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72
A Signal Processing Approach To Fair Surface Design
, 1995
"... In this paper we describe a new tool for interactive freeform fair surface design. By generalizing classical discrete Fourier analysis to twodimensional discrete surface signals  functions defined on polyhedral surfaces of arbitrary topology , we reduce the problem of surface smoothing, or fai ..."
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Cited by 523 (13 self)
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In this paper we describe a new tool for interactive freeform fair surface design. By generalizing classical discrete Fourier analysis to twodimensional discrete surface signals  functions defined on polyhedral surfaces of arbitrary topology , we reduce the problem of surface smoothing, or fairing, to lowpass filtering. We describe a very simple surface signal lowpass filter algorithm that applies to surfaces of arbitrary topology. As opposed to other existing optimizationbased fairing methods, which are computationally more expensive, this is a linear time and space complexity algorithm. With this algorithm, fairing very large surfaces, such as those obtained from volumetric medical data, becomes affordable. By combining this algorithm with surface subdivision methods we obtain a very effective fair surface design technique. We then extend the analysis, and modify the algorithm accordingly, to accommodate different types of constraints. Some constraints can be imposed without any modification of the algorithm, while others require the solution of a small associated linear system of equations. In particular, vertex location constraints, vertex normal constraints, and surface normal discontinuities across curves embedded in the surface, can be imposed with this technique. CR Categories and Subject Descriptors: I.3.3 [Computer Graphics]: Picture/image generation  display algorithms; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling  curve, surface, solid, and object representations;J.6[Com puter Applications]: ComputerAided Engineering  computeraided design General Terms: Algorithms, Graphics. 1
Variational Surface Modeling
 Computer Graphics
, 1992
"... We present a new approach to interactive modeling of freeform surfaces. Instead of a fixed mesh of control points, the model presented to the user is that of an infinitely malleable surface, with no fixed controls. The user is free to apply control points and curves which are then available as handl ..."
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Cited by 168 (4 self)
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We present a new approach to interactive modeling of freeform surfaces. Instead of a fixed mesh of control points, the model presented to the user is that of an infinitely malleable surface, with no fixed controls. The user is free to apply control points and curves which are then available as handles for direct manipulation. The complexity of the surface's shape may be increased by adding more control points and curves, without apparent limit. Within the constraints imposed by the controls, the shape of the surface is fully determined by one or more simple criteria, such as smoothness. Our method for solving the resulting constrained variational optimization problems rests on a surface representation scheme allowing nonuniform subdivision of Bspline surfaces. Automatic subdivision is used to ensure that constraints are met, and to enforce error bounds. Efficient numerical solutions are obtained by exploiting linearities in the problem formulation and the representation. Keywords: sur...
Shape Transformation Using Variational Implicit Functions
, 1999
"... Traditionally, shape transformation using implicit functions is performed in two distinct steps: 1) creating two implicit functions, and 2) interpolating between these two functions. We present a new shape transformation method that combines these two tasks into a single step. We create a transforma ..."
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Cited by 159 (7 self)
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Traditionally, shape transformation using implicit functions is performed in two distinct steps: 1) creating two implicit functions, and 2) interpolating between these two functions. We present a new shape transformation method that combines these two tasks into a single step. We create a transformation between two N dimensional objects by casting this as a scattered data interpolation problem in N + 1 dimensions. For the case of 2D shapes, we place all of our data constraints within two planes, one for each shape. These planes are placed parallel to one another in 3D. Zerovalued constraints specify the locations of shape boundaries and positivevalued constraints are placed along the normal direction in towards the center of the shape. We then invoke a variational interpolation technique (the 3D generalization of thinplate interpolation), and this yields a single implicit function in 3D. Intermediate shapes are simply the zerovalued contours of 2D slices through this 3D function. Shape transformation between 3D shapes can be performed similarly by solving a 4D interpolation problem. To our knowledge, ours is the first shape transformation method to unify the tasks of implicit function creation and interpolation. The transformations produced by this method appear smooth and natural, even between objects of differing topologies. If desired, one or more additional shapes may be introduced that influence the intermediate shapes in a sequence. Our method can also reconstruct surfaces from multiple slices that are not restricted to being parallel to one another.
FreeForm Shape Design Using Triangulated Surfaces
, 1994
"... We present an approach to modeling with truly mutable yet completely controllable freeform surfaces of arbitrary topology. Surfaces may be pinned down at points and along curves, cut up and smoothly welded back together, and faired and reshaped in the large. This style of control is formulated as a ..."
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Cited by 153 (0 self)
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We present an approach to modeling with truly mutable yet completely controllable freeform surfaces of arbitrary topology. Surfaces may be pinned down at points and along curves, cut up and smoothly welded back together, and faired and reshaped in the large. This style of control is formulated as a constrained shape optimization, with minimization of squared principal curvatures yielding graceful shapes that are free of the parameterization worries accompanying many patchbased approaches. Triangulated point sets are used to approximate these smooth variational surfaces, bridging the gap between patchbased and particlebased representations. Automatic refinement, mesh smoothing, and retriangulation maintain a good computational mesh as the surface shape evolves, and give sample points and surface features much of the freedom to slide around in the surface that oriented particles enjoy. The resulting surface triangulations are constructed and maintained in real time. 1 Introduction ...
Implicit, Nonparametric Shape Reconstruction from Unorganized Points Using A Variational Level Set Method
 Computer Vision and Image Understanding
, 1998
"... In this paper we consider a fundamental visualization problem which arises in computer vision, computer graphics and numerical simulation. The problem is to find a curve in two dimensions, or a surface in three dimensions which can be regarded as the shape represented by a set of unorganized points, ..."
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Cited by 131 (20 self)
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In this paper we consider a fundamental visualization problem which arises in computer vision, computer graphics and numerical simulation. The problem is to find a curve in two dimensions, or a surface in three dimensions which can be regarded as the shape represented by a set of unorganized points, and/or curves, and/or surface patches. We do not assume any knowledge of the ordering, connectivity or topology of the data sets or of the true shape. Only the location of each point or general Hausdorff distance to the data set is known. The key idea in our approach is to find an implicit nonparametric representation of the curve or surface on a fixed rectangular grid. With this representation of surfaces we can easily (a) find the closest point and distance from any point to the surface (useful in illumination and many other applications), (b) find the intersection curve of two surfaces which is guaranteed to lie on both surfaces in our representation, and (c) perform any Boolean operatio...
Dynamic NURBS with Geometric Constraints for Interactive Sculpting
, 1994
"... This article develops a dynamic generalization of the nonuniform rational Bspline (NURBS) model. NURBS have become a de facto standard in commercial modeling systems because of their power to represent freeform shapes as well as common analytic shapes. To date, however, they have been viewed as pu ..."
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Cited by 95 (27 self)
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This article develops a dynamic generalization of the nonuniform rational Bspline (NURBS) model. NURBS have become a de facto standard in commercial modeling systems because of their power to represent freeform shapes as well as common analytic shapes. To date, however, they have been viewed as purely geometric primitives that require the user to manually adjust multiple control points and associated weights in order to design shapes. Dynamic NURBS, or DNURBS, are physicsbased models that incorporate mass distributions, inertial deformation energies, and other physical quantities into the popular NURBS geometric substrate. Using DNURBS, a modeler can interactively sculpt curves and surfaces and design complex shapes to required specifications not only in the traditional indirect fashion, by adjusting control points and weights, but also through direct physical manipulation, by applying simulated forces and local and global shape constraints. DNURBS move and deform in a physically intuitive manner in response to the user's direct manipulations. Their dynamic behavior results from the numerical integration of a set of nonlinear differential equations that automatically evolve the control points and weights in response to the applied forces and constraints. To derive these equations, we employ Lagrangian mechanics and finiteelementlike discretization. Our approach supports the trimming of DNURBS surfaces using DNURBS curves. We demonstrate DNURBS models and constraints in applications including the rounding of solids, optimal surface fitting to unstructured data, surface design from crosssections, and freeform deformation. We also introduce a new technique for 2D shape metamorphosis using constrained DNURBS surfaces.
Modelling with Implicit Surfaces that Interpolate
 ACM Transactions on Graphics
, 2002
"... We introduce new techniques for modelling with interpolating implicit surfaces. This form of implicit surface was first used for problems of surface reconstruction [24] and shape transformation [30], but the emphasis of our work is on model creation. These implicit surfaces are described by specify ..."
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Cited by 88 (0 self)
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We introduce new techniques for modelling with interpolating implicit surfaces. This form of implicit surface was first used for problems of surface reconstruction [24] and shape transformation [30], but the emphasis of our work is on model creation. These implicit surfaces are described by specifying locations in 3D through which the surface should pass, and also identifying locations that are interior or exterior to the surface. A 3D implicit function is created from these constraints using a variational scattered data interpolation approach, and the isosurface of this function describes a surface. Like other implicit surface descriptions, these surfaces can be used for CSG and interference detection, may be interactively manipulated, are readily approximated by polygonal tilings, and are easy to ray trace. A key strength for model creation is that interpolating implicit surfaces allow the direct specification of both the location of points on the surface and the surface normals. These are two important manipulation techniques that are difficult to achieve using other implicit surface representations such as sums of spherical or ellipsoidal Gaussian functions ("blobbies"). We show that these properties make this form of implicit surface particularly attractive for interactive sculpting using the particle sampling technique introduced by Witkin and Heckbert in [32]. Our formulation also yields a simple method for converting a polygonal model to a smooth implicit model, as well as a new way to form blends between objects.
Geometric Signal Processing on Polygonal Meshes
, 2000
"... Very large polygonal models, which are used in more and more graphics applications today, are routinely generated by a variety of methods such as surface reconstruction algorithms from 3D scanned data, isosurface construction algorithms from volumetric data, and photogrametric methods from aerial ..."
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Cited by 72 (1 self)
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Very large polygonal models, which are used in more and more graphics applications today, are routinely generated by a variety of methods such as surface reconstruction algorithms from 3D scanned data, isosurface construction algorithms from volumetric data, and photogrametric methods from aerial photography. In this report we provide an overview of several closely related methods developed during the last few yers, to smooth, denoise, edit, compress, transmit, and animate very large polygonal models. 1.
A Multiresolution Framework for Dynamic Deformations
, 2002
"... We present a novel framework for dynamic simulation of elastically deformable solids. Our approach combines classical finite element methodology with subdivision wavelets to meet the needs of computer graphics applications. We represent deformations using a wavelet basis constructed from volumetric ..."
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Cited by 60 (2 self)
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We present a novel framework for dynamic simulation of elastically deformable solids. Our approach combines classical finite element methodology with subdivision wavelets to meet the needs of computer graphics applications. We represent deformations using a wavelet basis constructed from volumetric CatmullClark subdivision. CatmullClark subdivision solids allow the domain of deformation to be tailored to objects of arbitrary topology. The domain of deformation can correspond to the interior of a subdivision surface or can enclose an arbitrary surface mesh. Within the wavelet framework we develop the equations of motion for elastic deformations in the presence of external forces and constraints. We solve the resulting differential equations using an implicit method, which lends stability. Our framework allows tradeoff between speed and accuracy. For interactive applications, we accelerate the simulation by adaptively refining the wavelet basis while avoiding visual "popping" artifacts. Offline simulations can employ a fine basis for higher accuracy at the cost of more computation time. By exploiting the properties of smooth subdivision we can compute less expensive solutions using a trilinear basis yet produce a smooth result that meets the constraints.