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This volume constitutes the thoroughly refereed post-conference proceedings of the 7th International Conference on Curves and Surfaces, held in Avignon, in June 2010. The conference had the overall theme: "Representation and Approximation of Curves and Surfaces and Applications". The 39 revised full papers presented together with 9 invited talks were carefully reviewed and selected from 114 talks presented at the conference. The topics addressed by the papers range from mathematical foundations to practical implementation on modern graphics processing units and address a wide area of topics such as computer-aided geometric design, computer graphics and visualisation, computational geometry and topology, geometry processing, image and signal processing, interpolation and smoothing, scattered data processing and learning theory and subdivision, wavelets and multi-resolution methods.



Exact Medial Axis Computation for Triangulated Solids with Respect to Piecewise Linear Metrics

We propose a novel approach for the medial axis approximation of triangulated solids by using a polyhedral unit ball


instead of the standard Euclidean unit ball. By this means we compute the exact medial axis

$\mbox{\rm MA}(\Omega)$

of a triangulated solid Ω with respect to a piecewise linear (quasi-) metric



. The obtained representation of Ω by the medial axis transform

$\mbox{\rm MAT}(\Omega)$

allows for a convenient computation of the trimmed offset of Ω with respect to



. All calculations are performed within the field of rational numbers, resulting in a robust and efficient implementation of our approach. Adapting the properties of


provides an easy way to control the level of details captured by the medial axis, making use of the implicit pruning at flat boundary features.

Oswin Aichholzer, Wolfgang Aigner, Franz Aurenhammer, Bert Jüttler

Exact Medial Axis Computation for Circular Arc Boundaries

We propose a method to compute the algebraically correct medial axis for simply connected planar domains which are given by boundary representations composed of rational circular arcs. The algorithmic approach is based on the Divide-&-Conquer paradigm, as used in [12]. However, we show how to avoid inaccuracies in the medial axis computations arising from a non-algebraic biarc construction of the boundary. To this end we introduce the Exact Circular Arc Boundary representation (ECAB), which allows algebraically exact calculation of bisector curves. Fractions of these bisector curves are then used to construct the exact medial axis. We finally show that all necessary computations can be performed over the field of rational numbers with a small number of adjoint square-roots.

Oswin Aichholzer, Wolfgang Aigner, Thomas Hackl, Nicola Wolpert

Complex Bézier Curves and the Geometry of Polynomials

In this paper, we study the shape of the control polygon of a complex Bézier curve over a complex interval. We show that the location of the complex roots of the polynomial dictates geometrical constraints on the shape of the control polygon. Along the work, new proofs and generalizations of the Walsh coincidence Theorem and the Grace Theorem are given. Applications of the geometry of the control polygon of complex polynomials to Bernstein type inequalities are discussed.

Rachid Ait-Haddou, Taishin Nomura

The Shape of Conchoids to Plane Algebraic Curves

The classical conchoid construction, well-known in Mathematics since its introduction more than 2200 years ago, has been recently revisited in the context of algebraic curves. Partially, the motivation for this has been the analogy, up to a certain extent, between the conchoid and a well-known transformation in CAGD, namely the offset transformation. In this paper, we contribute to this study by addressing properties on the shape of conchoids to plane algebraic curves. For this purpose we introduce the notions of




conchoids, and we compare the shapes of these objects with that of the original curve.

Juan Gerardo Alcázar

Estimation of Integral Properties of a Planar Closed Curve Based on a Quadratic Spline Quasi-Interpolant

In this paper, we present a method based on a quadratic spline quasi-interpolant for the estimation of integral properties of a planar closed curve. The latter include the length, area, center of gravity and moment of inertia of the given curve. Then, we analyze the error estimates on the approximations of these properties and we validate the theoretical results by numerical examples.

C. Allouch, P. Sablonnière, D. Sbibih

Design of Multiresolution Operators Using Statistical Learning Tools: Application to Compression of Signals

Using multiresolution based on Harten’s framework [

J. Appl. Numer. Math.

, 12 (1993), pp. 153–192.] we introduce an alternative to construct a prediction operator using Learning statistical theory. This integrates two ideas: generalized wavelets and learning methods, and opens several possibilities in the compressed signal context. We obtain theoretical results which prove that this type of schemes (LMR schemes) are equal to or better than the classical schemes. Finally, we compare traditional methods with the algorithm that we present in this paper.

Francesc Aràndiga, Albert Cohen, Dionisio F. Yáñez

Weighted-Power p Nonlinear Subdivision Schemes

In this paper we present and analyze a generalization of the



subdivision schemes proposed in [3,12]. The



schemes are based on a harmonic weighted version of the



average considered in [12], and their development is motivated by the desire to generalize the nonlinear analysis in [3,5] to interpolatory subdivision schemes with higher than second order accuracy.

Francesc Aràndiga, Rosa Donat, Maria Santágueda

Tracking Level Set Representation Driven by a Stochastic Dynamics

We introduce a non-linear stochastic filtering technique to track the state of a free curve from image data. The approach we propose is implemented through a particle filter, which includes color measurements characterizing the target and the background respectively. We design a continuous-time dynamics that allows us to infer inter-frame deformations. The curve is defined by an implicit level-set representation and the stochastic dynamics is expressed on the level-set function. It takes the form of a stochastic partial differential equation with a Brownian motion of low dimension. Specific noise models lead to the traditional level set evolution law based on mean curvature motions, while other forms lead to new evolution laws with different smoothing behaviors. In these evolution models, we propose to combine local photometric information, some velocity induced by the curve displacement and an uncertainty modeling of the dynamics. The associated filter capabilities are demonstrated on various sequences with highly deformable objects.

Christophe Avenel, Etienne Mémin, Patrick Pérez

G 2 Hermite Interpolation with Curves Represented by Multi-valued Trigonometric Support Functions

It was recently proved in [27] that all rational hypocycloids and epicycloids are Pythagorean hodograph curves, i.e., rational curves with rational offsets. In this paper, we extend the discussion to a more general class of curves represented by trigonometric polynomial support functions. We show that these curves are offsets to translated convolutions of scaled and rotated hypocycloids and epicycloids. Using this result, we formulate a new and very simple



Hermite interpolation algorithm based on solving a small system of linear equations. The efficiency of the designed method is then presented on several examples. In particular, we show how to approximate general trochoids, which, as we prove, are not Pythagorean hodograph curves in general.

Bohumír Bastl, Miroslav Lávička, Zbyněk Šír

Approximating Algebraic Space Curves by Circular Arcs

We introduce a new method to approximate algebraic space curves. The algorithm combines a subdivision technique with local approximation of piecewise regular algebraic curve segments. The local technique computes pairs of polynomials with modified Taylor expansions and generates approximating circular arcs. We analyze the connection between the generated approximating arcs and the osculating circles of the algebraic curve.

Szilvia Béla, Bert Jüttler

Generating Series for Drawing the Output of Dynamical Systems

We provide the drawing of the output of dynamical system (Σ), particularly when the output is rough or near instability points. (Σ) being analytical in a neighborhood of the initial state


(0) and described by its state equations, its output




) in a neighborhood of


 = 0 is obtained by “evaluating” its generating series. Our algorithm consists in juxtaposing local approximating outputs on successive time intervals [






 + 1


0 ≤ 




 − 1

, to draw




) everywhere as far as possible. At every point



 + 1

we calculate at order


an approximated value of each component



of the state; on every interval [






 + 1


0 ≤ 




 − 1

we calculate an approximated output. These computings are obtained from the symbolic expressions of the generating series of





, truncated at order


, specified for





and “evaluated”. A Maple package is built, providing a suitable result for oscillating outputs or near instability points when a Runge-Kutta method is wrong.

Farida Benmakrouha, Christiane Hespel, Edouard Monnier

Practical Mixed-Integer Optimization for Geometry Processing

Solving mixed-integer problems, i.e., optimization problems where some of the unknowns are continuous while others are discrete, is NP-hard. Unfortunately, real-world problems like e.g., quadrangular remeshing usually have a large number of unknowns such that exact methods become unfeasible. In this article we present a greedy strategy to rapidly approximate the solution of large quadratic mixed-integer problems within a practically sufficient accuracy. The algorithm, which is freely available as an open source library implemented in


, determines the values of the discrete variables by successively solving relaxed problems. Additionally the specification of arbitrary linear equality constraints which typically arise as side conditions of the optimization problem is possible. The performance of the base algorithm is strongly improved by two novel extensions which are (1) simultaneously estimating sets of discrete variables which do not interfere and (2) a fill-in reducing reordering of the constraints. Exemplarily the solver is applied to the problem of quadrilateral surface remeshing, enabling a great flexibility by supporting different types of user guidance within a real-time modeling framework for input surfaces of moderate complexity.

David Bommes, Henrik Zimmer, Leif Kobbelt

Non-degenerate Developable Triangular Bézier Patches

In this talk we show a construction for characterising developable surfaces in the form of Bézier triangular patches. It is shown that constructions used for rectangular patches are not useful, since they provide degenerate triangular patches. Explicit constructions of non-degenerate developable triangular patches are provided.

Alicia Cantón, Leonardo Fernández-Jambrina

Stable Splitting of Bivariate Splines Spaces by Bernstein-Bézier Methods

We develop stable splitting of the minimal determining sets for the spaces of bivariate



splines on triangulations, including a modified Argyris space, Clough-Tocher, Powell-Sabin and quadrilateral macro-element spaces. This leads to the stable splitting of the corresponding bases as required in Böhmer’s method for solving fully nonlinear elliptic PDEs on polygonal domains.

Oleg Davydov, Abid Saeed

Mesh Segmentation and Model Extraction

High precision laser scanners deliver virtual surfaces of industrial objects whose accuracy must be evaluated. But this requires the automatic detection of reliable components such as facets, cylindric and spherical parts, etc. The method described here finds automatically parts in the surface to which geometric primitives can be fitted. Knowing certain properties of the input object, this primitive fitting helps quantifying the precision of an acquisition process and of the scanned mires. The method combines mesh segmentation with model fitting. The mesh segmentation method is based on the level set tree of a scalar function defined on the mesh. The method is applied with the simplest available intrinsic scalar function on the mesh, the mean curvature. In a first stage a fast algorithm extracts the level sets of the scalar function. Adapting to meshes a well known method for extracting Maximally Stable Extremal Regions from the level set tree on digital images, the method segments automatically the mesh into smooth parts separated by high curvature regions (the edges). This segmentation is followed by a model selection on each part permitting to fit planes, cylinders and spheres and to quantify the overall accuracy of the acquisition process.

Julie Digne, Jean-Michel Morel, Charyar Mehdi-Souzani, Claire Lartigue

Design of C 2 Spatial Pythagorean-Hodograph Quintic Spline Curves by Control Polygons

An intuitive approach to designing spatial



Pythagorean–hodograph (PH) quintic spline curves, based on given control polygons, is presented. Although PH curves can always be represented in Bézier or B–spline form, changes to their control polygons will usually compromise their PH nature. To circumvent this problem, an approach similar to that developed in [13] for the planar case is adopted. Namely, the “ordinary”



cubic B–spline curve determined by the given control polygon is first computed, and the



PH spline associated with that control polygon is defined so as to interpolate the nodal points of the cubic B–spline, with analogous end conditions. The construction of spatial PH spline curves is more challenging than the planar case, because of the residual degrees of freedom it entails. Two strategies for fixing these free parameters are presented, based on optimizing shape measures for the PH spline curves.

Rida T. Farouki, Carla Manni, Francesca Pelosi, Maria Lucia Sampoli

Shape Curvatures of Planar Rational Spirals

We discuss geometric conditions for a planar rational curve to be a spiral. The main used tool for characterizing a spiral is the so-called shape curvature which is a differential-geometric invariant of planar curves with respect to the group of orientation-preserving similarities. Different formulas for computation of shape curvature are given. Rational curves representing spirals and circular arcs possess a natural description in terms of the shape curvature. This description is applied to a complete classification of quadratic Bézier spirals.

Georgi H. Georgiev

Volumetric Geometry Reconstruction of Turbine Blades for Aircraft Engines

We present a framework for generating a trivariate B-spline parametrization of turbine blades from measurement data generated by optical scanners. This new representation replaces the standard patch-based representation of industrial blade designs. In a first step, the blade surface is represented by a smoothly varying family of B-spline curves. In a second step, the blade is parametrized by a trivariate B-spline volume. The resulting model is suitable for numerical simulation via isogeometric analysis, as well as for a fully automatic structured mesh generation with standard finite elements. We focus on the industrial applicability of the framework, by using standard turbine blade features throughout the process.

David Großmann, Bert Jüttler

Globally Convergent Adaptive Normal Multi-scale Transforms

In this paper we introduce a family of globally well-posed and convergent normal multi-scale transforms with high-order detail decay rate for smooth curves, based on adaptivity. For one of the members in the family, we propose a concrete algorithm what the adaptive criteria should be, and provide numerical evidence for the implementation. We compare the performance of our algorithm with other normal multi-scale transforms.

Stanislav Harizanov

Helmholtz-Hodge Decomposition on [0,1] d by Divergence-Free and Curl-Free Wavelets

This paper deals with the Helmholtz-Hodge decomposition of a vector field in bounded domain. We present a practical algorithm to compute this decomposition in the context of divergence-free and curl-free wavelets satisfying suitable boundary conditions. The method requires the inversion of divergence-free and curl-free wavelet Gram matrices. We propose an optimal preconditioning which allows to solve the systems with a small number of iterations. Finally, numerical examples prove the accuracy and the efficiency of the method.

Souleymane Kadri Harouna, Valérie Perrier

Finite Element Analysis with B-Splines: Weighted and Isogeometric Methods

Weighted and isogeometric methods use b-splines to construct bases for FEM. They combine the computational efficiency of regular grids with the geometric flexibility of CAD representations. We give a brief description of the key ideas of the two approaches, presenting them in a unified framework. In particular, we use b-spline nodes, to visualize the free parameters. Moreover, we explain how to combine features of both techniques by introducing weighted isogeometric finite elements. An error estimate for the resulting mixed method is given, and the performance of weighted approximations is illustrated by numerical examples.

Klaus Höllig, Jörg Hörner, Axel Hoffacker

$\sqrt{3}$ -Based 1-Form Subdivision

In this paper we construct an edge based, or


, subdivision scheme consistent with


subdivision. It produces smooth differential 1-forms in the limit. These can be identified with tangent vector fields, or viewed as edge elements in the sense of finite elements. In this construction, primal (0-form) and dual (2-form) subdivision schemes for surfaces are related through the exterior derivative with an edge (1-form) based subdivision scheme, amounting to a generalization of the well known formulé de commutation.

Starting with the classic


subdivision scheme as a 0-form subdivision scheme, we derive conditions for appropriate 1- and 2-form subdivision schemes without fixing the dual (2-form) subdivision scheme a priori. The resulting degrees of freedom are resolved through spectrum considerations and a conservation condition analogous to the usual moment condition for primal subdivision schemes.

Jinghao Huang, Peter Schröder

Curvature of Approximating Curve Subdivision Schemes

The promise of modeling by subdivision is to have simple rules that avoid cumbersome stitching-together of pieces. However, already in one variable, exactly reproducing a variety of basic shapes, such as conics and spirals, leads to non-stationary rules that are no longer as simple; and combining these pieces within the same curve by one set of rules is challenging. Moreover, basis functions, that allow reading off smoothness and computing curvature, are typically not available. Mimicking subdivision of splines with non-uniform knots allows us to combine the basic shapes. And to analyze non-uniform subdivision in general, the literature proposes interpolating the sequence of subdivision control points by circles. This defines a notion of discrete curvature for interpolatory subdivision. However, we show that this discrete curvature generically yields misleading information for non-interpolatory subdivision and typically does not converge, not even for non-uniform spline subdivision. Analyzing the causes yields three general approaches for solving or at least mitigating the problem: equalizing parameterizations, sampling subsequences and a new skip-interpolating subdivision approach.

Kȩstutis Karčiauskas, Jörg Peters

Fitting a Surface to One of Its Sectional Planar Curves Using Adaptive Trees in Spaces of Curves

We motivate from interventional medical imaging some geometric problems where one should identify how a curve could be generated from a known surface, asking for the pose of the surface from the outline of an orthogonal projection of it, or the position of a sectional plane from an isometric instance of the sectional curve. We describe a distance between curves that is efficiently implementable and use it to build stochastic trees in metric spaces towards subsets of curves of interest that either have strong convergence properties or permit quick searches on curves, and propose a real time application on a simplified version of such problem.

Yannick L. Kergosien

Verified Spatial Subdivision of Implicit Objects Using Implicit Linear Interval Estimations

In this paper we describe the LIETree, a new data structure for verified spatial decomposition of implicit objects. The LIETree is capable of utilizing implicit linear interval estimations for calculating a verified enclosure of the implicit function’s codomain. Furthermore, it uses consistency techniques to tighten the object enclosure. Overall, it delivers improved accuracy and uses fewer nodes than common uniform subdivision schemes using interval or affine arithmetic for enclosure.

Stefan Kiel

Image Separation Using Wavelets and Shearlets

In this paper, we present an image separation method for separating images into point- and curvelike parts by employing a combined dictionary consisting of wavelets and compactly supported shearlets utilizing the fact that they sparsely represent point and curvilinear singularities, respectively. Our methodology is based on the very recently introduced mathematical theory of geometric separation, which shows that highly precise separation of the morphologically distinct features of points and curves can be achieved by ℓ


minimization. Finally, we present some experimental results showing the effectiveness of our algorithm, in particular, the ability to accurately separate points from curves even if the curvature is relatively large due to the excellent localization property of compactly supported shearlets.

Gitta Kutyniok, Wang-Q Lim

On a Special Class of Polynomial Surfaces with Pythagorean Normal Vector Fields

Rational shapes with rational offsets, especially Pythagorean hodograph (PH) curves and Pythagorean normal vector (PN) surfaces, have been thoroughly studied for many years. However compared to PH curves, Pythagorean normal vector surfaces were introduced using dual approach only in their rational version and a complete characterization of polynomial surfaces with rational offsets, i.e., a polynomial solution of the well-known surface Pythagorean condition, still remains an open and challenging problem. In this contribution, we study a remarkable family of cubic polynomial PN surfaces with birational Gauss mapping, which represent a surface counterpart to the planar Tschirnhausen cubic. A full description of these surfaces is presented and their properties are discussed.

Miroslav Lávička, Jan Vršek

Convergence Rate of the Causal Jacobi Derivative Estimator

Numerical causal derivative estimators from noisy data are essential for real time applications especially for control applications or fluid simulation so as to address the new paradigms in solid modeling and video compression. By using an analytical point of view due to Lanczos [9] to this causal case, we revisit



order derivative estimators originally introduced within an algebraic framework by Mboup, Fliess and Join in [14,15]. Thanks to a given noise level


and a well-suitable integration length window, we show that the derivative estimator error can be

$\mathcal{O}(\delta ^{\frac{q+1}{n+1+q}})$



is the order of truncation of the Jacobi polynomial series expansion used. This so obtained bound helps us to choose the values of our parameter estimators. We show the efficiency of our method on some examples.

Da-yan Liu, Olivier Gibaru, Wilfrid Perruquetti

Continuous Deformations by Isometry Preserving Shape Integration

We introduce a novel continuous surface deformation method which relies on a time-dependent vector field over a triangular mesh. For every time step the piecewise linear vector field is obtained by least-squares minimization of the metric distortion induced by integration subject to boundary conditions. As an integral part of the approach, we introduce a new measure to describe local metric distortion which is invariant to the particular triangulation of the surface and which can incorporate smoothness of the field. Neither of these properties are met by previous work. A GPU implementation of the proposed algorithm enables fast deformations. The resulting deformations have lower metric distortions than deformations by existing (linear or non-linear) methods. This is shown for a number of representative test data sets.

Janick Martinez Esturo, Christian Rössl, Holger Theisel

On a (W)ENO-Type Multiscale Representation Based on Quincunx Refinement: Application to Image Compression

In this paper, we study nonlinear multiscale representations on ℝ


which are interpolatory and based on non-diagonal dilation matrices, such as the quincunx matrix. A compression procedure is then introduced for that kind of representations while numerical experiments conclude the paper.

Basarab Mateï, Sylvain Meignen, Anastasia Zakharova

OpenFlipper: An Open Source Geometry Processing and Rendering Framework

In this paper we present OpenFlipper, an extensible open source geometry processing and rendering framework. OpenFlipper is a free software toolkit and software development platform for geometry processing algorithms. It is mainly developed in the context of various academic research projects. Nevertheless some companies are already using it as a toolkit for commercial applications. This article presents the design goals for OpenFlipper, the central usability considerations and the important steps that were taken to achieve them. We give some examples of commercial applications which illustrate the flexibility of OpenFlipper. Besides software developers, end users also benefit from this common framework since all applications built on top of it share the same basic functionality and interaction metaphors.

Jan Möbius, Leif Kobbelt

Parameterization of Contractible Domains Using Sequences of Harmonic Maps

In this paper, we propose a new method for parameterizing a contractible domain (called the computational domain) which is defined by its boundary. Using a sequence of harmonic maps, we first build a mapping from the computational domain to the parameter domain, i.e., the unit square or unit cube. Then we parameterize the original domain by spline approximation of the inverse mapping. Numerical simulations of our method were performed with several shapes in 2D and 3D to demonstrate that our method is suitable for various shapes. The method is particular useful for isogeometric analysis because it provides an extension from a boundary representation of a model to a volume representation.

Thien Nguyen, Bert Jüttler

Nonlinear L 1 C 1 Interpolation: Application to Images

In this article, we address the problem of interpolating data points lying on a regular grid by






-bicubic spline surfaces. Our algorithm is based on a local univariate



minimization method which enable us to calculate first derivative values for



-cubic spline curves. In order to construct the interpolation surface, we calculate four derivative values at each data point using this local method. At is was shown in [17], our local interpolation



cubic spline curve algorithm preserves well the shape of the data even for abrupt changes.The sequential computational complexity of this local method is linear and the parallel computational complexity is O(1). Consequently, we can address in this manner data on large grids. In order to keep this linear complexity for spline surface interpolation, we define an interpolation scheme based on four linear directions so as to construct our



-bicubic surface. Some image interpolation examples show the efficiency of this non linear interpolation scheme.

E. Nyiri, O. Gibaru, Ph. Auquiert

Normal Multi-scale Transforms for Surfaces

We prove well-posedness, convergence, and detail decay estimates for the normal triangular mesh multi-scale transform for




graph surfaces given in the simplest case when the subdivision rule


used for base point prediction is given by edge midpoint insertion. A restrictive assumption is that the initial triangular mesh needs to be quasi-regular and of small enough mesh-size. We also provide numerical evidence with other


for dyadic refinement (Butterfly, Loop), and propose a modification of the normal scheme resulting in improved detail decay for smooth surfaces.

Peter Oswald

Generalized Dupin Cyclides with Rational Lines of Curvature

Dupin cyclides are algebraic surfaces of order three and four whose lines of curvature are circles. These surfaces have a variety of interesting properties and are aesthetic from a geometric and algebraic viewpoint. Besides their special property with respect to lines of curvature they appear as envelopes of one-parameter families of spheres in a twofold way. In the present article we study two families of canal surfaces with rational lines of curvature and rational principal curvatures, which contain the Dupin cyclides of order three and four as special instances in each family. The surfaces are constructed as anticaustics with respect to parallel illumination and reflection at tangent planes of curves on a cylinder of rotation.

Martin Peternell

Spline Volume Fairing

The use of trivariate NURBS in isogeometric analysis has put the quality of parametrization of NURBS volumes on the agenda. Sometimes a NURBS volume needs a better parametrization to meet requirements regarding smoothness, approximation or periodicity. In this paper we generalize various smoothing methods that already exist for bivariate parametric spline surfaces to trivariate parametric spline volumes. We will also address how rational and polynomial spline volumes create different challenges and solutions in the algorithms.

Kjell Fredrik Pettersen, Vibeke Skytt

Neuroelectric Current Localization from Combined EEG/MEG Data

EEG/MEG devices record external signals which are generated by the neuronal electric activity of the brain. The localization of the neuronal sources requires the solution of the neuroelectromagnetic inverse problem which is highly ill-posed and ill-conditioned. We provide an iterative thresholding algorithm for recovering neuroeletric current densities within the brain through combined EEG/MEG data. We use a joint sparsity constraint to promote solutions localized in small brain area, assuming that the vector components of the current densities possess the same sparse spatial pattern. At each iteration step, the EEG/MEG forward problem is numerically solved by a Galerkin boundary element method. Some numerical experiments on the localization of current dipole sources are also given. The numerical results show that joint sparsity constraints outperform classical regularization methods based on quadratic constraints.

Francesca Pitolli

Bootstrap-Based Normal Reconstruction

We propose a bootstrap-based method for normal estimation on an unorganised point set. Experimental results show that the accuracy of the method is comparable with the accuracy of the widely used Principal Component Analysis. The main advantage of our approach is that the variance of the normals over the bootstrap samples can be used as a confidence value for the estimated normal. In a proposed application, we use the confidence values to construct a bilateral Gaussian filter for normal smoothing.

Ahmad Ramli, Ioannis Ivrissimtzis

Couple Points – A Local Approach to Global Surface Analysis

We introduce the concept of couple points as a global feature of surfaces. Couple points are pairs of points

$({\mathbf x}_1,{\mathbf x}_2)$

on a surface with the property that the vector

${\mathbf x}_2 - {\mathbf x}_1$

is parallel to the surface normals both at

${\mathbf x}_1$


${\mathbf x}_2$

. In order to detect and classify them, we use higher order local feature detection methods, namely a Morse theoretic approach on a 4D scalar field. We apply couple points to a number of problems in Computer Graphics: the detection of maximal and minimal distances of surfaces, a fast approximation of the shortest geodesic path between two surface points, and the creation of stabilizing connections of a surface.

Christian Rössl, Holger Theisel

Chordal Cubic Spline Quasi Interpolation

This paper studies cubic spline quasi-interpolation of parametric curves through sequences of points in any space dimension. We show that if the parameter values are chosen by chord length, the order of accuracy is four. We also use this chordal cubic spline quasi interpolant to approximate the arc length derivatives and the length of the parametric curve.

P. Sablonnière, D. Sbibih, M. Tahrichi

Multiple Subdivision Schemes

Motivated by the concept of directionally adapted subdivision for the definition of shearlet multiresolution, the paper considers a generalized class of multivariate stationary subdivision schemes, where in each iteration step a scheme and a dilation matrix can be chosen from a given finite set. The standard questions of convergence and refinability will be answered as well as the continuous dependence of the resulting limit functions from the selection process. In addition, the concept of a

canonical factor

for multivariate subdivision schemes is introduced, which follows in a straightforward fashion from algebraic properties of the scaling matrix and takes the role of a smoothing factor for symbols.

Tomas Sauer

On a Linear Programming Approach to the Discrete Willmore Boundary Value Problem and Generalizations

We consider the problem of finding (possibly non connected) discrete surfaces spanning a finite set of discrete boundary curves in the three-dimensional space and minimizing (globally) a discrete energy involving mean curvature. Although we consider a fairly general class of energies, our main focus is on the Willmore energy, i.e. the total squared mean curvature. Most works in the literature have been devoted to the approximation of a surface evolving by the Willmore flow and, in particular, to the approximation of the so-called Willmore surfaces, i.e., the critical points of the Willmore energy. Our purpose is to address the delicate task of approximating


minimizers of the energy under boundary constraints. The main contribution of this work is to translate the nonlinear boundary value problem into an integer linear program, using a natural formulation involving pairs of elementary triangles chosen in a pre-specified dictionary and allowing self-intersection. The reason for such strategy is the well-known existence of algorithms that can compute

global minimizers

of a large class of linear optimization problems, however at a significant computational and memory cost. The case of integer linear programming is particularly delicate and usual strategies consist in relaxing the integral constraint


 ∈ {0,1} into


 ∈ [0,1] which is easier to handle. Our work focuses essentially on the connection between the integer linear program and its relaxation. We prove that:

One cannot guarantee the total unimodularity of the constraint matrix, which is a sufficient condition for the global solution of the relaxed linear program to be always integral, and therefore to be a solution of the integer program as well;

Furthermore, there are actually experimental evidences that, in some cases, solving the relaxed problem yields a fractional solution.

These facts indicate that the problem cannot be tackled with classical linear programming solvers, but only with pure integer linear solvers. Nevertheless, due to the very specific structure of the constraint matrix here, we strongly believe that it should be possible in the future to design ad-hoc integer solvers that yield high-definition approximations to solutions of several boundary value problems involving mean curvature, in particular the Willmore boundary value problem.

Thomas Schoenemann, Simon Masnou, Daniel Cremers

Interpolation Function of Generalized q −Bernstein-Type Basis Polynomials and Applications

The main aim of this paper is to construct a new generating function for the generalized


-Bernstein-type basis polynomials and to derive fundamental properties of these polynomials. We establish relations between the generalized


-Bernstein-type basis polynomials, the Bernoulli polynomials of higher-order and the generalized Stirling numbers of the second kind. By applying Mellin transform to this generating function, we also construct an interpolating function, which interpolates the generalized


-Bernstein-type basis polynomials at negative integers. Furthermore, we give applications on the generalized


-Bernstein-type basis polynomials and the Bézier curves.

Yilmaz Simsek

Differential Behaviour of Iteratively Generated Curves

The aim of our work is to specify and develop a geometric modeler, based on the formalism of iterated function systems with the following objectives: access to a new universe of original, various, aesthetic shapes, modeling of conventional shapes (smooth surfaces, solids) and unconventional shapes (rough surfaces, porous solids) by defining and controlling the relief (surface state) and lacunarity (size and distribution of holes). In this context we intend to develop differential calculus tools for fractal curves and surfaces defined by IFS. Using local fractional derivatives, we show that, even if most fractal curves are nowhere differentiable, they admit a left and right half-tangents, what gives us an additional parameter to characterize shapes.

Dmitry Sokolov, Christian Gentil, Hicham Bensoudane

Algebraic Curves of Low Convolution Degree

Studying convolutions of hypersurfaces (especially of curves and surfaces) has become an active research area in recent years. The main characterization from the point of view of convolutions is their convolution degree, which is an affine invariant associated to a hypersurface describing the complexity of the shape with respect to the operation of convolution. Extending the results from [1], we will focus on the two simplest classes of planar algebraic curves with respect to the operation of convolution, namely on the curves with the convolution degree one (so called LN curves) and two. We will present an algebraic analysis of these curves, provide their decomposition, and study their properties.

Jan Vršek, Miroslav Lávička

A Logistic Model for the Degradation of Triangle Mesh Normals

The performance of shading and ray-tracing algorithms depends heavily on the quality of the surface normal information. As a result, in many visual applications normal information turns out to be more important than spatial information. This paper proposes a logistic model for the degradation of the normal information resulting from the quantisation of the vertex coordinates. The mesh is degraded by the randomization of each vertex coordinate after its


-th significant bit. The normal degradation is computed as a weighted average of the angle differences between the normals of the original triangles and the corresponding degraded triangles. The proposed model is validated experimentally. As an application, we use the proposed logistic model to estimate suitable levels of quantisation for 3D triangle meshes.

Ying Yang, Ioannis Ivrissimtzis

On Single Image Scale-Up Using Sparse-Representations

This paper deals with the single image scale-up problem using sparse-representation modeling. The goal is to recover an original image from its blurred and down-scaled noisy version. Since this problem is highly ill-posed, a prior is needed in order to regularize it. The literature offers various ways to address this problem, ranging from simple linear space-invariant interpolation schemes (e.g., bicubic interpolation), to spatially-adaptive and non-linear filters of various sorts. We embark from a recently-proposed successful algorithm by Yang et. al. [1,2], and similarly assume a local


model on image patches, serving as regularization. Several important modifications to the above-mentioned solution are introduced, and are shown to lead to improved results. These modifications include a major simplification of the overall process both in terms of the computational complexity and the algorithm architecture, using a different training approach for the dictionary-pair, and introducing the ability to operate without a training-set by boot-strapping the scale-up task from the given low-resolution image. We demonstrate the results on true images, showing both visual and PSNR improvements.

Roman Zeyde, Michael Elad, Matan Protter

Periodic T-Splines and Tubular Surface Fitting

This paper discusses a special type of T-spline surfaces called periodic T-splines that are closed in one parameter direction, and their application in tubular surface fitting. First, a global representation is proposed for representing periodic T-splines. This representation does not require repeating control points, which facilitates surface fitting process. Then, an algorithm for adaptively fitting periodic T-splines to a tubular triangular mesh that has the same topology as a cylinder is presented. The resulting periodic T-spline is obtained respecting the geometric distribution of the input mesh. The use of periodic T-splines for tubular surface fitting has at least two advantages: 1) adaptive fitting is easily achieved due to the local refinement of T-splines; 2) the algorithm avoids cutting the mesh to make it a disk topologically for conventional B-spline fitting due to the periodic representation and this overcomes the drawback of finding a good cutting path, which is usually difficult.

Jianmin Zheng, Yimin Wang


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