Conics-based stereo,motion estimation,and pose determination

Stereo vision, motion and structure parameter estimation, and pose determination are three important problems in 3-D computer vision. The first step in all of these problems is to choose and to extract primitives and their features in images. In most of the previous work, people usually use edge points or straight line segments as primitives and their local properties as features. Few methods have been presented in the literature using more compact primitives and their global features. This article presents an approach using conics as primitives. For stereo vision, a closed-form solution is provided for both establishing the correspondence of conics in images and the reconstruction of conics in space. With this method, the correspondence is uniquely determined and the reconstruction is global. It is shown that the method can be extended for higher degree (degree3) planar curves.For motion and structure parameter estimation, it is shown that, in general, two sequential images of at least three conics are needed in order to determine the camera motion. A complicated nonlinear system must be solved in this case. In particular, if we are given two images of a pair of coplanar conics, a closed-form solution of camera motion is presented. In a CAD-based vision system, the object models are available, and this makes it possible to recognize 3-D objects and to determine their poses from a single image.For pose determination, it is shown that if there exist two conics on the surface of an object, the object's pose can be determined by an efficient one-dimensional search. In particular, if two conics are coplanar, a closed-form solution of the object's pose is presented.Uniqueness analysis and some experiments with real or synthesized data are presented in this article.     

Reconstructing camera projection matrices from multiple pairwise overlapping views

In this work we address the problem of projective reconstruction from multiple views with missing data. Factorization based algorithms require point correspondences across all the views. In many applications this is an unrealistic assumption. Current methods that solve the problem of projective reconstruction with missing data require correspondence information across triplets of images. We propose a projective reconstruction method that yields a consistent camera set given the fundamental matrices between pairs of views without directly using the image correspondences. The algorithm is based on breaking the reconstruction problem into small steps. In each step, we eliminate as much uncertainty as possible.     

空间二次曲线射影重建计算方法研究

利用二次曲线上点和直线的对应以及利用二次曲线光流对二次曲线进行解释,需要寻求多元方程组的最优解,计算过程较为复杂。将空间二次曲线表示为一个空间二次曲面和一个平面的交线,提出了一对对应二次曲线存在两个独立的多项式条件;利用两个二次曲面的基曲线推导出二次曲线重建的两个对应条件,在此基础上,给出了空间二次曲线射影重建的计算方法和计算步骤。实验验证表明该算法实用和可靠。     

The Geometry and Matching of Lines and Curves Over Multiple Views

This paper describes the geometry of imaged curves in two and three views. Multi-view relationships are developed for lines, conics and non-algebraic curves. The new relationships focus on determining the plane of the curve in a projective reconstruction, and in particular using the homography induced by this plane for transfer from one image to another. It is shown that given the fundamental matrix between two views, and images of the curve in each view, then the plane of a conic may be determined up to a two fold ambiguity, but local curvature of a curve uniquely determines the plane. It is then shown that given the trifocal tensor between three views, this plane defines a homography map which may be used to transfer a conic or the curvature from two views to a third. Simple expressions are developed for the plane and homography in each case.A set of algorithms are then described for automatically matching individual line segments and curves between images. The algorithms use both photometric information and the multiple view geometric relationships. For image pairs the homography facilitates the computation of a neighbourhood cross-correlation based matching score for putative line/curve correspondences. For image triplets cross-correlation matching scores are used in conjunction with line/curve transfer based on the trifocal geometry to disambiguate matches. Algorithms are developed for both short and wide baselines. The algorithms are robust to deficiencies in the segment extraction and partial occlusion.Experimental results are given for image pairs and triplets, for varying motions between views, and for different scene types. The methods are applicable to line/curve matching in stereo and trinocular rigs, and as a starting point for line/curve matching through monocular image sequences.     

Two-way ambiguity in 2D projective reconstruction from threeuncalibrated 1D images

We show that there is, in general, a two-way ambiguity for 2D projective reconstruction from three uncalibrated 1D views, independent of the number of point correspondences. The two distinct projective reconstructions are exactly related by a quadratic transformation with the three camera centers as fundamental points. Unique 2D reconstruction is possible only when the three camera centers are aligned. By Carlsson duality (1995), there is a dual two-way ambiguity for 2D projective reconstruction from six point correspondences, independent of the number of 1D views. The theoretical results are demonstrated on numerical examples     

Spatial-temporal consistent labeling of tracked pedestrians across non-overlapping camera views

Tracking people across multiple cameras with non-overlapping views is a challenging task, since their observations are separated in time and space and their appearances may vary significantly. This paper proposes a Bayesian model to solve the consistent labeling problem across multiple non-overlapping camera views. Significantly different from related approaches, our model assumes neither people are well segmented nor their trajectories across camera views are estimated. We formulate a spatial-temporal probabilistic model in the hypothesis space that consists the potentially matched objects between the exit field of view (FOV) of one camera and the entry FOV of another camera. A competitive major color spectrum histogram representation (CMCSHR) for appearance matching between two objects is also proposed. The proposed spatial-temporal and appearance models are unified by a maximum-a-posteriori (MAP) Bayesian model. Based on this Bayesian model, when a detected new object corresponds to a group hypothesis (more than one object), we further develop an online method for online correspondence update using optimal graph matching (OGM) algorithm. Experimental results on three different real scenarios validate the proposed Bayesian model approach and the CMCSHR method. The results also show that the proposed approach is able to address the occlusion problem/group problem, i.e. finding the corresponding individuals in another camera view for a group of people who walk together into the entry FOV of a camera.     

Self-Calibration Under the Cayley Framework

The Cayley framework here is meant to tackle the vision problems under the infinite Cayley transformation (ICT), its main advantage lies in its numerical stability. In this work, the stratified self-calibration under the Cayley framework is investigated. It is well known that the main difficulty of the stratified self-calibration in multiple view geometry is to upgrade a projective reconstruction to an affine one, in other words, to estimate the unknown 3-vector of the plane at infinity, called the normal vector. To our knowledge, without any prior knowledge about the scene or the camera motion, the only available constraint on a moving camera with constant intrinsic parameters is the well-known Modulus Constraint in the literature. Do other kinds of constraints exist? If yes, what they are? How could they be used? In this work, such questions will be systematically investigated under the Cayley framework. Our key contributions include: 1. The original projective expression of the ICT is simplified and a new projective expression is derived to make the upgrade easier from a projective reconstruction to a metric reconstruction. 2. The constraints on the normal vector are systematically investigated. For two views, two constraints on the normal vector are derived; one of them is the well-known modulus constraint, while the other is a new inequality constraint. There are only these two constraints for two views. For three views, besides the constraints for two views, two groups of new constraints are derived and each of them contains three constraints. In other words, there are 12 constraints in total for three views. 3. Based on our projective expression and these constraints, a stratified Cayley algorithm and a total Cayley algorithm are proposed for the metric reconstruction from images. It is experimentally shown that they both improve significantly the numerical stability of the classical algorithms. Compared with the global optimal algorithm under the infinite homography framework, the Cayley algorithms have comparable calibration accuracy, but substantially reduce the computational load.     

Correspondence-Free Activity Analysis and Scene Modeling in Multiple Camera Views

We propose a novel approach for activity analysis in multiple synchronized but uncalibrated static camera views. In this paper, we refer to activities as motion patterns of objects, which correspond to paths in far-field scenes. We assume that the topology of cameras is unknown and quite arbitrary, the fields of views covered by these cameras may have no overlap or any amount of overlap, and objects may move on different ground planes. Using low-level cues, objects are first tracked in each camera view independently, and the positions and velocities of objects along trajectories are computed as features. Under a probabilistic model, our approach jointly learns the distribution of an activity in the feature spaces of different camera views. Then, it accomplishes the following tasks: 1) grouping trajectories, which belong to the same activity but may be in different camera views, into one cluster; 2) modeling paths commonly taken by objects across multiple camera views; and 3) detecting abnormal activities. Advantages of this approach are that it does not require first solving the challenging correspondence problem, and that learning is unsupervised. Even though correspondence is not a prerequisite, after the models of activities have been learned, they can help to solve the correspondence problem, since if two trajectories in different camera views belong to the same activity, they are likely to correspond to the same object. Our approach is evaluated on a simulated data set and two very large real data sets, which have 22,951 and 14,985 trajectories, respectively.     

Multilinear Factorizations for Multi-Camera Rigid Structure from Motion Problems

Camera networks have gained increased importance in recent years. Existing approaches mostly use point correspondences between different camera views to calibrate such systems. However, it is often difficult or even impossible to establish such correspondences. But even without feature point correspondences between different camera views, if the cameras are temporally synchronized then the data from the cameras are strongly linked together by the motion correspondence: all the cameras observe the same motion. The present article therefore develops the necessary theory to use this motion correspondence for general rigid as well as planar rigid motions. Given multiple static affine cameras which observe a rigidly moving object and track feature points located on this object, what can be said about the resulting point trajectories? Are there any useful algebraic constraints hidden in the data? Is a 3D reconstruction of the scene possible even if there are no point correspondences between the different cameras? And if so, how many points are sufficient? Is there an algorithm which warrants finding the correct solution to this highly non-convex problem? This article addresses these questions and thereby introduces the concept of low-dimensional motion subspaces. The constraints provided by these motion subspaces enable an algorithm which ensures finding the correct solution to this non-convex reconstruction problem. The algorithm is based on multilinear analysis, matrix and tensor factorizations. Our new approach can handle extreme configurations, e.g. a camera in a camera network tracking only one single point. Results on synthetic as well as on real data sequences act as a proof of concept for the presented insights.     

Omnidirectional video

Omnidirectional video enables direct surround immersive viewing of a scene by warping the original image into the correct perspective given a viewing direction. However, novel views from viewpoints off the camera path can only be obtained if we solve the three-dimensional motion and calibration problem. In this paper we address the case of a parabolic catadioptric camera – a paraboloidal mirror in front of an orthographic lens – and we introduce a new representation, called the circle space, for points and lines in such images. In this circle space, we formulate an epipolar constraint involving a 4×4 fundamental matrix. We prove that the intrinsic parameters can be inferred in closed form from the two-dimensional subspace of the new fundamental matrix from two views if they are constant or from three views if they vary. Three-dimensional motion and structure can then be estimated from the decomposition of the fundamental matrix.     

Line Localization from Single Catadioptric Images

Indoor environments often contain several line segments. The 3D reconstruction of such environments can thus be reduced to the localization of lines in the 3D space. Multi-view reconstruction requires the solution of the correspondence problem. The use of a single image to localize space lines is attractive, since the correspondence problem can be avoided. However, using a central camera the line localization from single image is an ill-posed problem, because there are infinitely many lines sharing the same image.     

Multiple View Geometry of General Algebraic Curves

We introduce a number of new results in the context of multi-view geometry from general algebraic curves. We start with the recovery of camera geometry from matching curves. We first show how one can compute, without any knowledge on the camera, the homography induced by a single planar curve. Then we continue with the derivation of the extended Kruppa's equations which are responsible for describing the epipolar constraint of two projections of a general algebraic curve. As part of the derivation of those constraints we address the issue of dimension analysis and as a result establish the minimal number of algebraic curves required for a solution of the epipolar geometry as a function of their degree and genus.We then establish new results on the reconstruction of general algebraic curves from multiple views. We address three different representations of curves: (i) the regular point representation in which we show that the reconstruction from two views of a curve of degree d admits two solutions, one of degree d and the other of degree d(d – 1). Moreover using this representation, we address the problem of homography recovery for planar curves, (ii) dual space representation (tangents) for which we derive a lower bound for the number of views necessary for reconstruction as a function of the curve degree and genus, and (iii) a new representation (to computer vision) based on the set of lines meeting the curve which does not require any curve fitting in image space, for which we also derive lower bounds for the number of views necessary for reconstruction as a function of curve degree alone.     

A calibration method for paracatadioptric camera from sphere images

For paracatadioptric camera, the estimation of intrinsic parameters from sphere images is still an open and challenging problem. In this paper, we propose a calibration method for paracatadioptric camera based on sphere images, which only requires that the projected contour of parabolic mirror is visible on the image plane in one view. We have found that, under central catadioptric camera, a sphere is projected to two conics on the image plane, which are defined as a pair of antipodal sphere images. The conic that is visible on the image plane is called the sphere image, while the other invisible conic is called the antipodal sphere image. In the other aspect, according to the image formation of central catadioptric camera, these two conics can also be considered as the projections of two parallel circles on the viewing sphere by a virtue camera. That is to say, if three pairs of antipodal sphere images are known, central catadioptric camera can be directly calibrated by the calibration method based on two parallel circles. Therefore, the problem of calibrating central catadioptric camera is transferred to the estimations of sphere images and their antipodal sphere images. Based on this idea, we first initialize the intrinsic parameters of the camera by the projected contour of parabolic mirror, and use them to initialize the antipodal sphere images. Next, we study properties of several pairs of antipodal sphere images under paracatadioptric camera. Then, these properties are used to optimize sphere images and their antipodal sphere images, so as to calibrate the paracatadioptric camera. Experimental results on both simulated and real image data have demonstrated the effectiveness of our method.     

Catadioptric camera calibration using geometric invariants

Central catadioptric cameras are imaging devices that use mirrors to enhance the field of view while preserving a single effective viewpoint. In this paper, we propose a novel method for the calibration of central catadioptric cameras using geometric invariants. Lines and spheres in space are all projected into conics in the catadioptric image plane. We prove that the projection of a line can provide three invariants whereas the projection of a sphere can only provide two. From these invariants, constraint equations for the intrinsic parameters of catadioptric camera are derived. Therefore, there are two kinds of variants of this novel method. The first one uses projections of lines and the second one uses projections of spheres. In general, the projections of two lines or three spheres are sufficient to achieve catadioptric camera calibration. One important conclusion in this paper is that the method based on projections of spheres is more robust and has higher accuracy than that based on projections of lines. The performances of our method are demonstrated by both the results of simulations and experiments with real images.     

3D object pose form clustering with multiple views

A candidate pose algorithm is described which computes object pose from an assumed correspondence between a pair of 2D image points and a pair of 3D model points. By computing many pose candidates actual object pose can usually be determined by detecting a cluster in the space of all candidates. Cluster space can receive candidate pose parameters from independent computations in different camera views. It is shown that use of of geometric constraint can be sufficient for reliable pose detection, but use of other knowledge, such as edge presence and type, can be easily added for increased efficiency.     

一种基于无人机序列成像的地形地貌重建方法

以无人机平台上普通摄像机获取的序列图像为对象,提出了一种对地三维重建的自动化处理方法。首先提出了基于视差分析的序列图像关键帧选择方法,对关键帧图像特征点进行鲁棒性的提取与匹配;第二步用加权的RANSAC算法估计基础矩阵,同时获取准确匹配的内点集。根据已标定的像机内参数,解算相对运动并进行优化。最后对待重建的目标点提出几何约束和单应约束融合的方法实现快速准确匹配,通过三角交会完成目标形貌三维重建。仿真实验结果表明该算法对序列图像具有较好的自动化程度和鲁棒性。     

Projective reconstruction and invariants from multiple images

This correspondence investigates projective reconstruction of geometric configurations seen in two or more perspective views, and the computation of projective invariants of these configurations from their images. A basic tool in this investigation is the fundamental matrix that describes the epipolar correspondence between image pairs. It is proven that once the epipolar geometry is known, the configurations of many geometric structures (for instance sets of points or lines) are determined up to a collineation of projective 3-space 𝒫³ by their projection in two independent images. This theorem is the key to a method for the computation of invariants of the geometry. Invariants of six points in 𝒫³ and of four lines in 𝒫³ are defined and discussed. An example with real images shows that they are effective in distinguishing different geometrical configurations. Since the fundamental matrix is a basic tool in the computation of these invariants, new methods of computing the fundamental matrix from seven-point correspondences in two images or six-point correspondences in three images are given     

Lines and Points in Three Views and the Trifocal Tensor

This paper discusses the basic role of the trifocal tensor in scene reconstruction from three views. This 3× 3× 3 tensor plays a role in the analysis of scenes from three views analogous to the role played by the fundamental matrix in the two-view case. In particular, the trifocal tensor may be computed by a linear algorithm from a set of 13 line correspondences in three views. It is further shown in this paper, that the trifocal tensor is essentially identical to a set of coefficients introduced by Shashua to effect point transfer in the three view case. This observation means that the 13-line algorithm may be extended to allow for the computation of the trifocal tensor given any mixture of sufficiently many line and point correspondences. From the trifocal tensor the camera matrices of the images may be computed, and the scene may be reconstructed. For unrelated uncalibrated cameras, this reconstruction will be unique up to projectivity. Thus, projective reconstruction of a set of lines and points may be carried out linearly from three views.     

Self-Calibration of Turntable Sequences from Silhouettes

This paper addresses the problem of recovering both the intrinsic and extrinsic parameters of a camera from the silhouettes of an object in a turntable sequence. Previous silhouette-based approaches have exploited correspondences induced by epipolar tangents to estimate the image invariants under turntable motion and achieved a weak calibration of the cameras. It is known that the fundamental matrix relating any two views in a turntable sequence can be expressed explicitly in terms of the image invariants, the rotation angle, and a fixed scalar. It will be shown that the imaged circular points for the turntable plane can also be formulated in terms of the same image invariants and fixed scalar. This allows the imaged circular points to be recovered directly from the estimated image invariants, and provide constraints for the estimation of the imaged absolute conic. The camera calibration matrix can thus be recovered. A robust method for estimating the fixed scalar from image triplets is introduced, and a method for recovering the rotation angles using the estimated imaged circular points and epipoles is presented. Using the estimated camera intrinsics and extrinsics, a Euclidean reconstruction can be obtained. Experimental results on real data sequences are presented, which demonstrate the high precision achieved by the proposed method.