We propose a minimal solution for the similarity registration (rigid pose and scale) between two sets of 3D lines, and also between a set of co-planar points and a set of 3D lines. The first problem is solved up to 8 discrete solutions with a minimum of 2 line-line correspondences, while the second is solved up to 4 discrete solutions using 4 point-line correspondences. We use these algorithms to perform the extrinsic calibration between a pose tracking sensor and a 2D 3D ultrasound (US) curvilinear probe using a tracked needle as calibration target. The needle is tracked as a 3D line, and is scanned by the ultrasound as either a 3D line (3D US) or as a 2D point (2D US). Since the scale factor that converts US scan units to metric coordinates is unknown, the calibration is formulated as a similarity registration problem. We present results with both synthetic and real data and show that the minimum solutions outperform the correspondent non-minimal linear formulations.
Minimal solutions are a well established topic in computer vision literature @cite_31 @cite_14 @cite_12 @cite_22 @cite_7 . In most cases they require solving a system of polynomial equations, which can be achieved using Grobner basis methods @cite_14 @cite_9 @cite_7 . Although these methods provide a general framework to build numeric polynomial solvers, they require a certain amount of symbolic manipulation that often requires a case-by-case analysis. To address this issue an automatic generator of polynomial solvers have been proposed @cite_22 . In this paper we develop minimum solutions using the action matrix method as presented in @cite_9 .
@cite_14 @cite_22 @cite_7 @cite_9 @cite_31 @cite_12
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A method is presented for building solvers for classes of multivariate polynomial equations. The method is based on solving an analogous template problem over a finite field, and then using the elimination order established for this problem for the original class of problems. A strength of this method is that this permits pivoting in the elimination. Solvers for several minimal problems in computer vision are presented. Relative pose is solved both for a generalised camera, and for a camera with unknown focal length, both in two positions with six visible points. A solver for optimal triangulation in three images is presented. Model-free calibration for pinhole cameras is investigated. It is shown that for a smooth deformation of the image plane, the image plane can be projectively reconstructed from two flow-fields from purely translating cameras. Methods for hand-eye calibration using the multilinear constraints and vehicle-eye for laser-scanner based navigation systems are presented. (Less) Finding solutions to minimal problems for estimating epipolar geometry and camera motion leads to solving systems of algebraic equations. Often, these systems are not trivial and therefore special algorithms have to be designed to achieve numerical robustness and computational efficiency. The state of the art approach for constructing such algorithms is the Grobner basis method for solving systems of polynomial equations. Previously, the Grobner basis solvers were designed ad hoc for concrete problems and they could not be easily applied to new problems. In this paper we propose an automatic procedure for generating Grobner basis solvers which could be used even by non-experts to solve technical problems. The input to our solver generator is a system of polynomial equations with a finite number of solutions. The output of our solver generator is the Matlab or C code which computes solutions to this system for concrete coefficients. Generating solvers automatically opens possibilities to solve more complicated problems which could not be handled manually or solving existing problems in a better and more efficient way. We demonstrate that our automatic generator constructs efficient and numerically stable solvers which are comparable or outperform known manually constructed solvers. The automatic generator is available at http: cmp.felk.cvut.cz minimal We present a method for solving systems of polynomial equations appearing in computer vision. This method is based on polynomial eigenvalue solvers and is more straightforward and easier to implement than the state-of-the-art Grobner basis method since eigenvalue problems are well studied, easy to understand, and efficient and robust algorithms for solving these problems are available. We provide a characterization of problems that can be efficiently solved as polynomial eigenvalue problems (PEPs) and present a resultant-based method for transforming a system of polynomial equations to a polynomial eigenvalue problem. We propose techniques that can be used to reduce the size of the computed polynomial eigenvalue problems. To show the applicability of the proposed polynomial eigenvalue method, we present the polynomial eigenvalue solutions to several important minimal relative pose problems. This paper presents several new results on techniques for solving systems of polynomial equations in computer vision. Grobner basis techniques for equation solving have been applied successfully to several geometric computer vision problems. However, in many cases these methods are plagued by numerical problems. In this paper we derive a generalization of the Grobner basis method for polynomial equation solving, which improves overall numerical stability. We show how the action matrix can be computed in the general setting of an arbitrary linear basis for ?[x] I. In particular, two improvements on the stability of the computations are made by studying how the linear basis for ?[x] I should be selected. The first of these strategies utilizes QR factorization with column pivoting and the second is based on singular value decomposition (SVD). Moreover, it is shown how to improve stability further by an adaptive scheme for truncation of the Grobner basis. These new techniques are studied on some of the latest reported uses of Grobner basis methods in computer vision and we demonstrate dramatically improved numerical stability making it possible to solve a larger class of problems than previously possible.