Inlier Points Research Articles

Feature matching based on Gaussian kernel convolution and minimum relative motion

Feature matching is a necessary and important step for remote sensing image registration, intended to establish reliable point correspondences between two sets of features. In this paper, we propose a feature registration model based on local relative motion, which combines Gaussian kernel convolution with relative motion (GRM) vector to obtain better results by removing wrong matches and improving the inlier point accuracy. We first establish putative matching based on the similarity between local descriptors. Then, the preliminary hypothetical matching point set is filtered using consistency with nearest neighbors among the inlier points to obtain a more accurate motion vector, and to fit the real motion vector through the Gaussian convolution kernel. Finally, we find the displacement between the fitted motion vector and the matching generated motion vector. And combine the displacement with the optimization model to find the inlier point set. Experimental results show that our GRM method outperforms related work, achieving better matching results.

Energy-Based Detection of Adverse Weather Effects in LiDAR Data

Autonomous vehicles rely on LiDAR sensors to perceive the environment. Adverse weather conditions like rain, snow, and fog negatively affect these sensors, reducing their reliability by introducing unwanted noise in the measurements. In this work, we tackle this problem by proposing a novel approach for detecting adverse weather effects in LiDAR data. We reformulate this problem as an outlier detection task and use an energy-based framework to detect outliers in point clouds. More specifically, our method learns to associate low energy scores with inlier points and high energy scores with outliers allowing for robust detection of adverse weather effects. In extensive experiments, we show that our method performs better in adverse weather detection and has higher robustness to unseen weather effects than previous state-of-the-art methods. Furthermore, we show how our method can be used to perform simultaneous outlier detection and semantic segmentation. Finally, to help expand the research field of LiDAR perception in adverse weather, we release the SemanticSpray dataset, which contains labeled vehicle spray data in highway-like scenarios. The dataset is available at http://dx.doi.org/10.18725/OPARU-48815 .

DeepEgo: Deep Instantaneous Ego-Motion Estimation Using Automotive Radar

The problem of instantaneous ego-motion estimation with mm-wave automotive radar is studied. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">DeepEgo</i> , a deep learning-based method, is proposed for achieving robust and accurate ego-motion estimation. A hybrid approach that uses neural networks to extract complex features from input point clouds and applies weighted least squares (WLS) for motion estimation is utilized in <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">DeepEgo</i> . Additionally, a novel loss function, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Doppler loss</i> , is proposed to locate ‘inlier points’ originating from detected stationary objects without human annotation. Finally, a challenging real-world automotive radar dataset is selected for extensive performance evaluation. Compared to other methods selected from the literature, significant improvements in estimation accuracy, long-term stability, and runtime performance of DeepEgo in comparison to other methods are demonstrated.

Robust Detection and Modeling of the Major Temporal Arcade in Retinal Fundus Images

The Major Temporal Arcade (MTA) is a critical component of the retinal structure that facilitates clinical diagnosis and monitoring of various ocular pathologies. Although recent works have addressed the quantitative analysis of the MTA through parametric modeling, their efforts are strongly based on an assumption of symmetry in the MTA shape. This work presents a robust method for the detection and piecewise parametric modeling of the MTA in fundus images. The model consists of a piecewise parametric curve with the ability to consider both symmetric and asymmetric scenarios. In an initial stage, multiple models are built from random blood vessel points taken from the blood-vessel segmented retinal image, following a weighted-RANSAC strategy. To choose the final model, the algorithm extracts blood-vessel width and grayscale-intensity features and merges them to obtain a coarse MTA probability function, which is used to weight the percentage of inlier points for each model. This procedure promotes selecting a model based on points with high MTA probability. Experimental results in the public benchmark dataset Digital Retinal Images for Vessel Extraction (DRIVE), for which manual MTA delineations have been prepared, indicate that the proposed method outperforms existing approaches with a balanced Accuracy of 0.7067, Mean Distance to Closest Point of 7.40 pixels, and Hausdorff Distance of 27.96 pixels, while demonstrating competitive results in terms of execution time (9.93 s per image).

Fore-Net: Efficient inlier estimation network for large-scale indoor scenario

Point clouds scanned in the real-world environment, especially for cluttered indoor spaces, are often corrupted by extraneous points from sensor noise or missing points because of occlusion or partially overlapped situations. These problems reduce the performance of subsequent tasks, especially for point cloud registration. This paper proposes a learning-based network, Fore-Net, to efficiently filter out outlier points and retain only inlier points, which can promote the performance of subsequent algorithms by optimizing the quality of source data. We designed a dual space attention module in Fore-Net that can enrich the network's learning by aggregating information from both feature and cartesian spaces and realizing information exchange between point cloud pairs. In addition, the paper proposes two automatic labeling strategies that enable Fore-Net to be trained and tested to detect overlapping regions and denoise the point clouds using existing datasets without hand labeling. We comprehensively evaluated Fore-Net using two large-scale indoor datasets (the 3Dmatch and S3DIS datasets) and one synthetic dataset (the ModelNet40 dataset). The results showed that Fore-Net was more accurate and robust than the state-of-the-art approaches, and Fore-Net could generalize well in different types of indoor scenes. We further validated that by using Fore-Net as a pre-processing step, the performance of registration algorithms, whether classical methods or learning-based methods, could be promoted in the low-overlapping scenario; Fore-Net could also be used as an effective tool to denoise the point clouds of indoor scenes and objects. The results demonstrate Fore-Net's efficiency and compatibility, which will make it attractive for tasks relying on high-quality source data.

Multiview Point Cloud Registration Based on Minimum Potential Energy for Free-Form Blade Measurement

Point cloud registration is an essential step for free-form blade reconstruction in industrial measurement. Nonetheless, measuring defects of the 3-D acquisition system unavoidably result in noisy and incomplete point cloud data, which renders efficient and accurate registration challenging. In this article, we propose a novel global registration method that is based on the minimum potential energy (MPE) method to address these problems. The basic strategy is that the objective function is defined as the MPE optimization function of the physical registration system. The function distributes more weight to the majority of inlier points and less weight to the noise and outliers, which essentially reduces the influence of perturbations in the mathematical formulation. We decompose the solution into a globally optimal approximation procedure and a fine registration process with the trimmed iterative closest point algorithm to boost convergence. The approximation procedure consists of two main steps. First, according to the construction of the force traction operator, we can simply compute the position of the potential energy minimum. Second, to find the MPE point, we propose a new theory that employs two flags to observe the status of the registration procedure. We demonstrate the performance of the proposed algorithm on four types of blades. The proposed method outperforms the other global methods in terms of both accuracy and noise resistance.

Robust detection of non-overlapping ellipses from points with applications to circular target extraction in images and cylinder detection in point clouds

Detection of non-overlapping ellipses from 2-dimensional (2D) edge points is an essential step towards solving typical photogrammetry problems pertaining to feature detection, calibration, and registration of optical instruments. For instance, circular and spherical black and white calibration and registration targets are represented as ellipses in images. Furthermore, the intersection of a cut plane with cylindrical point clouds generates 2D points following elliptic patterns. To this end, this study proposes a collection of new methods for the automatic and robust detection of non-overlapping ellipses from 2D points. These methods will first be applied to detect circular and spherical targets in images and, second, to detect cylinders in 3D point clouds. The method utilizes the Euclidian ellipticity and a new systematic and generalizable threshold to decide if a set of connected points follow an elliptic pattern. When connected points include outliers, the newly proposed robust Monte Carlo-based ellipse fitting method will be deployed. This method includes three new developments: (i) selecting initial subsamples using a bucketing strategy based on the polar angle of the points; (ii) detecting inlier points by reducing the robust ellipse fitting to a robust circle fitting problem; and (iii) choosing the best inlier set amongst all subsamples using adaptive, systematic, and generalizable selection criteria. A new process is presented to extract cylinders from a point cloud by detecting non-overlapping ellipses from the points projected onto an intersecting cut plane. The proposed methods were compared to established state-of-the-art methods, using simulated and real-world datasets, through the design of four sets of original experiments. The experiments include (i) comparisons of robust ellipse fitting; (ii) sensitivity analysis of the ellipse validation criteria; (iii) comparison of non-overlapping ellipse detection; and (iv) detection of pipes from terrestrial laser scanner point clouds. It was found that the proposed robust ellipse detection was superior to four reliable robust methods, including the popular least median of squares, in both simulated and real-world datasets. The proposed process for detecting non-overlapping ellipses achieved F-measure of 99.3% on real images, compared to 42.4%, 65.6%, and 59.2%, obtained using the methods of Fornaciari, Patraucean, and Panagiotakis, respectively. The proposed cylinder extraction method identified all detectable mechanical pipes in two real-world point clouds collected in laboratory and industrial construction site conditions. The results of this investigation show promise for the application of the proposed methods for automatic extraction of circular targets from images and pipes from point clouds.

Voxel-Based Scene Representation for Camera Pose Estimation of a Single RGB Image

Deep learning has been utilized in end-to-end camera pose estimation. To improve the performance, we introduce a camera pose estimation method based on a 2D-3D matching scheme with two convolutional neural networks (CNNs). The scene is divided into voxels, whose size and number are computed according to the scene volume and the number of 3D points. We extract inlier points from the 3D point set in a voxel using random sample consensus (RANSAC)-based plane fitting to obtain a set of interest points consisting of a major plane. These points are subsequently reprojected onto the image using the ground truth camera pose, following which a polygonal region is identified in each voxel using the convex hull. We designed a training dataset for 2D–3D matching, consisting of inlier 3D points, correspondence across image pairs, and the voxel regions in the image. We trained the hierarchical learning structure with two CNNs on the dataset architecture to detect the voxel regions and obtain the location/description of the interest points. Following successful 2D–3D matching, the camera pose was estimated using n-point pose solver in RANSAC. The experiment results show that our method can estimate the camera pose more precisely than previous end-to-end estimators.

AUTOMATED UAV LIDAR STRIP ALIGNMENT IN FORESTED AREAS USING DENSITY-BASED CANOPY CLUSTERING

Abstract. Recently, LiDAR point cloud data acquired by Unmanned Aerial Vehicles (UAVs) are used in many scientific disciplines and like the former photogrammetric techniques these data are usually collected in overlapping strips. Generation of comprehensive models of the scanned areas requires these strips to be aligned together which is a challenging process due to the multi sensor scanning system including the scanning sensor, the GNSS receiver and the IMU sensor. The main errors result from the inaccurate GNSS locations and flight path shifts as well as failure of the GNSS signals in complex urban or forest environments. For that reasons, the development of an automatic feature-dependant method in urban areas or individual tree-based in forest areas where there are no distinct features for strip adjustment in these environments become a must. This research work focuses on automated co-registration/alignment multiple point cloud strips of forested areas acquired from UAV LiDAR (or referred to as ULS) lack of artificial ground control. The main limitations of ULS data of forests are the relatively low sampling density of near ground areas and stem nullity due to the top-view scanning mode of ULS. To obviate this, this work explicates the tree crowns shape to identify the key points required for co-registration by applying a density based clustering algorithm (DBSCAN) to the tree crowns and models resulting clusters with Gaussian mixture models by learning the best parameters using maximum likelihood estimation to define the key points. A feature vector is assigned to each point by quantifying its angular and linear relationship with respect to the local system origin. Next, the similarity score matrix is computed by a fixed geometric relationship between the distance and angle similarity. Then, the maximum weight matching problem is solved for the similarity score to gain point-to-point correspondence. Finally, the optimal 2D rigid transformation parameters (one rotation and two translations without scale factor ) are obtained using permutations to try out for all possible paired combinations and count the number of inlier points satisfying a tolerance of planimetric deviation after alignment within a user defined threshold. The results of two test forest plots with different tree species and ULS point densities show a mean planimetric enhancement from 1.79 m to 0.22 ± 0.13 m for plot one and from 2.33 ± 0.53 m to 0.61 ± 0.21 m.

Scalable KDE-based top-[formula omitted] local outlier detection over large-scale data streams

The detection of local outliers over high-volume data streams is critical for diverse real-time applications in the real world, where the distributions in different subsets of the data tend to be skewed. However, existing methods are not scalable to large-scale high-volume data streams owing to the high complexity of the re-detection of data updates. In this work, we propose a top-n local outlier detection method based on Kernel Density Estimation (KDE) over large-scale high-volume data streams. First, we define a KDE-based Outlier Factor (KOF) to measure the local outlierness score for the data points. Then, we propose the upper bounds of the KOF and an upper-bound-based pruning strategy to quickly eliminate the majority of the inlier points by leveraging the upper bounds without computing the expensive KOF scores. Moreover, we design an Upper-bound pruning-based top-nKOF detection method (UKOF) over data streams to efficiently address the data updates in a sliding window environment. Furthermore, we propose a Lazy update method of UKOF (LUKOF) for bulk updates in high-speed large-scale data streams to further minimize the computation cost. Our comprehensive experimental study demonstrates that the proposed method outperforms the state-of-the-art methods by up to 3,600 times in speed, while achieving the best performance in detecting local outliers over data streams.

A Digital Camera-Based Rotation-Invariant Fingerprint Verification Method

Fingerprint registration and verification is an active area of research in the field of image processing. Usually, fingerprints are obtained from sensors; however, there is recent interest in using images of fingers obtained from digital cameras instead of scanners. An unaddressed issue in the processing of fingerprints extracted from digital images is the angle of the finger during image capture. To match a fingerprint with 100% accuracy, the angles of the matching features should be similar. This paper proposes a rotation and scale-invariant decision-making method for the intelligent registration and recognition of fingerprints. A digital image of a finger is taken as the input and compared with a reference image for derotation. Derotation is performed by applying binary segmentation on both images, followed by the application of speeded up robust feature (SURF) extraction and then feature matching. Potential inliers are extracted from matched features by applying the M-estimator. Matched inlier points are used to form a homography matrix, the difference in the rotation angles of the finger in both the input and reference images is calculated, and finally, derotation is performed. Input fingerprint features are extracted and compared or stored based on the decision support system required for the situation.

3D iris model and reader for iris identification

SummaryIn the presence of eyelids and eyelashes movement, pupil dilation, poor lighting, blur due to movement during iris image acquisition, and factors that collectively cause distortion in the iris image, 2D image‐based iris identification techniques become limited and might lead to iris misclassification due to the dependency on the image appearance (texture) on the possibly distorted image. To alleviate this problem, we introduce a new 3D iris model and reader based upon which iris identification is performed. Using a set of at least two 2D images taken from different views, a small set of reliable and corresponding salient fiducial points (corner points taken from crypts, corona, and serpentine rings of iris pattern) in the two images are extracted, from which a set of 3D iris salient points are obtained using triangulation. Corresponding salient points in the 2D images are found using the Random Sampling Consensus (RANSAC) algorithm, which is robust in identifying the inlier points that correspond to each other in different views of the iris. From this small and reliable salient point set, a denser (high‐resolution) set of extra salient feature points is constructed at minimum cost using a loop subdivision method that yields corresponding extra salient points in the two images from which a high‐resolution 3D iris model is obtained. The points associated with the 3D high‐resolution model are tessellated to form a high‐resolution triangular mesh with the appearance of a triangular patch in the image imported onto the iris personalized 3D model rendering it a 3D iris surface with appearance. This 3D model construction method allows for a 3D classification of a test iris to one of many possible irises stored in the database. The classification is based on a geometric 3D point cloud error. For cross‐validation, the geometric‐based classification can be augmented by considering a mean squared error (MSE) based on the appearance of corresponding points on the test and base models to further disambiguate between irises with close geometric error values.

Moving object detection in aerial infrared images with registration accuracy prediction and feature points selection

Moving object detection in aerial infrared sequences is widely used in many applications, such as automatic traffic monitoring, border protection, and area surveillance. Feature-based image registration is the key step to compensate the background motion before detecting the moving objects. However, in practice it is often difficult to ensure the accurate registration, resulting in incomplete background compensation and thus decreasing detection performance. We find that the feature points in the object regions (FPOs) always impede the accurate registration of background. Besides, when the inlier points chosen by RANSAC excessively concentrate over the image, it will lead to inaccurate registration, especially in the case of insufficient feature points. These two issues are seldom studied before. In this paper, we propose a new framework of moving object detection, in which feature points selection and registration accuracy prediction are devised to improve detection accuracy. To do this, the detection information of previous frame is fed back to feature extraction of current frame for eliminating FPOs. Moreover, a quantitative metric is presented to measure the concentration of inlier points’ distribution over the image, with which one can predict the registration accuracy and determine whether to introduce additional cues for improving detection accuracy. Furthermore, we comprehensively investigate several classical feature extractors for aerial infrared image registration in terms of accuracy and running time, and suggest Speeded Up Robust Features (SURF) as feature detector and Local Difference Binary (LDB) as feature descriptor for image registration in aerial infrared sequences. Experimental results on benchmark infrared sequences show that our proposed method can greatly reduce false positives and also suppress false negatives, and performs favorably in terms of accuracy and efficiency.

RANSAC algorithm for matching inlier correspondences in video stabilisation

In order to stabilise jittery video, we need to find a transform which reduces the distortion between frames. To find this transformation, feature points must be identified in consecutive frames. In the existing methods, correspondences between feature points are found by considering sum of squared differences as matching cost between respective points but by this method, many of the point correspondences are obtained having limited accuracy. To rectify this problem, we proposed here, M-estimator SAmple Consensus (MSAC) algorithm which is variant of random sample consensus (RANSAC) algorithm. In our proposed method, inlier and outlier feature points are found by conventional RANSAC algorithm. Then to match these inlier feature points MSAC algorithm is used which give the robust estimate of transformation between consecutive video frames. The MSAC algorithm is repeated multiple times and at each run the cost of the end result is calculated via Sum of Absolute Differences between both image frames. Sum of absolute difference (SAD) measures the distortion between two frames by evaluating the similarity between image blocks. On the basis of SAD values, affine transform is derived. This transform gives details about the camera motion and is capable to improve the image plane. It is clear from simulation results, inliers correspondences get exactly coincident which gives more favourable results thus stabilising jittery videos.

A robust cylindrical fitting to point cloud data

ABSTRACTEnvironmental, engineering and industrial modelling of natural features (e.g. trees) and man-made features (e.g. pipelines) requires some form of fitting of geometrical objects such as cylinders, which is commonly undertaken using a least-squares method that—in order to get optimal estimation—assumes normal Gaussian distribution. In the presence of outliers, however, this assumption is violated leading to a Gaussian mixture distribution. This study proposes a robust parameter estimation method, which is an improved and extended form of vector algebraic modelling. The proposed method employs expectation maximisation and maximum likelihood estimation (MLE) to find cylindrical parameters in case of Gaussian mixture distribution. MLE computes the model parameters assuming that the distribution of model errors is a Gaussian mixture corresponding to inlier and outlier points. The parameters of the Gaussian mixture distribution and the membership functions of the inliers and outliers are computed using an expectation maximisation algorithm from the histogram of the model error distribution, and the initial guess values for the model parameters are obtained using total least squares. The method, illustrated by a practical example from a terrestrial laser scanning point cloud, is novel in that it is algebraic (i.e. provides a non-iterative solution to the global maximisation problem of the likelihood function), is practically useful for any type of error distribution model and is capable of separating points of interest and outliers.

SPEEDING UP COARSE POINT CLOUD REGISTRATION BY THRESHOLD-INDEPENDENT BAYSAC MATCH SELECTION

This paper presents an algorithm for the automatic registration of terrestrial point clouds by match selection using an efficiently conditional sampling method -- threshold-independent BaySAC (BAYes SAmpling Consensus) and employs the error metric of average point-to-surface residual to reduce the random measurement error and then approach the real registration error. BaySAC and other basic sampling algorithms usually need to artificially determine a threshold by which inlier points are identified, which leads to a threshold-dependent verification process. Therefore, we applied the LMedS method to construct the cost function that is used to determine the optimum model to reduce the influence of human factors and improve the robustness of the model estimate. Point-to-point and point-to-surface error metrics are most commonly used. However, point-to-point error in general consists of at least two components, random measurement error and systematic error as a result of a remaining error in the found rigid body transformation. Thus we employ the measure of the average point-to-surface residual to evaluate the registration accuracy. The proposed approaches, together with a traditional RANSAC approach, are tested on four data sets acquired by three different scanners in terms of their computational efficiency and quality of the final registration. The registration results show the st.dev of the average point-to-surface residuals is reduced from 1.4 cm (plain RANSAC) to 0.5 cm (threshold-independent BaySAC). The results also show that, compared to the performance of RANSAC, our BaySAC strategies lead to less iterations and cheaper computational cost when the hypothesis set is contaminated with more outliers.

SPEEDING UP COARSE POINT CLOUD REGISTRATION BY THRESHOLD-INDEPENDENT BAYSAC MATCH SELECTION

Abstract. This paper presents an algorithm for the automatic registration of terrestrial point clouds by match selection using an efficiently conditional sampling method -- threshold-independent BaySAC (BAYes SAmpling Consensus) and employs the error metric of average point-to-surface residual to reduce the random measurement error and then approach the real registration error. BaySAC and other basic sampling algorithms usually need to artificially determine a threshold by which inlier points are identified, which leads to a threshold-dependent verification process. Therefore, we applied the LMedS method to construct the cost function that is used to determine the optimum model to reduce the influence of human factors and improve the robustness of the model estimate. Point-to-point and point-to-surface error metrics are most commonly used. However, point-to-point error in general consists of at least two components, random measurement error and systematic error as a result of a remaining error in the found rigid body transformation. Thus we employ the measure of the average point-to-surface residual to evaluate the registration accuracy. The proposed approaches, together with a traditional RANSAC approach, are tested on four data sets acquired by three different scanners in terms of their computational efficiency and quality of the final registration. The registration results show the st.dev of the average point-to-surface residuals is reduced from 1.4 cm (plain RANSAC) to 0.5 cm (threshold-independent BaySAC). The results also show that, compared to the performance of RANSAC, our BaySAC strategies lead to less iterations and cheaper computational cost when the hypothesis set is contaminated with more outliers.

SPATIAL ANALYSIS FOR OUTLIER REMOVAL FROM LIDAR DATA

Abstract. Outlier detection in LiDAR point clouds is a necessary process before the subsequent modelling. So far, many studies have been done in order to remove the outliers from LiDAR data. Some of the existing algorithms require ancillary data such as topographic map, multiple laser returns or intensity data which may not be available, and some deal only with the single isolated outliers. This is an attempt to present an algorithm to remove both the single and cluster types of outliers, by exclusively use of the last return data. The outliers will be removed by spatial analyzing of LiDAR point clouds in a hierarchical scheme that is uses a cross-validation technique. The algorithm is tested on a dataset including many single and cluster outliers. Our algorithm can deal with both the irregular LiDAR point clouds and the regular grid data. Experimental results show that the presented algorithm almost completely detects both the single and cluster outliers, but some inlier points are wrongly removed as outlier. An accuracy assessment indicated 0.018% Error α and, 0.352% Error β that are very satisfactory.

Extraction of cylinders and estimation of their parameters from point clouds

In this paper, a novel algorithm is proposed for extracting cylinders and estimating their parameters from 3D point cloud data. First, normal vectors and curvature information are computed for each data point as a preprocessing step. Then potential points that could belong to cylindrical surfaces are extracted by using curvature information. For each potential cylinder point, its neighborhood is considered as inlier points and a robust cylinder fitting algorithm is applied on these inlier points. Inlier points are updated and the fitting process is applied iteratively to propagate to all remaining points belonging to a cylinder. A validation method is proposed to assess whether the detected cylinder is reliable or not. Finally, by applying mean shift clustering, final descriptive parameters of cylinders are estimated accurately. To demonstrate its robustness, the method is tested on both synthetic and complex point clouds with different levels of noise and outliers.

Robust point matching via vector field consensus.

In this paper, we propose an efficient algorithm, called vector field consensus, for establishing robust point correspondences between two sets of points. Our algorithm starts by creating a set of putative correspondences which can contain a very large number of false correspondences, or outliers, in addition to a limited number of true correspondences (inliers). Next, we solve for correspondence by interpolating a vector field between the two point sets, which involves estimating a consensus of inlier points whose matching follows a nonparametric geometrical constraint. We formulate this a maximum a posteriori (MAP) estimation of a Bayesian model with hidden/latent variables indicating whether matches in the putative set are outliers or inliers. We impose nonparametric geometrical constraints on the correspondence, as a prior distribution, using Tikhonov regularizers in a reproducing kernel Hilbert space. MAP estimation is performed by the EM algorithm which by also estimating the variance of the prior model (initialized to a large value) is able to obtain good estimates very quickly (e.g., avoiding many of the local minima inherent in this formulation). We illustrate this method on data sets in 2D and 3D and demonstrate that it is robust to a very large number of outliers (even up to 90%). We also show that in the special case where there is an underlying parametric geometrical model (e.g., the epipolar line constraint) that we obtain better results than standard alternatives like RANSAC if a large number of outliers are present. This suggests a two-stage strategy, where we use our nonparametric model to reduce the size of the putative set and then apply a parametric variant of our approach to estimate the geometric parameters. Our algorithm is computationally efficient and we provide code for others to use it. In addition, our approach is general and can be applied to other problems, such as learning with a badly corrupted training data set.