Algebraic and geometric tools to compute projective and permutationinvariants

Studies the computation of projective invariants in pairs of images from uncalibrated cameras and presents a detailed study of the projective and permutation invariants for configurations of points and/or lines. Two basic computational approaches are given, one algebraic and one geometric. In each case, invariants are computed in projective space or directly from image measurements. Finally, we develop combinations of those projective invariants which are insensitive to permutations of the geometric primitives of each of the basic configurations     

基于射影空间的视觉基础矩阵鲁棒估计

针对传统视觉基础矩阵计算中存在的噪音和误匹配问题,提出了基于射影空间的基础矩阵计算方法．首先定义了三视几何中射影标准基下基础矩阵只含有5个参数的特殊形式,并利用三视射影重建中空间点反投影图像误差最小为准则,消除了图像中误匹配的影响,然后基于RANSAC(random sampling consensus)技术寻找出最优7个匹配点（噪音最小）来进行对极几何估算．大量仿真模拟试验和真实图像表明此方法能够高精度地估计出基础矩阵．     

Theory and Practice of Projective Rectification

This paper gives a new method for image rectification, the process of resampling pairs of stereo images taken from widely differing viewpoints in order to produce a pair of matched epipolar projections. These are projections in which the epipolar lines run parallel with the x-axis and consequently, disparities between the images are in the x-direction only. The method is based on an examination of the fundamental matrix of Longuet-Higgins which describes the epipolar geometry of the image pair. The approach taken is consistent with that advocated by Faugeras (1992) of avoiding camera calibration. The paper uses methods of projective geometry to determine a pair of 2D projective transformations to be applied to the two images in order to match the epipolar lines. The advantages include the simplicity of the 2D projective transformation which allows very fast resampling as well as subsequent simplification in the identification of matched points and scene reconstruction.     

What you see is what you get [self-calibrating camera lens distortion]

A self-calibration method is presented for self-calibrating camera lens distortion by using only the image correspondences of two views. Two images of a single object are related by the epipolar geometry, which can be described by a 3 /spl times/ 3 singular matrix called fundamental matrix. It captures all geometric information contained in two images. An optimization method is applied to minimize the epipolar distances of the two images by adjusting the camera lens radial distortion coefficient. The merit of the method is that it does not rely on any ground truth data. Simulation and experimental results are given to demonstrate the applicability of the method.     

Invariants of six points and projective reconstruction from threeuncalibrated images

There are three projective invariants of a set of six points in general position in space. It is well known that these invariants cannot be recovered from one image, however an invariant relationship does exist between space invariants and image invariants. This invariant relationship is first derived for a single image. Then this invariant relationship is used to derive the space invariants, when multiple images are available. This paper establishes that the minimum number of images for computing these invariants is three, and the computation of invariants of six points from three images can have as many as three solutions. Algorithms are presented for computing these invariants in closed form. The accuracy and stability with respect to image noise, selection of the triplets of images and distance between viewing positions are studied both through real and simulated images. Applications of these invariants are also presented. Both the results of Faugeras (1992) and Hartley et al. (1992) for projective reconstruction and Sturm's method (1869) for epipolar geometry determination from two uncalibrated images with at least seven points are extended to the case of three uncalibrated images with only six points     

Recovering Multiple View Geometry from Mutual Projections of Multiple Cameras

In this paper, we analyze the computation of epipolar geometry in some special cases where multiple cameras are projected each other in their images. In such cases, epipoles can be obtained directly from images as the projection of cameras. As the result, the epipolar geometry can be computed from less image correspondences with higher stability. In this paper, we analyze the number of corresponding points required for computing bifocal, trifocal and quadrifocal tensors linearly in the case where cameras are projected mutually. We next show a practical linear method for computing multifocal tensors by using the mutual projection of cameras. The degenerate configurations of points and cameras is also studied, and it is shown that some degenerate configurations in general cases are no longer degenerate under the mutual projection of cameras.     

Stereo rectification of pushbroom satellite images by robustly estimating the fundamental matrix

ABSTRACT

Stereo rectification is one of the most important steps for stereo matching and subsequently for digital surface model generation from satellite stereo images. This study proposes a new framework to rectify two pushbroom images along the epipolar geometry in order to omit the vertical parallax between two images. Here, we assume the interior and relative parameters between the two pushbroom images are not known and the images can be taken at different dates. Traditional stereo rectification methods of pushbroom images require metadata such as rational polynomial coefficients (RPCs), parameters of physical sensor model or ground control points (GCPs). In this study, we develop an image-based framework for stereo rectification, which works without the need for such data. In the proposed framework, the correspondences are densely extracted by a tilling strategy, and then the fundamental matrix is robustly estimated by two geometric constraints. Both affine and projective fundamental matrices could be used for stereo rectification from pushbroom stereo images. The results on IRS P5, World view III, GeoEye and IKONOS stereo pairs as well as on multi-date stereo images demonstrate that the pushbroom images are rectified with sub-pixel accuracy.     

一种改进的MLESAC基本矩阵估计算法

为提高基本矩阵估计精度,提出一种改进的随机抽样最大似然估计算法.根据对极距离选择质量较好的原始数据,采用随机抽样一致性方法进行抽样,选择内点数最多的基本矩阵检验原始数据,剔除误差大的匹配点,结合约束条件对匹配集进行检验,以提高匹配集精度.实验结果表明,该算法的估计精度较高,稳定性较好.     

A Geometric Approach for the Theory and Applications of 3D Projective Invariants

A central task of computer vision is to automatically recognize objects in real-world scenes. The parameters defining image and object spaces can vary due to lighting conditions, camera calibration and viewing position. It is therefore desirable to look for geometric properties of the object which remain invariant under such changes in the observation parameters. The study of such geometric invariance is a field of active research. This paper presents the theory and computation of projective invariants formed from points and lines using the geometric algebra framework. This work shows that geometric algebra is a very elegant language for expressing projective invariants using n views. The paper compares projective invariants involving two and three cameras using simulated and real images. Illustrations of the application of such projective invariants in visual guided grasping, camera self-localization and reconstruction of shape and motion complement the experimental part.     

Constraint-based Correspondence Matching for Stereo-based Interactive Robotic Massage Machine

Locating the 3D positions of the points on the human back is an essential issue in stereo-based interactive robotic back massage machines. In stereoscopic 3D localization, the 3D positions are determined from the corresponding image points captured by calibrated stereo cameras. However, detecting these corresponding points on the human back is highly challenging due to the smooth and texture-less characteristics of human skin. In the present study, this problem is resolved by means of a novel correspondences detection scheme designated as Correspondences from Epipolar geometry and Contours via Triangle barycentric coordinates (CECT). In the proposed approach, reliable correspondences are extracted from the edge contours of the human back by applying epipolar geometry, and these correspondences are then used to compute the correspondences of the featureless points within the edge contour using a triangle barycentric coordinate approach. The accuracy and robustness of the estimated correspondences are ensured by applying three geometric constraints, namely a similarity constraint, a shape constraint and an epipolar constraint. The performance of the proposed approach is demonstrated by means of a series of experiments involving 28 subjects and four different testing conditions. In addition, the accuracy of the proposed localization scheme is evaluated by comparing the estimated 3D positions with those obtained using the cun-based measurement method in Traditional Chinese Medicine (TCM).     

基于遗传算法不同策略下的基础矩阵估计方法

在未定标系统中，对极几何约束给出了图像间的全部信息，成为解决许多视觉问题的关键环节，提出了一种基于遗传算法不同策略下的基础矩阵估计方法，它利用每个基因代表一个匹配点，每条染色体作为基础矩阵计算时的最小子集，并根据染色体长度决定采用何种策略估计基础矩阵，此方法在很大程度上减小了出格点对估计过程的影响，能够较好地汇聚到全局最优解，模拟数据和真实图像的实验结果都表明，所给出的方法能够有效地检测和删除错定位和误匹配点，提高了基础矩阵估计的鲁棒性和精度。     

一种无相机标定的极线校正新方法

提出一种无相机标定的立体图像对的极线校正新方法。该校正方法并不依赖基本矩阵F的精确求解,而是通过空间变换法分析校正前后图像点对应关系,依此分解并参数化描述极线变换矩阵,直接利用极线方程和图像的对应点集建立误差平方和函数,并运用非线性最小二乘法求解,使该函数取得最小值的变换参数。实验证明,该校正方法能够较好地消除垂直视差,图像产生的变形较小。     

Matching point features with ordered geometric, rigidity, anddisparity constraints

This correspondence presents a matching algorithm for obtaining feature point correspondences across images containing rigid objects undergoing different motions. First point features are detected using newly developed feature detectors. Then a variety of constraints are applied starting with simplest and following with more informed ones. First, an intensity-based matching algorithm is applied to the feature points to obtain unique point correspondences. This is followed by the application of a sequence of newly developed heuristic tests involving geometry, rigidity, and disparity. The geometric tests match two-dimensional geometrical relationships among the feature points, the rigidity test enforces the three dimensional rigidity of the object, and the disparity test ensures that no matched feature point in an image could be rematched with another feature, if reassigned another disparity value associated with another matched pair or an assumed match on the epipolar line. The computational complexity is proportional to the numbers of detected feature points in the two images. Experimental results with indoor and outdoor images are presented, which show that the algorithm yields only correct matches for scenes containing rigid objects     

Model-Based Object Recognition Using Geometric Invariants of Points and Lines

In this paper, we derive new geometric invariants for structured 3D points and lines from single image under projective transform, and we propose a novel model-based 3D object recognition algorithm using them. Based on the matrix representation of the transformation between space features (points and lines) and the corresponding projected image features, new geometric invariants are derived via the determinant ratio technique. First, an invariant for six points on two adjacent planes is derived, which is shown to be equivalent to Zhu's result [1], but in simpler formulation. Then, two new geometric invariants for structured lines are investigated: one for five lines on two adjacent planes and the other for six lines on four planes. By using the derived invariants, a novel 3D object recognition algorithm is developed, in which a hashing technique with thresholds and multiple invariants for a model are employed to overcome the over-invariant and false alarm problems. Simulation results on real images show that the derived invariants remain stable even in a noisy environment, and the proposed 3D object recognition algorithm is quite robust and accurate.     

Determining the Epipolar Geometry and its Uncertainty: A Review

Two images of a single scene/object are related by the epipolar geometry, which can be described by a 3×3 singular matrix called the essential matrix if images' internal parameters are known, or the fundamental matrix otherwise. It captures all geometric information contained in two images, and its determination is very important in many applications such as scene modeling and vehicle navigation. This paper gives an introduction to the epipolar geometry, and provides a complete review of the current techniques for estimating the fundamental matrix and its uncertainty. A well-founded measure is proposed to compare these techniques. Projective reconstruction is also reviewed. The software which we have developed for this review is available on the Internet.     

场景无关约束下的特征匹配算法

鉴于对极约束是立体图像中完全不依赖于场景的重要几何约束，因此在特征匹配中起着很重要的作用，而且由于同形映射描述了平面场景的立体图像之间的对应关系，故大量文献中利用它对平面场景的立体图像对进行特征匹配。为了提高立体图像匹配精度和速度，提出了一种改进的场景无关约束下的特征匹配算法，该算法针对用对极约束和同形映射来进行曲面场景匹配的过程中同形估计容易出现降阶的情况，通过引入区域面积检测法来避免降阶情况的发生，以改善匹配结果；同时，由于在同形矩阵估计中，通过加入基础矩阵和同形矩阵本质上的约束关系，可使得原本独立的同形约束和对极约柬关系很好地融入到匹配的整个过程中，从而快速有效地抑制了错误匹配的发生。对真实图像的实验分析证明，该改进算法具有迭代次数少、速度更快和匹配精度高的良好性能。     

Reconstruction-based recognition of scenes with translationallyrepeated quadrics

This paper addresses the problem of invariant-based recognition of quadric configurations from a single image. These configurations consist of a pair of rigidly connected translationally repeated quadric surfaces. This problem is approached via a reconstruction framework. A new mathematical framework, using relative affine structure, on the lines of Luong and Vieville (1996), has been proposed. Using this mathematical framework, translationally repeated objects have been projectively reconstructed, from a single image, with four image point correspondences of the distinguished points on the object and its translate. This has been used to obtain a reconstruction of a pair of translationally repeated quadrics. We have proposed joint projective invariants of a pair of proper quadrics. For the purpose of recognition of quadric configurations, we compute these invariants for the pair of reconstructed quadrics. Experimental results on synthetic and real images, establish the discriminatory power and stability of the proposed invariant-based recognition strategy. As a specific example, we have applied this technique for discriminating images of monuments which are characterized by translationally repeated domes modeled as quadrics