1035:
1021:
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5716:, it degrades gracefully, meaning the inside-outside segmentation would not change much if we poked holes in a closed mesh. For this reason, the Generalized Winding Number handles open meshes robustly. The boundary between inside and outside smoothly passes over holes in the mesh. In fact, in the limit, the Generalized Winding Number is equivalent to the ray-casting method as the number of rays goes to infinity.
549:
1429:. In order to sample points uniformly at random across the surface of the triangle mesh, the random sampling is broken into two stages: uniformly sampling points within a triangle; and non-uniformly sampling triangles, such that each triangle's associated probability is proportional to its surface area. Thereafter, the optimal transformation is calculated based on the difference between each
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and propagate these handle deformations to the rest of shape smoothly and without removing or distorting details. Some common forms of interactive deformations are point-based, skeleton-based, and cage-based. In point-based deformation, a user can apply transformations to small set of points, called handles, on the shape. Skeleton-based deformation defines a
530:
321:
308:), which has a position. This encodes the geometry of the shape. Directed edges connect these vertices into triangles, which by the right hand rule, then have a direction called the normal. Each triangle forms a face of the mesh. These are combinatoric in nature and encode the topology of the shape. In addition to triangles, a more general class of
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Deformation is concerned with transforming some rest shape to a new shape. Typically, these transformations are continuous and do not alter the topology of the shape. Modern mesh-based shape deformation methods satisfy user deformation constraints at handles (selected vertices or regions on the mesh)
316:
encode a coarse representation along with a sequence of transformations, which produce a fine or high resolution representation of the shape once applied. These meshes are useful in a variety of applications, including geomorphs, progressive transmission, mesh compression, and selective refinement.
458:. There is one connected component, 0 holes, and 3 connected components of the boundary (the waist and two leg holes). So in this case, the Euler characteristic is -1. To bring this into the discrete world, the Euler characteristic of a mesh is computed in terms of its vertices, edges, and faces.
160:. At the final stage of the shape's "life," it is consumed. This can mean it is consumed by a viewer as a rendered asset in a game or movie, for instance. The end of a shape's life can also be defined by a decision about the shape, like whether or not it satisfies some criteria. Or it can even be
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17:
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In the limit of shooting many, many rays, this method handles open meshes, however it in order to become accurate, far too many rays are required for this method to be computationally ideal. Instead, a more robust approach is the
Generalized Winding Number. Inspired by the 2D
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Depending on how a shape is initialized or "birthed," the shape might exist only as a nebula of sampled points that represent its surface in space. To transform the surface points into a mesh, the
Poisson reconstruction strategy can be employed. This method states that the
533:
This image shows a mesh of a pair of pants, with Euler characteristic -1. This is explained by the equation to compute the characteristic: 2c - 2h - b. The mesh has 1 connected component, 0 topological holes, and 3 boundaries (the waist hole and each leg hole): 2 - 0 - 3 =
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with a
Laplacian-based energy. Applying the Laplace operator to these mappings allows us to measure how the position of a point changes relative to its neighborhood, which keeps the handles smooth. Thus, the energy we would like to minimize can be written as:
5676:
3452:
for the shape, which allows a user to move the bones and rotate the joints. Cage-based deformation requires a cage to be drawn around all or part of a shape so that, when the user manipulates points on the cage, the volume it encloses changes accordingly.
2748:
One way to measure the distortion accrued in the mapping process is to measure how much the length of the edges on the 2D mapping differs from their lengths in the original 3D surface. In more formal terms, the objective function can be written as:
140:, usually in 2D or 3D, although the shape can live in a space of arbitrary dimensions. The processing of a shape involves three stages, which is known as its life cycle. At its "birth," a shape can be instantiated through one of three methods: a
1625:
2702:{\displaystyle M_{ij}={\begin{cases}{\frac {1}{3}}\sum \limits _{t=1}^{m}{\begin{cases}Area(t)&{\text{if triangle t contains vertex i}}\\0&{\text{otherwise}}\end{cases}}&{\text{if i=j}}\\0&{\text{otherwise}}\end{cases}}}
3073:
2036:{\displaystyle {\begin{aligned}\sum _{i}M_{i}\delta f_{i}{\bar {f}}_{i}&=\sum _{i}M_{i}\delta f_{i}\sum _{j}(\mathbf {I} +\lambda \nabla ^{2})f_{j}=\sum _{i}\delta f_{i}\sum _{j}(M+\lambda M\nabla ^{2})f_{j},\end{aligned}}}
152:. After a shape is born, it can be analyzed and edited repeatedly in a cycle. This usually involves acquiring different measurements, such as the distances between the points of the shape, the smoothness of the shape, or its
3910:
is a translation vector. Unfortunately, there's no way to know the rotations in advance, so instead we pick a “best” rotation that minimizes displacements. To achieve local rotation invariance, however, requires a function
3759:
2905:
of their neighbours. The Tutte
Mapping, however, still suffers from severe distortions as it attempts to make the edge lengths equal, and hence does not correctly account for the triangle sizes on the actual surface mesh.
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To avoid the problem of having all the vertices mapped to a single point, we also require that the solution to the optimization problem must have a non-zero norm and that it is orthogonal to the trivial solution.
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804:
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onto which we can map the surface so that distortions are minimized. In this manner, parameterization can be seen as an optimization problem. One of the major applications of mesh parameterization is
4629:
565:, a function that determines which points in space belong to the surface of the shape, can actually be computed from the sampled points. The key concept is that gradient of the indicator function is
1449:
and its projection. In the following iteration, the projections are calculated based on the result of applying the previous transformation on the samples. The process is repeated until convergence.
3957:
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1315:
648:
2445:{\displaystyle L_{ij}={\begin{cases}{\frac {1}{2}}(\cot(\alpha _{ij})+\cot(\beta _{ij}))&{\text{edge ij exists}}\\-\sum \limits _{i\neq j}L_{ij}&i=j\\0&{\text{otherwise}}\end{cases}}}
1810:
5137:
3873:
3258:{\displaystyle {\begin{bmatrix}{\dfrac {\partial u}{\partial x}}&{\dfrac {\partial u}{\partial y}}\\{\dfrac {\partial v}{\partial x}}&{\dfrac {\partial v}{\partial y}}\end{bmatrix}}=1}
1682:
1457:
When shapes are defined or scanned, there may be accompanying noise, either to a signal acting upon the surface or to the actual surface geometry. Reducing noise on the former is known as
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is closed. Take the pair of pants example from the top of this article. This mesh clearly has a semantic inside-and-outside, despite there being holes at the waist and the legs.
2113:
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everywhere, except at the sampled points, where it is equal to the inward surface normal. More formally, suppose the collection of sampled points from the surface is denoted by
524:
3523:
1047:
One common problem encountered in geometry processing is how to merge multiple views of a single object captured from different angles or positions. This problem is known as
3674:
1675:
556:. Sometimes shapes are initialized only as "point clouds," a collection of sampled points from the shape's surface. Often, these point clouds need to be converted to meshes.
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of a shape is a collection of properties that do not change even after smooth transformations have been applied to the shape. It concerns dimensions such as the number of
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An animation depicting the same registration procedure as above, but with piecewise linear approximation of the projection function. Note that it converges much faster.
54:, manipulation, simulation and transmission of complex 3D models. As the name implies, many of the concepts, data structures, and algorithms are directly analogous to
2185:
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term is to map the image of the
Laplacian from areas to points. Because the variation is free, this results in a self-adjoint linear problem to solve with a parameter
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A comparison of the Tutte
Embedding and Least-Squares-Conformal-Mapping parameterization. Notice how the LSCM parameterization is smooth on the side of the beetle.
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While this method is translation invariant, it is unable to account for rotations. The As-Rigid-As-Possible deformation scheme applies a rigid transformation
3120:. In addition, we would also like the mapping to have proportionally similar sized regions as the original. This results to setting the Jacobian of the
2901:. Doing so prevents the vertices from collapsing into a single vertex when the mapping is applied. The non-boundary vertices are then positioned at the
4148:
While seemingly trivial, in many cases, determining the inside from the outside of a triangle mesh is not an easy problem. In general, given a surface
2889:
is the set of vertices. However, optimizing this objective function would result in a solution that maps all of the vertices to a single vertex in the
3682:
3635:. Ideally, the transformed shape adds as little distortion as possible to the original. One way to model this distortion is in terms of displacements
1385:
is therefore employed to solve for small transformations iteratively, instead of solving for the potentially large transformation in one go. In ICP,
3421:{\displaystyle {\underset {u,v}{\text{min}}}\int _{S}{\frac {1}{2}}||\nabla u||^{2}+{\frac {1}{2}}||\nabla v||^{2}-\nabla u\cdot \nabla v^{\perp }}
2754:
4959:
324:
A mesh of the famous
Stanford bunny. Shapes are usually represented as a mesh, a collection of polygons that delineate the contours of the shape.
6067:
5873:
1026:
An animation depicting registration of a partial mesh onto a complete mesh, with piecewise constant approximation of the projection function
5396:
1319:
While rotations are non-linear in general, small rotations can be linearized as skew-symmetric matrices. Moreover, the distance function
4131:{\displaystyle \min _{{\textbf {x,R}}\in SO(3)}\int _{\Omega }||\nabla {\textbf {x}}-{\textbf {R}}\nabla {\hat {\textbf {x}}}||^{2}dA}
770:
5937:
3528:
6316:
4797:{\displaystyle isInside_{r}(q)=\left\{{\begin{array}{ll}1&count_{r}\ is\ odd\\0&count_{r}\ is\ even\\\end{array}}\right.}
161:
117:
6364:
6119:
734:{\displaystyle \triangledown g={\begin{cases}{\textbf {n}}_{i},&\forall p_{i}\in S\\0,&{\text{otherwise}}\end{cases}}}
6369:
5858:
4140:
Note that the translation vector is not present in the final objective function because translations have constant gradient.
3461:
Handles provide a sparse set of constraints for the deformation: as the user moves one point, the others must stay in place.
3914:
2550:
M as an operator computes the local integral of a function's value and is often set for a mesh with m triangles as follows:
5281:{\displaystyle isInside(q)=\left\{{\begin{array}{ll}1&rayTest(q)\geq 0.5\\0&rayTest(q)<0.5\\\end{array}}\right.}
2934:
coordinate functions. The wobbliness and distortion apparent in the mass springs methods are due to high variations in the
2190:
1220:
6354:
2902:
1790:{\displaystyle 0=\delta {\mathcal {L}}(f)=\int _{\Omega }\delta f(\mathbf {I} +\lambda \nabla ^{2})f-\delta f{\bar {f}}dx}
6344:
5801:
3825:
278:
1465:. The task of geometric smoothing is analogous to signal noise reduction, and consequently employs similar approaches.
455:
3959:
which outputs the best rotation for every point on the surface. The resulting energy, then, must optimize over both
5999:
5791:
6282:
5888:
5787:
5671:{\displaystyle isInside(q)=\left\{{\begin{array}{ll}1&wn(q)\geq 0.5\\0&wn(q)<0.5\\\end{array}}\right.}
3267:
Putting these requirements together, we can augment the
Dirichlet energy so that our objective function becomes:
79:
3084:
454:
is the number of connected components of the boundary of the surface. A concrete example of this is a mesh of a
5893:
5851:
5832:
5781:
4900:
1497:
and the smoothness of the resulting signal, which approximated by the magnitude of the gradient with a weight
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Approximating inside-outside segmentation by shooting rays from a query point for varying number of rays
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94:
75:
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851:
461:
273:
In computers, everything must be discretized. Shapes in geometry processing are usually represented as
16:
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1468:
The pertinent
Lagrangian to be minimized is derived by recording the conformity to the initial signal
255:
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339:
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102:
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98:
71:
31:
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The naive attempt to solve this problem is to shoot many rays in random directions, and classify
4443:
4171:
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830:
is the vector field defined by the samples. As a variational problem, one can view the minimizer
562:
313:
172:
Like any other shape, the shapes used in geometry processing have properties pertaining to their
3467:
1800:
By discretizing this onto piecewise-constant elements with our signal on the vertices we obtain
1620:{\displaystyle {\mathcal {L}}(f)=\int _{\Omega }\|f-{\bar {f}}\|^{2}+\lambda \|\nabla f\|^{2}dx}
1471:
1322:
4528:
4344:
132:
A mesh of a cactus showing the
Gaussian Curvature at each vertex, using the angle defect method
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range scanners in airport security, autonomous vehicles, medical scanner data reconstruction
3077:
There are a few other things to consider. We would like to minimize the angle distortion to
2140:
67:
59:
35:
5683:
5503:
5391:
is proportional to the sum of the solid angle contribution from each triangle in the mesh:
5362:
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990:
857:
833:
750:
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592:
284:
226:
199:
128:
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it must pass through twice, because S is bounded, so any ray entering it must exit. So if
2894:
2729:
2716:
Occasionally, we need to flatten a 3D surface onto a flat plane. This process is known as
2245:
When working with triangle meshes one way to determine the values of the Laplacian matrix
1462:
1458:
181:
63:
3068:{\displaystyle {\underset {u,v}{\text{min}}}\int _{S}||\nabla u||^{2}+||\nabla v||^{2}dA}
267:
85:
Applications of geometry processing algorithms already cover a wide range of areas from
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The Computational Geometry Algorithms Library (see section on Polygon Mesh Processing)
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274:
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90:
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is inside, the same logic applies to the previous case, but the ray must intersect
548:
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4301:
In the simplest case, the shape is closed. In this case, to determine if a point
2744:
The Tutte Embedding shows non-smooth parameterizations on the side of the beetle.
5296:
3754:{\displaystyle \min _{\textbf {d}}\int _{\Omega }||\Delta {\textbf {d}}||^{2}dA}
2547:
747:
problem. To find the indicator function of the surface, we must find a function
553:
141:
39:
6168:
6152:
5602:
5182:
4681:
149:
86:
2942:
coordinate functions. With this approach, the objective function becomes the
2840:{\displaystyle {\underset {U}{\text{min}}}\sum _{ij\in E}||u_{i}-u_{j}||^{2}}
6239:
6155:(2003). "Differential coordinates for local mesh morphing and deformation".
6138:
5929:
4829:
2914:
193:
43:
6016:
5532:
is equivalent to the characteristic function for the volume represented by
5077:{\displaystyle rayTest(q)={\frac {1}{k}}\sum _{i=1}^{k}isInside_{r_{i}}(q)}
2897:
and restrict the boundary vertices of the mesh onto a unit circle or other
6328:
6086:
2045:
1007:, which can then be applied in subsequent computer graphics applications.
312:
can also be used to represent a shape. More advanced representations like
164:
in the real world, through a method such as 3D printing or laser cutting.
6278:
6202:
Proceedings of EUROGRAPHICS/ACM SIGGRAPH Symposium on Geometry Processing
5901:
3449:
1361:
is non-linear, but is amenable to linear approximations if the change in
251:
177:
173:
109:
51:
23:
by Mario Botsch et al. is a textbook on the topic of Geometry Processing.
6273:
5974:
5878:
529:
320:
185:
66:
might convolve an intensity signal with a blur kernel formed using the
6221:"Robust Inside-Outside Segmentation using Generalized Winding Numbers"
2265:
is through analyzing the geometry of connected triangles on the mesh.
6287:
6264:
6068:"Least squares conformal maps for automatic texture atlas generation"
5339:
is inside or outside. The value of the Generalized Winding Number at
3595:
is a 2D parametric domain. The same can be done with another mapping
6269:
5491:{\displaystyle wn(q)={\frac {1}{4\pi }}\sum _{t\in F}solidAngle(t)}
4341:
in any direction from a query point, and count the number of times
5924:
Botsch, Mario; Kobbelt, Leif; Pauly, Mark; Alliez, Pierre (2010).
4828:
2913:
2044:
547:
528:
431:
319:
137:
127:
15:
2893:-coordinates. Borrowing an idea from graph theory, we apply the
6301:
799:{\displaystyle \lVert \triangledown \chi -{\textbf {V}}\rVert }
196:. It also includes the dimension in which the shape lives (ex.
180:. The geometry of a shape concerns the position of the shape's
3568:{\displaystyle {\hat {x}}:\Omega \rightarrow \mathbb {R} ^{3}}
3142:
4877:
an odd number of times. To quantify this, let us say we cast
643:. Then the gradient of the indicator function is defined as:
6322:
4857:
as being inside if and only if most of the rays intersected
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of the shape. One example of a non-orientable shape is the
156:. Editing may involve denoising, deforming, or performing
2922:
Another way to measure the distortion is to consider the
338:
One particularly important property of a 3D shape is its
6307:
Discrete Differential Geometry: An Applied Introduction
6120:"Bounded Biharmonic Weights for Real-Time Deformation"
3952:{\displaystyle {\textbf {R}}:\Omega \rightarrow SO(3)}
116:
academic conference, and the main topic of the annual
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2238:{\displaystyle {\bar {f}}=(M+\lambda \mathbf {L} )f.}
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1310:{\displaystyle \int _{x\in X}||Rx+t-P_{Y}(x)||^{2}dx}
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342:, which can alternatively be defined in terms of its
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is inside or outside the surface, we can cast a ray
5807:muscle and bone modelling, real-time hand tracking
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3868:{\displaystyle R\in SO(3)\subset \mathbb {R} ^{3}}
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544:Poisson reconstruction from surface points to mesh
518:
446:
422:
402:
382:
346:. The formula for this in the continuous sense is
300:
242:
215:
108:Geometry processing is a common research topic at
281:. Each node in the graph is a vertex (usually in
4017:
3687:
1215:that minimize the following objective function:
616:, and the corresponding normal at that point by
6042:"Intrinsic Parameterizations of Surface Meshes"
4168:we pose this problem as determining a function
30:is an area of research that uses concepts from
5864:Glossary of differential geometry and topology
3443:An example of as-rigid-as-possible deformation
1175:, we want to find the optimal rotation matrix
1051:. In registration, we wish to find an optimal
6114:Jacobson, Alec; Baran, Ilya; Popović, Jovan;
5319:of each triangle in the mesh to determine if
4806:Now, oftentimes we cannot guarantee that the
78:geometry with a blur kernel formed using the
8:
4604:one extra time for the first time it leaves
1602:
1592:
1577:
1555:
979:lie on the surface to be reconstructed, the
793:
774:
136:Geometry processing involves working with a
5732:
854:. After obtaining a good approximation for
430:is number of holes (as in donut holes, see
4420:then the ray must either not pass through
3113:{\displaystyle \nabla u=\nabla v^{\perp }}
743:The task of reconstruction then becomes a
6000:"Efficient Variants of the ICP Algorithm"
5919:
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4862:
4842:
4811:
4757:
4705:
4680:
4658:
4631:
4609:
4589:
4569:
4548:
4530:
4510:
4490:
4463:
4445:
4425:
4405:
4385:
4364:
4346:
4326:
4306:
4283:
4263:
4243:
4223:
4173:
4153:
4116:
4111:
4105:
4094:
4092:
4091:
4082:
4081:
4072:
4071:
4063:
4058:
4052:
4022:
4021:
4020:
4014:
3991:
3990:
3988:
3967:
3966:
3964:
3919:
3918:
3916:
3894:
3890:
3889:
3880:
3859:
3855:
3854:
3827:
3794:
3788:
3787:
3775:
3769:
3739:
3734:
3728:
3722:
3721:
3713:
3708:
3702:
3691:
3690:
3684:
3655:
3654:
3640:
3620:
3600:
3580:
3559:
3555:
3554:
3533:
3532:
3530:
3509:
3505:
3504:
3501:
3472:
3471:
3469:
3412:
3387:
3382:
3376:
3365:
3360:
3350:
3341:
3336:
3330:
3319:
3314:
3304:
3298:
3276:
3274:
3219:
3195:
3169:
3145:
3137:
3135:
3104:
3086:
3053:
3048:
3042:
3031:
3026:
3017:
3012:
3006:
2995:
2990:
2984:
2962:
2960:
2874:
2854:
2831:
2826:
2820:
2814:
2801:
2792:
2787:
2772:
2758:
2756:
2687:
2673:
2659:
2645:
2614:
2608:
2597:
2583:
2575:
2563:
2557:
2519:
2495:
2489:
2465:
2459:
2430:
2401:
2385:
2369:
2352:
2324:
2298:
2290:
2278:
2272:
2250:
2221:
2195:
2194:
2192:
2172:
2148:
2142:
2122:
2120:
2100:
2091:
2085:
2064:
2058:
2049:A noisy sphere being iteratively smoothed
2020:
2007:
1982:
1972:
1959:
1946:
1933:
1918:
1909:
1899:
1886:
1876:
1859:
1848:
1847:
1840:
1827:
1817:
1809:
1807:
1770:
1769:
1748:
1733:
1718:
1696:
1695:
1684:
1661:
1660:
1658:
1635:
1605:
1580:
1565:
1564:
1549:
1527:
1526:
1524:
1502:
1476:
1475:
1473:
1434:
1414:
1394:
1366:
1336:
1324:
1295:
1290:
1284:
1269:
1245:
1240:
1228:
1222:
1200:
1180:
1160:
1140:
1106:
1100:
1080:
1060:
992:
937:
899:
879:
859:
835:
814:
813:
811:
787:
786:
772:
752:
719:
695:
677:
671:
670:
661:
650:
627:
621:
600:
594:
574:
511:
503:
495:
487:
479:
471:
463:
439:
415:
395:
351:
292:
286:
234:
228:
207:
201:
4949:{\displaystyle r_{1},r_{2},\dots ,r_{k}}
3438:
2739:
1381:is small. An iterative solution such as
552:A triangle mesh is constructed out of a
6195:"As-Rigid-As-Possible Surface Modeling"
5913:
3815:{\displaystyle x_{i}=R{\hat {x_{i}}}+t}
777:
652:
410:is the number of connected components,
6279:Mathematical Geometry Processing Group
6017:"Chris Tralie : Laplacian Meshes"
6309:, course notes by Keenan Crane et al.
5874:List of interactive geometry software
3903:{\displaystyle t\in \mathbb {R} ^{3}}
983:algorithm can be used to construct a
7:
5969:
5967:
972:{\displaystyle \chi (x,y,z)=\sigma }
4095:
4083:
4073:
4023:
3992:
3968:
3920:
3723:
3692:
2594:
2382:
815:
788:
672:
124:Geometry processing as a life cycle
4380:it passes through the surface. If
4088:
4068:
4053:
3928:
3718:
3703:
3582:
3547:
3405:
3396:
3370:
3324:
3231:
3223:
3207:
3199:
3181:
3173:
3157:
3149:
3097:
3088:
3036:
3000:
2720:. The goal is to find coordinates
2108:{\displaystyle M^{-1}\mathbf {L} }
2061:
2004:
1930:
1745:
1719:
1595:
1550:
688:
168:Discrete Representation of a Shape
74:might be achieved by convolving a
14:
6215:Jacobson, Alec; Ladislav, Kavan;
5998:Szymon Rusinkiewicz, Marc Levoy.
2514:are the angles opposite the edge
519:{\displaystyle \chi =|V|-|E|+|F|}
6265:Symposium on Geometry Processing
5975:"Poisson surface reconstruction"
3525:can be described with a mapping
3518:{\displaystyle \mathbb {R} ^{3}}
2910:Least-squares conformal mappings
2222:
2123:
2101:
1919:
1734:
1033:
1019:
118:Symposium on Geometry Processing
6302:Polygon Mesh Processing Library
2647:if triangle t contains vertex i
5859:Discrete differential geometry
5699:
5693:
5652:
5646:
5622:
5616:
5591:
5585:
5519:
5513:
5485:
5479:
5412:
5406:
5378:
5372:
5262:
5256:
5217:
5211:
5171:
5165:
5084:which is the average value of
5071:
5065:
4990:
4984:
4670:
4664:
4205:
4199:
4112:
4106:
4099:
4064:
4059:
4043:
4037:
3946:
3940:
3931:
3847:
3841:
3800:
3735:
3729:
3714:
3709:
3669:{\displaystyle d=x-{\hat {x}}}
3660:
3550:
3538:
3477:
3383:
3377:
3366:
3361:
3337:
3331:
3320:
3315:
3049:
3043:
3032:
3027:
3013:
3007:
2996:
2991:
2827:
2821:
2793:
2788:
2640:
2634:
2533:
2521:
2364:
2361:
2345:
2333:
2317:
2308:
2226:
2209:
2200:
2013:
1988:
1939:
1915:
1853:
1775:
1754:
1730:
1708:
1702:
1677:emits the necessary condition
1670:{\displaystyle {\mathcal {L}}}
1570:
1539:
1533:
1481:
1409:are chosen and projected onto
1348:
1342:
1291:
1285:
1281:
1275:
1246:
1241:
1118:
1112:
960:
942:
919:
901:
512:
504:
496:
488:
480:
472:
1:
5742:Image-to-world Registration,
4000:{\displaystyle {\textbf {R}}}
3976:{\displaystyle {\textbf {x}}}
2869:is the set of mesh edges and
1383:Iterative Closest Point (ICP)
1131:is the projection of a point
823:{\displaystyle {\textbf {V}}}
589:, each point in the space by
383:{\displaystyle \chi =2c-2h-b}
6297:Polygon Mesh Processing Book
6228:ACM Transactions on Graphics
6127:ACM Transactions on Graphics
6075:ACM Transactions on Graphics
5802:Modelling biological systems
5125:{\displaystyle isInside_{r}}
3615:for the transformed surface
2477:{\displaystyle \alpha _{ij}}
2130:{\displaystyle \mathbf {L} }
2115:for the cotangent Laplacian
6325:geometry processing library
5792:mathematical visualizations
4478:{\displaystyle count_{r}=0}
4211:{\displaystyle isInside(q)}
4144:Inside-Outside Segmentation
3128:coordinate functions to 1.
3081:. That means we would like
2507:{\displaystyle \beta _{ij}}
2073:{\displaystyle \nabla ^{2}}
146:mathematical representation
97:, to biomedical computing,
6386:
5788:Information visualizations
5132:from each ray. Therefore:
4485:) or, each time it enters
3486:{\displaystyle {\hat {S}}}
1490:{\displaystyle {\bar {f}}}
1389:random sample points from
1354:{\displaystyle x-P_{Y}(x)}
6283:Free University of Berlin
6169:10.1007/s00371-002-0180-0
6040:Desbrun, Mathieu (2002).
5889:Digital signal processing
5295:, this approach uses the
4557:{\displaystyle count_{r}}
4373:{\displaystyle count_{r}}
3875:is a rotation matrix and
2903:barycentric interpolation
277:, which can be seen as a
80:Laplace-Beltrami operator
6270:Multi-Res Modeling Group
5894:Digital signal processor
5852:Digital image processing
5833:Graphics processing unit
5782:Visualization (graphics)
4956:. We associate a number
3822:to each handle i, where
2180:{\displaystyle \lambda }
1646:{\displaystyle \delta f}
1510:{\displaystyle \lambda }
1124:{\displaystyle P_{Y}(x)}
1055:that will align surface
6288:Computer Graphics Group
6240:10.1145/2461912.2461916
6139:10.1145/2010324.1964973
5926:Polygon Mesh Processing
3588:{\displaystyle \Omega }
3457:Point-based deformation
1195:and translation vector
925:{\displaystyle (x,y,z)}
887:{\displaystyle \sigma }
21:Polygon Mesh Processing
6365:Computational geometry
6292:RWTH Aachen University
6193:; Alexa, Marc (2007).
5869:Industrial CT scanning
5818:Calculus of variations
5733:Surface Reconstruction
5706:
5672:
5546:
5526:
5492:
5385:
5353:
5333:
5313:
5282:
5126:
5078:
5026:
4950:
4891:
4871:
4851:
4834:
4820:
4798:
4618:
4598:
4578:
4558:
4519:
4499:
4479:
4434:
4414:
4394:
4374:
4335:
4315:
4292:
4272:
4252:
4232:
4212:
4162:
4132:
4001:
3977:
3953:
3904:
3869:
3816:
3755:
3670:
3629:
3609:
3589:
3569:
3519:
3487:
3444:
3422:
3259:
3114:
3079:preserve orthogonality
3069:
2919:
2883:
2863:
2841:
2745:
2703:
2613:
2540:
2508:
2478:
2446:
2259:
2239:
2181:
2161:
2160:{\displaystyle M^{-1}}
2131:
2109:
2074:
2050:
2037:
1791:
1671:
1647:
1621:
1511:
1491:
1443:
1423:
1403:
1375:
1355:
1311:
1209:
1189:
1169:
1149:
1125:
1089:
1069:
1001:
973:
926:
888:
868:
844:
824:
800:
761:
735:
637:
610:
583:
557:
539:Surface reconstruction
535:
520:
448:
424:
404:
384:
325:
302:
244:
217:
133:
24:
6370:Differential geometry
6217:Sorkine-Hornung, Olga
6087:10.1145/566654.566590
5841:Computer-aided design
5726:Computer-aided design
5707:
5705:{\displaystyle wn(q)}
5673:
5552:. Therefore, we say:
5547:
5527:
5525:{\displaystyle wn(q)}
5493:
5386:
5384:{\displaystyle wn(q)}
5354:
5334:
5314:
5283:
5127:
5079:
5006:
4951:
4892:
4872:
4852:
4832:
4821:
4799:
4619:
4599:
4579:
4564:is even. Likewise if
4559:
4520:
4500:
4480:
4435:
4415:
4395:
4375:
4336:
4316:
4293:
4273:
4253:
4233:
4213:
4163:
4133:
4002:
3978:
3954:
3905:
3870:
3817:
3756:
3671:
3630:
3610:
3590:
3570:
3520:
3488:
3442:
3423:
3260:
3115:
3070:
2917:
2884:
2864:
2842:
2743:
2704:
2593:
2541:
2539:{\displaystyle (i,j)}
2509:
2479:
2447:
2260:
2240:
2182:
2162:
2132:
2110:
2075:
2048:
2038:
1792:
1672:
1648:
1622:
1512:
1492:
1444:
1424:
1404:
1376:
1356:
1312:
1210:
1190:
1170:
1150:
1126:
1090:
1070:
1002:
1000:{\displaystyle \chi }
974:
927:
894:for which the points
889:
869:
867:{\displaystyle \chi }
845:
843:{\displaystyle \chi }
825:
801:
762:
760:{\displaystyle \chi }
736:
638:
636:{\displaystyle n_{i}}
611:
609:{\displaystyle p_{i}}
584:
551:
532:
521:
449:
425:
405:
385:
329:Properties of a shape
323:
303:
301:{\displaystyle R^{3}}
245:
243:{\displaystyle R^{3}}
218:
216:{\displaystyle R^{2}}
158:rigid transformations
131:
95:computer-aided design
62:. For example, where
46:for the acquisition,
19:
6355:3D computer graphics
6066:Levy, Bruno (2002).
5955:"Progressive Meshes"
5828:3D computer graphics
5747:Image-guided surgery
5684:
5558:
5536:
5504:
5397:
5363:
5343:
5323:
5303:
5138:
5088:
4960:
4901:
4881:
4861:
4841:
4810:
4630:
4608:
4588:
4568:
4529:
4509:
4489:
4444:
4424:
4404:
4384:
4345:
4325:
4305:
4282:
4262:
4242:
4222:
4172:
4152:
4013:
3987:
3963:
3915:
3879:
3826:
3768:
3683:
3639:
3619:
3599:
3579:
3529:
3500:
3468:
3273:
3134:
3085:
2959:
2873:
2853:
2755:
2556:
2518:
2488:
2458:
2271:
2249:
2191:
2171:
2141:
2119:
2084:
2057:
2053:where our choice of
1806:
1683:
1657:
1634:
1523:
1501:
1472:
1433:
1413:
1393:
1365:
1323:
1221:
1199:
1179:
1159:
1139:
1099:
1095:. More formally, if
1079:
1059:
1053:rigid transformation
991:
936:
898:
878:
858:
834:
810:
806:is minimized, where
771:
751:
649:
620:
593:
573:
462:
438:
414:
394:
350:
340:Euler characteristic
334:Euler Characteristic
285:
227:
200:
154:Euler characteristic
103:scientific computing
42:to design efficient
6345:Geometry processing
6157:The Visual Computer
5772:collision detection
5764:Physics simulations
5760:reverse engineering
5500:For a closed mesh,
2736:Mass springs method
1630:Taking a variation
987:from the function
99:reverse engineering
72:geometric smoothing
32:applied mathematics
28:Geometry processing
5777:Geologic modelling
5702:
5668:
5663:
5542:
5522:
5488:
5448:
5381:
5349:
5329:
5309:
5278:
5273:
5122:
5074:
4946:
4887:
4867:
4847:
4835:
4816:
4794:
4789:
4614:
4594:
4574:
4554:
4515:
4495:
4475:
4430:
4410:
4390:
4370:
4331:
4311:
4288:
4268:
4248:
4228:
4218:which will return
4208:
4158:
4128:
4047:
3997:
3973:
3949:
3900:
3865:
3812:
3751:
3697:
3666:
3625:
3605:
3585:
3565:
3515:
3483:
3445:
3418:
3292:
3255:
3243:
3239:
3215:
3189:
3165:
3110:
3065:
2978:
2920:
2879:
2859:
2837:
2786:
2766:
2746:
2699:
2694:
2666:
2536:
2504:
2474:
2442:
2437:
2396:
2255:
2235:
2177:
2157:
2127:
2105:
2070:
2051:
2033:
2031:
1987:
1964:
1914:
1881:
1822:
1787:
1667:
1643:
1617:
1507:
1487:
1439:
1419:
1399:
1371:
1351:
1307:
1205:
1185:
1165:
1145:
1121:
1085:
1065:
997:
969:
922:
884:
864:
852:Poisson's equation
840:
820:
796:
757:
731:
726:
633:
606:
579:
563:indicator function
558:
536:
516:
444:
420:
400:
380:
326:
314:progressive meshes
298:
240:
213:
134:
25:
5884:Signal processing
5823:Computer graphics
5714:harmonic function
5545:{\displaystyle S}
5433:
5431:
5352:{\displaystyle q}
5332:{\displaystyle q}
5312:{\displaystyle q}
5004:
4890:{\displaystyle k}
4870:{\displaystyle S}
4850:{\displaystyle q}
4819:{\displaystyle S}
4774:
4765:
4722:
4713:
4617:{\displaystyle S}
4597:{\displaystyle S}
4577:{\displaystyle q}
4518:{\displaystyle q}
4498:{\displaystyle S}
4433:{\displaystyle S}
4413:{\displaystyle S}
4393:{\displaystyle q}
4334:{\displaystyle r}
4314:{\displaystyle q}
4291:{\displaystyle 0}
4271:{\displaystyle S}
4251:{\displaystyle q}
4231:{\displaystyle 1}
4161:{\displaystyle S}
4102:
4097:
4085:
4075:
4025:
4016:
3994:
3970:
3922:
3803:
3725:
3694:
3686:
3663:
3628:{\displaystyle S}
3608:{\displaystyle x}
3541:
3480:
3358:
3312:
3280:
3277:
3238:
3214:
3188:
3164:
2966:
2963:
2882:{\displaystyle U}
2862:{\displaystyle E}
2768:
2762:
2759:
2690:
2676:
2662:
2648:
2591:
2433:
2381:
2372:
2306:
2258:{\displaystyle L}
2203:
1978:
1955:
1905:
1872:
1856:
1813:
1778:
1573:
1484:
1442:{\displaystyle x}
1422:{\displaystyle Y}
1402:{\displaystyle X}
1374:{\displaystyle X}
1208:{\displaystyle t}
1188:{\displaystyle R}
1168:{\displaystyle Y}
1148:{\displaystyle X}
1088:{\displaystyle Y}
1068:{\displaystyle X}
850:as a solution of
817:
790:
722:
674:
582:{\displaystyle S}
447:{\displaystyle b}
423:{\displaystyle h}
403:{\displaystyle c}
262:, as well as the
114:computer graphics
56:signal processing
6377:
6252:
6251:
6225:
6212:
6206:
6205:
6199:
6187:
6181:
6180:
6149:
6143:
6142:
6124:
6111:
6105:
6104:
6102:
6101:
6095:
6089:. Archived from
6072:
6063:
6057:
6056:
6046:
6037:
6031:
6030:
6028:
6027:
6013:
6007:
6006:
6004:
5995:
5989:
5988:
5986:
5985:
5971:
5962:
5961:
5959:
5950:
5944:
5943:
5921:
5711:
5709:
5708:
5703:
5677:
5675:
5674:
5669:
5667:
5664:
5551:
5549:
5548:
5543:
5531:
5529:
5528:
5523:
5497:
5495:
5494:
5489:
5447:
5432:
5430:
5419:
5390:
5388:
5387:
5382:
5358:
5356:
5355:
5350:
5338:
5336:
5335:
5330:
5318:
5316:
5315:
5310:
5287:
5285:
5284:
5279:
5277:
5274:
5131:
5129:
5128:
5123:
5121:
5120:
5083:
5081:
5080:
5075:
5064:
5063:
5062:
5061:
5025:
5020:
5005:
4997:
4955:
4953:
4952:
4947:
4945:
4944:
4926:
4925:
4913:
4912:
4896:
4894:
4893:
4888:
4876:
4874:
4873:
4868:
4856:
4854:
4853:
4848:
4825:
4823:
4822:
4817:
4803:
4801:
4800:
4795:
4793:
4790:
4772:
4763:
4762:
4761:
4720:
4711:
4710:
4709:
4663:
4662:
4623:
4621:
4620:
4615:
4603:
4601:
4600:
4595:
4583:
4581:
4580:
4575:
4563:
4561:
4560:
4555:
4553:
4552:
4524:
4522:
4521:
4516:
4504:
4502:
4501:
4496:
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3998:
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3030:
3022:
3021:
3016:
3010:
2999:
2994:
2989:
2988:
2979:
2977:
2964:
2944:Dirichlet energy
2888:
2886:
2885:
2880:
2868:
2866:
2865:
2860:
2846:
2844:
2843:
2838:
2836:
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2830:
2824:
2819:
2818:
2806:
2805:
2796:
2791:
2785:
2767:
2760:
2718:parameterization
2712:Parameterization
2708:
2706:
2705:
2700:
2698:
2697:
2691:
2688:
2677:
2674:
2670:
2669:
2663:
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2607:
2592:
2584:
2571:
2570:
2545:
2543:
2542:
2537:
2513:
2511:
2510:
2505:
2503:
2502:
2483:
2481:
2480:
2475:
2473:
2472:
2451:
2449:
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2441:
2440:
2434:
2431:
2409:
2408:
2395:
2373:
2370:
2360:
2359:
2332:
2331:
2307:
2299:
2286:
2285:
2264:
2262:
2261:
2256:
2244:
2242:
2241:
2236:
2225:
2205:
2204:
2196:
2186:
2184:
2183:
2178:
2166:
2164:
2163:
2158:
2156:
2155:
2136:
2134:
2133:
2128:
2126:
2114:
2112:
2111:
2106:
2104:
2099:
2098:
2080:is chosen to be
2079:
2077:
2076:
2071:
2069:
2068:
2042:
2040:
2039:
2034:
2032:
2025:
2024:
2012:
2011:
1986:
1977:
1976:
1963:
1951:
1950:
1938:
1937:
1922:
1913:
1904:
1903:
1891:
1890:
1880:
1864:
1863:
1858:
1857:
1849:
1845:
1844:
1832:
1831:
1821:
1796:
1794:
1793:
1788:
1780:
1779:
1771:
1753:
1752:
1737:
1723:
1722:
1701:
1700:
1676:
1674:
1673:
1668:
1666:
1665:
1652:
1650:
1649:
1644:
1626:
1624:
1623:
1618:
1610:
1609:
1585:
1584:
1575:
1574:
1566:
1554:
1553:
1532:
1531:
1516:
1514:
1513:
1508:
1496:
1494:
1493:
1488:
1486:
1485:
1477:
1448:
1446:
1445:
1440:
1428:
1426:
1425:
1420:
1408:
1406:
1405:
1400:
1380:
1378:
1377:
1372:
1360:
1358:
1357:
1352:
1341:
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1308:
1300:
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1274:
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1214:
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1206:
1194:
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1191:
1186:
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1151:
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1130:
1128:
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1111:
1110:
1094:
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1074:
1072:
1071:
1066:
1037:
1023:
1006:
1004:
1003:
998:
978:
976:
975:
970:
931:
929:
928:
923:
893:
891:
890:
885:
873:
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870:
865:
849:
847:
846:
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829:
827:
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821:
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805:
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802:
797:
792:
791:
766:
764:
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758:
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738:
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732:
730:
729:
723:
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682:
681:
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675:
642:
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631:
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588:
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525:
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522:
517:
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499:
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483:
475:
453:
451:
450:
445:
429:
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426:
421:
409:
407:
406:
401:
389:
387:
386:
381:
307:
305:
304:
299:
297:
296:
249:
247:
246:
241:
239:
238:
222:
220:
219:
214:
212:
211:
68:Laplace operator
60:image processing
36:computer science
6385:
6384:
6380:
6379:
6378:
6376:
6375:
6374:
6335:
6334:
6313:Video tutorials
6261:
6256:
6255:
6223:
6214:
6213:
6209:
6197:
6189:
6188:
6184:
6151:
6150:
6146:
6122:
6113:
6112:
6108:
6099:
6097:
6093:
6070:
6065:
6064:
6060:
6044:
6039:
6038:
6034:
6025:
6023:
6021:www.ctralie.com
6015:
6014:
6010:
6002:
5997:
5996:
5992:
5983:
5981:
5973:
5972:
5965:
5957:
5952:
5951:
5947:
5940:
5923:
5922:
5915:
5910:
5814:
5797:Texture mapping
5767:Computer games
5722:
5682:
5681:
5662:
5661:
5638:
5632:
5631:
5608:
5597:
5556:
5555:
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4654:
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4606:
4605:
4586:
4585:
4566:
4565:
4544:
4527:
4526:
4507:
4506:
4487:
4486:
4459:
4442:
4441:
4440:(in which case
4422:
4421:
4402:
4401:
4382:
4381:
4360:
4343:
4342:
4323:
4322:
4303:
4302:
4280:
4279:
4260:
4259:
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3637:
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3597:
3596:
3577:
3576:
3553:
3527:
3526:
3503:
3498:
3497:
3466:
3465:
3464:A rest surface
3459:
3437:
3408:
3381:
3335:
3294:
3281:
3271:
3270:
3242:
3241:
3230:
3222:
3217:
3206:
3198:
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3172:
3167:
3156:
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3138:
3132:
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3083:
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3047:
3011:
2980:
2967:
2957:
2956:
2912:
2871:
2870:
2851:
2850:
2825:
2810:
2797:
2753:
2752:
2738:
2730:texture mapping
2714:
2693:
2692:
2685:
2679:
2678:
2671:
2665:
2664:
2657:
2651:
2650:
2643:
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2576:
2559:
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2274:
2269:
2268:
2247:
2246:
2189:
2188:
2169:
2168:
2144:
2139:
2138:
2117:
2116:
2087:
2082:
2081:
2060:
2055:
2054:
2030:
2029:
2016:
2003:
1968:
1942:
1929:
1895:
1882:
1865:
1846:
1836:
1823:
1804:
1803:
1744:
1714:
1681:
1680:
1655:
1654:
1632:
1631:
1601:
1576:
1545:
1521:
1520:
1499:
1498:
1470:
1469:
1463:surface fairing
1455:
1431:
1430:
1411:
1410:
1391:
1390:
1363:
1362:
1332:
1321:
1320:
1289:
1265:
1224:
1219:
1218:
1197:
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1176:
1157:
1156:
1137:
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1102:
1097:
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1045:
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1038:
1029:
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1024:
1013:
989:
988:
934:
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895:
876:
875:
856:
855:
832:
831:
808:
807:
769:
768:
749:
748:
725:
724:
717:
708:
707:
691:
686:
669:
662:
647:
646:
623:
618:
617:
596:
591:
590:
571:
570:
546:
541:
460:
459:
436:
435:
412:
411:
392:
391:
348:
347:
336:
331:
288:
283:
282:
275:triangle meshes
230:
225:
224:
203:
198:
197:
182:points in space
170:
126:
64:image smoothing
12:
11:
5:
6383:
6381:
6373:
6372:
6367:
6362:
6357:
6352:
6347:
6337:
6336:
6333:
6332:
6326:
6320:
6310:
6304:
6299:
6294:
6285:
6276:
6267:
6260:
6259:External links
6257:
6254:
6253:
6207:
6182:
6163:(2): 105–114.
6144:
6106:
6081:(3): 362–371.
6058:
6032:
6008:
5990:
5963:
5953:Hugues Hoppe.
5945:
5938:
5912:
5911:
5909:
5906:
5905:
5904:
5899:
5898:
5897:
5891:
5881:
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5509:
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5411:
5408:
5405:
5402:
5380:
5377:
5374:
5371:
5368:
5348:
5328:
5308:
5293:winding number
5276:
5270:
5267:
5264:
5261:
5258:
5255:
5252:
5249:
5246:
5243:
5240:
5237:
5234:
5232:
5229:
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5051:
5047:
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5019:
5016:
5013:
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5003:
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3507:
3479:
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3411:
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3404:
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3333:
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2970:
2911:
2908:
2899:convex polygon
2878:
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2125:
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2063:
2028:
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2019:
2015:
2010:
2006:
2002:
1999:
1996:
1993:
1990:
1985:
1981:
1975:
1971:
1967:
1962:
1958:
1954:
1949:
1945:
1941:
1936:
1932:
1928:
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1538:
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1506:
1483:
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1459:data denoising
1454:
1451:
1438:
1418:
1398:
1370:
1350:
1347:
1344:
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1231:
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1204:
1184:
1164:
1144:
1120:
1117:
1114:
1109:
1105:
1084:
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1039:
1032:
1031:
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1018:
1017:
1016:
1015:
1014:
1012:
1009:
996:
981:marching cubes
968:
965:
962:
959:
956:
953:
950:
947:
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883:
863:
839:
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728:
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710:
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703:
698:
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630:
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603:
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578:
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537:
514:
510:
506:
502:
498:
494:
490:
486:
482:
478:
474:
470:
467:
443:
419:
399:
379:
376:
373:
370:
367:
364:
361:
358:
355:
335:
332:
330:
327:
310:polygon meshes
295:
291:
237:
233:
210:
206:
169:
166:
125:
122:
112:, the premier
93:and classical
48:reconstruction
13:
10:
9:
6:
4:
3:
2:
6382:
6371:
6368:
6366:
6363:
6361:
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6356:
6353:
6351:
6348:
6346:
6343:
6342:
6340:
6330:
6327:
6324:
6321:
6318:
6314:
6311:
6308:
6305:
6303:
6300:
6298:
6295:
6293:
6289:
6286:
6284:
6280:
6277:
6275:
6271:
6268:
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6249:
6245:
6241:
6237:
6233:
6229:
6222:
6218:
6211:
6208:
6203:
6196:
6192:
6191:Sorkine, Olga
6186:
6183:
6178:
6174:
6170:
6166:
6162:
6158:
6154:
6148:
6145:
6140:
6136:
6132:
6128:
6121:
6117:
6116:Sorkine, Olga
6110:
6107:
6096:on 2017-03-15
6092:
6088:
6084:
6080:
6076:
6069:
6062:
6059:
6054:
6050:
6043:
6036:
6033:
6022:
6018:
6012:
6009:
6001:
5994:
5991:
5980:
5976:
5970:
5968:
5964:
5956:
5949:
5946:
5941:
5939:9781568814261
5935:
5931:
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5920:
5918:
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5907:
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5872:
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5862:
5860:
5857:
5853:
5850:
5849:
5848:
5847:Digital image
5845:
5842:
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5824:
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4361:
4357:
4354:
4351:
4348:
4328:
4308:
4299:
4285:
4265:
4245:
4238:if the point
4225:
4202:
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4193:
4190:
4187:
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4178:
4175:
4155:
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3309:
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3252:
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3202:
3184:
3176:
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3139:
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3127:
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3018:
3003:
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2954:
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2949:
2945:
2941:
2937:
2933:
2929:
2925:
2916:
2909:
2907:
2904:
2900:
2896:
2895:Tutte Mapping
2892:
2876:
2856:
2847:
2832:
2815:
2811:
2807:
2802:
2798:
2782:
2779:
2776:
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2723:
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2709:
2682:
2654:
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2622:
2616:
2609:
2604:
2601:
2598:
2588:
2585:
2577:
2572:
2567:
2564:
2560:
2551:
2549:
2530:
2527:
2524:
2499:
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2452:
2425:
2418:
2415:
2412:
2405:
2402:
2398:
2392:
2389:
2386:
2378:
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2353:
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2342:
2339:
2336:
2328:
2325:
2321:
2314:
2311:
2303:
2300:
2292:
2287:
2282:
2279:
2275:
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2252:
2232:
2229:
2218:
2215:
2212:
2206:
2197:
2174:
2152:
2149:
2145:
2095:
2092:
2088:
2065:
2047:
2043:
2026:
2021:
2017:
2008:
2000:
1997:
1994:
1991:
1983:
1979:
1973:
1969:
1965:
1960:
1956:
1952:
1947:
1943:
1934:
1926:
1923:
1910:
1906:
1900:
1896:
1892:
1887:
1883:
1877:
1873:
1869:
1867:
1860:
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1801:
1798:
1784:
1781:
1772:
1766:
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1760:
1757:
1749:
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1738:
1727:
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1715:
1711:
1705:
1692:
1689:
1686:
1678:
1640:
1637:
1628:
1614:
1611:
1606:
1598:
1589:
1586:
1581:
1567:
1561:
1558:
1546:
1542:
1536:
1518:
1504:
1478:
1466:
1464:
1460:
1452:
1450:
1436:
1416:
1396:
1388:
1384:
1368:
1345:
1337:
1333:
1329:
1326:
1317:
1304:
1301:
1296:
1278:
1270:
1266:
1262:
1259:
1256:
1253:
1250:
1235:
1232:
1229:
1225:
1216:
1202:
1182:
1162:
1155:onto surface
1142:
1135:from surface
1134:
1115:
1107:
1103:
1082:
1075:with surface
1062:
1054:
1050:
1036:
1022:
1010:
1008:
994:
986:
985:triangle mesh
982:
966:
963:
957:
954:
951:
948:
945:
939:
916:
913:
910:
907:
904:
881:
861:
853:
837:
783:
780:
754:
746:
741:
714:
711:
704:
701:
696:
692:
683:
678:
663:
658:
655:
644:
628:
624:
601:
597:
576:
568:
564:
555:
550:
543:
538:
531:
527:
508:
500:
492:
484:
476:
468:
465:
457:
456:pair of pants
441:
433:
417:
397:
377:
374:
371:
368:
365:
362:
359:
356:
353:
345:
341:
333:
328:
322:
318:
315:
311:
293:
289:
280:
276:
271:
269:
265:
264:orientability
261:
257:
253:
235:
231:
208:
204:
195:
191:
187:
183:
179:
175:
167:
165:
163:
159:
155:
151:
147:
143:
139:
130:
123:
121:
119:
115:
111:
106:
104:
100:
96:
92:
91:entertainment
88:
83:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
29:
22:
18:
6231:
6227:
6210:
6201:
6185:
6160:
6156:
6147:
6130:
6126:
6109:
6098:. Retrieved
6091:the original
6078:
6074:
6061:
6052:
6049:Eurographics
6048:
6035:
6024:. Retrieved
6020:
6011:
5993:
5982:. Retrieved
5978:
5948:
5925:
5804:
5784:
5768:
5755:
5752:Architecture
5743:
5736:
5720:Applications
5679:
5554:
5499:
5393:
5289:
5134:
4836:
4805:
4626:
4525:is outside,
4400:was outside
4300:
4147:
4139:
4009:
3763:
3679:
3463:
3460:
3446:
3429:
3269:
3266:
3130:
3125:
3121:
3076:
2955:
2951:
2947:
2939:
2935:
2931:
2927:
2921:
2890:
2848:
2751:
2747:
2725:
2721:
2715:
2552:
2453:
2267:
2052:
1802:
1799:
1679:
1629:
1519:
1467:
1456:
1386:
1318:
1217:
1132:
1049:registration
1046:
1011:Registration
874:and a value
742:
645:
566:
559:
337:
272:
268:Mobius strip
171:
135:
107:
84:
27:
26:
20:
6319:grad school
6153:Marc, Alexa
5297:solid angle
4298:otherwise.
3435:Deformation
2548:mass matrix
745:variational
554:point cloud
40:engineering
6350:3D imaging
6339:Categories
6204:: 109–116.
6100:2017-03-14
6026:2017-03-16
5984:2017-01-26
5979:hhoppe.com
5908:References
5758:creating,
4258:is inside
2924:variations
767:such that
260:boundaries
162:fabricated
87:multimedia
44:algorithms
6248:207202533
5930:CRC Press
5626:≥
5442:∈
5435:∑
5428:π
5221:≥
5008:∑
4931:…
4100:^
4089:∇
4079:−
4069:∇
4054:Ω
4050:∫
4029:∈
3932:→
3929:Ω
3886:∈
3851:⊂
3833:∈
3801:^
3719:Δ
3704:Ω
3700:∫
3661:^
3652:−
3583:Ω
3551:→
3548:Ω
3539:^
3478:^
3414:⊥
3406:∇
3403:⋅
3397:∇
3394:−
3371:∇
3325:∇
3296:∫
3232:∂
3224:∂
3208:∂
3200:∂
3182:∂
3174:∂
3158:∂
3150:∂
3106:⊥
3098:∇
3089:∇
3037:∇
3001:∇
2982:∫
2808:−
2780:∈
2770:∑
2689:otherwise
2661:otherwise
2595:∑
2493:β
2463:α
2432:otherwise
2390:≠
2383:∑
2379:−
2350:β
2343:
2322:α
2315:
2219:λ
2201:¯
2175:λ
2150:−
2093:−
2062:∇
2005:∇
1998:λ
1980:∑
1966:δ
1957:∑
1931:∇
1927:λ
1907:∑
1893:δ
1874:∑
1854:¯
1834:δ
1815:∑
1776:¯
1764:δ
1761:−
1746:∇
1742:λ
1725:δ
1720:Ω
1716:∫
1693:δ
1638:δ
1603:‖
1596:∇
1593:‖
1590:λ
1578:‖
1571:¯
1562:−
1556:‖
1551:Ω
1547:∫
1505:λ
1482:¯
1453:Smoothing
1330:−
1263:−
1233:∈
1226:∫
995:χ
967:σ
940:χ
882:σ
862:χ
838:χ
794:‖
784:−
781:χ
778:▽
775:‖
755:χ
721:otherwise
702:∈
689:∀
653:▽
485:−
466:χ
375:−
366:−
354:χ
194:curvature
6360:Geometry
6317:SGP 2017
6234:(4): 1.
6219:(2013).
6133:(4): 1.
6118:(2011).
5902:Topology
5812:See also
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186:tangents
178:topology
174:geometry
110:SIGGRAPH
52:analysis
6274:Caltech
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190:normals
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