Summary. This paper provides a tutorial and survey of methods for parameterizing surfaces with a view to applications in geometric modelling and computer graphics. We gather various concepts from differential geometry which are relevant to surface mapping and use them to understand the strengths and weaknesses of the many methods for parameterizing piecewise linear surfaces and their relationship to one another. 1

We propose a new definition of the total variation norm for vector valued functions which can be applied to restore color and other vector valued images. The new TV norm has the desirable properties of (i) not penalizing discontinuities (edges) in the image, (ii) rotationally invariant in the image space, and (iii) reduces to the usual TV norm in the scalar case. Some numerical experiments on denoising simple color images in RGB color space are presented. 1 Introduction During gathering and transfer of image data some noise and blur is usually introduced into the image. Several reconstruction methods based on the total variation (TV) norm have been proposed and studied for intensity (gray scale) images, see [9, 14, 21, 26, 29]. Since these methods have been successful in reducing noise and blur without smearing sharp edges for intensity images, it is natural to extend the TV norm to handle color and other vector valued images. Why do we need color restoration? It can be argued that si...

An expression-invariant 3D face recognition approach is presented. Our basic assumption is that facial expressions can be modelled as isometries of the facial surface. This allows to construct expression-invariant representations of faces using the bending-invariant canonical forms approach. The result is an efficient and accurate face recognition algorithm, robust to facial expressions, that can distinguish between identical twins (the first two authors). We demonstrate a prototype system based on the proposed algorithm and compare its performance to classical face recognition methods. The numerical methods employed by our approach do not require the facial surface explicitly. The surface gradients field, or the surface metric, are sufficient for constructing the expression-invariant representation of any given face. It allows us to perform the 3D face recognition task while avoiding the surface reconstruction stage.

Many differential methods for the recovery of the optic flow field from an image sequence can be expressed in terms of a variational problem where the optic flow minimizes some energy. Typically, these energy functionals consist of two terms: a data term, which requires e.g. that a brightness constancy assumption holds, and a regularizer that encourages global or piecewise smoothness of the flow field. In this paper we present a systematic classification of rotation invariant convex regularizers by exploring their connection to diffusion filters for multichannel images. This taxonomy provides a unifying framework for data-driven and flow-driven, isotropic and anisotropic, as well as spatial and spatio-temporal regularizers. While some of these techniques are classic methods from the literature, others are derived here for the first time. We prove that all these methods are well-posed: they posses a unique solution that depends in a continuous way on the initial data. An interesting structural relation between isotropic and anisotropic flow-driven regularizers is identified, and a design criterion is proposed for constructing anisotropic flow-driven regularizers in a simple and direct way from isotropic ones. Its use is illustrated by several examples.

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Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools like global parameterization and inter-surface mapping, and demonstrates a variety of parameterization applications.

Abstract. Polynomial Pythagorean hodograph (PH) curves form a remarkable subclass of polynomial parametric curves; they are distinguished by having a polynomial arc length function and rational offsets (parallel curves). Many related references can be found in the article by Farouki and Neff on C 1 Hermite interpolation with PH quintics. We extend the C 1 Hermite interpolation scheme by taking additional curvature information at the segment boundaries into account. As a result we obtain a new construction of curvature continuous polynomial PH spline curves. We discuss Hermite interpolation of G 2 [C 1] boundary data (points, first derivatives, and curvatures) with PH curves of degree 7. It is shown that up to eight possible solutions can be found by computing the roots of two quartic polynomials. With the help of the canonical Taylor expansion of planar curves, we analyze the existence and shape of the solutions. More precisely, for Hermite data which are taken from an analytical curve, we study the behaviour of the solutions for decreasing stepsize ∆. It is shown that a regular solution is guaranteed to exist for sufficiently small stepsize ∆, provided that certain technical assumptions are satisfied. Moreover, this solution matches the shape of the original curve; the approximation order is 6. As a consequence, any given curve, which is assumed to be G 2 (curvature continuous) and to consist of analytical segments can approximately be converted into polynomial PH form. The latter assumption is automatically satisfied by the standard curve representations of Computer Aided Geometric Design, such as Bézier or B-spline curves. The conversion procedure acts locally, without any need for solving a global system of equations. It produces G 2 polynomial PH spline curves of degree 7. 1.

In this paper we propose a new model, Frenet-Serret motion, for the motion of an observer in a stationary environment. This model relates the motion parameters of the observer to the curvature and torsion of the path along which the observer moves. Screw-motion equations for Frenet-Serret motion are derived and employed for geometrical analysis of the motion. Normal ow is used to derive constraints on the rotational and translational velocity of the observer and to compute egomotion by intersecting these constraints in the manner proposed in [6]. The accuracy of egomotion estimation is analyzed for di erentcombinations of observer motion and feature distance. We explain the advantages of controlling feature distance to analyze egomotion and derive the constraints on depth whichmake either rotation or translation dominant in the perceived normal ow eld. The results of experiments on real image sequences are presented. 1

This article is devoted to the rotation minimizing frames that are associated with spatial curves. Firstly we summarize some results concerning the differential geometry of the sweeping surfaces which are generated by these frames (the so-called profile or moulding surfaces). In the second part of the article we describe a rational approximation scheme. This scheme is based on the use of spatial Pythagorean hodograph (PH) cubics (also called cubic helices) as spine curves. We discuss the existence of solutions and the approximation order of G 1 Hermite interpolation with PH cubics. It is shown that any spatial curve can approximately be converted into cubic PH spline form. By composing the rational Frenet-Serret frame of these curves with suitable rotations around the tangent we develop a highly accurate rational approximation of the rotation minimizing frame. This leads to an approximate rational representation of profile surfaces.

Leading–order equations governing the dynamics of a two–dimensional thin viscous sheet are derived. The inclusion of inertia effects is found to result in an ill– posed model when the sheet is compressed, and the resulting paradox is resolved by rescaling the equations over new length – and timescales which depend on the Reynolds number of the flow and the aspect ratio of the sheet. Physically this implies a dominant lengthscale for transverse displacements during viscous buckling. The theory is generalised to give new models for fully three–dimensional sheets. 1