Search

This book contains material for an introductory course on homogeneous representations for geometry in 2 and 3 dimensions, camera projections, representations of 3D rotations, epipolar geometry, and estimation of various type of geometric objects. Based on these results, a set of applications are presented. It also contains a toolbox of general results that are useful for the presented material. The book is intended for undergraduate studies at advanced level in master programs, or in PhD-courses at introductory level.

We introduce a simple and efficient procedure for the segmentation of rigidly moving objects, imaged under an affine camera model. For this purpose we revisit the theory of "linear combination of views" (LCV), proposed by Ullman and Basri [20], which states that the set of 2d views of an object undergoing 3d rigid transformations, is embedded in a low-dimensional linear subspace that is spanned by a small number of basis views. Our work shows, that one may use this theory for motion segmentation, and cluster the trajectories of 3d objects using only two 2d basis views. We therefore propose a practical motion segmentation method, built around LCV, that is very simple to implement and use, and in addition is very fast, meaning it is well suited for real-time SfM and tracking applications. We have experimented on real image sequences, where we show good segmentation results, comparable to the state-of-the-art in literature. If we also consider computational complexity, our proposed method is one of the best performers in combined speed and accuracy. © 2011. The copyright of this document resides with its authors.

We propose a method for segmenting an arbitrary number of moving objects using the geometry of 6 points in 2D images to infer motion consistency. This geometry allows us to determine whether or not observations of 6 points over several frames are consistent with a rigid 3D motion. The matching between observations of the 6 points and an estimated model of their configuration in 3D space, is quantified in terms of a geometric error derived from distances between the points and 6 corresponding lines in the image. This leads to a simple motion inconsistency score, based on the geometric errors of 6points that in the ideal case should be zero when the motion of the points can be explained by a rigid 3D motion. Initial point clusters are determined in the spatial domain and merged in motion trajectory domain based on this score. Each point is then assigned to the cluster, which gives the lowest score.Our algorithm has been tested with real image sequences from the Hopkins155 database with very good results, competing withthe state of the art methods, particularly for degenerate motion sequences. In contrast to the motion segmentation methods basedon multi-body factorization, that assume an affine camera model, the proposed method allows the mapping from 3D space to the 2D image to be fully projective.

A local descriptor for 3D data, the scene tensor, is presentedtogether with novel applications. It can describe multiple planarsegments in a local 3D region; for the case of up to three segments itis possible to recover the geometry of the local region in terms of thesize, position and orientation of each of the segments from thedescriptor. In the setting of range data, this property makes thedescriptor unique compared to other popular local descriptors, such asspin images or point signatures. The estimation of the descriptor canbe based on 3D orientation tensors that, for example, can be computeddirectly from surface normals but the representation itself does notdepend on a specific estimation method and can also be applied to othertypes of 3D data, such as motion stereo. A series of experiments onboth real and synthetic range data show that the proposedrepresentation can be used as a interest point detector with highrepeatability. Further, the experiments show that, at such detectedpoints, the local geometric structure can be robustly recovered, evenin the presence of noise. Last we expand a framework for object poseestimation, based on the scene tensor and previously appliedsuccessfully on 2D image data, to work also on range data. Poseestimation from real range data shows that there are advantages oversimilar descriptors in 2D and that use of range data gives superiorperformance., See also the addendum which is found at http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-57329

This document is an addendum to the main text in A local geometry-based descriptor for 3D data applied to object pose estimation by Fredrik Viksten and Klas Nordberg. This addendum gives proofs for propositions stated in the main document. This addendum also details how to extract information from the fourth order tensor refered to as S22 in the main document.

DIPLECS

The paper describes a minimal set of 18 parameters that can representany trifocal tensor consistent with the internal constraints. 9parameters describe three orthogonal matrices and 9 parameters describe10 elements of a sparse tensor T' with 17 elements in well-defined positions equal to zero. Any valid trifocal tensor isthen given as some specific T' transformed by the orthogonalmatrices in the respective image domain. The paper also describes asimple approach for estimating the three orthogonal matrices in thecase of a general 3 x 3 x 3 tensor, i.e., when the internalconstraints are not satisfied. This can be used to accomplish a leastsquares approximation of a general tensor to a tensor that satisfies the internal constraints. This type of constraint enforcement, inturn, can be used to obtain an improved estimate of the trifocal tensorbased on the normalized linear algorithm, with the constraintenforcement as a final step. This makes the algorithm more similar tothe corresponding algorithm for estimation of the fundamental matrix. An experiment on synthetic data shows that the constraint enforcementof the trifocal tensor produces a significantly better result thanwithout enforcement, expressed by the positions of the epipoles, giventhat the constraint enforcement is made in normalized image coordinates., VISCOS

This paper presents a method for triangulation of 3D points given their projections in two images. Recent results show that the triangulation mapping can be represented as a linear operator K applied to the outer product of corresponding homogeneous image coordinates, leading to a triangulation of very low computational complexity. K can be determined from the camera matrices, together with a so-called blind plane, but we show here that it can be further refined by a process similar to Gold Standard methods for camera matrix estimation. In particular it is demonstrated that K can be adjusted to minimize the Euclidean L, residual 3D error, bringing it down to the same level as the optimal triangulation by Hartley and Sturm. The resulting K optimally fits a set of 2D+2D+3D data where the error is measured in the 3D space. Assuming that this calibration set is representative for a particular application, where later only the 2D points are known, this K can be used for triangulation of 3D points in an optimal way, which in addition is very efficient since the optimization need only be made once for the point set. The refinement of K is made by iteratively reducing errors in the 3D and 2D domains, respectively. Experiments on real data suggests that very few iterations are needed to accomplish useful results., VISCOS

Point-of-interest detection is a way of reducing the amount of data that needs to be processed in a certain application and is widely used in 2D image analysis. In 2D image analysis, point-of-interest detection is usually related to extraction of local descriptors for object recognition, classification, registration or pose estimation. In analysis of range data however, some local descriptors have been published in the last decade or so, but most of them do not mention any kind of point-of-interest detection. We here show how to use an extended Harris detector on range data and discuss variants of the Harris measure. All described variants of the Harris detector for 3D should also be usable in medical image analysis, but we focus on the range data case. We do present a performance evaluation of the described variants of the Harris detector on range data.

Triangulation of a 3D point from two or more views can be solved inseveral ways depending on how perturbations in the image coordinatesare dealt with. A common approach is optimal triangulation which minimizes the total L2 reprojection error in the images,corresponding to finding a maximum likelihood estimate of the 3Dpoint assuming independent Gaussian noise in the image spaces.Computational approaches for optimal triangulation have beenpublished for the stereo case and, recently, also for the three-viewcase. In short, they solve an independent optimization problem foreach 3D point, using relatively complex computations such as findingroots of high order polynomials or matrix decompositions. Thispaper discuss three-view triangulation and reports the followingresults: (1) the 3D point can be computed as multi-linear mapping(tensor) applied on the homogeneous image coordinates, (2) the set of triangulation tensors forms a 7-dimensional space determined by the camera matrices, (3) given a set of corresponding 3D/2Dcalibration points, the 3D residual L1 errors can be optimized over the elements in the 7-dimensional space, (4) using theresulting tensor as initial value, the error can be further reducedby tuning the tensor in a two-step iterative process, (5) the 3Dresidual L1 error for a set of evaluation points which lieclose to the calibration set is comparable to the three-view optimalmethod. In summary, three-view triangulation can be done by firstperforming an optimization of the triangulation tensor and once this is done, triangulation can be made with 3D residual error at thesame level as the optimal method, but at a much lower computationalcost. This makes the proposed method attractive for real-time three-view triangulation of large data sets provided that thenecessary calibration process can be performed., VISCOS

This Chapter presents a vision-based method for unmanned aerial vehicle (UAV) motion estimation that uses as input an image motion field obtained from matches of point-like features. The Chapter enhances visionbased techniques developed for single UAV localization and demonstrates how they can be modified to deal with the problem of multi-UAV relative position estimation. The proposed approach is built upon the assumption that if different UAVs identify, using their cameras, common objects in a scene, the relative pose displacement between the UAVs can be computed from these correspondences under certain assumptions. However, although point-like features are suitable for local UAV motion estimation, finding matches between images collected using different cameras is a difficult task that may be overcome using blob features. Results justify the proposed approach., COMETS: “Real-time coordination and control of multiple heterogeneous unmanned aerial vehicles”, IST-2001-34304

We present a novel local descriptor for range data that can describe one or more planes or lines in a local region. It is possible to recover the geometry of the described local region and extract the size, position and orientation of each local plane or line-like structure from the descriptor. This gives the descriptor a property that other popular local descriptors for range data, such as spin images or point signatures, does not have. The estimation of the descriptor is dependant on estimation of surface normals but does not depend on the specific normal estimation method used. It is shown that is possible to extract how many planar surface regions the descriptor represents and that this could be used as a point-of-interest detector.

A runtime system for implementation of image processing operations is presented. It is designed for working in a flexible and distributed environment related to the software architecture of a newly developed UAV system. The software architecture can be characterized at a coarse scale as a layered system, with a deliberative layer at the top, a reactive layer in the middle, and a processing layer at the bottom. At a finer scale each of the three levels is decomposed into sets of modules which communicate using CORBA, allowing system development and deployment on the UAV to be made in a highly flexible way. Image processing takes place in a dedicated module located in the process layer, and is the main focus of the paper. This module has been designed as a runtime system for data flow graphs, allowing various processing operations to be created online and on demand by the higher levels of the system. The runtime system is implemented in Java, which allows development and deployment to be made on a wide range of hardware/software configurations. Optimizations for particular hardware platforms have been made using Java's native interface.

We have developed a system which integrates the information output from several pose estimation algorithms and from several views of the scene. It is tested in a real setup with a robotic manipulator. It is shown that integrating pose estimates from several algorithms increases the overall performance of the pose estimation accuracy as well as the robustness as compared to using only a single algorithm. It is shown that increased robustness can be achieved by using pose estimation algorithms based on complementary features, so called algorithmic multi-cue integration (AMC). Furthermore it is also shown that increased accuracy can be achieved by integrating pose estimation results from different views of the scene, so-called temporal multi-cue integration (TMC). Temporal multi-cue integration is the most interesting aspect of this paper., VISATEC

This paper describes a method for vision-based unmanned aerial vehicle (UAV) motion estimation from multiple planar homographies. The paper also describes the determination of the relative displacement between different UAVs employing techniques for blob feature extraction and matching. It then presents and shows experimental results of the application of the proposed technique to multi-UAV detection of forest fires.