Jan Heller

Structure-from-motion based hand-eye calibration using L ∞ minimization

This paper presents a novel method for so-called hand-eye calibration. Using a calibration target is not possible for many applications of hand-eye calibration. In such situations Structure-from-Motion approach of hand-eye calibration is commonly used to recover the camera poses up to scaling. The presented method takes advantage of recent results in the L∞-norm optimization using Second-Order Cone Programming (SOCP) to recover the correct scale. Further, the correctly scaled displacement of the hand-eye transformation is recovered solely from the image correspondences and robot measurements, and is guaranteed to be globally optimal with respect to the L∞-norm. The method is experimentally validated using both synthetic and real world datasets.

A branch-and-bound algorithm for globally optimal hand-eye calibration

This paper introduces a novel solution to hand-eye calibration problem. It is the first method that uses camera measurements directly and at the same time requires neither prior knowledge of the external camera calibrations nor a known calibration device. Our algorithm uses branch-and-bound approach to minimize an objective function based on the epipolar constraint. Further, it employs Linear Programming to decide the bounding step of the algorithm. The presented technique is able to recover both the unknown rotation and translation simultaneously and the solution is guaranteed to be globally optimal with respect to the L∞-norm.

Hand-Eye and Robot-World Calibration by Global Polynomial Optimization

The need to relate measurements made by a camera to a different known coordinate system arises in many engineering applications. Historically, it appeared for the first time in the connection with cameras mounted on robotic systems. This problem is commonly known as hand-eye calibration. In this paper, we present several formulations of hand-eye calibration that lead to multivariate polynomial optimization problems. We show that the method of convex linear matrix inequality (LMI) relaxations can be used to effectively solve these problems and to obtain globally optimal solutions. Further, we show that the same approach can be used for the simultaneous hand-eye and robot-world calibration. Finally, we validate the proposed solutions using both synthetic and real datasets.

Radial distortion homography

The importance of precise homography estimation is often underestimated even though it plays a crucial role in various vision applications such as plane or planarity detection, scene degeneracy tests, camera motion classification, image stitching, and many more. Ignoring the radial distortion component in homography estimation-even for classical perspective cameras-may lead to significant errors or totally wrong estimates. In this paper, we fill the gap among the homography estimation methods by presenting two algorithms for estimating homography between two cameras with different radial distortions. Both algorithms can handle planar scenes as well as scenes where the relative motion between the cameras is a pure rotation. The first algorithm uses the minimal number of five image point correspondences and solves a nonlinear system of polynomial equations using Gröbner basis method. The second algorithm uses a non-minimal number of six image point correspondences and leads to a simple system of two quadratic equations in two unknowns and one system of six linear equations. The proposed algorithms are fast, stable, and can be efficiently used inside a RANSAC loop.

Stereographic rectification of omnidirectional stereo pairs

We present a general technique for rectification of a stereo pair acquired by a calibrated omnidirectional camera. Using this technique we formulate a new stereographic rectification method. Our rectification does not map epipolar curves onto lines as common rectification methods, but rather maps epipolar curves onto circles. We show that this rectification in a certain sense minimizes the distortion of the original omnidirectional images. We formulate the rectification for multiple images and show that the choice of the optimal projection center of the rectification is under certain circumstances equivalent to the classical problem of spherical minimax location. We demonstrate the behaviour and the quality of the rectification in real experiments with images from 180 degree field of view fish eye lenses.

Efficient Intersection of Three Quadrics and Applications in Computer Vision

In this paper, we present a new algorithm for finding all intersections of three quadrics. The proposed method is algebraic in nature and it is considerably more efficient than the Gröbner basis and resultant-based solutions previously used in computer vision applications. We identify several computer vision problems that are formulated and solved as systems of three quadratic equations and for which our algorithm readily delivers considerably faster results. Also, we propose new formulations of three important vision problems: absolute camera pose with unknown focal length, generalized pose-and-scale, and hand-eye calibration with known translation. These new formulations allow our algorithm to significantly outperform the state-of-the-art in speed.

Efficient Solution to the Epipolar Geometry for Radially Distorted Cameras

The estimation of the epipolar geometry of two cameras from image matches is a fundamental problem of computer vision with many applications. While the closely related problem of estimating relative pose of two different uncalibrated cameras with radial distortion is of particular importance, none of the previously published methods is suitable for practical applications. These solutions are either numerically unstable, sensitive to noise, based on a large number of point correspondences, or simply too slow for real-time applications. In this paper, we present a new efficient solution to this problem that uses 10 image correspondences. By manipulating ten input polynomial equations, we derive a degree 10 polynomial equation in one variable. The solutions to this equation are efficiently found using the Sturm sequences method. In the experiments, we show that the proposed solution is stable, noise resistant, and fast, and as such efficiently usable in a practical Structure-from-Motion pipeline.

Hand-Eye calibration without hand orientation measurement using minimal solution

In this paper we solve the problem of estimating the relative pose between a robots gripper and a camera mounted rigidly on the gripper in situations where the rotation of the gripper w.r.t. the robot global coordinate system is not known. It is a variation of the so called hand-eye calibration problem. We formulate it as a problem of seven equations in seven unknowns and solve it using the Grobner basis method for solving systems of polynomial equations. This enables us to calibrate from the minimal number of two relative movements and to provide the first exact algebraic solution to the problem. Further, we describe a method for selecting the geometrically correct solution among the algebraically correct ones computed by the solver. In contrast to the previous iterative methods, our solution works without any initial estimate and has no problems with error accumulation. Finally, by evaluating our algorithm on both synthetic and real scene data we demonstrate that it is fast, noise resistant, and numerically stable.

Singly-Bordered Block-Diagonal Form for Minimal Problem Solvers

The Grobner basis method for solving systems of polynomial equations became very popular in the computer vision community as it helps to find fast and numerically stable solutions to difficult problems. In this paper, we present a method that potentially significantly speeds up Grobner basis solvers. We show that the elimination template matrices used in these solvers are usually quite sparse and that by permuting the rows and columns they can be transformed into matrices with nice block-diagonal structure known as the singly-bordered block-diagonal (SBBD) form. The diagonal blocks of the SBBD matrices constitute independent subproblems and can therefore be solved, i.e. eliminated or factored, independently. The computational time can be further reduced on a parallel computer by distributing these blocks to different processors for parallel computation. The speedup is visible also for serial processing since we perform \(O(n^3)\) Gauss-Jordan eliminations on smaller (usually two, approximately \({n \over 2} \times {n \over 2}\) and one \({n \over 3} \times {n \over 3}\)) matrices. We propose to compute the SBBD form of the elimination template in the preprocessing offline phase using hypergraph partitioning. The final online Grobner basis solver works directly with the permuted block-diagonal matrices and can be efficiently parallelized. We demonstrate the usefulness of the presented method by speeding up solvers of several important minimal computer vision problems.

World-Base Calibration by Global Polynomial Optimization

This paper presents a novel solution to the world-base calibration problem. It is applicable in situations where a known calibration target is observed by a camera attached to the end effector of a robotic manipulator. The presented method works by minimizing geometrically meaningful error function based on image projections. Our formulation leads to a non-convex multivariate polynomial optimization problem of a constant size. However, we show how such a problem can be relaxed using linear matrix inequality (LMI) relaxations and effectively solved using Semi definite Programming. Although the technique of LMI relaxations guaranties only a lower bound on the global minimum of the original problem, it can provide a certificate of optimality in cases when the global minimum is reached. Indeed, we reached the global minimum for all calibration tasks in our experiments with both synthetic and real data. The experiments also show that the presented method is fast and noise resistant.