensures the latent distortion map varies smoothly throughout the image. The proposed model is effectively parameter-free and allows us to optimize the 6 degree-of-freedom camera pose without explicitly knowing the intrinsic calibration. We show that the method is applicable to a wide selection of cameras with varying distortion and in multiple applications, such as visual localization and structure-from-motion.

Neural implicit representations have recently shown encouraging results in various domains, including promising progress in simultaneous localization and mapping (SLAM). Nevertheless, existing methods produce over- smoothed scene reconstructions and have difficulty scaling up to large scenes. These limitations are mainly due to their simple fully-connected network architecture that does not incorporate local information in the observations. In this paper, we present NICE-SLAM, a dense SLAM system that incorporates multi-level local information by introducing a hierarchical scene representation. Optimizing this representation with pre-trained geometric priors enables detailed reconstruction on large indoor scenes. Compared to recent neural implicit SLAM systems, our approach is more scalable, efficient, and robust. Experiments on five challenging datasets demonstrate competitive results of NICE-SLAM in both mapping and tracking quality. Project page: https://pengsongyou.github.io/nice-slam.

We consider the problem of stitching image sequences with cameras undergoing pure rotational motion. We leverage the assumption of a locally constant rotation axis, i.e., neighboring frames have a shared but unknown rotation axis. This assumption holds in many common image capturing scenarios, e.g., panoramic sweeping motions. Using this additional constraint, we develop techniques for three-view camera rotation estimation; a minimal solver for the two-view estimation with a known rotation axis; and a globally optimal robust estimator for the two-view case. We show on publicly available datasets that the proposed methods lead to camera rotation estimation superior to the state-of-the-art in terms of accuracy with comparable run-time. The source code will be made available.

Recent works on localization and mapping from privacy preserving line features have made significant progress towards addressing the privacy concerns arising from cloud-based solutions in mixed reality and robotics. The requirement for calibrated cameras is a fundamental limitation for these approaches, which prevents their application in many crowd-sourced mapping scenarios. In this paper, we propose a solution to the uncalibrated privacy preserving localization and mapping problem. Our approach simultaneously recovers the intrinsic and extrinsic calibration of a camera from line-features only. This enables uncalibrated devices to both localize themselves within an existing map as well as contribute to the map, while preserving the privacy of the image contents. Furthermore, we also derive a solution to bootstrapping maps from scratch using only uncalibrated devices. Our approach provides comparable performance to the calibrated scenario and the privacy compromising alternatives based on traditional point features.

In this paper we study structure from motion problems for 1D radial cameras. Under this model the projection of a 3D point is a line in the image plane going through the principal point, which makes the model invariant to radial distortion and changes in focal length. It can therefore effectively be applied to uncalibrated image collections without the need for explicit estimation of camera intrinsics.We show that the reprojection errors of 1D radial cameras are examples of quasiconvex functions. This opens up the possibility to solve a general class of relevant reconstruction problems globally optimally using tools from convex optimization. In fact, our resulting algorithm is based on solving a series of LP problems. We perform an extensive experimental evaluation, on both synthetic and real data, showing that a whole class of multiview geometry problems across a range of different cameras models with varying and unknown intrinsic calibration can be reliably and accurately solved within the same framework.1

Compared to feature point detection and description, detecting and matching line segments offer additional challenges. Yet, line features represent a promising complement to points for multi-view tasks. Lines are indeed well-defined by the image gradient, frequently appear even in poorly textured areas and offer robust structural cues. We thus hereby introduce the first joint detection and description of line segments in a single deep network. Thanks to a self-supervised training, our method does not require any annotated line labels and can therefore generalize to any dataset. Our detector offers repeatable and accurate localization of line segments in images, departing from the wireframe parsing approach. Leveraging the recent progresses in descriptor learning, our proposed line descriptor is highly discriminative, while remaining robust to viewpoint changes and occlusions. We evaluate our approach against previous line detection and description methods on several multi-view datasets created with homographic warps as well as real-world viewpoint changes. Our full pipeline yields higher repeatability, localization accuracy and matching metrics, and thus represents a first step to bridge the gap with learned feature points methods. Code and trained weights are available at https://github.com/cvg/SOLD2.

Camera pose estimation in known scenes is a 3D geometry task recently tackled by multiple learning algorithms. Many regress precise geometric quantities, like poses or 3D points, from an input image. This either fails to generalize to new viewpoints or ties the model parameters to a specific scene. In this paper, we go Back to the Feature: we argue that deep networks should focus on learning robust and invariant visual features, while the geometric estimation should be left to principled algorithms. We introduce PixLoc, a scene-agnostic neural network that estimates an accurate 6-DoF pose from an image and a 3D model. Our approach is based on the direct alignment of multiscale deep features, casting camera localization as metric learning. PixLoc learns strong data priors by end-to-end training from pixels to pose and exhibits exceptional generalization to new scenes by separating model parameters and scene geometry. The system can localize in large environments given coarse pose priors but also improve the accuracy of sparse feature matching by jointly refining keypoints and poses with little overhead. The code will be publicly available at github.com/cvg/pixloc.

In this paper we consider the epipolar geometry between orthographic and perspective cameras. We generalize many of the classical results for the perspective essential matrix to this setting and derive novel minimal solvers, not only for the calibrated case, but also for partially calibrated and non-central camera setups. While orthographic cameras might seem exotic, they occur naturally in many applications. They can e.g. model 2D maps (such as floor plans), aerial/satellite photography and even approximate narrow field-of-view cameras (e.g. from telephoto lenses). In our experiments we highlight various applications of the developed theory and solvers, including Radar-Camera calibration and aligning Structure-from-Motion models to aerial or satellite images.

Finding local features that are repeatable across multiple views is a cornerstone of sparse 3D reconstruction. The classical image matching paradigm detects keypoints per-image once and for all, which can yield poorly-localized features and propagate large errors to the final geometry. In this paper, we refine two key steps of structure-from-motion by a direct alignment of low-level image information from multiple views: we first adjust the initial keypoint locations prior to any geometric estimation, and subsequently refine points and camera poses as a post-processing. This refinement is robust to large detection noise and appearance changes, as it optimizes a featuremetric error based on dense features predicted by a neural network. This significantly improves the accuracy of camera poses and scene geometry for a wide range of keypoint detectors, challenging viewing conditions, and off-the-shelf deep features. Our system easily scales to large image collections, enabling pixel-perfect crowd-sourced localization at scale. Our code is publicly available at github.com/cvg/pixel-perfect-sfm as an add-on to the popular SfM software COLMAP.

Motion blur is one of the major challenges remaining for visual odometry methods. In low-light conditions where longer exposure times are necessary, motion blur can appear even for relatively slow camera motions. In this paper we present a novel hybrid visual odometry pipeline with direct approach that explicitly models and estimates the camera’s local trajectory within the exposure time. This allows us to actively compensate for any motion blur that occurs due to the camera motion. In addition, we also contribute a novel benchmarking dataset for motion blur aware visual odometry. In experiments we show that by directly modeling the image formation process, we are able to improve robustness of the visual odometry, while keeping comparable accuracy as that for images without motion blur. Both the code and the datasets can be found from https://github.com/ethliup/MBA-VO.

This paper introduces minimal solvers that jointly solve for radial lens undistortion and affine-rectification using local features extracted from the image of coplanar translated and reflected scene texture, which is common in man-made environments. The proposed solvers accommodate different types of local features and sampling strategies, and three of the proposed variants require just one feature correspondence. State-of-the-art techniques from algebraic geometry are used to simplify the formulation of the solvers. The generated solvers are stable, small and fast. Synthetic and real-image experiments show that the proposed solvers have superior robustness to noise compared to the state of the art. The solvers are integrated with an automated system for rectifying imaged scene planes from coplanar repeated texture. Accurate rectifications on challenging imagery taken with narrow to wide field-of-view lenses demonstrate the applicability of the proposed solvers.

Most consumer cameras are equipped with electronic rolling shutter, leading to image distortions when the camera moves during image capture. We explore a surprisingly simple camera configuration that makes it possible to undo the rolling shutter distortion: two cameras mounted to have different rolling shutter directions. Such a setup is easy and cheap to build and it possesses the geometric constraints needed to correct rolling shutter distortion using only a sparse set of point correspondences between the two images. We derive equations that describe the underlying geometry for general and special motions and present an efficient method for finding their solutions. Our synthetic and real experiments demonstrate that our approach is able to remove large rolling shutter distortions of all types without relying on any specific scene structure.

We present a novel network for rolling shutter effect correction. Our network takes two consecutive rolling shutter images and estimates the corresponding global shutter image of the latest frame. The dense displacement field from a rolling shutter image to its corresponding global shutter image is estimated via a motion estimation network. The learned feature representation of a rolling shutter image is then warped, via the displacement field, to its global shutter representation by a differentiable forward warping block. An image decoder recovers the global shutter image based on the warped feature representation. Our network can be trained end-to-end and only requires the global shutter image for supervision. Since there is no public dataset available, we also propose two large datasets: the Carla-RS dataset and the Fastec-RS dataset. Experimental results demonstrate that our network outperforms the state-of-the-art methods. We make both our code and datasets available at https://github.com/ethliup/DeepUnrollNet.

Camera calibration is an essential first step in setting up 3D Computer Vision systems. Commonly used parametric camera models are limited to a few degrees of freedom and thus often do not optimally fit to complex real lens distortion. In contrast, generic camera models allow for very accurate calibration due to their flexibility. Despite this, they have seen little use in practice. In this paper, we argue that this should change. We propose a calibration pipeline for generic models that is fully automated, easy to use, and can act as a drop-in replacement for parametric calibration, with a focus on accuracy. We compare our results to parametric calibrations. Considering stereo depth estimation and camera pose estimation as examples, we show that the calibration error acts as a bias on the results. We thus argue that in contrast to current common practice, generic models should be preferred over parametric ones whenever possible. To facilitate this, we released our calibration pipeline at https://github.com/puzzlepaint/camera_calibration, making both easy-to-use and accurate camera calibration available to everyone.

We present minimal, non-iterative solutions to the absolute pose problem for images from rolling shutter cameras. The absolute pose problem is a key problem in computer vision and rolling shutter is present in a vast majority of today's digital cameras. We discuss several camera motion models and propose two feasible rolling shutter camera models for a polynomial solver. In previous work a linearized camera model was used that required an initial estimate of the camera orientation. We show how to simplify the system of equations and make this solver faster. Furthermore, we present a first solution of the non-linearized camera orientation model using the Cayley parameterization. The new solver does not require any initial camera orientation estimate and therefore serves as a standalone solution to the rolling shutter camera pose problem from six 2D-to-3D correspondences. We show that our algorithms outperform P3P followed by a non-linear refinement using a rolling shutter model.

This paper presents the first optimal, maximal likelihood, solution to the triangulation problem for radially distorted cameras. The proposed solution to the two-view triangulation problem minimizes the L2-norm of the reprojection error in the distorted image space. We cast the problem as the search for corrected distorted image points, and we use a Lagrange multiplier formulation to impose the epipolar constraint for undistorted points. For the one-parameter division model, this formulation leads to a system of five quartic polynomial equations in five unknowns, which can be exactly solved using the Groebner basis method. While the proposed Groebner basis solution is provably optimal; it is too slow for practical applications. Therefore, we developed a fast iterative solver to this problem. Extensive empirical tests show that the iterative algorithm delivers the optimal solution virtually every time, thus making it an L2-optimal algorithm de facto. It is iterative in nature, yet in practice, it converges in no more than five iterations. We thoroughly evaluate the proposed method on both synthetic and real-world data, and we show the benefits of performing the triangulation in the distorted space in the presence of radial distortion.

In this paper we study the problem of estimating receiver and sender positions from time-difference-of-arrival measurements, assuming an unknown constant time-difference-of-arrival offset. This problem is relevant for example for repetitive sound events. In this paper it is shown that there are three minimal cases to the problem. One of these (the five receiver, five sender problem) is of particular importance. A fast solver (with run-time under 4 µs) is given. We show how this solver can be used in robust estimation algorithms, based on RANSAC, for obtaining an initial estimate followed by local optimization using a robust error norm. The system is verified on both real and synthetic data.

The problem of estimating receiver or sender node positions from measured receiver-sender distances is a key issue in different applications such as microphone array calibration, radio antenna array calibration, mapping and positioning using UWB or using round-trip-time measurements between mobile phones and WiFi-units. In this paper we address the problem of optimally estimating a receiver position given a number of distance measurements to known sender positions, so called trilateration. We show that this problem can be rephrased as an eigenvalue problem. We also address different error models and the multilateration setting where an additional offset is also unknown, and show that these problems can be modeled using the same framework.

In this paper we consider the problem of relative pose estimation from two images with per-pixel polarimetric information. Using these additional measurements we derive a simple minimal solver for the essential matrix which only requires two point correspondences. The polarization constraints allow us to pointwise recover the 3D surface normal up to a two-fold ambiguity for the diffuse reflection. Since this ambiguity exists per point, there is a combinatorial explosion of possibilities. However, since our solver only requires two point correspondences, we only need to consider 16 configurations when solving for the relative pose. Once the relative orientation is recovered, we show that it is trivial to resolve the ambiguity for the remaining points. For robustness, we also propose a joint optimization between the relative pose and the refractive index to handle the refractive distortion. In experiments, on both synthetic and real data, we demonstrate that by leveraging the additional information available from polarization cameras, we can improve over classical methods which only rely on the 2D-point locations to estimate the geometry. Finally, we demonstrate the practical applicability of our approach by integrating it into a state-of-the-art global Structure-from-Motion pipeline.

To model radial distortion there are two main approaches; either the image points are undistorted such that they correspond to pinhole projections, or the pinhole projections are distorted such that they align with the image measurements. Depending on the application, either of the two approaches can be more suitable. For example, distortion models are commonly used in Structure-from-Motion since they simplify measuring the reprojection error in images. Surprisingly, all previous minimal solvers for pose estimation with radial distortion use undistortion models. In this paper we aim to fill this gap in the literature by proposing the first minimal solvers which can jointly estimate distortion models together with camera pose. We present a general approach which can handle rational models of arbitrary degree for both distortion and undistortion.

To estimate the 6-DoF extrinsic pose of a pinhole camera with partially unknown intrinsic parameters is a critical sub-problem in structure-from-motion and camera localization. In most of existing camera pose estimation solvers, the principal point is assumed to be in the image center. Unfortunately, this assumption is not always true, especially for asymmetrically cropped images. In this paper, we develop the first exactly minimal solver for the case of unknown principal point and focal length by using four and a half point correspondences (P4.5Pfuv). We also present an extremely fast solver for the case of unknown aspect ratio (P5Pfuva). The new solvers outperform the previous state-of-the-art in terms of stability and speed. Finally, we explore the extremely challenging case of both unknown principal point and radial distortion, and develop the first practical non-minimal solver by using seven point correspondences (P7Pfruv). Experimental results on both simulated data and real Internet images demonstrate the usefulness of our new solvers.

Many computer vision applications require robust estimation of the underlying geometry, in terms of camera motion and 3D structure of the scene. These robust methods often rely on running minimal solvers in a RANSAC framework. In this paper we show how we can make polynomial solvers based on the action matrix method faster, by careful selection of the monomial bases. These monomial bases have traditionally been based on a Grobner basis for the polynomial ideal. Here we describe how we can enumerate all such bases in an efficient way. We also show that going beyond Grobner bases leads to more efficient solvers in many cases. We present a novel basis sampling scheme that we evaluate on a number of problems.

This paper introduces the first minimal solvers that jointly solve for affine-rectification and radial lens distortion from coplanar repeated patterns. Even with imagery from moderately distorted lenses, plane rectification using the pinhole camera model is inaccurate or invalid. The proposed solvers incorporate lens distortion into the camera model and extend accurate rectification to wide-angle imagery, which is now common from consumer cameras. The solvers are derived from constraints induced by the conjugate translations of an imaged scene plane, which are integrated with the division model for radial lens distortion. The hidden-variable trick with ideal saturation is used to reformulate the constraints so that the solvers generated by the Gröbner-basis method are stable, small and fast. Rectification and lens distortion are recovered from either one conjugately translated affine-covariant feature or two independently translated similarity-covariant features. The proposed solvers are used in a RANSAC-based estimator, which gives accurate rectifications after few iterations. The proposed solvers are evaluated against the state-of-the-art and demonstrate significantly better rectifcations on noisy measurements. Qualitative results on diverse imagery demonstrate high-accuracy undistortion and rectification.

In this paper we study the problem of automatically generating polynomial solvers for minimal problems. The main contribution is a new method for finding small elimination templates by making use of the syzygies (i.e. the polynomial relations) that exist between the original equations. Using these syzygies we can essentially parameterize the set of possible elimination templates. We evaluate our method on a wide variety of problems from geometric computer vision and show improvement compared to both handcrafted and automatically generated solvers. Furthermore we apply our method on two previously unsolved relative orientation problems.

Traditional matrix factorization methods approximate high dimensional data with a low dimensional subspace. This imposes constraints on the matrix elements which allow for estimation of missing entries. A lower rank provides stronger constraints and makes estimation of the missing entries less ambiguous at the cost of measurement fit. In this paper we propose a new factorization model that further constrains the matrix entries. Our approach can be seen as a unification of traditional low-rank matrix factorization and the more recent union-of-subspace approach. It adaptively finds clusters that can be modeled with low dimensional local subspaces and simultaneously uses a global rank constraint to capture the overall scene interactions. For inference we use an energy that penalizes a trade-off between data fit and degrees-of-freedom of the resulting factorization. We show qualitatively and quantitatively that regularizing both local and global dynamics yields significantly improved missing data estimation.

In this paper we present a new method for creating polynomial solvers for problems where a (possibly infinite) subset of the solutions are undesirable or uninteresting. These solutions typically arise from simplifications made during modeling, but can also come from degeneracies which are inherent to the geometry of the original problem. The proposed approach extends the standard action matrix method to saturated ideals. This allows us to add constraints that some polynomials should be non-zero on the solutions. This does not only offer the possibility of improved performance by removing superfluous solutions, but makes a larger class of problems tractable. Previously, problems with infinitely many solutions could not be solved directly using the action matrix method as it requires a zero-dimensional ideal. In contrast we only require that after removing the unwanted solutions only finitely many remain. We evaluate our method on three applications, optimal triangulation, time-of-arrival self-calibration and optimal vanishing point estimation.

In this paper we present new techniques for constructing compact and robust minimal solvers for absolute pose estimation. We focus on the P4Pfr problem, but the methods we propose are applicable to a more general setting. Previous approaches to P4Pfr suffer from artificial degeneracies which come from their formulation and not the geometry of the original problem. In this paper we show how to avoid these false degeneracies to create more robust solvers. Combined with recently published techniques for Gröbner basis solvers we are also able to construct solvers which are significantly smaller. We evaluate our solvers on both real and synthetic data, and show improved performance compared to competing solvers. Finally we show that our techniques can be directly applied to the P3.5Pf problem to get a non-degenerate solver, which is competitive with the current state-of-the-art.

This paper considers the problem of recovering either a low rank matrix or a sparse vector from observations of linear combinations of the vector or matrix elements. Recent methods replace the non-convex regularization with ℓ1 or nuclear norm relaxations. It is well known that this approach recovers near optimal solutions if a so called restricted isometry property (RIP) holds. On the other hand it also has a shrinking bias which can degrade the solution. In this paper we study an alternative non-convex regularization term that does not suffer from this bias. Our main theoretical results show that if a RIP holds then the stationary points are often well separated, in the sense that their differences must be of high cardinality/rank. Thus, with a suitable initial solution the approach is unlikely to fall into a bad local minimum. Our numerical tests show that the approach is likely to converge to a better solution than standard ℓ1/nuclear-norm relaxation even when starting from trivial initializations. In many cases our results can also be used to verify global optimality of our method.

Previous work on estimating the epipolar geometry of two views relies on being able to reliably match feature points based on appearance. In this paper, we go one step further and show that it is feasible to compute both the epipolar geometry and the correspondences at the same time based on geometry only. We do this in a globally optimal manner. Our approach is based on an efficient branch and bound technique in combination with bipartite matching to solve the correspondence problem. We rely on several recent works to obtain good bounding functions to battle the combinatorial explosion of possible matchings. It is experimentally demonstrated that more difficult cases can be handled and that more inlier correspondences can be obtained by being less restrictive in the matching phase.

Despite of having no explicit shape model, multi-atlas approaches to image segmentation have proved to be a top-performer for several diverse datasets and imaging modalities. In this paper, we show how one can directly incorporate shape regularization into the multi-atlas framework. Unlike traditional methods, our proposed approach does not rely on label fusion on the voxel level. Instead, each registered atlas is viewed as an estimate of the position of a shape model. We evaluate and compare our method on two public benchmarks: (i) the VISCERAL Grand Challenge on multi-organ segmentation of whole-body CT images and (ii) the Hammers brain atlas of MR images for segmenting the hippocampus and the amygdala. For this wide spectrum of both easy and hard segmentation tasks, our experimental quantitative results are on par or better than state-of-the-art. More importantly, we obtain qualitatively better segmentation boundaries, for instance, preserving fine structures.

Outliers pose a problem in all real structure from motion systems. Due to the use of automatic matching methods one has to expect that a (sometimes very large) portion of the detected correspondences can be incorrect. In this paper we propose a method that estimates the relative translation between two cameras and simultaneously maximizes the number of inlier correspondences. Traditionally, outlier removal tasks have been addressed using RANSAC approaches. However, these are random in nature and offer no guarantees of finding a good solution. If the amount of mismatches is large, the approach becomes costly because of the need to evaluate a large number of random samples. In contrast, our approach is based on the branch and bound methodology which guarantees that an optimal solution will be found. While most optimal methods trade speed for optimality, the proposed algorithm has competitive running times on problem sizes well beyond what is common in practice. Experiments on both real and synthetic data show that the method outperforms state-of-the-art alternatives, including RANSAC, in terms of solution quality. In addition, the approach is shown to be faster than RANSAC in settings with a large amount of outliers.

Optimal energy management of hybrid electric vehicles requires a priori information regarding future driving conditions; the acquisition and processing of this information is nevertheless often neglected in academic research. This paper introduces a commuter route optimized energy management system, where the bulk of the computations are performed on a server. The idea is to identify commuter routes from historical driving data, using hierarchical agglomerative clustering, and then precompute an optimal solution to the energy management control problem with dynamic programming; the obtained solution can then be transmitted to the vehicle in the form of a lookup table. To investigate the potential of such a system, a simulation study is performed using a detailed vehicle model implemented in the Autonomie simulation environment for MATLAB/Simulink. The simulation results for a plug-in hybrid electric vehicle indicate that the average fuel consumption along the commuter route(s) can be reduced by 4%-9% and battery usage by 10%-15%.