FLOATER, M. S. 1997. Parameterization and Smooth Approximation of Surface Triangulations. Computer Aided Geometric Design 14, 3, 231-- 250.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and is removed at the end of the remeshing process (this is possible because its connectivity matches the connectivity of the underlying regular level) Parameterization and layout relaxation. The parameterizations within patches are computed using Floater s parameterization scheme [10] and a biconjugate gradient solver [12] We relax the patch layout boundaries and corners similarly to the approach taken in [15] The same relaxation procedure ensures a smooth parametric transition from repatched regions to surrounding areas when the patch layout in the buffer zone is relaxed. ....

M. S. Floater. Parameterization and smooth approximation of surface triangulations. Computer Aided Geometric Design, 14:231--250, 1997.

....by this map of a straight line in the paxameter domain. Praun, Sweldens and SchrSder [14] use the same method to straighten the PL, but with a different parameterization. While Eck et al. use a paxameterization based on haxmonic maps, Praun et al. use the pa rameterization introduced by Floater [6]. By contrast our algorithm does not depend on any parameterization of the triangle mesh, and is much simpler to implement. The paper is organized as follows. In section 2, the notations for a PL on a 2D manifold triangulation are given. Section 3 outlines the algorithm. Section 4 is dedicated to ....

FLOATER, M. Parameterization and smooth approximation of surface triangulations. CAGD 16 (1997), 231-250.

....But these weights can be negative and there exist triangulations for which the harmonic parameterization is not valid. v w w v Figure 4: Notation for defining various weights. Figure 5: A point cloud with 1,042 samples and the connectivity graph for k = 16. The shape preserving weights [4] were the first known to result in parameterizations that meet both requirements, but also the mean value weights [6] #vw = tan(# v 2) tan(# w 2) w# do. In addition, they depend smoothly on the v i . The drawback of these linear methods is that they require at least some of the ....

M. S. Floater. Parameterization and smooth approximation of surface triangulations. Computer Aided Geometric Design, 14:231--250, 1997.

.... which minimizes some energy functional [18, 37, 62, 29, 27, 64, 25, 55, 39] The energy serves to encode measures such as low distortion [59, 60] conformality [33, 13, 47, 28] area preservation [13] or elastic energy [50] Others define the solution to satisfy barycentric coordinate conditions [19, 21] which take the original triangle shapes into account. A basic building block of all of these parameterization algorithms is the solution of sparse linear systems which are defined relative to the input mesh, i.e. they contain as many degrees of freedom (DOFs) as vertices. Such systems tend to ....

M. S. Floater, Parameterization and Smooth Approximation of Surface Triangulations, Computer Aided Geometric Design, 14 (1997), pp. 231--250.

....a planar triangulation that best matches the geometry of the 3D mesh, minimizing some measure of distortion, yet is still valid. In this context, valid means that the individual planar triangles are disjoint and do not overlap. Most of the recent works on the subject of parametrization (e.g. [6,9,21,26,30]) have focused on defining the distortion, and showing how to minimize it. The different parametrization methods published may be classified into two categories: Those that require that the boundary parameter values be pre defmed and form a convex shape (e.g. 9,26] and those who impose no ....

....of parametrization (e.g. 6,9,21,26,30] have focused on defining the distortion, and showing how to minimize it. The different parametrization methods published may be classified into two categories: Those that require that the boundary parameter values be pre defmed and form a convex shape (e.g. [9,26]) and those who impose no special shape on the boundary (e.g. 6,30] While parametrizing to the plane is the most natural way to perform texture mapping, this is less natural for other mesh processing operations which also require a parametrization. For applications such as morphing [1,17,27] ....

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M.S. Floater. Parameterization and smooth approximation of surface triangulations. Computer Aided Geometric Design, 14:231-250, 1997.

.... algorithms targeting surface texturing [3] 4] geometry approximation with semi regular approximations [5] 6] as well as general mesh parameterization techniques [7] 8] In this paper we introduce a modification to a wellknown shape preserving parameterization scheme of Michael Floater [9]. We work in a setting useful to traditional remeshing algorithms that split the original surface mesh into topologically simple patches, and map each patch onto a simple planar region. We therefore restrict our attention to the case of a single mesh region mapped onto a square. The goal of this ....

....local parameterization on a small neighborhood of the mesh, we shall reserve # = # , # ) for such parameterization. A typical local parameterization of the umbrella of faces adjacent to a given vertex can be obtained by flattening such a neighborhood via a polar map as described in [12] [9]. Parameterization bijectively maps a mesh region onto a planar region. In remeshing applications a regularly sampled mesh is the goal, and it is therefore typical that the boundary of a mesh patch is mapped onto the boundary of a simple plane region (e.g. a square) in a fixed way. The ....

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Floater M.S. "Parameterization and Smooth Approximation of Surface Triangulations." Computer Aided Geometric Design, vol. 14, 231--250, 1997

....we want the following quality criteria: 1. equal distribution of surface area amongst patches; 2. smooth patch boundaries; 3. fair patch boundaries; in particular they should not swirl. The first two criteria are easy to understand and intuitively clear and can be achieved through relaxation [3]. The third one is more difficult. In essence we want to avoid unnecessary winding or swirling of the curves. The swirling phenomenon leads to particularly nasty patches that cannot be fixed through relaxation. A simple example of the swirl operator, a bijective map from a mesh onto itself ....

....p f are already associated with the coarsest level vertices b f : # f = b f . Parameter values for the other vertices of are computed by solving a linear system, using conjugate gradients: # i = j #V(i) w ij # j . The weights w ij are computed using Floater s shape preserving scheme [3]. As a result w ij 0 and their sum over j equals 1 for each i, implying that the # i are convex combinations of the # j . Taken together with the fact that the boundary conditions # f = b f for the linear system are F vectors with one non zero component equal to 1, the components of each of ....

FLOATER, M. S. Parameterization and Smooth Approximation of Surface Triangulations. Computer Aided Geometric Design 14 (1997), 231--250.

....base points are then connected by geodesic curves. The collection of all geodesics between base points splits the input mesh into a set of triangular patches. For each patch, a parameterization over a unit triangle is computed. To minimize distortion, harmonic parameterizations are preferred [15]. One way to think of harmonic parameterizations is to consider the mesh as a massspring system: each edge of the original mesh is replaced by a spring with the remainder of the length proportional to the actual length of the edge. To find the parameter values for each vertex of the triangular ....

Floater, M. (1997) Parameterization and smooth approximation of surface triangulations, Comp. Aided Geom. Design, 14, 231-250

....for continuous morphing between two given compatible planar triangulations with a common convex boundary. The method is based on the convex representation of triangulations using barycentric coordinates, rst introduced by Tutte [15] for graph drawing purposes, and later generalized by Floater [16] for parameterization of 3D meshes. In a nutshell, this means that a planar triangulation on n points may be (non uniquely) represented by a sparse n#n stochastic matrix (i.e. a non negative matrix with unit row sums) where the ith row contains non zero entries only at columns associated with ....

Floater MS. Parameterization and smooth approximation of surface triangulation. Computer Aided Geometric Design 1997;14:231}50.

....are mapped into the interior of the face in the coarse mesh, corresponding to the patch they reside in. To find good initial locations for the interior vertices of the fine connectivity patch, we fix the remaining vertices of the fine connectivity and perform a few shape preserving iterations [4] based on the target edge lengths. The resulting locations are shown in Figure 7 (d) The accompanying video shows the convergence of the horse mesh using the hierarchical solver. We compared the hierarchical embedding times of various meshes with that of the standard solver. For small meshes ....

M. S. Floater. Parameterization and smooth approximation of surface triangulation. Computer Aided Geometric Design, 14:231--250, 1997.

....a general method for morphing between two given compatible triangulations with a common convex boundary. The method is based on the convex representation of triangulations using barycentric coordinates, first introduced by Tutte [20] for graph drawing purposes, and later generalized by Floater [7] for parameterization of 3D meshes. In a nutshell, this means that a legal planar triangulation on n points may be (nonuniquely) represented by a sparse n n stochastic matrix (i.e. a non negative matrix with unit row sums) where the i th row contains non zero entries only at columns associated ....

M. S. Floater. Parameterization and smooth approximation of surface triangulation. Computer Aided Geometric Design, 14:231--250, 1997.

....we want the following quality criteria: 1. equal distribution of surface area amongst patches; 2. smooth patch boundaries; 3. fair patch boundaries; in particular they should not swirl. The first two criteria are easy to understand and intuitively clear and can be achieved through relaxation [3]. The third one is more difficult. In essence we want to avoid unnecessary winding or swirling of the curves. The swirling phenomenon leads to particularly nasty patches that cannot be fixed through relaxation. A simple example of the swirl operator, a bijective map from a mesh onto itself ....

....p f are already associated with the coarsest level vertices b f : # f = b f . Parameter values for the other vertices of M are computed by solving a linear system, using conjugate gradients: # i = # j #V(i) w ij # j . The weights w ij are computed using Floater s shape preserving scheme [3]. As a result w ij # 0 and their sum over j equals 1 for each i, implying that the # i are convex combinations of the # j . Taken together with the fact that the boundary conditions # f = b f for the linear system are F vectors with one non zero component equal to 1, the components of each of ....

FLOATER, M. S. Parameterization and Smooth Approximation of Surface Triangulations. Computer Aided Geometric Design 14 (1997), 231--250.

....level. All the remaining fine vertices are mapped into the interior of the coarse faces, corresponding to the patches they reside in. To find good initial locations for the latter fine vertices, we fix the fine vertices on coarse vertices or edges and perform a few shape preserving iterations [5] based on the target edge lengths. The resulting locations are shown in Figure 7(d) We varied the patch size for building the hierarchy between # and ##. It turned out that the hierarchical solver converged faster with small patch size. A patch size of # outperformed the patch size of ## by ....

M. S. Floater. Parameterization and smooth approximation of surface triangulation. Computer Aided Geometric Design, 14:231--250, 1997.

....This is a basic building block of our algorithm. Several parameterization methods have been proposed and our method takes components from each of them: mesh simplification and polar maps from MAPS [18] patchwise relaxation from [9] and a specific smoothness functional similar to the one used in [10] and [20] The algorithm will use local parameterizations which need to be computed fast and robustly. Most of them are temporary and are quickly discarded unless they can be used as a starting guess for another parameterization. Consider a region of the mesh homeomorphic to a disc that we want ....

....we want to parameterize onto a convex planar region , i.e. find a bijective map . The map is fixed by a boundary condition and minimizes a certain energy functional. Several functionals can be used leading to, e.g. conformal or harmonic mappings. We take an approach based on the work of Floater [10]. In short, the function needs to satisfy the following equation in the interior: p i ) # i i (p ) 1) where (i) is the 1 ring neighborhood of the vertex i and the weights i come from the shape preserving parameterization scheme [10] The main advantage of the Floater weights is that they ....

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FLOATER, M. S. Parameterization and Smooth Approximation of Surface Triangulations. Computer Aided Geometric Design 14 (1997), 231--250.

....regions is a fundamental part of many remeshing, texturing, and other mesh processing algorithms. The Figure 7 shows the parameterized mesh region of the David s ear. The texture coordinates are assigned with the (u, v) coordinates computed with the shape preserving parameterization of Floater [14]. The original unfiltered region of this mesh contained twelve tunnels and could not be properly parameterized onto the unit square. Our algorithm removes all of these tunnels in fifteen seconds, and produces a mesh that is homeomorphic to a square, allowing it to be parameterized. Additionally, ....

Michael S. Floater. Parameterization and smooth approximation of surface triangulations. Computer Aided Geometric Design, 14:231--250, 1997.

....This is a basic building block of our algorithm. Several parameterization methods have been proposed and our method takes components from each of them: mesh simplification and polar maps from MAPS [18] patchwise relaxation from [9] and a specific smoothness functional similar to the one used in [10] and [20] The algorithm will use local parameterizations which need to be computed fast and robustly. Most of them are temporary and are quickly discarded unless they can be used as a starting guess for another parameterization. Consider a region R of the mesh homeomorphic to a disc that we ....

....onto a convex planar region B, i.e. find a bijective map u: R#B. The map u is fixed by a boundary condition #R##B and minimizes a certain energy functional. Several functionals can be used leading to, e.g. conformal or harmonic mappings. We take an approach based on the work of Floater [10]. In short, the function u needs to satisfy the following equation in the interior: u(p i ) # k#V(i) # ik u(pk ) 1) where V(i) is the 1 ring neighborhood of the vertex i and the weights # ik come from the shape preserving parameterization scheme [10] The main advantage of the Floater ....

[Article contains additional citation context not shown here]

FLOATER, M. S. Parameterization and Smooth Approximation of Surface Triangulations. Computer Aided Geometric Design 14 (1997), 231--250.

....also related to the work of Dale and his colleagues (Dale et al. 1999; 1993; Fischl et al. 1999) This work, in turn, draws on ideas from the seminal work of Schwartz (1990) and his colleagues. There are a number of very closely related developments in the graphics literature (Eck et al. 1995; Floater, 1997; Levy Mallet, 1998) as well as the mathematical literature on surface properties and topological graphs referenced above. In this section we mainly discuss the general principles and the work directly pertaining to cortical flattening. Drury et al. 1996) describe two related algorithms for ....

.... node at a position equal to the average position of its neighbors (the sample nodes connected by an edge) When the neighbors form a convex shape, placing the node at the average position of the neighbors will not introduce any edge intersections and the mesh will continue to be a planar graph (Floater, 1997; Levy Mallet, 1998; Tutte, 1960) If the node s neighbors are a concave shape, this is not guaranteed. But, in our experience over hundreds of unfolds, the mesh generation algorithm combined with the initial placement procedure introduces no edge intersections among the edges of interior ....

Floater, M. S. (1997). Parameterization and smooth approximation of surface triangulations. Computer Aided Geometric Design, 14(3), 231-250.

....by researchers in several different areas. These include classical subdivision [22] which we generalize to the irregular setting with the help of mesh simplification [13] and careful attention to the role of smooth parameterizations. Parameterizations were examined in the context of remeshing [19, 8, 9], texture mapping (e.g. 20] and variational modeling [16, 28, 21] One area which employs these elements is hierarchical editing for semi regular [29] and irregular meshes [18] Signal processing as an approach to surface fairing in the irregular setting was first considered by Taubin [26, ....

....is important to treat all three coordinates x, y,andz as dependent variables with independent parameters u and v,giv ing us three functional settings. The independent parameters are typically unknown and must be estimated. Algorithms to estimate global smooth parameterizations are described in [19, 8, 9, 20]. We require only local parameterizations which are consistent over the support of a small filter stencil. Triangulations To talk about local neighborhoods of vertices within the mesh it is convenient to describe the topological and geometric aspects of a mesh separately. We use notation inspired ....

FLOATER, M. S. Parameterization and Smooth Approximation of Surface Triangulations. Computer Aided Geometric Design 14 (1997), 231--250.

....for any triangle # i (# 1 v 1 # 2 v 2 # 3 v 3 ) # 1 # i (v 1 ) # 2 # i (v 2 ) # 3 # i (v 3 ) 2) for any real numbers # 1 , # 2 , # 3 0 which sum to one. Thus # i will be completely determined by the parameter points # i (v) for vertices v i . We will use the method of [6]. First of all, due to construction, corresponding to the vertices v j , v k , v # of T i , there must be three corresponding corner vertices x j , x k , x # in the boundary of i . We therefore set # i (x r ) v r , for r # j, k, # . Next we determine # i at the remaining boundary ....

....the interior vertices of i by solving a sparse linear system of equations. For each interior vertex v of i , we demand that # i (v) w#Nv # vw # i (w) where the weights # vw are strictly positive and sum to one. Here N v denotes the set of neighbouring vertices of v. It was shown in [6] that this linear system always has a unique solution. We take the weights # vw to be the shape preserving weights of [6] which were shown to have a reproduction property and tend to minimize distortion. Since the equations force each point # i (v) to be a convex combination of its neighbouring ....

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Floater, M. S., Parameterization and smooth approximation of surface triangulations, Comput. Aided Geom. Design 14 (1997), 231--250.

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FLOATER, M. S. 1997. Parameterization and Smooth Approximation of Surface Triangulations. Computer Aided Geometric Design 14, 3, 231-- 250.

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Floater, M.: Parameterization and smooth approximation of surface triangulations. Computer Aided Geometric Design 14--3 (1997) 231--250.

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M. S. Floater. Parameterization and smooth approximation of surface triangulations. Computer Aided Geometric Design, 14(3):231--250, 1997.

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M. Floater. Parameterization and smooth approximation of surface triangulations. Computer Aided Geometric Design, 14:231--250, 1997.

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M. Floater. Parameterization and smooth approximation of surface triangulations. Computer Aided Geometric Design, 14(3):321--250, 1997.

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FLOATER M.: Parameterization and smooth approximation of surface triangulation. Computer Aided Geometric Design 14 (1997), 231--250.

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Michael S. Floater. Parameterization and smooth approximation of surface triangulations. Comput. Aided Geom. Design, 14(4):231-250, 1997.

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Michael S. Floater. Parameterization and smooth approximation of surface triangulations. Computer Aided Geometric Design, 14(4):231--250, 1997. 5

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FLOATER, M. S. Parameterization and smooth approximation of surface triangulations. Computer Aided Geometric Design 14 (1997), 231--250.

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M. S. Floater. Parameterization and smooth approximation of surface triangulation. Computer Aided Geometric Design, 14:231--250, 1997.

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Michael S. Floater. Parameterization and smooth approximation of surface triangulations. Computer Aided Geometric Design, 14:231--250, 1997.

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M. Floater. Parameterization and smooth approximation of surface triangulations. Computer Aided Geometric Design, 14(3):321--250, 1997.

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Floater, M. S.: Parameterization and smooth approximation of surface triangulation. Computer Aided Geometric Design 14, 231--250 (1997).

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M. Floater. Parameterization and smooth approximation of surface triangulations. Computer Aided Geometric Design, 14(3):231--250, Apr. 1997. 3

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Floater M.S., Parameterization and smooth approximation of surface triangulations, Computer Aided Geometric Design 14:231--250, 1997. 231--250

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M. S. Floater. Parameterization and smooth approximation of surface triangulations. Computer Aided Geometric Design, 14:231--250, 1997.

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M. S. Floater. Parameterization and smooth approximation of surface triangulations. Computer Aided Geometric Design, 14:231--250, 1997.

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