Hugh M. Hilden

Bio: Hugh M. Hilden is an academic researcher from University of Hawaii at Manoa. The author has contributed to research in topics: Log-polar coordinates & Centroid. The author has an hindex of 3, co-authored 3 publications receiving 1143 citations.

Closed-form solution of absolute orientation using orthonormal matrices

Abstract: Finding the relationship between two coordinate systems by using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task. The solution has applications in stereophotogrammetry and in robotics. We present here a closed-form solution to the least-squares problem for three or more points. Currently, various empirical, graphical, and numerical iterative methods are in use. Derivation of a closed-form solution can be simplified by using unit quaternions to represent rotation, as was shown in an earlier paper [ J. Opt. Soc. Am. A4, 629 ( 1987)]. Since orthonormal matrices are used more widely to represent rotation, we now present a solution in which 3 × 3 matrices are used. Our method requires the computation of the square root of a symmetric matrix. We compare the new result with that obtained by an alternative method in which orthonormality is not directly enforced. In this other method a best-fit linear transformation is found, and then the nearest orthonormal matrix is chosen for the rotation. We note that the best translational offset is the difference between the centroid of the coordinates in one system and the rotated and scaled centroid of the coordinates in the other system. The best scale is equal to the ratio of the root-mean-square deviations of the coordinates in the two systems from their respective centroids. These exact results are to be preferred to approximate methods based on measurements of a few selected points.

Erratum to `Isotopies of homeomorphisms of Riemann surfaces’

Abstract: There is an error in the statement and proof of Lemma 5.1 of [1]. The Lemma in question is true in some cases and false in others. The error does not affect the main body of [1], that is Theorems 1,2,3 and 4, but it does imply that Theorem 5, the proof of which uses Lemma 5.1, is true precisely when Lemma 5.1 is true and must be modified when it is false. Theorem 5 of [1] is well-known to be true in the case of hyperelliptic covering spaces of the punctured sphere. That application has been used widely in the mathematical literature. Since [1] was published 43 years ago and has been used by many authors, we checked through the 42 papers that, according to MathSciNet, cited our work, and verified that they had all used Theorems 1-4 but not Theorem 5.

Efficient variants of the ICP algorithm

Abstract: The ICP (Iterative Closest Point) algorithm is widely used for geometric alignment of three-dimensional models when an initial estimate of the relative pose is known. Many variants of ICP have been proposed, affecting all phases of the algorithm from the selection and matching of points to the minimization strategy. We enumerate and classify many of these variants, and evaluate their effect on the speed with which the correct alignment is reached. In order to improve convergence for nearly-flat meshes with small features, such as inscribed surfaces, we introduce a new variant based on uniform sampling of the space of normals. We conclude by proposing a combination of ICP variants optimized for high speed. We demonstrate an implementation that is able to align two range images in a few tens of milliseconds, assuming a good initial guess. This capability has potential application to real-time 3D model acquisition and model-based tracking.

A tutorial on visual servo control

Abstract: This article provides a tutorial introduction to visual servo control of robotic manipulators. Since the topic spans many disciplines our goal is limited to providing a basic conceptual framework. We begin by reviewing the prerequisite topics from robotics and computer vision, including a brief review of coordinate transformations, velocity representation, and a description of the geometric aspects of the image formation process. We then present a taxonomy of visual servo control systems. The two major classes of systems, position-based and image-based systems, are then discussed in detail. Since any visual servo system must be capable of tracking image features in a sequence of images, we also include an overview of feature-based and correlation-based methods for tracking. We conclude the tutorial with a number of observations on the current directions of the research field of visual servo control.

Shape and motion from image streams under orthography: a factorization method

Abstract: Inferring scene geometry and camera motion from a stream of images is possible in principle, but is an ill-conditioned problem when the objects are distant with respect to their size. We have developed a factorization method that can overcome this difficulty by recovering shape and motion under orthography without computing depth as an intermediate step. An image stream can be represented by the 2FxP measurement matrix of the image coordinates of P points tracked through F frames. We show that under orthographic projection this matrix is of rank 3. Based on this observation, the factorization method uses the singular-value decomposition technique to factor the measurement matrix into two matrices which represent object shape and camera rotation respectively. Two of the three translation components are computed in a preprocessing stage. The method can also handle and obtain a full solution from a partially filled-in measurement matrix that may result from occlusions or tracking failures. The method gives accurate results, and does not introduce smoothing in either shape or motion. We demonstrate this with a series of experiments on laboratory and outdoor image streams, with and without occlusions.

EPnP: An Accurate O(n) Solution to the PnP Problem

Abstract: We propose a non-iterative solution to the PnP problem--the estimation of the pose of a calibrated camera from n 3D-to-2D point correspondences--whose computational complexity grows linearly with n This is in contrast to state-of-the-art methods that are O(n 5) or even O(n 8), without being more accurate Our method is applicable for all n?4 and handles properly both planar and non-planar configurations Our central idea is to express the n 3D points as a weighted sum of four virtual control points The problem then reduces to estimating the coordinates of these control points in the camera referential, which can be done in O(n) time by expressing these coordinates as weighted sum of the eigenvectors of a 12×12 matrix and solving a small constant number of quadratic equations to pick the right weights Furthermore, if maximal precision is required, the output of the closed-form solution can be used to initialize a Gauss-Newton scheme, which improves accuracy with negligible amount of additional time The advantages of our method are demonstrated by thorough testing on both synthetic and real-data