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Assessing Losses for Point Set Registration
Abstract
This letter introduces a framework for evaluation of the losses used in point set registration. In order for a loss to be useful with a local optimizer, such as e.g. Levenberg-Marquardt, or expectation maximization (EM), it must be monotonic with respect to the sought transformation. This motivates us to introduce
monotonicity violation probability
(MVP) curves, and use these to assess monotonicity empirically for many different local distances, such as point-to-point, point-to-plane, and plane-to-plane. We also introduce a local shape-to-shape distance, based on the Wasserstein distance of the local normal distributions. Evaluation is done on a comprehensive benchmark of terrestrial lidar scans from two publicly available datasets. It demonstrates that matching robustness can be improved significantly, by using kernel versions of local distances together with inverse density based sample weighting.
... Another branch of methods employs kernel density estimation by setting ρ in (2.5) to a smooth kernel function. To their benefit, losses based on kernel density have improved monotonicity properties [106] compared to the L 2 loss used in ICP. In [108] the kernel is set to an isotropic Gaussian. ...
... The intuition is that for aligned scans, nearby surface points are expected to lie within this distance. The Huber loss L δ , defined in (7), makes the cost less sensitive to outliers [70], [71]. This is done by piecewise reshaping the cost function to increase quadratically for small values and linearly for larger values: ...
This article presents an accurate, highly efficient, and learning-free method for large-scale odometry estimation using spinning radar, empirically found to generalize well across very diverse environments—outdoors, from urban to woodland, and indoors in warehouses and mines—without changing parameters. Our method integrates motion compensation within a sweep with one-to-many scan registration that minimizes distances between nearby oriented surface points and mitigates outliers with a robust loss function. Extending our previous approach conservative filtering for efficient and accurate radar odometry (CFEAR), we present an in-depth investigation on a wider range of datasets, quantifying the importance of filtering, resolution, registration cost and loss functions, keyframe history, and motion compensation. We present a new solving strategy and configuration that overcomes previous issues with sparsity and bias, and improves our state-of-the-art by 38%, thus, surprisingly, outperforming radar simultaneous localization and mapping (SLAM) and approaching lidar SLAM. The most accurate configuration achieves 1.09% error at 5 Hz on the Oxford benchmark, and the fastest achieves 1.79% error at 160 Hz.
... The intuition is that for aligned scans, nearby surface points are expected to lie within this distance. The Huber loss L δ , defined in (7), makes the cost less sensitive to outliers [70], [71]. This is done by piecewise reshaping the cost function to increase quadratically for small values and linearly for larger values: ...
This paper presents an accurate, highly efficient, and learning-free method for large-scale odometry estimation using spinning radar, empirically found to generalize well across very diverse environments -- outdoors, from urban to woodland, and indoors in warehouses and mines - without changing parameters. Our method integrates motion compensation within a sweep with one-to-many scan registration that minimizes distances between nearby oriented surface points and mitigates outliers with a robust loss function. Extending our previous approach CFEAR, we present an in-depth investigation on a wider range of data sets, quantifying the importance of filtering, resolution, registration cost and loss functions, keyframe history, and motion compensation. We present a new solving strategy and configuration that overcomes previous issues with sparsity and bias, and improves our state-of-the-art by 38%, thus, surprisingly, outperforming radar SLAM and approaching lidar SLAM. The most accurate configuration achieves 1.09% error at 5Hz on the Oxford benchmark, and the fastest achieves 1.79% error at 160Hz.
... Many perception tasks, including odometry estimation [1], localization [2], and sensor calibration [3], rely on point cloud registration. However, registration sometimes provides incorrect estimates; e.g., due to local minima of the registration cost function [4], uncompensated motion distortion [5], rapid rotation (leading to poor initial pose estimates) [6], or when the registration problem is geometrically underconstrained [7], [8]. ...
Robust perception is an essential component to enable long-term operation of mobile robots. It depends on failure resilience through reliable sensor data and preprocessing, as well as failure awareness through introspection, for example the ability to self-assess localization performance. This paper presents CorAl: a principled, intuitive, and generalizable method to measure the quality of alignment between pairs of point clouds, which learns to detect alignment errors in a self-supervised manner. CorAl compares the differential entropy in the point clouds separately with the entropy in their union to account for entropy inherent to the scene. By making use of dual entropy measurements, we obtain a quality metric that is highly sensitive to small alignment errors and still generalizes well to unseen environments. In this work, we extend our previous work on lidar-only CorAl to radar data by proposing a two-stage filtering technique that produces high-quality point clouds from noisy radar scans. Thus we target robust perception in two ways: by introducing a method that introspectively assesses alignment quality, and applying it to an inherently robust sensor modality. We show that our filtering technique combined with CorAl can be applied to the problem of alignment classification, and that it detects small alignment errors in urban settings with up to 98% accuracy, and with up to 96% if trained only in a different environment. Our lidar and radar experiments demonstrate that CorAl outperforms previous methods both on the ETH lidar benchmark, which includes several indoor and outdoor environments, and the large-scale Oxford and MulRan radar data sets for urban traffic scenarios The results also demonstrate that CorAl generalizes very well across substantially different environments without the need of retraining.
... Many perception tasks, including localization [1], scene understanding and sensor calibration [2], rely on point cloud registration. However, registration may provide incorrect estimates due to local minima of the registration cost function [3], uncompensated motion distortion [4], noise or when the registration problem is geometrically under-constrained [5], [6]. Consequently, it is essential to measure alignment quality and to reject or re-estimate alignment when quality is low. ...
In robotics perception, numerous tasks rely on point cloud registration. However, currently there is no method that can automatically detect misaligned point clouds reliably and without environment-specific parameters. We propose "CorAl", an alignment quality measure and alignment classifier for point cloud pairs, which facilitates the ability to introspectively assess the performance of registration. CorAl compares the joint and the separate entropy of the two point clouds. The separate entropy provides a measure of the entropy that can be expected to be inherent to the environment. The joint entropy should therefore not be substantially higher if the point clouds are properly aligned. Computing the expected entropy makes the method sensitive also to small alignment errors, which are particularly hard to detect, and applicable in a range of different environments. We found that CorAl is able to detect small alignment errors in previously unseen environments with an accuracy of 95% and achieve a substantial improvement to previous methods.
... These methods find the relative transformation between point sets by modeling them as densities using a kernel, and then minimize a loss based on correlation or divergence [18]. The density based losses have improved monotonicity compared to the L 2 loss used in ICP [30]. Feature matching: In order to increase the accuracy of the correspondence set, it is beneficial to use geometric features [28]. ...
Probabilistic methods for point set registration have interesting theoretical properties, such as linear complexity in the number of used points, and they easily generalize to joint registration of multiple point sets. In this work, we improve their recognition performance to match state of the art. This is done by incorporating learned features, by adding a von Mises-Fisher feature model in each mixture component, and by using learned attention weights. We learn these jointly using a registration loss learning strategy (RLL) that directly uses the registration error as a loss, by back-propagating through the registration iterations. This is possible as the probabilistic registration is fully differentiable, and the result is a learning framework that is truly end-to-end. We perform extensive experiments on the 3DMatch and Kitti datasets. The experiments demonstrate that our approach benefits significantly from the integration of the learned features and our learning strategy, outperforming the state-of-the-art on Kitti. Code is available at https://github.com/felja633/RLLReg.
Registration of point cloud data containing both depth and color information is critical for a variety of applications, including in-field robotic plant manipulation, crop growth modeling, and autonomous navigation. However, current state-of-the-art registration methods often fail in challenging agricultural field conditions due to factors such as occlusions, plant density, and variable illumination. To address these issues, we propose the NDT-6D registration method, which is a color-based variation of the Normal Distribution Transform (NDT) registration approach for point clouds. Our method computes correspondences between pointclouds using both geometric and color information and minimizes the distance between these correspondences using only the three-dimensional (3D) geometric dimensions. We evaluate the method using the GRAPES3D data set collected with a commercial-grade RGB-D sensor mounted on a mobile platform in a vineyard. Results show that registration methods that only rely on depth information fail to provide quality registration for the tested data set. The proposed color-based variation outperforms state-of-the-art methods with a root mean square error (RMSE) of 1.1–1.6 cm for NDT-6D compared with 1.1–2.3 cm for other color-information-based methods and 1.2–13.7 cm for noncolor-information-based methods. The proposed method is shown to be robust against noises using the TUM RGBD data set by artificially adding noise present in an outdoor scenario. The relative pose error (RPE) increased
14% for our method compared to an increase of
75% for the best-performing registration method. The obtained average accuracy suggests that the NDT-6D registration methods can be used for in-field precision agriculture applications, for example, crop detection, size-based maturity estimation, and growth modeling.
This paper presents CorAl: a principled, intuitive, and generalizable method to measure the quality of alignment between pairs of point clouds, which learns to detect alignment errors in a self-supervised manner. CorAl compares the differential entropy in the point clouds separately with the entropy in their union to account for entropy inherent to the scene. By making use of dual entropy measurements, we obtain a quality metric that is highly sensitive to small alignment errors and still generalizes well to unseen environments. In this work, we extend our previous work on lidar-only CorAl to radar data by proposing a two-step filtering technique that produces highquality point clouds from noisy radar scans. Thus we target robust perception in two ways: by introducing a method that introspectively assesses alignment quality, and applying it to inherently robust sensor modalities. We show that our filtering technique combined with CorAl can be applied to the problem of alignment classification, and detect small alignment errors in urban settings with up to 98% accuracy, and with up to 96% if trained only in a different environment. Our lidar and radar experiments demonstrate that CorAl outperforms previous methods both on the ETH lidar benchmark, which includes several indoor and outdoor environments, and the large-scale Oxford and MulRan radar data sets for urban traffic scenarios, and also enables a high degree of generalization across substantially different environments without the need of retraining.
Probabilistic methods for point set registration have demonstrated competitive results in recent years. These techniques estimate a probability distribution model of the point clouds. While such a representation has shown promise, it is highly sensitive to variations in the density of 3D points. This fundamental problem is primarily caused by changes in the sensor location across point sets. We revisit the foundations of the probabilistic registration paradigm. Contrary to previous works, we model the underlying structure of the scene as a latent probability distribution, and thereby induce invariance to point set density changes. Both the probabilistic model of the scene and the registration parameters are inferred by minimizing the Kullback-Leibler divergence in an Expectation Maximization based framework. Our density-adaptive registration successfully handles severe density variations commonly encountered in terrestrial Lidar applications. We perform extensive experiments on several challenging real-world Lidar datasets. The results demonstrate that our approach outperforms state-of-the-art probabilistic methods for multi-view registration, without the need of re-sampling.
3D laser scanners are frequently used sensors for mobile robots or autonomous cars and they are often used to perceive the static as well as dynamic aspects in the scene. In this context, matching 3D point clouds of objects is a crucial capability. Most matching methods such as numerous flavors of ICP provide little information about the quality of the match, i.e. how well do the matched objects correspond to each other, which goes beyond point-to-point or point-to-plane distances. In this paper, we propose a projective method that yields a probabilistic measure for the quality of matched scans. It not only considers the differences in the point locations but can also take free-space information into account. Our approach provides a probabilistic measure that is meaningful enough to evaluate scans and to cluster real-world data such as scans taken with Velodyne scanner in urban scenes in an unsupervised manner.
Transport-based techniques for signal and data analysis have recently received increased interest. Given their ability to provide accurate generative models for signal intensities and other data distributions, they have been used in a variety of applications, including content-based retrieval, cancer detection, image superresolution, and statistical machine learning, to name a few, and they have been shown to produce state-of-the-art results. Moreover, the geometric characteristics of transport-related metrics have inspired new kinds of algorithms for interpreting the meaning of data distributions. Here, we provide a practical overview of the mathematical underpinnings of mass transport-related methods, including numerical implementation, as well as a review, with demonstrations, of several applications. Software accompanying this article is available from [43].
This course covers algorithms and tools for measuring, calibrating, and registering the information required to build models of real scenes with the intent of photo-realistic image synthesis. We focus on the pipeline, i.e. how it all fits together, and practical artist-driven methods and algorithms for capturing and fusing all captured data into a Virtual Photo Set (VPS): a geometric 3D-model of the scene, the lighting (radiance distribution) in the scene, and information describing the materials (color, reflectance and texture) of objects within the scene so that photo-realistic renderings of virtual objects can be produced.
This paper describes a probabilistic generative model and its associated algorithm to jointly register multiple point sets. The vast majority of state-of-the-art registration techniques select one of the sets as the “model” and perform pairwise alignments between the other sets and this set. The main drawback of this mode of operation is that there is no guarantee that the model-set is free of noise and outliers, which contaminates the estimation of the registration parameters. Unlike previous work, the proposed method treats all the point sets on an equal footing: they are realizations of a Gaussian mixture (GMM) and the registration is cast into a clustering problem. We formally derive an EM algorithm that estimates both the GMM parameters and the rotations and translations that map each individual set onto the “central” model. The mixture means play the role of the registered set of points while the variances provide rich information about the quality of the registration. We thoroughly validate the proposed method with challenging datasets, we compare it with several state-of-the-art methods, and we show its potential for fusing real depth data.
The topic of this review is geometric registration in robotics. Registration algorithms associate sets of data into a common coordinate system. They have been used extensively in object reconstruction, inspection, medical application, and localization of mobile robotics. We focus on mobile robotics applications in which point clouds are to be registered. While the underlying principle of those algorithms is simple, many variations have been proposed for many different applications. In this review, we give a historical perspective of the registration problem and show that the plethora of solutions can be organized and differentiated according to a few elements. Accordingly, we present a formalization of geometric registration and cast algorithms proposed in the literature into this framework. Finally, we review a few applications of this framework in mobile robotics that cover different kinds of platforms, environments, and tasks. These examples allow us to study the specific requirements of each use case and the necessary configuration choices leading to the registration implementation. Ultimately, the objective of this review is to provide guidelines for the choice of geometric registration configuration.
Many modern sensors used for mapping produce 3D point clouds, which are typically registered together using the iterative closest point (ICP) algorithm. Because ICP has many variants whose performances depend on the environment and the sensor, hundreds of variations have been published. However, no comparison frameworks are available, leading to an arduous selection of an appropriate variant for particular experimental conditions. The first contribution of this paper consists of a protocol that allows for a comparison between ICP variants, taking into account a broad range of inputs. The second contribution is an open-source ICP library, which is fast enough to be usable in multiple real-world applications, while being modular enough to ease comparison of multiple solutions. This paper presents two examples of these field applications. The last contribution is the comparison of two baseline ICP variants using data sets that cover a rich variety of environments. Besides demonstrating the need for improved ICP methods for natural, unstructured and information-deprived environments, these baseline variants also provide a solid basis to which novel solutions could be compared. The combination of our protocol, software, and baseline results demonstrate convincingly how open-source software can push forward the research in mapping and navigation.
In this paper, we present a unified framework for the rigid and nonrigid point set registration problem in the presence of significant amounts of noise and outliers. The key idea of this registration framework is to represent the input point sets using Gaussian mixture models. Then, the problem of point set registration is reformulated as the problem of aligning two Gaussian mixtures such that a statistical discrepancy measure between the two corresponding mixtures is minimized. We show that the popular iterative closest point (ICP) method and several existing point set registration methods in the field are closely related and can be reinterpreted meaningfully in our general framework. Our instantiation of this general framework is based on the the L2 distance between two Gaussian mixtures, which has the closed-form expression and in turn leads to a computationally efficient registration algorithm. The resulting registration algorithm exhibits inherent statistical robustness, has an intuitive interpretation, and is simple to implement. We also provide theoretical and experimental comparisons with other robust methods for point set registration.
We present a system for accurate real-time mapping of complex and arbitrary indoor scenes in variable lighting conditions, using only a moving low-cost depth camera and commodity graphics hardware. We fuse all of the depth data streamed from a Kinect sensor into a single global implicit surface model of the observed scene in real-time. The current sensor pose is simultaneously obtained by tracking the live depth frame relative to the global model using a coarse-to-fine iterative closest point (ICP) algorithm, which uses all of the observed depth data available. We demonstrate the advantages of tracking against the growing full surface model compared with frame-to-frame tracking, obtaining tracking and mapping results in constant time within room sized scenes with limited drift and high accuracy. We also show both qualitative and quantitative results relating to various aspects of our tracking and mapping system. Modelling of natural scenes, in real-time with only commodity sensor and GPU hardware, promises an exciting step forward in augmented reality (AR), in particular, it allows dense surfaces to be reconstructed in real-time, with a level of detail and robustness beyond any solution yet presented using passive computer vision.
With the advent of new, low-cost 3D sensing hardware such as the Kinect, and continued efforts in advanced point cloud processing, 3D perception gains more and more importance in robotics, as well as other fields. In this paper we present one of our most recent initiatives in the areas of point cloud perception: PCL (Point Cloud Library - http://pointclouds.org). PCL presents an advanced and extensive approach to the subject of 3D perception, and it's meant to provide support for all the common 3D building blocks that applications need. The library contains state-of- the art algorithms for: filtering, feature estimation, surface reconstruction, registration, model fitting and segmentation. PCL is supported by an international community of robotics and perception researchers. We provide a brief walkthrough of PCL including its algorithmic capabilities and implementation strategies.
Scan registration is an essential subtask when building maps based on range finder data from mobile robots. The problem is to deduce how the robot has moved between con- secutive scans, based on the shape of overlapping portions of the scans. This paper pre- sents a new algorithm for registration of 3D data. The algorithm is a generalization and improvement of the normal distributions transform NDT for 2D data developed by Biber and Strasser, which allows for accurate registration using a memory-efficient rep- resentation of the scan surface. A detailed quantitative and qualitative comparison of the new algorithm with the 3D version of the popular ICP iterative closest point algorithm is presented. Results with actual mine data, some of which were collected with a new prototype 3D laser scanner, show that the presented algorithm is faster and slightly more reliable than the standard ICP algorithm for 3D registration, while using a more memory- efficient scan surface representation. © 2007 Wiley Periodicals, Inc.
The problem of creating a complete model of a physical object is studied. Although this may be possible using intensity images, the authors use range images which directly provide access to three-dimensional information. The first problem that needs to be solved is to find the transformation between the different views. Previous approaches have either assumed this transformation to be known (which is extremely difficult for a complete model) or computed it with feature matching (which is not accurate enough for integration. The authors propose an approach that works on range data directly and registers successive views with enough overlapping area to get an accurate transformation between views. This is performed by minimizing a functional that does not require point-to-point matches. Details are given of the registration method and modeling procedure, and they are illustrated on range images of complex objects
Point set registration is a key component in many computer vision tasks. The goal of point set registration is to assign correspondences between two sets of points and to recover the transformation that maps one point set to the other. Multiple factors, including an unknown nonrigid spatial transformation, large dimensionality of point set, noise, and outliers, make the point set registration a challenging problem. We introduce a probabilistic method, called the Coherent Point Drift (CPD) algorithm, for both rigid and nonrigid point set registration. We consider the alignment of two point sets as a probability density estimation problem. We fit the Gaussian mixture model (GMM) centroids (representing the first point set) to the data (the second point set) by maximizing the likelihood. We force the GMM centroids to move coherently as a group to preserve the topological structure of the point sets. In the rigid case, we impose the coherence constraint by reparameterization of GMM centroid locations with rigid parameters and derive a closed form solution of the maximization step of the EM algorithm in arbitrary dimensions. In the nonrigid case, we impose the coherence constraint by regularizing the displacement field and using the variational calculus to derive the optimal transformation. We also introduce a fast algorithm that reduces the method computation complexity to linear. We test the CPD algorithm for both rigid and nonrigid transformations in the presence of noise, outliers, and missing points, where CPD shows accurate results and outperforms current state-of-the-art methods.
The authors describe a general-purpose, representation-independent method for the accurate and computationally efficient registration of 3-D shapes including free-form curves and surfaces. The method handles the full six degrees of freedom and is based on the iterative closest point (ICP) algorithm, which requires only a procedure to find the closest point on a geometric entity to a given point. The ICP algorithm always converges monotonically to the nearest local minimum of a mean-square distance metric, and the rate of convergence is rapid during the first few iterations. Therefore, given an adequate set of initial rotations and translations for a particular class of objects with a certain level of `shape complexity', one can globally minimize the mean-square distance metric over all six degrees of freedom by testing each initial registration. One important application of this method is to register sensed data from unfixtured rigid objects with an ideal geometric model, prior to shape inspection. Experimental results show the capabilities of the registration algorithm on point sets, curves, and surfaces
In this paper we present a framework for the automatic registration of multiple terrestrial laser scans. The proposed method can handle arbitrary point clouds with reasonable pairwise overlap, without knowledge about their initial orientation and without the need for artificial markers or other specific objects. The framework is divided into a coarse and a fine registration part, which each start with pairwise registration and then enforce consistent global alignment across all scans. While we put forward a complete, functional registration system, the novel contribution of the paper lies in the coarse global alignment step. Merging multiple scans into a consistent network creates loops along which the relative transformations must add up. We pose the task of finding a global alignment as picking the best candidates from a set of putative pairwise registrations, such that they satisfy the loop constraints. This yields a discrete optimization problem that can be solved efficiently with modern combinatorial methods. Having found a coarse global alignment in this way, the framework proceeds by pairwise refinement with standard ICP, followed by global refinement to evenly spread the residual errors. The framework was tested on six challenging, real-world datasets. The discrete global alignment step effectively detects, removes and corrects failures of the pairwise registration procedure, finally producing a globally consistent coarse scan network which can be used as initial guess for the highly non-convex refinement. Our overall system reaches success rates close to 100% at acceptable runtimes <1 h, even in challenging conditions such as scanning in the forest. © 2015 International Society for Photogrammetry and Remote Sensing, Inc. (ISPRS).
The set of rigid-body motions forms a Lie group called SE(3), the special Euclidean group in three dimensions. In this paper, we investigate possilble choices of Riemannian metrics and affine connections on SE(3) for applications to kinematic analysis and robot-trajectory planning. In the first part of the paper, we study metrics whose geodesics are screw motions. We prove that no Riemannian metrics call have such geodesics, and we show that the metrics whose geodesics are screw motions form a two-parameter family of semi-Riemannan metrics. In the second part of the paper, we investigate affine connections which through the covariant derivative give the correct expression for the acceleration of a rigid body. We prove that there is a unique symmetric connection with this property. Furthermore, we show that there is a family of Riemannian metrics that are compatible with such a connection. These metrics are products of the bi-invariant metric on the group of rotations and a positive-definite constant metric on the group of translations.
For interval estimation of a proportion, coverage probabilities tend to be too large for "exact" confidence intervals based on inverting the binomial test and too small for the interval based on inverting the Wald large-sample normal test (i.e., sample proportion ± z-score × estimated standard error). Wilson's suggestion of inverting the related score test with null rather than estimated standard error yields coverage probabilities close to nominal confidence levels, even for very small sample sizes. The 95% score interval has similar behavior as the adjusted Wald interval obtained after adding two "successes" and two "failures" to the sample. In elementary courses, with the score and adjusted Wald methods it is unnecessary to provide students with awkward sample size guidelines.
This paper is conceived as a tutorial on rotation averaging, summarizing the research that has been carried out in this area; it discusses methods for single-view and multiple-view rotation averaging, as well as providing proofs of convergence and convexity in many cases. However, at the same time it contains many new results, which were developed to fill gaps in knowledge, answering fundamental questions such as radius of convergence of the algorithms, and existence of local minima. These matters, or even proofs of correctness have in many cases not been considered in the Computer Vision literature. We consider three main problems: single rotation averaging, in which a single rotation is computed starting from several measurements; multiple-rotation averaging, in which absolute orientations are computed from several relative orientation measurements; and conjugate rotation averaging, which relates a pair of coordinate frames. This last is related to the hand-eye coordination problem and to multiple-camera calibration.
A heuristic method has been developed for registering two sets of 3-D curves obtained by using an edge-based stereo system, or two dense 3-D maps obtained by using a correlation-based stereo system. Geometric matching in general is a difficult unsolved problem in computer vision. Fortunately, in many practical applications, some a priori knowledge exists which considerably simplifies the problem. In visual navigation, for example, the motion between successive positions is usually approximately known. From this initial estimate, our algorithm computes observer motion with very good precision, which is required for environment modeling (e.g., building a Digital Elevation Map). Objects are represented by a set of 3-D points, which are considered as the samples of a surface. No constraint is imposed on the form of the objects. The proposed algorithm is based on iteratively matching points in one set to the closest points in the other. A statistical method based on the distance distribution is used to deal with outliers, occlusion, appearance and disappearance, which allows us to do subset-subset matching. A least-squares technique is used to estimate 3-D motion from the point correspondences, which reduces the average distance between points in the two sets. Both synthetic and real data have been used to test the algorithm, and the results show that it is efficient and robust, and yields an accurate motion estimate.
Optimal transportation distances are a fundamental family of parameterized
distances for histograms. Despite their appealing theoretical properties,
excellent performance in retrieval tasks and intuitive formulation, their
computation involves the resolution of a linear program whose cost is
prohibitive whenever the histograms' dimension exceeds a few hundreds. We
propose in this work a new family of optimal transportation distances that look
at transportation problems from a maximum-entropy perspective. We smooth the
classical optimal transportation problem with an entropic regularization term,
and show that the resulting optimum is also a distance which can be computed
through Sinkhorn-Knopp's matrix scaling algorithm at a speed that is several
orders of magnitude faster than that of transportation solvers. We also report
improved performance over classical optimal transportation distances on the
MNIST benchmark problem.
This paper introduces a novel similarity measure for 3D rigid registration algorithms that use comparison between image-based descriptors in order to find correspondences between two partial 3D point clouds belonging to the same object. Unlike the similarity measures based on correlation coefficient, joint entropy, mutual information or others that have been used by the most popular 3D registration algorithms this similarity measure is based on distance between pixels and takes into account the problems of clutter and occlusion that can appear in real situations that need 3D registration or object recognition.
In this paper we combine the Iterative Closest Point ('ICP') and 'point-to-plane ICP' algorithms into a single probabilistic framework. We then use this framework to model locally planar surface structure from both scans instead of just the "model" scan as is typically done with the point-to-plane method. This can be thought of as 'plane-to-plane'. The new approach is tested with both simulated and real-world data and is shown to outperform both standard ICP and point-to-plane. Furthermore, the new approach is shown to be more robust to incorrect correspondences, and thus makes it easier to tune the maximum match distance parameter present in most variants of ICP. In addition to the demonstrated performance improvement, the proposed model allows for more expressive probabilistic models to be incorporated into the ICP framework. While maintaining the speed and simplicity of ICP, the Generalized-ICP could also allow for the addition of outlier terms, measurement noise, and other probabilistic techniques to increase robustness. I. INTRODUCTION
Correlation is a very effective way to align intensity images. We extend the correlation technique to point set registration using a method we call kernel correlation. Kernel correlation is an affinity measure, and it is also a function of the point set entropy. We define the point set registration problem as finding the maximum kernel correlation configuration of the the two point sets to be registered. The new registration method has intuitive interpretations, simple to implement algorithm and easy to prove convergence property. Our method shows favorable performance when compared with the iterative closest point (ICP) and EM-ICP methods.KeywordsCost FunctionRegistration Method
Iterative Close PointRegistration AlgorithmIterative Close PointThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
We investigate in this article the rigid registration of large sets of points, generally sampled from surfaces. We formulate this problem as a general Maximum-Likelihood (ML) estimation of the transformation and the matches. We show that, in the specific case of a Gaussian noise, it corresponds to the Iterative Closest Point algorithm (ICP) with the Mahalanobis distance.
Then, considering matches as a hidden variable, we obtain a slightly more complex criterion that can be efficiently solved using Expectation-Maximization (EM) principles. In the case of a Gaussian noise, this new methods corresponds to an ICP with multiple matches weighted by normalized Gaussian weights, giving birth to the EM-ICP acronym of the method.
The variance of the Gaussian noise is a new parameter that can be viewed as a “scale or blurring factor” on our point clouds. We show that EM-ICP robustly aligns the barycenters and inertia moments with a high variance, while it tends toward the accurate ICP for a small variance. Thus, the idea is to use a multi-scale approach using an annealing scheme on this parameter to combine robustness and accuracy. Moreover, we show that at each “scale”, the criterion can be efficiently approximated using a simple decimation of one point set, which drastically speeds up the algorithm.
Experiments on real data demonstrate a spectacular improvement of the performances of EM-ICP w.r.t. the standard ICP algorithm in terms of robustness (a factor of 3 to 4) and speed (a factor 10 to 20), with similar performances in precision. Though the multiscale scheme is only justified with EM, it can also be used to improve ICP, in which case the performances reaches then the one of EM when the data are not too noisy.
This paper describes a general purpose, representation independent
method for the accurate and computationally efficient registration of
3-D shapes including free-form curves and surfaces. The method handles
the full six-degrees of freedom and is based on the iterative closest
point (ICP) algorithm, which requires only a procedure to find the
closest point on a geometric entity to a given point. The ICP algorithm
always converges monotonically to the nearest local minimum of a
mean-square distance metric, and experience shows that the rate of
convergence is rapid during the first few iterations. Therefore, given
an adequate set of initial rotations and translations for a particular
class of objects with a certain level of 'shape complexity', one can
globally minimize the mean-square distance metric over all six degrees
of freedom by testing each initial registration. For examples, a given
'model' shape and a sensed 'data' shape that represents a major portion
of the model shape can be registered in minutes by testing one initial
translation and a relatively small set of rotations to allow for the
given level of model complexity. One important application of this
method is to register sensed data from unfixtured rigid objects with an
ideal geometric model prior to shape inspection. The described method is
also useful for deciding fundamental issues such as the congruence
(shape equivalence) of different geometric representations as well as
for estimating the motion between point sets where the correspondences
are not known. Experimental results show the capabilities of the
registration algorithm on point sets, curves, and surfaces.
A number of techniques have been developed for reconstructing surfaces by integrating groups of aligned range images. A desirable set of properties for such algorithms includes: incremental updating, representation of directional uncertainty, the ability to fill gaps in the reconstruction, and robustness in the presence of outliers. Prior algorithms possess subsets of these properties. In this paper, we present a volumetric method for integrating range images that possesses all of these properties.
Robust registration of 2D and 3D point sets
- A W Fitzgibbon
Optimal mass transport: Signal processing and machine-learning applications
- S Koluri
- S R Park
- M Thorpe
- D Slepcev
- G K Rohde
Platonic solid,” From MathWorld–A Wolfram Web Resource
- weisstein