## Semi-Regular Mesh Extraction from Volumes (2000)

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Citations: | 91 - 11 self |

### BibTeX

@MISC{Wood00semi-regularmesh,

author = {Zoë J. Wood and Mathieu Desbrun and Peter Schröder and David Breen},

title = {Semi-Regular Mesh Extraction from Volumes},

year = {2000}

}

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### Abstract

We present a novel method to extract iso-surfaces from distance volumes. It generates high quality semi-regular multiresolution meshes of arbitrary topology. Our technique proceeds in two stages. First, a very coarse mesh with guaranteed topology is extracted. Subsequently an iterative multi-scale force-based solver refines the initial mesh into a semi-regular mesh with geometrically adaptive sampling rate and good aspect ratio triangles. The coarse mesh extraction is performed using a new approach we call surface wavefront propagation. A set of discrete iso-distance ribbons are rapidly built and connected while respecting the topology of the iso-surface implied by the data. Subsequent multi-scale refinement is driven by a simple force-based solver designed to combine good iso-surface fit and high quality sampling through reparameterization. In contrast to the Marching Cubes technique our output meshes adapt gracefully to the iso-surface geometry, have a natural multiresolution structure and good aspect ratio triangles, as demonstrated with a number of examples.

### Citations

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Citation Context ...e force-based multi-scale solver. 4.2 Setup To solve for the iso-surface one may consider the signed distance function of the volume as a potential field and search for the minimum potential solution =-=[27, 26, 25, 46, 42]-=-. Employing the calculus of variations this results in an energy minimization problem intended to bring the current mesh representation to the final desired shape by following the gradient of distance... |

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Citation Context ...iewing the exterior boundary of the liver. This surface can be extracted from the volume data by finding a particular iso-surface. The predominant algorithm for iso-surface extraction, Marching Cubes =-=[40]-=-, extracts a surface in the form of a triangle mesh. The algorithm computes a local triangulation within each voxel of the volume that contains the surface, resulting in a uniform resolution mesh. Uni... |

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Citation Context ...al fine mesh. This approach could follow the application of Marching Cubes, to address the oversampling problem [23]. Unfortunately, common mesh simplification algorithms have large memory footprints =-=[24, 16]-=-, and are impractical for decimating meshes with millions of polygons (see [38, 37] for an approach to tackle this situation). Alternatively one may apply classic iso-surface extraction techniques suc... |

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Citation Context ...al fine mesh. This approach could follow the application of Marching Cubes, to address the oversampling problem [23]. Unfortunately, common mesh simplification algorithms have large memory footprints =-=[24, 16]-=-, and are impractical for decimating meshes with millions of polygons (see [38, 37] for an approach to tackle this situation). Alternatively one may apply classic iso-surface extraction techniques suc... |

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Citation Context ...rs of our surface that correctly encode the topology of the surface (see Chapter 3 for more information). 2.4 Signed Distance Volumes Signed distance volumes are utilized in a variety of applications =-=[8, 7, 17, 45, 57]-=-. A distance volume is a volume dataset that stores the shortest distance to the surface at the vertices of each voxel. Whether a vertex is inside or outside of the surface is encoded in the sign of t... |

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Citation Context ...ce. We discuss this further in Chapter 3. 2.3.2 Distance Iso-contours Our coarse mesh extraction approach was also inspired by work on computing level sets on manifolds, specifically polygonal meshes =-=[29, 48]-=-. Particularly of interest is the computation of the geodesic graph used to extract skeletal curves [35, 55]. Skeletal curves are another method to encode a surfaces topology, however, they suffer fro... |

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Citation Context ...ead we define reparameterization forces which act similarly, but only along the local parameter plane, not in space. 4.4.1 Decoupling Smoothing and Reparameterization In recent work on mesh smoothing =-=[54, 10]-=-, the Laplacian operator has been extensively used to denoise triangulated surfaces, using the approximation: L(xi) = 1 m � j∈N1(i) xj − xi (4.1) where xj are the neighbors of vertex xi, andm =#N1(i) ... |

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Citation Context ...of large meshes is a multiresolution approach. The multiresolution paradigm encompasses a class of surface representations that scale well under increasing geometric complexity and arbitrary topology =-=[12]-=-. As their name implies, multiresolution representations organize data into different levels of reso1 Volume data is defined here as a discrete three dimensional field; it encodes samples of a functio... |

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Citation Context ... our approach is focused on directly extracting a good mesh, we have replaced the Marching Cubes extraction algorithm. Our approach is closely related to surface extraction based on deformable models =-=[43, 26, 42, 46, 31]-=-. In these approaches a potential function is defined and the surface is found by formulating a finite element problem whose minimum energy solution is the desired surface. Aside from the interpolatio... |

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Citation Context ...ead we define reparameterization forces which act similarly, but only along the local parameter plane, not in space. 4.4.1 Decoupling Smoothing and Reparameterization In recent work on mesh smoothing =-=[54, 10]-=-, the Laplacian operator has been extensively used to denoise triangulated surfaces, using the approximation: L(xi) = 1 m � j∈N1(i) xj − xi (4.1) where xj are the neighbors of vertex xi, andm =#N1(i) ... |

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Citation Context ...ppear in wavelets and subdivision; and (b) fine to coarse: constructions based on mesh simplification. The former comes with a rich mathematical structure which can be leveraged for many applications =-=[52, 53, 47]-=-. The latter is applied when a very fine mesh has already been constructed. Such meshes arise from 3D scanning or Marching Cubes iso-surface extractions. As mentioned in the Introduction to this thesi... |

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279 |
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Citation Context ...n of the desired surface. As volume data has grown in size, the resulting storage concerns have been addressed through volume compression, streaming extraction techniques, and hierarchical structures =-=[59, 39, 4]-=-. However, these techniques do not solve the problem that the resulting mesh will still be excessively large and inefficiently represented due to uniform sampling. While some algorithms have focused o... |

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Citation Context ... Finally, all three neighbors of T0 can be subdivided and T0 is not subdivided. In this case we add three conforming edges between the refined triangles. We also use the second Laplacian operator L 2 =-=[30, 10]-=- to ensure a smoother variation of sampling rate over the surface. As in the case of the Laplacian, we use the same weights for the conforming edges, and suppress the normal component in the same way.... |

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Citation Context ...e force-based multi-scale solver. 4.2 Setup To solve for the iso-surface one may consider the signed distance function of the volume as a potential field and search for the minimum potential solution =-=[27, 26, 25, 46, 42]-=-. Employing the calculus of variations this results in an energy minimization problem intended to bring the current mesh representation to the final desired shape by following the gradient of distance... |

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Citation Context ...resentations [56, 2]. Unfortunately, it is difficult to guarantee the topology of the mesh extracted from the simplified volume. An alternative to building a hierarchy through decimation is remeshing =-=[12, 32, 36, 31, 21]-=-. Although previous work is compelling, it is inefficient and inelegant in the setting of iso-surface extraction from volumes: such approaches first extract a poor, oversampled mesh, and then repair t... |

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Citation Context ...resentations [56, 2]. Unfortunately, it is difficult to guarantee the topology of the mesh extracted from the simplified volume. An alternative to building a hierarchy through decimation is remeshing =-=[12, 32, 36, 31, 21]-=-. Although previous work is compelling, it is inefficient and inelegant in the setting of iso-surface extraction from volumes: such approaches first extract a poor, oversampled mesh, and then repair t... |

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Citation Context .... One approach to obtaining a multiresolution surface is repeated decimation of an initial fine mesh. This approach could follow the application of Marching Cubes, to address the oversampling problem =-=[23]-=-. Unfortunately, common mesh simplification algorithms have large memory footprints [24, 16], and are impractical for decimating meshes with millions of polygons (see [38, 37] for an approach to tackl... |

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181 | Interactive multiresolution mesh editing - Zorin, Schroder, et al. |

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Citation Context ...n of the desired surface. As volume data has grown in size, the resulting storage concerns have been addressed through volume compression, streaming extraction techniques, and hierarchical structures =-=[59, 39, 4]-=-. However, these techniques do not solve the problem that the resulting mesh will still be excessively large and inefficiently represented due to uniform sampling. While some algorithms have focused o... |

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Citation Context ...ssist in the creation of triangles for the coarse mesh. In order to come up with a consistent ordering within the rings, we use an idea very similar to work done on encoding a digital region boundary =-=[14]-=- and digital surface tracking [18]. Since we want to traverse the ring in an ordered manner, we need to pick a consistent orientation in space and an ordering for traversing that orientation. Luckily,... |

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Citation Context ...At this point, we have a list of all cross sections of the surface which are required for tiling a good coarse approximation of the final surface. This final step is related to contour stitching (see =-=[1, 15, 13]-=-). However, since we work within the framework of the volume data with the additional information stored in the Ring Master, we do not facesRing n 1 1 Ring n+1 2 3 2 3.6 Coarse Mesh Construction 21 4 ... |

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Citation Context ...ess the oversampling problem [23]. Unfortunately, common mesh simplification algorithms have large memory footprints [24, 16], and are impractical for decimating meshes with millions of polygons (see =-=[38, 37]-=- for an approach to tackle this situation). Alternatively one may apply classic iso-surface extraction techniques such as Marching Cubes to hierarchical (filtered and down-sampled) volumetric represen... |

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Citation Context ... our approach is focused on directly extracting a good mesh, we have replaced the Marching Cubes extraction algorithm. Our approach is closely related to surface extraction based on deformable models =-=[43, 26, 42, 46, 31]-=-. In these approaches a potential function is defined and the surface is found by formulating a finite element problem whose minimum energy solution is the desired surface. Aside from the interpolatio... |

104 | Speeding up isosurface extraction using interval trees
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Citation Context ...n of the desired surface. As volume data has grown in size, the resulting storage concerns have been addressed through volume compression, streaming extraction techniques, and hierarchical structures =-=[59, 39, 4]-=-. However, these techniques do not solve the problem that the resulting mesh will still be excessively large and inefficiently represented due to uniform sampling. While some algorithms have focused o... |

99 | Guaranteeing the topology of an implicit surface polygonization for interactive modeling
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Citation Context ...resents the same topology as a Marching Cubes mesh rely on the equivalence of Surfels and Lachaud’s loops. Other works concerned with coding the topology of a surface are Morse Theory and Reeb graphs =-=[51, 49, 50]-=-. Morse Theory is used to describe the minima of functionalss8 2 Related Work and Algorithm Overview on an infinite dimensional space of paths. By applying this theory to differentiable manifolds, the... |

98 | Three-dimensional distance field metamorphosis
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Citation Context ...rs of our surface that correctly encode the topology of the surface (see Chapter 3 for more information). 2.4 Signed Distance Volumes Signed distance volumes are utilized in a variety of applications =-=[8, 7, 17, 45, 57]-=-. A distance volume is a volume dataset that stores the shortest distance to the surface at the vertices of each voxel. Whether a vertex is inside or outside of the surface is encoded in the sign of t... |

89 | Real-time Exploration of Regular Volume Data by Adaptive Reconstruction of IsoSurfaces
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Citation Context ...approach to tackle this situation). Alternatively one may apply classic iso-surface extraction techniques such as Marching Cubes to hierarchical (filtered and down-sampled) volumetric representations =-=[56, 2]-=-. Unfortunately, it is difficult to guarantee the topology of the mesh extracted from the simplified volume. An alternative to building a hierarchy through decimation is remeshing [12, 32, 36, 31, 21]... |

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Citation Context ...resents the same topology as a Marching Cubes mesh rely on the equivalence of Surfels and Lachaud’s loops. Other works concerned with coding the topology of a surface are Morse Theory and Reeb graphs =-=[51, 49, 50]-=-. Morse Theory is used to describe the minima of functionalss8 2 Related Work and Algorithm Overview on an infinite dimensional space of paths. By applying this theory to differentiable manifolds, the... |

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Citation Context ...rs of our surface that correctly encode the topology of the surface (see Chapter 3 for more information). 2.4 Signed Distance Volumes Signed distance volumes are utilized in a variety of applications =-=[8, 7, 17, 45, 57]-=-. A distance volume is a volume dataset that stores the shortest distance to the surface at the vertices of each voxel. Whether a vertex is inside or outside of the surface is encoded in the sign of t... |

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Citation Context ...eoretical, applied, and implementation benefits: Theory Specific connectivity structure improves compression efficiency, data structure compactness, and analytic error estimates in various algorithms =-=[28, 21, 61, 5]-=-. Applications Many applications that are “downstream” with respect to iso-surface extraction are more efficient when using semi-regular meshes (these include editing [61], finite element simulations ... |

76 | A shrink wrapping approach to remeshing polygonal surfaces, Computer Graphics Forum(Proc
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Citation Context ...resentations [56, 2]. Unfortunately, it is difficult to guarantee the topology of the mesh extracted from the simplified volume. An alternative to building a hierarchy through decimation is remeshing =-=[12, 32, 36, 31, 21]-=-. Although previous work is compelling, it is inefficient and inelegant in the setting of iso-surface extraction from volumes: such approaches first extract a poor, oversampled mesh, and then repair t... |

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72 | Anisotropic feature-preserving denoising of height fields and bivariate data
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Citation Context ...ing shape smoothing • and a component in the tangent plane, fairing the parameterization of the mesh. The normal vector to the surface can be found easily by normalizing the curvature normal vector K =-=[10, 11]-=-: K(xi) = 1 2A � j∈N1(i) (cot αij +cotβij)(xi − xj). (4.2) For arbitrary connectivity meshes numerical evidence shows that no spurious drifting artifacts appear when the surface is modified only in th... |

69 |
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Citation Context ...opology. Designing a solver which does not require the corrects2.3 Topology 7 topology initially and instead topologically modifies the mesh as the algorithm proceeds is possible, but rather delicate =-=[34]-=-. Instead we opt for a robust algorithm which initially extracts a topologically correct coarse mesh from the volume data. Subsequent refinement is always performed through quadrisection, giving us th... |

68 | K.N.: Arbitrary Topology Shape Reconstruction from Planar Cross
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Citation Context ...At this point, we have a list of all cross sections of the surface which are required for tiling a good coarse approximation of the final surface. This final step is related to contour stitching (see =-=[1, 15, 13]-=-). However, since we work within the framework of the volume data with the additional information stored in the Ring Master, we do not facesRing n 1 1 Ring n+1 2 3 2 3.6 Coarse Mesh Construction 21 4 ... |

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54 |
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Citation Context ...resents the same topology as a Marching Cubes mesh rely on the equivalence of Surfels and Lachaud’s loops. Other works concerned with coding the topology of a surface are Morse Theory and Reeb graphs =-=[51, 49, 50]-=-. Morse Theory is used to describe the minima of functionalss8 2 Related Work and Algorithm Overview on an infinite dimensional space of paths. By applying this theory to differentiable manifolds, the... |

48 |
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Citation Context ...ss of a minimum distance for non convexsets. Consequently, the problem must be regularized to ensure convergence and a unique solution. Following the practice in variational geometric modeling (e.g., =-=[19]-=-) this is typically done by adding potential energy terms which are functions of first and second derivative magnitudes of the surface. Such thin-plate approaches have been used, for example, by Qin [... |

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Citation Context ...At this point, we have a list of all cross sections of the surface which are required for tiling a good coarse approximation of the final surface. This final step is related to contour stitching (see =-=[1, 15, 13]-=-). However, since we work within the framework of the volume data with the additional information stored in the Ring Master, we do not facesRing n 1 1 Ring n+1 2 3 2 3.6 Coarse Mesh Construction 21 4 ... |

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Citation Context ...ess the oversampling problem [23]. Unfortunately, common mesh simplification algorithms have large memory footprints [24, 16], and are impractical for decimating meshes with millions of polygons (see =-=[38, 37]-=- for an approach to tackle this situation). Alternatively one may apply classic iso-surface extraction techniques such as Marching Cubes to hierarchical (filtered and down-sampled) volumetric represen... |