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86
Surface Parameterization: a Tutorial and Survey
 In Advances in Multiresolution for Geometric Modelling, Mathematics and Visualization
, 2005
"... 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 ..."
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Cited by 175 (5 self)
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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
Color TV: Total Variation Methods for Restoration of Vector Valued Images
 IEEE Trans. Image Processing
, 1996
"... 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 ..."
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Cited by 112 (13 self)
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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...
ThreeDimensional Face Recognition
, 2005
"... An expressioninvariant 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 expressioninvariant representations of faces using the bendinginvariant canonical forms approach. The re ..."
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Cited by 105 (22 self)
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An expressioninvariant 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 expressioninvariant representations of faces using the bendinginvariant 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 expressioninvariant representation of any given face. It allows us to perform the 3D face recognition task while avoiding the surface reconstruction stage.
A Theoretical Framework for Convex Regularizers in PDEBased Computation of Image Motion
, 2000
"... 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 consta ..."
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Cited by 79 (20 self)
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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 datadriven and flowdriven, isotropic and anisotropic, as well as spatial and spatiotemporal 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 wellposed: they posses a unique solution that depends in a continuous way on the initial data. An interesting structural relation between isotropic and anisotropic flowdriven regularizers is identified, and a design criterion is proposed for constructing anisotropic flowdriven regularizers in a simple and direct way from isotropic ones. Its use is illustrated by several examples.
Mesh Parameterization: Theory and Practice
 SIGGRAPH ASIA 2008 COURSE NOTES
, 2008
"... 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 ..."
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Cited by 35 (2 self)
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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 intersurface mapping, and demonstrates a variety of parameterization applications.
Estimation of 3D surface shape and smooth radiance from 2D images: A level set approach
 JOURNAL OF SCIENTIFIC COMPUTING
, 2003
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C 2 Hermite interpolation by Pythagorean hodograph space curves
 Mathematics of Computation
"... 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 ..."
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Cited by 18 (7 self)
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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 Bspline 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.
Models for thin viscous sheets
"... 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 timescal ..."
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Cited by 11 (1 self)
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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
Rational approximation of rotation minimizing frames using Pythagoreanhodograph cubics
 J. GEOM. GRAPHICS
, 1999
"... 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 socalled profile or moulding surfaces). In the second part of t ..."
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Cited by 11 (3 self)
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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 socalled 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 FrenetSerret 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.
Unsupervised segmentation based on robust estimation and color active contour models
 IEEE TRANS. ON INFORMATION TECHNOLOGY IN BIOMEDICINE
, 2005
"... One of the most commonly utilized clinical tests performed today is the routine evaluation of peripheral blood smears. In this paper, we investigate the design, development, and implementation of a robust color GVF active contour model for performing segmentation using a database of 1, 791 imaged ce ..."
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Cited by 10 (3 self)
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One of the most commonly utilized clinical tests performed today is the routine evaluation of peripheral blood smears. In this paper, we investigate the design, development, and implementation of a robust color GVF active contour model for performing segmentation using a database of 1, 791 imaged cells. The algorithms developed for this research operate in Luv color space and introduce a color gradient and L2E robust estimation into the traditional GVF snake. The accuracy of the new model was compared with the segmentation results utilizing a meanshift approach, the traditional color GVF snake and several other commonly utilized segmentation strategies. The unsupervised robust color snake with L2E robust estimation was shown to provide results which were superior to the other unsupervised approaches and was comparable with supervised segmentation as judged by a panel of human experts.