Fairing Scalar Fields by Variational Modeling of Contours
|
|
- Kristina Welch
- 6 years ago
- Views:
Transcription
1 Fairing Scalar Fields by Variational Modeling of Contours Martin Bertram University of Kaiserslautern, Germany Abstract Volume rendering and isosurface extraction from three-dimensional scalar fields are mostly based on piecewise trilinear representations. In regions of high geometric complexity such visualization methods often exhibit artefacts, due to trilinear interpolation. In this work, we present an iterative fairing method for scalar fields interpolating function values associated with grid points while smoothing the contours inside the grid cells based on variational principles. We present a local fairing method providing a piecewise bicubic representation of two-dimensional scalar fields. Our algorithm generalizes to the trivariate case and can be used to increase the resolution of data sets either locally or globally, reducing interpolation artefacts. In contrast to filtering methods, our algorithm does not reduce geometric detail supported by the data. CR Categories: G.1. [Numerical Analysis]: Approximation Approximation of Surfaces and Contours; G.1.6 [Numerical Analysis]: Optimization Constrained Optimization; I.4.3 [Image Processing and Computer Vision]: Enhancement Smoothing Keywords: Contours, Fairing, Variational Modeling. 1 Introduction Visualization of two- and three-dimensional scalar fields, like terrain models and computer tomography images, is mostly based on bilinear (trilinear) interpolation of function values sampled on regular rectangular (hexahedral) grids. Multilinear interpolation is efficiently evaluated, but it often provides contours of poor quality due to lacking smoothness and incorrect topology when reconstructing small features, see figure 1. (Contours denote level curves f (x, y) = const. in two-dimensional scalar fields and isosurfaces f (x,y,z) = const. in volume data sets.) Artefacts of this kind can be reduced by gaussian filtering of the discrete data set, but this also will corrupt geometric details of fine resolution. Smooth interpolation using, for example cubic B-splines, will provide smoother contour components without attempting to preserve topological features. A smooth scalar field may still have many contour components of high curvature that could be merged to larger and smoother contours. An approach is needed for fairing each individual contour such that it is consistent with the discrete data (located at the grid points) and does not intersect other contours. Fairing parametric curves and surfaces is well known, but in our case the geometry is implicitly defined and cannot be processed using the same techniques. In the present work we attempt to overcome this problem by performing an optimization process on the entire domain of the scalar field. Therefore, we minimize the variation of the scalar field s gradient along all individual contours. This gradient defines the normal vector of a contour multiplied with the local slope of the scalar field. Minimizing its variation along the tangent vector (plane) of a contour in every point of the volume provides a representation with smoother contours, emphasizing regions of great slope. The contents of the paper are organized as follows. In section, we summarize related work. Section 3 presents the theory of our optimization method, independent of the choice of basis functions. We develop a concrete algorithm based on a bicubic B-spline representation and local fairing with linear time complexity in section 4. Sections 5 and 6 provide numerical examples and conclusions, respectively. Related Work A variety of different contouring schemes exist for the reconstruction of isosurfaces from piecewise trilinear scalar fields. The original marching cubes algorithm [Lorensen and Cline 1987] has been adapted to hierarchical data representations [Westermann et al. 1999]. Feature lines can be recognized in the extracted contours [Kobbelt et al. 001]. Efficient methods extract multiple contours for volume rendering purposes [Gerstner 001]. An important breakthrough is the extraction of topologically correct isosurfaces with respect to the trilinear interpolant [Nielson 003]. Topological analysis of scalar fields provides critical points where the topology of contours changes when a passing a certain isovalue [Weber et al. 003]. Unfortunately, the topology of a trilinear interpolant is often different from the topology of an original scalar field prior to discretization. The question arises how to find the best reconstruction of the original shape consistent with the discrete data. Image processing techniques like anisotropic diffusion [Weickert 1998; Diewald et al. 000] are capable of recognizing local features, but they modify the data. Such approaches are mostly useful when the data is contaminated with noise. Fairing techniques of this kind are also applicable to the fairing of geometric shapes [Desbrun et al. 1999; Clarenz et al. 000]. Variational modeling [Welch and Witkin 199; Hagen et al. 1993] provides a useful tool for fairing individual surfaces. An efficient method extracting and fairing a single isosurface consistent with the discrete data has recently been described [Nielson et al. 003]. In the present work, we extend the variational methods to fairing the entire set of contours in a scalar field. We present a local fairing algorithm blending local estimates into a global representation, in analogy to a fitting method developed earlier for scattered data approximation [Bertram et al. 003]. 3 Variational Modeling of Contours In this section, we first summarize related work on variational modeling. Then, we introduce our new approach for fairing contours. Fairness is measured in terms of low curvature providing visually pleasant curves and surfaces. We obtain fairness by minimizing the
2 (a) (b) (c) Figure 1: (a) Bilinear interpolation of a regularly sampled feature; (b) bicubic interpolation with local minimization of thin-plate energy; (c) local fairing of contours. overall curvature of a geometric shape within a prescribed space of basis functions representing the geometry. s f(s,t) t a i (s i, t i) Figure : Interpolating parametric surface. 3.1 Classical Optimization Methods Consider a parametric surface f (s,t) satisfying some interpolation constraints f (s i,t i ) = a i, (1) Where a i are interpolation values with corresponding parameter locations (s i,t i ), see figure. The goal is to modify f, such that it minimizes some smoothness criterion, like thin-plate energy, and satisfies above interpolation constraints. We search this optimum in the space spanned by a certain basis, for example using B-splines. In order to simplify the optimization process, we use two sets of basis functions, Φ and Ψ, where Φ contains one basis function for every interpolation constraint and Ψ provides the remaining degrees of freedom used for fairing. One can easily construct these bases such that f (s,t) = φ i (s j,t j ) = δ i j, and ψ k (s j,t j ) = 0. a i φ i (s,t) + c k ψ k (s,t), i:φ i Φ k:ψ k Ψ For the fairing process, the coefficients c k need to be determined, minimizing a certain criterion. Note that the coefficients a i coincide with the given interpolation values, due to equation (). As fairness criterion, often thin-plate energy is minimized: f ss + f st + f tt ds dt min, (3) () where f ss, f st, and f tt denote second partial derivatives of f. If the norm to be minimized is induced by a scalar product, for example, < f,g > T P = f ss g ss + f st g st + f tt g tt ds dt, (4) ck then the constraints < f, f >= 0 necessary for optimization provide a linear system of equations, c i < ψ i,ψ k > = a i < φ i,ψ k > k : ψ k Ψ, (5) i:ψ i Ψ i:φ i Φ in matrix notation: Ac = b. The matrix A is positive definite and its rank equals the number of degrees of freedom Ψ. The solution of this system provides the remaining coefficients c representing f in equation (), minimizing the norm (3) among all choices of c. Optimization methods of this kind are known as variational modeling, see [Welch and Witkin 199; Hagen et al. 1993]. An example for thin-plate energy minimization is shown in figure 1(b). 3. Fairing Contours Now, we consider a two-dimensional scalar field f (x,y) bound to some interpolation constraints f (x i,y i ) = a i, as in equation (1). Selecting a certain isovalue α, we want to fair the corresponding contour line composed of all points (x,y) satisfying f (x,y) = α. We note that the following deliberations also generalize to volumes, where contours are surfaces. Suppose that we get hold of a parametric representation of the contour associated with isovalue α, say g(s) such that f (g(s)) = α. (6) Using this parametric form, fairing the contour can be achieved by minimizing its second derivative, g (s) ds min. (7) In the case of an arc-length parametrization, this is equivalent to minimizing the variation n s := n of the contour s normal vector n s along g(s), see figure 3: n s (s) ds min. (8)
3 n 4 Local-optimization Algorithm n g g s g(s) Figure 3: Deviation of g and n are equal in absolute value. The method developed in the previous section can be applied to smooth B-spline surfaces and volumes providing additional modeling degrees compared to multilinear interpolation. The remaining issues addressed in the following are constructing the bases Φ and Ψ satisfying equation () and localizing the fairing process to obtain linear computation time. Given a rectangular grid with associated function values, we consider every inner vertex with its 8-neighborhood for local fairing. We create four bicubic patches interpolating these function values. Finally, the computed patches are blended into one single, piecewise bicubic representation. Generalization to volumes is straight forward. To emphasize regions of steep gradient in the fairing process, we replace the normal n(s) = f (g(s)) by the gradient f in equation (8). The variation of this gradient has two components: n f f (g(s)) orthogonal to g(s) and n f parallel to g(s). Hence, we have N 04 N 14 N 4 N 34 N 44 s f (g(s)) = n f, (9) since the derivative g has unit length (due to arc-length parametrization) and is orthogonal to n. The optimization problem for a single contour g(s) becomes n f ds min. (10) 0 1 Figure 4: B-splines used for local fairing. Finally, we eliminate the need for constructing an arc-length parametrization for every contour g. Since we intend fairing all contours of the scalar field in one single process, we need to integrate equation (10) with respect to α, which is equivalent to n f dx dy min. (11) a 0 a Unfortunately, both n and f depend on the scalar field f, turning this optimization problem into a non-linear problem requiring an iterative solution. In every iteration of this process, we use the normalized gradient field n(x, y) of the previous estimate for f. With the fixed normal field n = (n 1,n,0) T, our scalar product inducing the norm for the optimization process becomes < f,g > n = (n f xx n 1 f xy )(n g xx n 1 g xy ) +(n f xy n 1 f yy )(n g xy n 1 g yy ) dx dy. (1) 1 0 a 00 a 0 Inserting this norm into equation (5) and solving this system of equations provides the next estimate for f. The complete iterative algorithm works as follows: First, we compute f (0) using thin-plate energy minimization based on the scalar product <, > T P from equation (4). In order to compute f (i+1) from f (i), we first sample the normalized gradient field n (i) from f (i). Then, we compute f (i+1) from the optimization process using < f,g > = (1 w) < f,g > n (i) +w < f,g > T P, (13) where w is a small number, say We added a small portion of thin-plate energy minimization to increase numerical stability (note that our criterion is fairing all contours while ignoring what isovalues are associated with these). All scalar products between basis functions are computed by numerical integration. 0 1 Figure 5: Every local B-spline surface is defined by 5 coefficients (de Boor points) satisfying 9 interpolation constraints. 4.1 Local Fairing Considering a -matrix of scalar values a i j, we construct an interpolating bicubic B-spline surface composed of four bicubic patches. The two knot vectors are both {0,0,0,0,1,,,,}, providing interpolation of the four corner values and C -continuity at the inner patch boundaries. The set of one-dimensional B-splines is illustrated in figure 4. Figure 5 depicts the bivariate case. Interpolation of the inner scalar values is obtained by a simple
4 s [0,1) ϕ 0 (s) = s 3 + 3s 3s + 1 ϕ 1 (s) =.5s 3 5.5s + 3s ϕ (s) = s 3 + 3s ϕ 3 (s) = 0.75s s ϕ 4 (s) = 0 s [1,] ϕ 0 (s) = 0 ϕ 1 (s) = 0.75s s 6s + 3 ϕ (s) = s 3 9s + 1s 4 ϕ 3 (s) =.5s s 9s + 3 ϕ 4 (s) = s 3 3s + 3s 1 Table 1: One-dimensional basis functions. ϕ ϕ 0 ϕ 4 For merging the B-spline surfaces located around the inner grid points, we transform every surface into Bézier patches. Starting with the surface representation 4 4 h(s,t) = c i j ϕ i (s)ϕ j (t), (16) i=0 j=0 the basis transform provides a 7 7-matrix of Bézier points defining the four patches. Considering only one row (column) of coefficients c i, the representation 4 c i ϕ i (s), s [0,] (17) i=0 is transformed into two cubic Bézier segments 3 b i B 3 i (s), s [0,1) and i=0 (18) 3 b i+3 B 3 i (s 1), s [1,], i=0 where B 3 i are the Bernstein polynomials. ϕ 1 ϕ Figure 6: Transformed B-splines used for interpolation (φ 0, φ, φ 4 ) and optimization (φ 1, φ 3 ). d 0=c0 b 0 d 1=c1 b 1 b d d =c 3 3 b =c b 3 4 b 5 d =c 4 4 b 6 basis transform: ϕ 0 = N 04, ϕ 1 = N N 4, ϕ = N 4, ϕ 3 = N N 4, ϕ 4 = N 44. (14) The coefficients of the basis functions ϕ i depicted in figure 6 are provided in table 1. The reader may verify the following properties: ϕ 0 (0) = 1, ϕ i (0) = 0, i = 1,,3,4; ϕ (1) = 1, ϕ i (1) = 0, i = 0,1,3,4; ϕ 4 () = 1, ϕ i () = 0, i = 0,1,,3. (15) The basis functions used for the optimization process are tensor products ϕ i j (s,t) = ϕ i (s)ϕ j (t). The set Φ of interpolating functions is composed of ϕ i j : i, j {0,,4}. The remaining functions provide degrees of freedom for the optimization process and are thus elements of Ψ. In the bivariate case, we obtain Φ = 9 and Ψ = 16. (In the trivariate case: Φ = 7 and Ψ = 98.) Using the bases Φ and Ψ, the optimization process described in section 3 is performed. We found that the iterative process converges very fast, such that two or three iterations are sufficient. After computing local representations of the scalar field, we need to blend the overlapping B-spline patches into a global representation. 4. Merging Surface Components 0 1 Figure 7: Relation between coefficients c i, de Boor points d i, and Bézier points b i. The relation to the coefficients c i, the corresponding de Boor points d i, and the Bézier points b i is shown in figure 7. The two Bézier segments composed of the points b 0,b 1,b,b 3 and b 3,b 4,b 5,b 6, are obtained by b 0 = c 0, b 1 = c 1, b = c + 0.5c 1 0.5c 3, b 3 = c, b 4 = c 0.5c c 3, b 5 = c 3, b 6 = c 4. (19) This one-dimensional transform is applied to the rows and then to the columns providing the Bézier points of the four patches. For every inner grid cell, we have four Bézier patches computed from the local fairing around each of the four corner points. From the fairing process around a grid point p, we keep only those Bézier points that are closer to p than to any other inner grid point, see figure 8. The final representation of the scalar field is now C 1 -continuous, since only the first partial derivatives are continuous across patch boundaries. C -continuity is approximated, since we merge multiple overlapping B-spline surfaces. We note that a B-spline representation of the final scalar field using double knots is more memory efficient than the Bézier representation.
5 poorly sampled data, eliminating artefacts due to trilinear interpolation. References Figure 8: Grid with 6 5 interpolation points and 5 4 Bézier patches. The shaded regions indicate which Bézier points are determined by each B-spline surface. If a multilinear representation is preferred, our algorithm can be used to increase the resolution in regions of high geometric complexity. The smooth scalar field representation is then sampled at a finer resolution with respect to the initial data. Hierarchical representations can be stored in a quadtree (octree), see [Westermann et al. 1999]. We note that a refined representation will not carry more geometric detail than provided by the data at its initial resolution. Our objective is removing geometric detail which is due to interpolation artefacts. 5 Numerical Examples The examples in figures 1, 9, and 10 were computed on a PC equipped with a 1466 MHz AMD Athlon Processor. The computation times for fairing a 7 7 and a data set are summarized in table. These examples show that the algorithm is still time consuming, despite of its linear time complexity. Hence, it may be used in a preprocessing step for data preparation. Our algorithm is easily implemented in parallel. no. vertices 49 (5) 100 (64) thin-plate one iteration Table : Computation times in seconds for thin-plate energy minimization and for one iteration of the fairing process. Note that the number of inner vertices (in brackets) determines the number of local surfaces to be computed. The local fairing examples in figure 9 suggest a fast convergence rate. The global examples in figures 1 and 10 show the improved contours with respect to the bilinear interpolant. In particular, diagonal features are better represented after fairing. BERTRAM, M., TRICOCHE, X., AND HAGEN, H Adaptive smooth scattered-data approximation. In Proceedings of VisSym 03, Joint Eurographics and IEEE TCVG Symposium on Visualization, IEEE, Data Visualisation, & 97. CLARENZ, U., DIEWALD, U., AND RUMPF, M Nonlinear anisotropic diffusion in surface processing. In Proceedings of Visualization 00, IEEE, & 580. DESBRUN, M., MEYER, M., SCHROEDER, P., AND BARR, A Implicit fairing of irregular meshes using diffusion and curvature flow. In Proceedings of Siggraph 99, ACM, Computer Graphics, DIEWALD, U., PREUSSER, T., AND RUMPF, M Anisotropic diffusion in vector field visualization on euclidean domains and surfaces. In Transactions on Visualization and Computer Graphics, IEEE, vol. 6, GERSTNER, T Fast multiresolution extraction of multiple transparent isosurfaces. In Proceedings of VisSym 03, Joint Eurographics and IEEE TCVG Symposium on Visualization, IEEE, Data Visualisation, & 336. HAGEN, H., BRUNNETT, G., AND SANTARELLI, P Variational principles in curve and surface design. In Surveys on Mathematics for Industry, vol. 3, 1 7. KOBBELT, L., BOTSCH, M., SCHWANECKE, U., AND SEIDEL, H.-P Feature-sensitive surface extraction from volume data. In Proceedings of Siggraph 01, ACM, Computer Graphics, LORENSEN, W., AND CLINE, H Marching cubes: a high resolution 3d surface construction algorithm. In Proceedings of Siggraph 87, ACM, Computer Graphics, NIELSON, G., GRAF, G., HOLMES, R., HUANG, A., AND PHIELIPP, M Shrouds: optimal separating surfaces for enumerated volumes. In Proceedings of VisSym 03, Joint Eurographics and IEEE TCVG Symposium on Visualization, 003, Data Visualisation, & 87. NIELSON, G On marching cubes. In Transactions on Visualization and Computer Graphics, IEEE, vol. 9, WEBER, G., SCHEUERMANN, G., AND HAMANN, B Detecting critical regions in scalar fields. In Proceedings of VisSym 03, Joint Eurographics and IEEE TCVG Symposium on Visualization, IEEE, Data Visualisation, & 88. WEICKERT, J In Anisotropic diffusion in image processing, Teubner Stuttgart, ECMI Series. WELCH, W., AND WITKIN, A Variational surface modeling. In Proceedings of Siggraph 9, ACM, Computer Graphics, WESTERMANN, R., KOBBELT, L., AND ERTL, T Real-time exploration of regular volume data by adaptive reconstruction of isosurfaces. In The Visual Computer, vol. 15, Conclusions and Future Work We presented a new variational modeling method for fairing contours of scalar fields. Despite of linear time complexity, our algorithm is computationally expensive. However, it is easily implemented in parallel for processing large data sets. We have presented bivariate examples for our technique, but we have not yet explored its behavior in trivariate applications. We expect that the proposed technique will significantly improve volume rendering of
6 (a) (b) (c) (d) Figure 9: Iterative fairing for two different local situations. (a) Initial thin-plate minimization; (b) first iteration; (c) second iteration; (d) result after ten iterations. (a) (b) Figure 10: Circle sampled by points. (a) Bilinear contours; (b) fair contours after two iterations.
implicit surfaces, approximate implicitization, B-splines, A- patches, surface fitting
24. KONFERENCE O GEOMETRII A POČÍTAČOVÉ GRAFICE ZBYNĚK ŠÍR FITTING OF PIECEWISE POLYNOMIAL IMPLICIT SURFACES Abstrakt In our contribution we discuss the possibility of an efficient fitting of piecewise
More informationIsosurface Rendering. CSC 7443: Scientific Information Visualization
Isosurface Rendering What is Isosurfacing? An isosurface is the 3D surface representing the locations of a constant scalar value within a volume A surface with the same scalar field value Isosurfaces form
More informationSurface Reconstruction
Eurographics Symposium on Geometry Processing (2006) Surface Reconstruction 2009.12.29 Some methods for surface reconstruction Classification 1. Based on Delaunay triangulation(or Voronoi diagram) Alpha
More informationA Global Laplacian Smoothing Approach with Feature Preservation
A Global Laplacian Smoothing Approach with Feature Preservation hongping Ji Ligang Liu Guojin Wang Department of Mathematics State Key Lab of CAD&CG hejiang University Hangzhou, 310027 P.R. China jzpboy@yahoo.com.cn,
More informationInterpolating and approximating scattered 3D-data with hierarchical tensor product B-splines
Interpolating and approximating scattered 3D-data with hierarchical tensor product B-splines Günther Greiner Kai Hormann Abstract In this note we describe surface reconstruction algorithms based on optimization
More information9. Three Dimensional Object Representations
9. Three Dimensional Object Representations Methods: Polygon and Quadric surfaces: For simple Euclidean objects Spline surfaces and construction: For curved surfaces Procedural methods: Eg. Fractals, Particle
More informationNormals of subdivision surfaces and their control polyhedra
Computer Aided Geometric Design 24 (27 112 116 www.elsevier.com/locate/cagd Normals of subdivision surfaces and their control polyhedra I. Ginkel a,j.peters b,,g.umlauf a a University of Kaiserslautern,
More informationSmoothing & Fairing. Mario Botsch
Smoothing & Fairing Mario Botsch Motivation Filter out high frequency noise Desbrun, Meyer, Schroeder, Barr: Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow, SIGGRAPH 99 2 Motivation
More informationIntroduction to Computer Graphics. Modeling (3) April 27, 2017 Kenshi Takayama
Introduction to Computer Graphics Modeling (3) April 27, 2017 Kenshi Takayama Solid modeling 2 Solid models Thin shapes represented by single polygons Unorientable Clear definition of inside & outside
More informationDesign considerations
Curves Design considerations local control of shape design each segment independently smoothness and continuity ability to evaluate derivatives stability small change in input leads to small change in
More informationDual Marching Cubes. Gregory M. Nielson. Arizona State University
Dual Marching Cubes Gregory M. Nielson Arizona State University Figure 1. March Cubes Surface MC-Patch surface, S MC-Dual surface, S. ABSTRACT We present the definition and computational algorithms for
More informationShape Modeling and Geometry Processing
252-0538-00L, Spring 2018 Shape Modeling and Geometry Processing Discrete Differential Geometry Differential Geometry Motivation Formalize geometric properties of shapes Roi Poranne # 2 Differential Geometry
More informationCSC Computer Graphics
// CSC. Computer Graphics Lecture Kasun@dscs.sjp.ac.lk Department of Computer Science University of Sri Jayewardanepura Polygon Filling Scan-Line Polygon Fill Algorithm Span Flood-Fill Algorithm Inside-outside
More informationApproximation of 3D-Parametric Functions by Bicubic B-spline Functions
International Journal of Mathematical Modelling & Computations Vol. 02, No. 03, 2012, 211-220 Approximation of 3D-Parametric Functions by Bicubic B-spline Functions M. Amirfakhrian a, a Department of Mathematics,
More informationScalar Visualization
Scalar Visualization Visualizing scalar data Popular scalar visualization techniques Color mapping Contouring Height plots outline Recap of Chap 4: Visualization Pipeline 1. Data Importing 2. Data Filtering
More informationFall CSCI 420: Computer Graphics. 4.2 Splines. Hao Li.
Fall 2014 CSCI 420: Computer Graphics 4.2 Splines Hao Li http://cs420.hao-li.com 1 Roller coaster Next programming assignment involves creating a 3D roller coaster animation We must model the 3D curve
More informationContouring and Isosurfaces. Ronald Peikert SciVis Contouring 2-1
Contouring and Isosurfaces Ronald Peikert SciVis 2007 - Contouring 2-1 What are contours? Set of points where the scalar field s has a given value c: Examples in 2D: height contours on maps isobars on
More informationInteractive Graphics. Lecture 9: Introduction to Spline Curves. Interactive Graphics Lecture 9: Slide 1
Interactive Graphics Lecture 9: Introduction to Spline Curves Interactive Graphics Lecture 9: Slide 1 Interactive Graphics Lecture 13: Slide 2 Splines The word spline comes from the ship building trade
More informationTopology Preserving Tetrahedral Decomposition of Trilinear Cell
Topology Preserving Tetrahedral Decomposition of Trilinear Cell Bong-Soo Sohn Department of Computer Engineering, Kyungpook National University Daegu 702-701, South Korea bongbong@knu.ac.kr http://bh.knu.ac.kr/
More informationSpline Curves. Spline Curves. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 1
Spline Curves Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 1 Problem: In the previous chapter, we have seen that interpolating polynomials, especially those of high degree, tend to produce strong
More informationINF3320 Computer Graphics and Discrete Geometry
INF3320 Computer Graphics and Discrete Geometry More smooth Curves and Surfaces Christopher Dyken, Michael Floater and Martin Reimers 10.11.2010 Page 1 More smooth Curves and Surfaces Akenine-Möller, Haines
More informationVariational Design and Fairing of Spline Surfaces
EUROGRAPHICS 94 / M. Daehlen and L. Kjelldahl (Guest Editors), Blackwell Publishers Eurographics Association, 1994 Volume 13, (1994), number 3 Variational Design and Fairing of Spline Surfaces Günther
More informationCOMPUTER AIDED GEOMETRIC DESIGN. Thomas W. Sederberg
COMPUTER AIDED GEOMETRIC DESIGN Thomas W. Sederberg January 31, 2011 ii T. W. Sederberg iii Preface This semester is the 24 th time I have taught a course at Brigham Young University titled, Computer Aided
More informationLevel Set Extraction from Gridded 2D and 3D Data
Level Set Extraction from Gridded 2D and 3D Data David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International
More information3D Modeling Parametric Curves & Surfaces
3D Modeling Parametric Curves & Surfaces Shandong University Spring 2012 3D Object Representations Raw data Point cloud Range image Polygon soup Solids Voxels BSP tree CSG Sweep Surfaces Mesh Subdivision
More informationFree-Form Deformation and Other Deformation Techniques
Free-Form Deformation and Other Deformation Techniques Deformation Deformation Basic Definition Deformation: A transformation/mapping of the positions of every particle in the original object to those
More informationG 2 Interpolation for Polar Surfaces
1 G 2 Interpolation for Polar Surfaces Jianzhong Wang 1, Fuhua Cheng 2,3 1 University of Kentucky, jwangf@uky.edu 2 University of Kentucky, cheng@cs.uky.edu 3 National Tsinhua University ABSTRACT In this
More informationAn introduction to interpolation and splines
An introduction to interpolation and splines Kenneth H. Carpenter, EECE KSU November 22, 1999 revised November 20, 2001, April 24, 2002, April 14, 2004 1 Introduction Suppose one wishes to draw a curve
More informationSurfaces: notes on Geometry & Topology
Surfaces: notes on Geometry & Topology 1 Surfaces A 2-dimensional region of 3D space A portion of space having length and breadth but no thickness 2 Defining Surfaces Analytically... Parametric surfaces
More informationSurface Modeling. Polygon Tables. Types: Generating models: Polygon Surfaces. Polygon surfaces Curved surfaces Volumes. Interactive Procedural
Surface Modeling Types: Polygon surfaces Curved surfaces Volumes Generating models: Interactive Procedural Polygon Tables We specify a polygon surface with a set of vertex coordinates and associated attribute
More informationIndirect Volume Rendering
Indirect Volume Rendering Visualization Torsten Möller Weiskopf/Machiraju/Möller Overview Contour tracing Marching cubes Marching tetrahedra Optimization octree-based range query Weiskopf/Machiraju/Möller
More information3D Modeling Parametric Curves & Surfaces. Shandong University Spring 2013
3D Modeling Parametric Curves & Surfaces Shandong University Spring 2013 3D Object Representations Raw data Point cloud Range image Polygon soup Surfaces Mesh Subdivision Parametric Implicit Solids Voxels
More informationNormals of subdivision surfaces and their control polyhedra
Normals of subdivision surfaces and their control polyhedra I. Ginkel, a, J. Peters b, and G. Umlauf a, a University of Kaiserslautern, Germany b University of Florida, Gainesville, FL, USA Abstract For
More informationThe Level Set Method. Lecture Notes, MIT J / 2.097J / 6.339J Numerical Methods for Partial Differential Equations
The Level Set Method Lecture Notes, MIT 16.920J / 2.097J / 6.339J Numerical Methods for Partial Differential Equations Per-Olof Persson persson@mit.edu March 7, 2005 1 Evolving Curves and Surfaces Evolving
More informationUsing Isosurface Methods for Visualizing the Envelope of a Swept Trivariate Solid
Using Isosurface Methods for Visualizing the Envelope of a Swept Trivariate Solid Jason Conkey Kenneth I. Joy Center for Image Processing and Integrated Computing Department of Computer Science University
More informationSplines. Parameterization of a Curve. Curve Representations. Roller coaster. What Do We Need From Curves in Computer Graphics? Modeling Complex Shapes
CSCI 420 Computer Graphics Lecture 8 Splines Jernej Barbic University of Southern California Hermite Splines Bezier Splines Catmull-Rom Splines Other Cubic Splines [Angel Ch 12.4-12.12] Roller coaster
More informationLecture 2.2 Cubic Splines
Lecture. Cubic Splines Cubic Spline The equation for a single parametric cubic spline segment is given by 4 i t Bit t t t i (..) where t and t are the parameter values at the beginning and end of the segment.
More informationComputergrafik. Matthias Zwicker Universität Bern Herbst 2016
Computergrafik Matthias Zwicker Universität Bern Herbst 2016 Today Curves NURBS Surfaces Parametric surfaces Bilinear patch Bicubic Bézier patch Advanced surface modeling 2 Piecewise Bézier curves Each
More informationGeometric Modeling and Processing
Geometric Modeling and Processing Tutorial of 3DIM&PVT 2011 (Hangzhou, China) May 16, 2011 6. Mesh Simplification Problems High resolution meshes becoming increasingly available 3D active scanners Computer
More informationMotivation. Freeform Shape Representations for Efficient Geometry Processing. Operations on Geometric Objects. Functional Representations
Motivation Freeform Shape Representations for Efficient Geometry Processing Eurographics 23 Granada, Spain Geometry Processing (points, wireframes, patches, volumes) Efficient algorithms always have to
More informationCurve Representation ME761A Instructor in Charge Prof. J. Ramkumar Department of Mechanical Engineering, IIT Kanpur
Curve Representation ME761A Instructor in Charge Prof. J. Ramkumar Department of Mechanical Engineering, IIT Kanpur Email: jrkumar@iitk.ac.in Curve representation 1. Wireframe models There are three types
More informationCS 523: Computer Graphics, Spring Differential Geometry of Surfaces
CS 523: Computer Graphics, Spring 2009 Shape Modeling Differential Geometry of Surfaces Andrew Nealen, Rutgers, 2009 3/4/2009 Recap Differential Geometry of Curves Andrew Nealen, Rutgers, 2009 3/4/2009
More informationUntil now we have worked with flat entities such as lines and flat polygons. Fit well with graphics hardware Mathematically simple
Curves and surfaces Escaping Flatland Until now we have worked with flat entities such as lines and flat polygons Fit well with graphics hardware Mathematically simple But the world is not composed of
More informationImplicit Surfaces & Solid Representations COS 426
Implicit Surfaces & Solid Representations COS 426 3D Object Representations Desirable properties of an object representation Easy to acquire Accurate Concise Intuitive editing Efficient editing Efficient
More informationCurve and Surface Fitting with Splines. PAUL DIERCKX Professor, Computer Science Department, Katholieke Universiteit Leuven, Belgium
Curve and Surface Fitting with Splines PAUL DIERCKX Professor, Computer Science Department, Katholieke Universiteit Leuven, Belgium CLARENDON PRESS OXFORD 1995 - Preface List of Figures List of Tables
More informationComputergrafik. Matthias Zwicker. Herbst 2010
Computergrafik Matthias Zwicker Universität Bern Herbst 2010 Today Curves NURBS Surfaces Parametric surfaces Bilinear patch Bicubic Bézier patch Advanced surface modeling Piecewise Bézier curves Each segment
More informationSURFACE CONSTRUCTION USING TRICOLOR MARCHING CUBES
SURFACE CONSTRUCTION USING TRICOLOR MARCHING CUBES Shaojun Liu, Jia Li Oakland University Rochester, MI 4839, USA Email: sliu2@oakland.edu, li4@oakland.edu Xiaojun Jing Beijing University of Posts and
More informationCS 4620 Final Exam. (a) Is a circle C 0 continuous?
CS 4620 Final Exam Wednesday 9, December 2009 2 1 2 hours Prof. Doug James Explain your reasoning for full credit. You are permitted a double-sided sheet of notes. Calculators are allowed but unnecessary.
More informationAlmost Curvature Continuous Fitting of B-Spline Surfaces
Journal for Geometry and Graphics Volume 2 (1998), No. 1, 33 43 Almost Curvature Continuous Fitting of B-Spline Surfaces Márta Szilvási-Nagy Department of Geometry, Mathematical Institute, Technical University
More informationSurface Reconstruction. Gianpaolo Palma
Surface Reconstruction Gianpaolo Palma Surface reconstruction Input Point cloud With or without normals Examples: multi-view stereo, union of range scan vertices Range scans Each scan is a triangular mesh
More informationGeometric Modeling of Curves
Curves Locus of a point moving with one degree of freedom Locus of a one-dimensional parameter family of point Mathematically defined using: Explicit equations Implicit equations Parametric equations (Hermite,
More informationCentral issues in modelling
Central issues in modelling Construct families of curves, surfaces and volumes that can represent common objects usefully; are easy to interact with; interaction includes: manual modelling; fitting to
More information05 - Surfaces. Acknowledgements: Olga Sorkine-Hornung. CSCI-GA Geometric Modeling - Daniele Panozzo
05 - Surfaces Acknowledgements: Olga Sorkine-Hornung Reminder Curves Turning Number Theorem Continuous world Discrete world k: Curvature is scale dependent is scale-independent Discrete Curvature Integrated
More informationCurves and Curved Surfaces. Adapted by FFL from CSE167: Computer Graphics Instructor: Ronen Barzel UCSD, Winter 2006
Curves and Curved Surfaces Adapted by FFL from CSE167: Computer Graphics Instructor: Ronen Barzel UCSD, Winter 2006 Outline for today Summary of Bézier curves Piecewise-cubic curves, B-splines Surface
More informationTopological Equivalence between a 3D Object and the Reconstruction of its Digital Image
to appear in IEEE PAMI Topological Equivalence between a 3D Object and the Reconstruction of its Digital Image Peer Stelldinger, Longin Jan Latecki and Marcelo Siqueira Abstract Digitization is not as
More informationweighted minimal surface model for surface reconstruction from scattered points, curves, and/or pieces of surfaces.
weighted minimal surface model for surface reconstruction from scattered points, curves, and/or pieces of surfaces. joint work with (S. Osher, R. Fedkiw and M. Kang) Desired properties for surface reconstruction:
More informationScalar Algorithms: Contouring
Scalar Algorithms: Contouring Computer Animation and Visualisation Lecture tkomura@inf.ed.ac.uk Institute for Perception, Action & Behaviour School of Informatics Contouring Scaler Data Last Lecture...
More informationTwo Algorithms for Adaptive Approximation of Bivariate Functions by Piecewise Linear Polynomials on Triangulations
Two Algorithms for Adaptive Approximation of Bivariate Functions by Piecewise Linear Polynomials on Triangulations Nira Dyn School of Mathematical Sciences Tel Aviv University, Israel First algorithm from
More informationPlotting technologies for the study of functions of two real variables
Plotting technologies for the study of functions of two real variables David Zeitoun 1 and Thierry Dana-Picard 2 1 Department of Mathematics, Orot College of Education, Rehovot, Israel, ed.technologie@gmail.com
More informationGL9: Engineering Communications. GL9: CAD techniques. Curves Surfaces Solids Techniques
436-105 Engineering Communications GL9:1 GL9: CAD techniques Curves Surfaces Solids Techniques Parametric curves GL9:2 x = a 1 + b 1 u + c 1 u 2 + d 1 u 3 + y = a 2 + b 2 u + c 2 u 2 + d 2 u 3 + z = a
More informationComputer Graphics I Lecture 11
15-462 Computer Graphics I Lecture 11 Midterm Review Assignment 3 Movie Midterm Review Midterm Preview February 26, 2002 Frank Pfenning Carnegie Mellon University http://www.cs.cmu.edu/~fp/courses/graphics/
More informationComputer Graphics CS 543 Lecture 13a Curves, Tesselation/Geometry Shaders & Level of Detail
Computer Graphics CS 54 Lecture 1a Curves, Tesselation/Geometry Shaders & Level of Detail Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) So Far Dealt with straight lines
More informationVariational Geometric Modeling with Wavelets
Variational Geometric Modeling with Wavelets Steven J. Gortler and Michael F. Cohen Microsoft Research Redmond, WA (excerpted from Hierarchical and Variational Geometric Modeling with Wavelets, by Steven
More informationComputational Physics PHYS 420
Computational Physics PHYS 420 Dr Richard H. Cyburt Assistant Professor of Physics My office: 402c in the Science Building My phone: (304) 384-6006 My email: rcyburt@concord.edu My webpage: www.concord.edu/rcyburt
More informationHierarchical Iso-Surface Extraction
Hierarchical Iso-Surface Extraction Ulf Labsik Kai Hormann Martin Meister Günther Greiner Computer Graphics Group University of Erlangen Multi-Res Modeling Group Caltech Computer Graphics Group University
More informationAn Efficient, Geometric Multigrid Solver for the Anisotropic Diffusion Equation in Two and Three Dimensions
1 n Efficient, Geometric Multigrid Solver for the nisotropic Diffusion Equation in Two and Three Dimensions Tolga Tasdizen, Ross Whitaker UUSCI-2004-002 Scientific Computing and Imaging Institute University
More informationCurves and Surfaces. CS475 / 675, Fall Siddhartha Chaudhuri
Curves and Surfaces CS475 / 675, Fall 26 Siddhartha Chaudhuri Klein bottle: surface, no edges (Möbius strip: Inductiveload@Wikipedia) Möbius strip: surface, edge Curves and Surfaces Curve: D set Surface:
More informationCover Page. Title: Surface Approximation Using Geometric Hermite Patches Abstract:
Cover Page Title: Surface Approximation Using Geometric Hermite Patches Abstract: A high-order-of-approximation surface patch is used to construct continuous, approximating surfaces. This patch, together
More informationLocal Modification of Subdivision Surfaces Based on Curved Mesh
Local Modification of Subdivision Surfaces Based on Curved Mesh Yoshimasa Tokuyama Tokyo Polytechnic University tokuyama@image.t-kougei.ac.jp Kouichi Konno Iwate University konno@cis.iwate-u.ac.jp Junji
More informationCurves and Surfaces for Computer-Aided Geometric Design
Curves and Surfaces for Computer-Aided Geometric Design A Practical Guide Fourth Edition Gerald Farin Department of Computer Science Arizona State University Tempe, Arizona /ACADEMIC PRESS I San Diego
More information1.7.1 Laplacian Smoothing
1.7.1 Laplacian Smoothing 320491: Advanced Graphics - Chapter 1 434 Theory Minimize energy functional total curvature estimate by polynomial-fitting non-linear (very slow!) 320491: Advanced Graphics -
More informationEdge and local feature detection - 2. Importance of edge detection in computer vision
Edge and local feature detection Gradient based edge detection Edge detection by function fitting Second derivative edge detectors Edge linking and the construction of the chain graph Edge and local feature
More informationParameterization. Michael S. Floater. November 10, 2011
Parameterization Michael S. Floater November 10, 2011 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to generate from point
More informationNear-Optimum Adaptive Tessellation of General Catmull-Clark Subdivision Surfaces
Near-Optimum Adaptive Tessellation of General Catmull-Clark Subdivision Surfaces Shuhua Lai and Fuhua (Frank) Cheng (University of Kentucky) Graphics & Geometric Modeling Lab, Department of Computer Science,
More informationScalar Visualization
Scalar Visualization 5-1 Motivation Visualizing scalar data is frequently encountered in science, engineering, and medicine, but also in daily life. Recalling from earlier, scalar datasets, or scalar fields,
More informationEXPLOITING DATA COHERENCY IN MULTIPLE DATASET VISUALIZATION
EXPLOITING DATA COHERENCY IN MULTIPLE DATASET VISUALIZATION Gaurav Khanduja and Bijaya B. Karki Department of Computer Science, Louisiana State University Baton Rouge, LA USA gkhand1@lsu.edu and karki@csc.lsu.edu
More informationScientific Visualization Example exam questions with commented answers
Scientific Visualization Example exam questions with commented answers The theoretical part of this course is evaluated by means of a multiple- choice exam. The questions cover the material mentioned during
More informationSurfaces, meshes, and topology
Surfaces from Point Samples Surfaces, meshes, and topology A surface is a 2-manifold embedded in 3- dimensional Euclidean space Such surfaces are often approximated by triangle meshes 2 1 Triangle mesh
More information(Refer Slide Time: 00:02:24 min)
CAD / CAM Prof. Dr. P. V. Madhusudhan Rao Department of Mechanical Engineering Indian Institute of Technology, Delhi Lecture No. # 9 Parametric Surfaces II So these days, we are discussing the subject
More informationDIGITAL TERRAIN MODELLING. Endre Katona University of Szeged Department of Informatics
DIGITAL TERRAIN MODELLING Endre Katona University of Szeged Department of Informatics katona@inf.u-szeged.hu The problem: data sources data structures algorithms DTM = Digital Terrain Model Terrain function:
More informationBackground for Surface Integration
Background for urface Integration 1 urface Integrals We have seen in previous work how to define and compute line integrals in R 2. You should remember the basic surface integrals that we will need to
More informationCOMPUTER AIDED ENGINEERING DESIGN (BFF2612)
COMPUTER AIDED ENGINEERING DESIGN (BFF2612) BASIC MATHEMATICAL CONCEPTS IN CAED by Dr. Mohd Nizar Mhd Razali Faculty of Manufacturing Engineering mnizar@ump.edu.my COORDINATE SYSTEM y+ y+ z+ z+ x+ RIGHT
More informationApproximation of Isosurface in the Marching Cube: Ambiguity Problem.
Approximation of Isosurface in the Marching Cube: Ambiguity Problem Sergey V Matveyev Computer Science Department Institute for High Energy Physics 142284, Protvino, Moscow Region, Russia E-mail: matveyev@desertihepsu
More informationVolume Illumination, Contouring
Volume Illumination, Contouring Computer Animation and Visualisation Lecture 0 tkomura@inf.ed.ac.uk Institute for Perception, Action & Behaviour School of Informatics Contouring Scaler Data Overview -
More informationMathematical Tools in Computer Graphics with C# Implementations Table of Contents
Mathematical Tools in Computer Graphics with C# Implementations by Hardy Alexandre, Willi-Hans Steeb, World Scientific Publishing Company, Incorporated, 2008 Table of Contents List of Figures Notation
More informationMesh Processing Pipeline
Mesh Smoothing 1 Mesh Processing Pipeline... Scan Reconstruct Clean Remesh 2 Mesh Quality Visual inspection of sensitive attributes Specular shading Flat Shading Gouraud Shading Phong Shading 3 Mesh Quality
More informationShape optimization of smooth surfaces with arbitrary topology
International conference on Innovative Methods in Product Design June 15 th 17 th, 2011, Venice, Italy Shape optimization of smooth surfaces with arbitrary topology Przemysław Kiciak (a) (a) Institut Matematyki
More informationAn Intuitive Framework for Real-Time Freeform Modeling
An Intuitive Framework for Real-Time Freeform Modeling Leif Kobbelt Shape Deformation Complex shapes Complex deformations User Interaction Very limited user interface 2D screen & mouse Intuitive metaphor
More informationElement Quality Metrics for Higher-Order Bernstein Bézier Elements
Element Quality Metrics for Higher-Order Bernstein Bézier Elements Luke Engvall and John A. Evans Abstract In this note, we review the interpolation theory for curvilinear finite elements originally derived
More informationing and enhancing operations for shapes represented by level sets (isosurfaces) and applied unsharp masking to the level set normals. Their approach t
Shape Deblurring with Unsharp Masking Applied to Mesh Normals Hirokazu Yagou Λ Alexander Belyaev y Daming Wei z Λ y z ; ; Shape Modeling Laboratory, University of Aizu, Aizu-Wakamatsu 965-8580 Japan fm50534,
More informationScaling the Topology of Symmetric, Second-Order Planar Tensor Fields
Scaling the Topology of Symmetric, Second-Order Planar Tensor Fields Xavier Tricoche, Gerik Scheuermann, and Hans Hagen University of Kaiserslautern, P.O. Box 3049, 67653 Kaiserslautern, Germany E-mail:
More informationCurves and Surfaces Computer Graphics I Lecture 9
15-462 Computer Graphics I Lecture 9 Curves and Surfaces Parametric Representations Cubic Polynomial Forms Hermite Curves Bezier Curves and Surfaces [Angel 10.1-10.6] February 19, 2002 Frank Pfenning Carnegie
More information3D Modeling: Surfaces
CS 430/536 Computer Graphics I 3D Modeling: Surfaces Week 8, Lecture 16 David Breen, William Regli and Maxim Peysakhov Geometric and Intelligent Computing Laboratory Department of Computer Science Drexel
More information03 - Reconstruction. Acknowledgements: Olga Sorkine-Hornung. CSCI-GA Geometric Modeling - Spring 17 - Daniele Panozzo
3 - Reconstruction Acknowledgements: Olga Sorkine-Hornung Geometry Acquisition Pipeline Scanning: results in range images Registration: bring all range images to one coordinate system Stitching/ reconstruction:
More informationcoding of various parts showing different features, the possibility of rotation or of hiding covering parts of the object's surface to gain an insight
Three-Dimensional Object Reconstruction from Layered Spatial Data Michael Dangl and Robert Sablatnig Vienna University of Technology, Institute of Computer Aided Automation, Pattern Recognition and Image
More informationCurves and Surfaces Computer Graphics I Lecture 10
15-462 Computer Graphics I Lecture 10 Curves and Surfaces Parametric Representations Cubic Polynomial Forms Hermite Curves Bezier Curves and Surfaces [Angel 10.1-10.6] September 30, 2003 Doug James Carnegie
More informationKnow it. Control points. B Spline surfaces. Implicit surfaces
Know it 15 B Spline Cur 14 13 12 11 Parametric curves Catmull clark subdivision Parametric surfaces Interpolating curves 10 9 8 7 6 5 4 3 2 Control points B Spline surfaces Implicit surfaces Bezier surfaces
More informationSung-Eui Yoon ( 윤성의 )
CS480: Computer Graphics Curves and Surfaces Sung-Eui Yoon ( 윤성의 ) Course URL: http://jupiter.kaist.ac.kr/~sungeui/cg Today s Topics Surface representations Smooth curves Subdivision 2 Smooth Curves and
More informationGeometric Modeling in Graphics
Geometric Modeling in Graphics Part 10: Surface reconstruction Martin Samuelčík www.sccg.sk/~samuelcik samuelcik@sccg.sk Curve, surface reconstruction Finding compact connected orientable 2-manifold surface
More informationAdaptive and Smooth Surface Construction by Triangular A-Patches
Adaptive and Smooth Surface Construction by Triangular A-Patches Guoliang Xu Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, China Abstract
More information