| Vision allows humans to perceive and understand the world surrounding them. Computer vision aims at to duplicate the effect of human vision by electronically perceiving and understanding an image. Giving computers the ability to see is not an easy task. We live in a three-dimensional (3D) world, and when computers try to analyze objects in 3D space, the visual sensors available usually give two-dimensional (2D) image, and this projection to a lower number of dimensions incurs an enormous loss of information. Dynamic scenes such as those to which we are accustomed, with moving objects or a moving camera, make computer vision even more complicated. Inferring three-dimensional information from images take from different viewpoints is a central problem in computer vision. At present, there are two approaches to accomplish this task. The first approach is based on previous camera calibration. However, calibration can not be used in active systems due to its lack of flexibility. Note that in active system, the optical and geometrical characterstics of the cameras might change dynamically depending on the imaging scene and camera motion. The second approach is based on computing the epipolar geometry between both imaging sensors. Two perspective images of a single rigid object/scene are related by the so-called epipolar geometry-the intrinsic projective geometry between two views, which can be described by a 3×3 matrix of rank-2. It is independent of scene structure, and only depends on the cameras'internal parameters and relative pose. The matrix is called fundamental matrix. The fundamental matrix can be computed from correspondences of imaged scene points alone, without requiring knowledge of the cameras' internal parameters or relative pose. It contains all geometric information that is necessary for establishing correspondences between two images, from which three-dimensional structure of the perceived scene can be inferred. The importance of the fundamental matrix has been stressed in the last few years and it is now believed that it will have an significant role in future applications of computer vision.Applications using the fundamental matrix include 3D-structure recovery, motion segmentation and camera self-calibration, depth from stereo, structure from motion, ego-motion and image coding. et al. Therefore an accurate and robust estimation of the fundamental matrix is crucial in computation vision. Consider two images of the same three-dimensional scene taken from different viewpoints. If a point m =(u ,v,1)Tin the first image and a point m ′=(u ′,v′,1)T in the second image correspond to the same physical 3D point in space, then m ′Fm=0, where F is a 3×3 matrix, called fundamental matrix. Although matrix F has nine elements, it has only seven degrees of freedom. Fundamental mtrix is the basic tool in 3D reconstruction from two images. In a stereovision system where the camera geometry is calibrated, it is possible to calculate such a matrix from the camera perspective projection matrices through calibration. When the intrinsic parameters are known but the extrinsic ones (the rotation and translation between the two images) are not, the problem is known as motion and structure from motion, and has been extensively studied in computer vision. We are interested here in different techniques for estimating the fundamental matrix from two uncalibrated images, i.e., the case where both the intrinsic and extrinsic parameters of the images are unknown. Up to now, many methods have been proposed to compute the fundamental matrix from correspondences. These methods can be divided into three classes: linear methods, iteration methods and robust methods.We have studied the theory of the fundamental matrix, and proposed several new methods for estimating fundamental matrix. 1. A new linear method for estimating fundamental matrix Linear methods can largely reduce the computing time. Iteration methods and robust methods are more accurate but they are time-consuming. Linear methods can supply good initial values for iteration methods. Iteration and robust methods are just repeated applications of linear methods in general. Thus linear methods are foundation of various methods of estimating the fundamental matrix. Linear methods are mainly based on various least-squares methods, and the method using eigen analysis can obtain the best results because an orthogonal least-squares minimization is more realistic than the classic one. The matrix obtained by the method, however, is of rank 3, which does not satisfy the requirement that all the epipolar lines should intersect in a unique epipole. One has found that the method is sensitive to noise. One reason is that the rank-2 constraint is not satisfied. Although a mathematical method, which transforms a rank-3 square matrix to the closest rank-2 matrix by singular value decomposition, has been proposed, the transformation will yield worse results. We propose a new linear approach to estimating the fundamental matrix. Instead of using the eigenvector corresponding to the smallest eigenvalue from the orthogonal least-squares method to form the fundamental matrix, we make use of two eigenvectors corresponding to the two smallest eigenvalues to construct a 3×3 generalized eigenvalue problem. The solutions to the problem give not only the fundamental matrix with rank of 2 but also the corresponding epipoles. The new approach is compared with other linear techniques in literature for estimation of the fundamental matrix and good performance is illustrated. 2. The Golden section method for consistent fundamental matrix estimationAs estimation of the fundamental matrix is based on 2D image measurements, the parameters in the fundamental matrix can be over-sensitive to the accuracy of the 2D measurements. In order to combat their undesirable influence, Hartley recently demonstrated that the effect of measurement errors can be contained to some degree by means of a suitable normalization after which the fundamental matrix can be estimated using a linear algorithm. This preprocessing has found a theoretical justification limited to the assumption of noise confined only in one of the two images. Because these assumptions are not realistic, the corresponding estimations are inconsistent and biased. A consistent fundamental matrix estimation based on a quadratic measurement error model has recently been established. The more realistic assumptions in the estimation are adopted. We propose a Golden section method for consistent fundamental matrix estimation. Note that in the estimation, the key is solving a non-smooth optimization problem. The proposed method requires calculating values of a function only. The computation of derivatives is avoided. The new approach is compared with the total least-squares method by using synthetic and real images. The effectiveness of the proposed approach is verified and illustrated. The convergence velocity of the method is linear. 3. An extended system method for consistent fundamental matrix estimation The proposed Golden section method above requires solving total eigenvalue problems for each trial point. In addition, the convergence velocity of the method is linear, therefore the method is time-consuming. We propose an extended system method for consistent fundamental matrix estimation in the quadratic measurement error model. First we transform the non-smooth optimization problem to a smooth one. Then an extended system for determining the consistent estimator is proposed, and an efficient implementation for solving the system-a continuation method has been developed to fix on an interval in which the minimum of the objectfunction belongs. An optimization method using a quadratic interpolation is used to exactly locate the minimum. The convergence velocity of the method is superlinear. The proposed method avoids solving total eigenvalue problems. Thus the computational cost is significantly reduced. Experiments with synthetic and real images have shown the effectiveness of the proposed method. 4. A robust method for consistent fundamental matrix estimation The previous proposed methods for consistent fundamental matrix estimation are not concerned with the issues of robustness. In many practical applications, data of matched points are not only noisy, but also contain outliers, data that are in gross disagreement with a postulated model. Outliers, which are inevitably included in an initial set of matched points, can so distort an estimating process that the estimated parameters become arbitrary. In such circumstances, more emphasis should be placed on improving the quality of data used for estimation. Therefore the deployment of robust estimation methods is essential. Robust methods continue to recover meaningful descriptions of a statistical population even when the data contain outlying elements belonging to a different population. A new method is presented for robustly estimating fundamental matrix from matched points. Here, "robust"means that the method gives good results even when the data contain outliers, i.e., data that do not conform to the assumed probability distribution. The method comprises two parts. The first part of the method uses a robust technique-the random sample consensus (RANSAC) to discard outliers in an initial set of matched points. It adopts the sampling strategy to generate inliers of putative matched points. The second part of the method is an algorithm for computing fundamental matrix, using the output of RANSAC. This algorithm is based on the consistent fundamental matrix estimation in a quadratic measurement error model. An extended system method for determining the estimator is proposed. Results for both synthetic and real images are used to show the effectiveness of the proposed method. |
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