Lyle Noakes

Null cubics and Lie quadratics

Riemannian cubics are curves in a Riemannian manifold M satisfying a variational condition. They arise in computer graphics and in trajectory planning problems for rigid body motion, where M is the group SO(3) of rotations of Euclidean three-space E3. Riemannian cubics on a Lie group correspond to Lie quadratics in the Lie algebra. There are only a few cases where closed-form expressions are available for Lie quadratics. The present article is a qualitative analysis of null quadratics in so(3), focusing on long term dynamics and internal symmetries. Conclusions are drawn for asymptotics and symmetries of null cubics in SO(3).

Nonlinearities and Noise Reduction in 3-Source Photometric Stereo

Abstract1-D Leap-Frog (L. Noakes, J. Math. Australian Soc. A, Vol. 64, pp. 37–50, 1999) is an iterative scheme for solving a class of nonquadratic optimization problems. In this paper a 2-D version of Leap-Frog is applied to a non optimization problem in computer vision, namely the recovery (so far as possible) of an unknown surface from 3 noisy camera images. This contrasts with previous work on photometric stereo, in which noise is added to the gradient of the height function rather than camera images. Given a suitable initial guess, 2-D Leap-Frog is proved to converge to the maximum-likelihood estimate for the vision problem. Performance is illustrated by examples.

Nonlinear corner‐cutting

This begins the study of a Riemannian generalization of a special case of algorithm I of Lane and Riesenfeld (1980), closely related to the de Casteljau algorithm (Goldman, 1989) for generating cubic polynomial curves. In our version, as in Shoemakes (1985), straight lines are replaced by geodesic segments. Our construction differs from Shoemakes in that it is a kind of stationary subdivision algorithm, defined by a recursive procedure, and it is not at all clear from the construction that a limiting curve q∞ exists, much less that it is differentiable. Indeed, the aim of the present paper is to prove that q∞ is differentiable and that the derivative is Lipschitz. The result is nontrivial: it is well‐known that stationary subdivision typically defines non‐differentiable curves (Cavaretta et al., 1991). On the other hand Shoemakes algorithm is non‐recursive and evidently defines a C∞ curve. Other approaches to splines on curved spaces are considered in (Barr et al., 1992; Chapman and Noakes, 1991; Duff, 1985; Gabriel and Kajiya, 1985; Noakes et al., 1989).

Geometric Properties for Incomplete Data

Contributors. Preface. I Continuous Geometry: Representation of Free-form Objects. Spheres and Conics. Algorithms for Spatial Pythagoreanhodograph Curves. Cumulative Chords, Piecewise-Quadratics and Piecewise-Cubics. Spherical Splines. Graph-Spectral Methods for Surface Height Recovery from Gauss Maps.- II Discrete Geometry: Segmentation of Boundaries into Convex and Concave Parts. Convex and Concave Parts of Digital Curves. Polygonalisation and Polyhedralisation by Optimisation. Binary Tomography by Iterating Linear Programs. Cascade of dual LDA Operators for Face Recognition. Precision of Geometric Moments in Picture Analysis. Shape-from-Shading by Iterative Fast Marching for Vertical and Oblique Light Sources. Shape from Shadows.- III Approximation and Regularization: A Confidence Measure for Variational Optic Flow Methods. Video Image Sequence Analysis: Estimating Missing Data and Segmenting Multiple Motions. Robust Local Approximation of Scattered Data. On Robust Estimation and Smoothing with Spatial and Tonal Kernels. Subspace Estimation with Uncertain and Correlated Data. On the use of Dual Norms in Bounded Variation Type Regularization.- Index.

Bézier curves and C2 interpolation in Riemannian manifolds

In a connected Riemannian manifold, generalised Bezier curves are C^~ curves defined by a generalisation, in which line segments are replaced by minimal geodesics, of the classical de Casteljau algorithm. As in Euclidean space, these curves join their first and last control points. We compute the endpoint velocities and (covariant) accelerations of a generalised Bezier curve of arbitrary degree and use the formulae to express the curves control points in terms of these quantities. These results allow generalised Bezier curves to be pieced together into C^2 splines, and thereby allow C^2 interpolation of a sequence of data points. For the case of uniform splines in symmetric spaces, we show that C^2 continuity is equivalent to a simple relationship, involving the global symmetries at knot points, between the control points of neighbouring curve segments. We also present some examples in hyperbolic 2-space.

Non-null Lie quadratics in E3

Interpolation problems in the space SO(3) of rotations of Euclidean 3-space E3 are reviewed in Secs. I s2 as background and motivation to a study of curves in E3 called Lie quadratics. Except for a special class called null, Lie quadratics have resisted analysis until now. The rest of the present paper is devoted to new results showing non-null Lie quadratics have rich analytical, geometrical, and asymptotic structures: rates of growth are studied using differential equations and inequalities, Lie quadratics are proved to be extendible over the whole of R, and existence of axes is proved under fairly general conditions. Examples show sharpness of many results.

Duality and Riemannian cubics

AbstractRiemannian cubics are curves used for interpolation in Riemannian manifolds. Applications in trajectory planning for rigid bodiy motion emphasise the group SO(3) of rotations of Euclidean 3-space. It is known that a Riemannian cubic in a Lie group G with bi-invariant Riemannian metric defines a Lie quadraticV in the Lie algebra, and satisfies a linking equation. Results of the present paper include explicit solutions of the linking equation by quadrature in terms of the Lie quadratic, when G is SO(3) or SO(1,2). In some cases we are able to give examples where the Lie quadratic is also given in closed form. A basic tool for constructing solutions is a new duality theorem. Duality is also used to study asymptotics of differential equations of the form

Length estimation for curves with different samplings

\dot{x}(t)=(\beta_{0}+t\beta_{1})x(t)

2D leapfrog algorithm for optimal surface reconstruction

, where β0,β1 are skew-symmetric 3×3 matrices, and x :ℝ→ SO(3). This is done by showing that the dual of β0+tβ1 is a null Lie quadratic. Then results on asymptotics of x follow from known properties of null Lie quadratics.

Piecewise-quadratics and exponential parameterization for reduced data

This paper* looks at the problem of approximating the length of the unknown parametric curve ?: [0, 1] ? IRn from points qi = ?(ti), where the parameters ti are not given. When the ti are uniformly distributed Lagrange interpolation by piecewise polynomials provides efficient length estimates, but in other cases this method can behave very badly [15]. In the present paper we apply this simple algorithm when the ti are sampled in what we call an ?-uniform fashion, where 0 ? ? ? 1. Convergence of length estimates using Lagrange interpolants is not as rapid as for uniform sampling, but better than for some of the examples of [15]. As a side-issue we also consider the task of approximating ? up to parameterization, and numerical experiments are carried out to investigate sharpness of our theoretical results. The results may be of interest in computer vision, computer graphics, approximation and complexity theory, digital and computational geometry, and digital image analysis.