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/.

Clustering multiple motions from observed point trajectories is a fundamental task in understanding dynamic scenes. Most motion models require multiple tracks to estimate their parameters, hence identifying clusters when multiple motions are observed is a very challenging task. This is even aggravated for high-dimensional motion models. The starting point of our work is that this high-dimensionality of motion model can actually be leveraged to our advantage as sufficiently long trajectories identify the underlying motion uniquely in practice. Consequently, we propose to learn a mapping from trajectories to embedding vectors that represent the generating motion. The obtained trajectory embeddings are useful for clustering multiple observed motions, but are also trained to contain sufficient information to recover the parameters of the underlying motion by utilizing a geometric loss. We therefore are able to use only weak supervision from given motion segmentation to train this mapping. The entire algorithm consisting of trajectory embedding, clustering and motion parameter estimation is highly efficient. We conduct experiments on the Hopkins155, Hopkins12, and KT3DMoSeg datasets and show state-of-the-art performance of our proposed method for trajectory-based motion segmentation on full sequences and its competitiveness on the occluded sequences. Project page: https://ylochman.github.io/trajectory-embedding.

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.

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

Monocular depth estimation is a highly challenging problem that is often addressed with deep neural networks. While these use recognition of high level image features to predict reasonably looking depth maps, the result often has poor metric accuracy. Moreover, the standard feed forward architecture does not allow modification of the prediction based on cues other than the image.In this paper we relax the monocular depth estimation task by proposing a network that allows us to complement image features with a set of auxiliary variables. These allow disambiguation when image features are not enough to accurately pinpoint the exact depth map and can be thought of as a low dimensional parameterization of the surfaces that are reasonable monocular predictions. By searching the parameterization we can combine monocular estimation with traditional photoconsistency or geometry based methods to achieve both visually appealing and metrically accurate surface estimations. Since we relax the problem we are able to work with smaller networks than current architectures. In addition we design a self-supervised training scheme, eliminating the need for ground truth image depth-map pairs. Our experimental evaluation shows that our method generates more accurate depth maps and generalizes better than competing state-of-the-art approaches.

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

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 explore the role of duality principles within the problem of rotation averaging, a fundamental task in a wide range of applications. In its conventional form, rotation averaging is stated as a minimization over multiple rotation constraints. As these constraints are non-convex, this problem is generally considered challenging to solve globally. We show how to circumvent this difficulty through the use of Lagrangian duality. While such an approach is well-known it is normally not guaranteed to provide a tight relaxation. Based on spectral graph theory, we analytically prove that in many cases there is no duality gap unless the noise levels are severe. This allows us to obtain certifiably global solutions to a class of important non-convex problems in polynomial time. We also propose an efficient, scalable algorithm that outperforms general purpose numerical solvers by a large margin and compares favourably to current state-of-the-art. Further, our approach is able to handle the large problem instances commonly occurring in structure from motion settings and it is trivially parallelizable. Experiments are presented for a number of different instances of both synthetic and real-world data.

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 study the convex envelopes of a new class of functions. Using this approach, we are able to unify two important classes of regularizers from unbiased non-convex formulations and weighted nuclear norm penalties. This opens up for possibilities of combining the best of both worlds, and to leverage each methods contribution to cases where simply enforcing one of the regularizers are insufficient. We show that the proposed regularizers can be incorporated in standard splitting schemes such as Alternating Direction Methods of Multipliers (ADMM), and other sub-gradient methods. This can be implemented efficiently since the the proximal operator can be computed fast. Furthermore, we show on real non-rigid structure from motion datasets, the issues that arise from using weighted nuclear norm penalties, and how this can be remedied using our proposed prior-free method.

In this paper we explore the role of duality principles within the problem of rotation averaging, a fundamental task in a wide range of computer vision applications. In its conventional form, rotation averaging is stated as a minimization over multiple rotation constraints. As these constraints are non-convex, this problem is generally considered challenging to solve globally. We show how to circumvent this difficulty through the use of Lagrangian duality. While such an approach is well-known it is normally not guaranteed to provide a tight relaxation. Based on spectral graph theory, we analytically prove that in many cases there is no duality gap unless the noise levels are severe. This allows us to obtain certifiably global solutions to a class of important non-convex problems in polynomial time. We also propose an efficient, scalable algorithm that outperforms general purpose numerical solvers and is able to handle the large problem instances commonly occurring in structure from motion settings. The potential of this proposed method is demonstrated on a number of different problems, consisting of both synthetic and real-world data.

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.

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.

In computer vision, many problems can be formulated as finding a low rank approximation of a given matrix. Ideally, if all elements of the measurement matrix are available, this is easily solved in the L2-norm using factorization. However, in practice this is rarely the case. Lately, this problem has been addressed using different approaches, one is to replace the rank term by the convex nuclear norm, another is to derive the convex envelope of the rank term plus a data term. In the latter case, matrices are divided into sub-matrices and the envelope is computed for each subblock individually. In this paper a new convex envelope is derived which takes all sub-matrices into account simultaneously. This leads to a simpler formulation, using only one parameter to control the trade-of between rank and data fit, for applications where one seeks low rank approximations of multiple matrices with the same rank. We show in this paper how our general framework can be used for manifold denoising of several images at once, as well as just denoising one image. Experimental comparisons show that our method achieves results similar to state-of-the-art approaches while being applicable for other problems such as linear shape model estimation.

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.

Online 3D reconstruction has been an active research area for a long time. Since the release of the Microsoft Kinect Camera and publication of KinectFusion [11] attention has been drawn how to acquire dense models in real-time. In this paper we present a method to make online 3D reconstruction which increases robustness for scenes with little structure information and little texture information. It is shown empirically that our proposed method also increases robustness when the distance between the camera positions becomes larger than what is commonly assumed. Quantitative and qualitative results suggest that this approach can handle situations where other well-known methods fail. This is important in, for example, robotics applications like when the camera position and the 3D model must be created online in real-time.

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.

Many standard optimization methods for segmentation and reconstruction compute ML model estimates for appearance or geometry of segments, e.g. Zhu-Yuille [23], Torr [20], Chan-Vese [6], GrabCut [18], Delong et al. [8]. We observe that the standard likelihood term in these formu-lations corresponds to a generalized probabilistic K-means energy. In learning it is well known that this energy has a strong bias to clusters of equal size [11], which we express as a penalty for KL divergence from a uniform distribution of cardinalities. However, this volumetric bias has been mostly ignored in computer vision. We demonstrate signif- icant artifacts in standard segmentation and reconstruction methods due to this bias. Moreover, we propose binary and multi-label optimization techniques that either (a) remove this bias or (b) replace it by a KL divergence term for any given target volume distribution. Our general ideas apply to continuous or discrete energy formulations in segmenta- tion, stereo, and other reconstruction problems.

One of the major research areas in computer vision is scene reconstruction from image streams. The advent of RGB-D cameras, such as the Microsoft Kinect, has lead to new possibilities for performing accurate and dense 3D reconstruction. There are already well-working algorithms to acquire 3D models from depth sensors, both for large and small scale scenes. However, these methods often break down when the scene geometry is not so informative, for example, in the case of planar surfaces. Similarly, standard image-based methods fail for texture-less scenes. We combine both color and depth measurements from an RGB-D sensor to simultaneously reconstruct both the camera motion and the scene geometry in a robust manner. Experiments on real data show that we can accurately reconstruct large-scale 3D scenes despite many planar surfaces.

Stereo matching is an inherently difficult problem due to ambiguous and noisy texture. The non-convexity and non-differentiability makes local linear (or quadratic) approximations poor, thereby preventing the use of standard local descent methods. Therefore recent methods are predominantly based on discretization and/or random sampling of some class of approximating surfaces (e.g. planes). While these methods are very efficient in generating a rough surface estimate, via either fusion of proposals or label propagation, the end result is usually not as smooth as desired. In this paper we show that, if the objective function is decomposed correctly, local refinement of candidate solutions can be performed using an ADMM approach. This allows searching over more general function classes, thereby resulting in visually more appealing smooth surface estimations.

It is commonly believed that higher order smoothness should be modeled using higher order interactions. For example, 2nd order derivatives for deformable (active) contours are represented by triple cliques. Similarly, the 2nd order regularization methods in stereo predominantly use MRF models with scalar (1D) disparity labels and triple clique interactions. In this paper we advocate a largely overlooked alternative approach to stereo where 2nd order surface smoothness is represented by pairwise interactions with 3D-labels, e.g. tangent planes. This general paradigm has been criticized due to perceived computational complexity of optimization in higher-dimensional label space. Contrary to popular beliefs, we demonstrate that representing 2nd order surface smoothness with 3D labels leads to simpler optimization problems with (nearly) sub modular pairwise interactions. Our theoretical and experimental results demonstrate advantages over state-of-the-art methods for 2nd order smoothness stereo.

Energies with high-order non-sub modular interactions have been shown to be very useful in vision due to their high modeling power. Optimization of such energies, however, is generally NP-hard. A naive approach that works for small problem instances is exhaustive search, that is, enumeration of all possible labelings of the underlying graph. We propose a general minimization approach for large graphs based on enumeration of labelings of certain small patches. This partial enumeration technique reduces complex high-order energy formulations to pair wise Constraint Satisfaction Problems with unary costs (uCSP), which can be efficiently solved using standard methods like TRW-S. Our approach outperforms a number of existing state-of-the-art algorithms on well known difficult problems (e.g. curvature regularization, stereo, deconvolution), it gives near global minimum and better speed. Our main application of interest is curvature regularization. In the context of segmentation, our partial enumeration technique allows to evaluate curvature directly on small patches using a novel integral geometry approach.

We propose an energy-based framework for approximating surfaces from a cloud of point measurements corrupted by noise and outliers. Our energy assigns a tangent plane to each (noisy) data point by minimizing the squared distances to the points and the irregularity of the surface implicitly defined by the tangent planes. In order to avoid the well-known ”shrinking” bias associated with first-order surface regularization, we choose a robust smoothing term that approximates curvature of the underlying surface. In contrast to a number of recent publications estimating curvature using discrete (e.g. binary) labellings with triple-cliques we use higher-dimensional labels that allows modeling curvature with only pair-wise interactions. Hence, many standard optimization algorithms (e.g. message passing, graph cut, etc) can minimize the proposed curvature-based regularization functional. The accuracy of our approach for representing curvature is demonstrated by theoretical and empirical results on synthetic and real data sets from multiview reconstruction and stereo.

In this paper we consider the problem of outlier removal for large scale multiview reconstruction problems. An efficient and very popular method for this task is RANSAC. However, as RANSAC only works on a subset of the images, mismatches in longer point tracks may go undetected. To deal with this problem we would like to have, as a post processing step to RANSAC, a method that works on the entire (or a larger) part of the sequence. In this paper we consider two algorithms for doing this. The first one is related to a method by Sim & Hartley where a quasiconvex problem is solved repeatedly and the error residuals with the largest error is removed. Instead of solving a quasiconvex problem in each step we show that it is enough to solve a single LP or SOCP which yields a significant speedup. Using duality we show that the same theoretical result holds for our method. The second algorithm is a faster version of the first, and it is related to the popular method of L1-optimization. While it is faster and works very well in practice, there is no theoretical guarantee of success. We show that these two methods are related through duality, and evaluate the methods on a number of data sets with promising results.

We study geometric reconstruction problems in one-dimensional retina vision. In such problems, the scene is modeled as a two-dimensional plane, and the camera sensor produces one-dimensional images of the scene. Our main contribution is an efficient method for computing the global optimum to the structure and motion problem with respect to the $L_{\infty}$ norm of the reprojection errors. One-dimensional cameras have proven useful in several applications, most prominently for autonomous vehicles, where they are used to provide inexpensive and reliable navigational systems. Previous results on one-dimensional vision are limited to the classification and solving of minimal cases, bundle adjustment for finding local optima, and linear algorithms for algebraic cost functions. In contrast, we present an approach for finding globally optimal solutions with respect to the $L_{\infty}$ norm of the angular reprojection errors. We show how to solve intersection and resection problems as well as the problem of simultaneous localization and mapping (SLAM). The algorithm is robust to use when there are missing data, which means that all points are not necessarily seen in all images. Our approach has been tested on a variety of different scenarios, both real and synthetic. The algorithm shows good performance for intersection and resection and for SLAM with up to five views. For more views the high dimension of the search space tends to give long running times. The experimental section also gives interesting examples showing that for one-dimensional cameras with limited field of view the SLAM problem is often inherently ill-conditioned.

Work in multiple view geometry has focused on obtaining globally optimal solutions at the price of computational time efficiency. On the other hand, traditional bundle adjustment algorithms have been found to provide good solutions even though there may be multiple local minima. In this paper we justify this observation by giving a simple sufficient condition for global optimality that can be used to verify that a solution obtained from any local method is indeed global. The method is tested on numerous problem instances of both synthetic and real data sets. In the vast majority of cases we are able to verify that the solutions are optimal, in particular for small-scale problems. We also develop a branch and bound procedure that goes beyond verification. In cases where the sufficient condition does not hold, the algorithm returns either of the following two results: (i) a certificate of global optimality for the local solution or (ii) the global solution.

The concept of graph cuts is by now a standard method for all sorts of low level vision problems. Its popularity is largely due to the fact that globally or near globally optimal solutions can be computed using efficient max flow algorithms. On the other hand it has been observed that this method may suffer from metrication errors. Recent work has begun studying continuous versions of graph cuts, which give smaller metrication errors. Another advantage is that continuous cuts are straightforward to parallelize. In this paper we extend the class of functionals that can be optimized in the continuous setting to include anisotropic TV-norms. We show that there is a so called coarea formula for these functionals making it possible to minimize them by solving a convex problem. We also show that the concept of a-expansion moves can be reformulated to fit the continuous formulation, and we derive approximation bounds in analogy with the discrete case. A continuous version of the Potts model for multi-class segmentation problems is presented, and it is shown how to obtain provably good solutions using continuous α-expansions.

In this paper, we propose a practical and efficient method for finding the globally optimal solution to the problem of determining the pose of an object. We present a framework that allows us to use point-to-point, point-to-line, and point-to-plane correspondences for solving various types of pose and registration problems involving euclidean (or similarity) transformations. Traditional methods such as the iterative closest point algorithm or bundle adjustment methods for camera pose may get trapped in local minima due to the nonconvexity of the corresponding optimization problem. Our approach of solving the mathematical optimization problems guarantees global optimality. The optimization scheme is based on ideas from global optimization theory, in particular convex underestimators in combination with branch-and-bound methods. We provide a provably optimal algorithm and demonstrate good performance on both synthetic and real data. We also give examples of where traditional methods fail due to the local minima problem.

We present a framework for computing optimal transformations, aligning one point set to another, in the presence of outliers. Example applications include shape matching and registration (using, for example, similarity, affine or projective transformations) as well as multiview reconstruction problems (triangulation, camera pose etc.). While standard methods like RANSAC essentially use heuristics to cope with outliers, we seek to find the largest possible subset of consistent correspondences and the globally optimal transformation aligning the point sets. Based on theory from computational geometry, we show that this is indeed possible to accomplish in polynomial-time. We develop several algorithms which make efficient use of convex programming. The scheme has been tested and evaluated on both synthetic and real data for several applications.

In this paper we consider the problem of solving different pose and registration problems under rotational constraints. Traditionally, methods such as the iterative closest point algorithm have been used to solve these problems. They may however get stuck in local minima due to the non-convexity of the problem. In recent years methods for finding the global optimum, based on Branch and Bound and convex under-estimators, have been developed. These methods are provably optimal, however since they are based on global optimization methods they are in general more time consuming than local methods. In this paper we adopt a dual approach. Rather than trying to find the globally optimal solution we investigate the quality of the solutions obtained using Lagrange duality. Our approach allows us to formulate a single convex semidefinite program that approximates the original problem well.

In this paper we introduce two new methods for solving binary quadratic problems. While spectral relaxation methods have been the workhorse subroutine for a wide variety of computer vision problems - segmentation, clustering, image restoration to name a few - it has recently been challenged by semidefinite programming (SDP) relaxations. In fact, it can be shown that SDP relaxations produce better lower bounds than spectral relaxations on binary problems with a quadratic objective function. On the other hand, the computational complexity for SDP increases rapidly as the number of decision variables grows making them inapplicable to large scale problems. Our methods combine the merits of both spectral and SDP relaxations -better (lower) bounds than traditional spectral methods and considerably faster execution times than SDP. The first method is based on spectral subgradients and can be applied to large scale SDPs with binary decision variables and the second one is based on the trust region problem. Both algorithms have been applied to several large scale vision problems with good performance.

The structure and motion problem of multiple one-dimensional projections of a two-dimensional environment is studied. One-dimensional cameras have proven useful in several different applications, most prominently for autonomous guided vehicles, but also in ordinary vision for analysing planar motion and the projection of lines. Previous results on one-dimensional vision are limited to classifying and solving minimal cases, bundle adjustment for finding local minima to the structure and motion problem and linear algorithms based on algebraic cost functions. In this paper, we present a method for finding the global minimum to the structure and motion problem using the max norm of reprojection errors. We show how the optimal solution can be computed efficiently using simple linear programming techniques. The algorithms have been tested on a variety of different scenarios, both real and synthetic, with good performance. In addition, we show how to solve the multiview triangulation problem, the camera pose problem and how to dualize the algorithm in the Carlsson duality sense, all within the same framework.

Indisputably Normalized Cuts is one of the most popular segmentation algorithms in computer vision. It has been applied to a wide range of segmentation tasks with great success. A number of extensions to this approach have also been proposed, ones that can deal with multiple classes or that can incorporate a priori information in the form of grouping constraints. However, what is common for all these suggested methods is that they are noticeably limited and can only address segmentation problems on a very specific form. In this paper, we present a reformulation of Normalized Cut segmentation that in a unified way can handle all types of linear equality constraints for an arbitrary number of classes. This is done by restating the problem and showing how linear constraints can be enforced exactly through duality. This allows us to add group priors, for example, that certain pixels should belong to a given class. In addition, it provides a principled way to perform multi-class segmentation for tasks like interactive segmentation. The method has been tested on real data with convincing results.

In this paper we consider the problem of solving geometric reconstruction problems with the L∞-norm. Previous work has shown that globally optimal solutions can be computed reliably for a series of such problems. The methods for computing the solutions have relied on the property of quasiconvexity. For quasiconvex problems, checking if there exists a solution below a certain objective value can be posed as a convex feasibility problem. To solve the L∞-problem one typically employs a bisection algorithm, generating a sequence of convex problems. In this paper we present more efficient ways of computing the solutions. We derive necessary and sufficient conditions for a global optimum. A key property is that of pseudoconvexity, which is a stronger condition than quasiconvexity. The results open up the possibility of using local optimization methods for more efficient computations. We present two such algorithms. The first one is an interior point method that uses the KKT conditions and the second one is similar to the bisection method in the sense it solves a sequence of SOCP problems. Results are presented and compared to the standard bisection algorithm on real data for various problems and scenarios with improved performance.

In this paper we propose a practical and efficient method for finding the globally optimal solution to the problem of pose estimation of a known object. We present a framework that allows us to use both point-to-point, point-to-line and point-to-plane correspondences in the optimization algorithm. Traditional methods such as the iterative closest point algorithm may get trapped in local minima due to the non-convexity of the problem, however, our approach guarantees global optimality. The approach is based on ideas from global optimization theory, in particular, convex under-estimators in combination with branch and bound. We provide a provably optimal algorithm and demonstrate good performance on both synthetic and real data.