Results 1  10
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65
ExampleBased 3D Scan Completion
 EUROGRAPHICS SYMPOSIUM ON GEOMETRY PROCESSING
, 2005
"... Optical acquisition devices often produce noisy and incomplete data sets, due to occlusion, unfavorable surface reflectance properties, or geometric restrictions in the scanner setup. We present a novel approach for obtaining a complete and consistent 3D model representation from such incomplete sur ..."
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Cited by 87 (24 self)
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Optical acquisition devices often produce noisy and incomplete data sets, due to occlusion, unfavorable surface reflectance properties, or geometric restrictions in the scanner setup. We present a novel approach for obtaining a complete and consistent 3D model representation from such incomplete surface scans, using a database of 3D shapes to provide geometric priors for regions of missing data. Our method retrieves suitable context models from the database, warps the retrieved models to conform with the input data, and consistently blends the warped models to obtain the final consolidated 3D shape. We define a shape matching penalty function and corresponding optimization scheme for computing the nonrigid alignment of the context models with the input data. This allows a quantitative evaluation and comparison of the quality of the shape extrapolation provided by each model. Our algorithms are explicitly designed to accommodate uncertain data and can thus be applied directly to raw scanner output. We show on a variety of real data sets how consistent models can be obtained from highly incomplete input. The information gained during the shape completion process can be utilized for future scans, thus continuously simplifying the creation of complex 3D models.
Contextbased surface completion
 ACM Transactions on Graphics
"... Figure 1: Completing a hole in a pointbased model. In the darker colored region we removed sample points to demonstrate the surface completion technique. In the middle right the region is filled with a smooth patch conforming with the densely sampled areas, and the result of our contextbased surfa ..."
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Cited by 79 (5 self)
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Figure 1: Completing a hole in a pointbased model. In the darker colored region we removed sample points to demonstrate the surface completion technique. In the middle right the region is filled with a smooth patch conforming with the densely sampled areas, and the result of our contextbased surface completion is on the right. Sampling complex, realworld geometry with range scanning devices almost always yields imperfect surface samplings. These “holes ” in the surface are commonly filled with a smooth patch that conforms with the boundary. We introduce a contextbased method: the characteristics of the given surface are analyzed, and the hole is iteratively filled by copying patches from valid regions of the given surface. In particular, the method needs to determine best matching patches, and then, fit imported patches by aligning them with the surrounding surface. The completion process works top down, where details refine intermediate coarser approximations. To align an imported patch with the existing surface, we apply a rigid transformation followed by an iterative closest point procedure with nonrigid transformations. The surface is essentially treated as a point set, and local implicit approximations aid in measuring the similarity between two point set patches. We demonstrate the method at several pointsampled surfaces, where the holes either result from imperfect sampling during range scanning or manual removal.
Discrete Willmore flow
 IN EUROGRAPHICS SYMPOSIUM ON GEOMETRY PROCESSING
, 2005
"... The Willmore energy of a surface, � (H 2 − K)dA, as a function of mean and Gaussian curvature, captures the deviation of a surface from (local) sphericity. As such this energy and its associated gradient flow play an important role in digital geometry processing, geometric modeling, and physical si ..."
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Cited by 51 (0 self)
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The Willmore energy of a surface, � (H 2 − K)dA, as a function of mean and Gaussian curvature, captures the deviation of a surface from (local) sphericity. As such this energy and its associated gradient flow play an important role in digital geometry processing, geometric modeling, and physical simulation. In this paper we consider a discrete Willmore energy and its flow. In contrast to traditional approaches it is not based on a finite element discretization, but rather on an ab initio discrete formulation which preserves the Möbius symmetries of the underlying continuous theory in the discrete setting. We derive the relevant gradient expressions including a linearization (approximation of the Hessian), which are required for nonlinear numerical solvers. As examples we demonstrate the utility of our approach for surface restoration, nsided hole filling, and nonshrinking surface smoothing.
An adaptive finite element method for the Laplace–Beltrami operator on implicitly defined surfaces
 SIAM J. Numer. Anal
"... Abstract. We present an adaptive finite element method for approximating solutions to the LaplaceBeltrami equation on surfaces in R3 which may be implicitly represented as level sets of smooth functions. Residualtype a posteriori error bounds which show that the error may be split into a “residual ..."
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Cited by 49 (4 self)
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Abstract. We present an adaptive finite element method for approximating solutions to the LaplaceBeltrami equation on surfaces in R3 which may be implicitly represented as level sets of smooth functions. Residualtype a posteriori error bounds which show that the error may be split into a “residual part ” and a “geometric part ” are established. In addition, implementation issues are discussed and several computational examples are given. Key words. LaplaceBeltrami operator, adaptive finite element methods, a posteriori error estimation, boundary value problems on surfaces AMS subject classification. 58J32, 65N15, 65N30 1. Introduction. In
A level set formulation for Willmore flow
 INTERFACES FREE BOUNDARIES
, 2004
"... A level set formulation of Willmore flow is derived using the gradient flow perspective. Starting from single embedded surfaces and the corresponding gradient flow, the metric is generalized to sets of level set surfaces using the identification of normal velocities and variations of the level set f ..."
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Cited by 41 (10 self)
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A level set formulation of Willmore flow is derived using the gradient flow perspective. Starting from single embedded surfaces and the corresponding gradient flow, the metric is generalized to sets of level set surfaces using the identification of normal velocities and variations of the level set function in time via the level set equation. The approach in particular allows to identify the natural dependent quantities of the derived variational formulation. Furthermore, spatial and temporal discretization are discussed and some numerical simulations are presented.
Higherorder finite element methods and pointwise error estimates for elliptic problems on surfaces
 SIAM J. Numer. Anal
"... Abstract. We define higherorder analogs to the piecewise linear surface finite element method studied in [Dz88] and prove error estimates in both pointwise and L2based norms. Using the LaplaceBeltrami problem on an implicitly defined surface Γ as a model PDE, we define Lagrange finite element met ..."
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Cited by 29 (2 self)
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Abstract. We define higherorder analogs to the piecewise linear surface finite element method studied in [Dz88] and prove error estimates in both pointwise and L2based norms. Using the LaplaceBeltrami problem on an implicitly defined surface Γ as a model PDE, we define Lagrange finite element methods of arbitrary degree on polynomial approximations to Γ which likewise are of arbitrary degree. Then we prove a priori error estimates in the L2, H1, and corresponding pointwise norms that demonstrate the interaction between the “PDE error ” that arises from employing a finitedimensional finite element space and the “geometric error ” that results from approximating Γ. We also consider parametric finite element approximations that are defined on Γ and thus induce no geometric error. Computational examples confirm the sharpness of our error estimates. Key words. LaplaceBeltrami operator, surface finite element methods, a priori error estimates, boundary value problems on surfaces, pointwise and maximum norm error estimates AMS subject classification. 58J32, 65N15, 65N30 1. Introduction. The
Completion and Reconstruction with Primitive Shapes
 Computer Graphics Forum (Proc. of Eurographics
, 2009
"... Figure 1: Reconstruction of the fandisk model. Orange color signifies completed surface parts. (a) The input pointcloud with holes (b) Final result (c) Result without the connectivity enforcement algorithm of Sec. 5. The disconnected primitive highlighted in red cuts off part of the model. (d) Clos ..."
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Cited by 26 (0 self)
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Figure 1: Reconstruction of the fandisk model. Orange color signifies completed surface parts. (a) The input pointcloud with holes (b) Final result (c) Result without the connectivity enforcement algorithm of Sec. 5. The disconnected primitive highlighted in red cuts off part of the model. (d) Closeup views of result without consistent edge labels and final result (see Sec. 7) We consider the problem of reconstruction from incomplete pointclouds. To find a closed mesh the reconstruction is guided by a set of primitive shapes which has been detected on the input pointcloud (e.g. planes, cylinders etc.). With this guidance we not only continue the surrounding structure into the holes but also synthesize plausible edges and corners from the primitives ’ intersections. To this end we give a surface energy functional that incorporates the primitive shapes in a guiding vector field. The discretized functional can be minimized with an efficient graphcut algorithm. A novel greedy optimization strategy is proposed to minimize the functional under the constraint that surface parts corresponding to a given primitive must be connected. From the primitive shapes our method can also reconstruct an idealized model that is suitable for use in a CAD system. Categories and Subject Descriptors (according to ACM CCS): Computer Graphics [I.3.5]: Curve, surface, solid, and object representations— 1.
Computing discrete shape operators on general meshes
 EUROGRAPHICS
, 2006
"... Discrete curvature and shape operators, which capture complete information about directional curvatures at a point, are essential in a variety of applications: simulation of deformable twodimensional objects, variational modeling and geometric data processing. In many of these applications, objects ..."
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Cited by 25 (5 self)
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Discrete curvature and shape operators, which capture complete information about directional curvatures at a point, are essential in a variety of applications: simulation of deformable twodimensional objects, variational modeling and geometric data processing. In many of these applications, objects are represented by meshes. Currently, a spectrum of approaches for formulating curvature operators for meshes exists, ranging from highly accurate but computationally expensive methods used in engineering applications to efficient but less accurate techniques popular in simulation for computer graphics. We propose a simple and efficient formulation for the shape operator for variational problems on general meshes, using degrees of freedom associated with normals. On the one hand, it is similar in its simplicity to some of the discrete curvature operators commonly used in graphics; on the other hand, it passes a number of important convergence tests and produces consistent results for different types of meshes and mesh refinement.
Fixing Geometric Errors on Polygonal Models: A Survey
 J. COMPUT SCI
"... Polygonal models are popular representations of 3D objects. The use of polygonal models in computational applications often requires a model to properly bound a 3D solid. That is, the polygonal model needs to be closed, manifold, and free of selfintersections. This paper surveys a sizeable literatu ..."
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Cited by 23 (1 self)
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Polygonal models are popular representations of 3D objects. The use of polygonal models in computational applications often requires a model to properly bound a 3D solid. That is, the polygonal model needs to be closed, manifold, and free of selfintersections. This paper surveys a sizeable literature for repairing models that do not satisfy this criteria, focusing on categorizing them by their methodology and capability. We hope to offer pointers to further readings for researchers and practitioners, and suggestions of promising directions for future research endeavors.
Computational parametric Willmore flow
, 2008
"... We propose a new algorithm for the computation of Willmore flow. This is the L²gradient flow for the Willmore functional, which is the classical bending energy of a surface. Willmore flow is described by a highly nonlinear system of PDEs of fourth order for the parametrization of the surface. The ..."
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Cited by 21 (1 self)
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We propose a new algorithm for the computation of Willmore flow. This is the L²gradient flow for the Willmore functional, which is the classical bending energy of a surface. Willmore flow is described by a highly nonlinear system of PDEs of fourth order for the parametrization of the surface. The spatially discrete numerical scheme is stable and consistent. The discretization relies on an adequate calculation of the first variation of the Willmore functional together with a derivation of the second variation of the area functional which is well adapted to discretization techniques with finite elements. The algorithm uses finite elements on surfaces. We give numerical examples and tests for piecewise linear finite elements. A convergence proof for the full algorithm remains an open question.