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Laplacian Smoothing
Laplacian smoothing is an algorithm to smooth a polygonal mesh In 3D computer graphics and solid modeling, a polygon mesh is a collection of , s and s that defines the shape of a polyhedral object. The faces usually consist of triangles (triangle mesh), quadrilaterals (quads), or other simple convex polyg .... For each vertex in a mesh, a new position is chosen based on local information (such as the position of neighbours) and the vertex is moved there. In the case that a mesh is topologically a rectangular grid (that is, each internal vertex is connected to four neighbours) then this operation produces the Laplacian of the mesh. More formally, the smoothing operation may be described per-vertex as: :\bar_= \frac \sum_^\bar_j Where N is the number of adjacent vertices to node i, \bar_ is the position of the j-th adjacent vertex and \bar_ is the new position for node i. See also * Tutte embedding, an embedding of a planar mesh in which each vertex is already at the aver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smoothing
In statistics and image processing, to smooth a data set is to create an approximating function (mathematics), function that attempts to capture important patterns in the data, while leaving out noise or other fine-scale structures/rapid phenomena. In smoothing, the data points of a signal are modified so individual points higher than the adjacent points (presumably because of noise) are reduced, and points that are lower than the adjacent points are increased leading to a smoother signal. Smoothing may be used in two important ways that can aid in data analysis (1) by being able to extract more information from the data as long as the assumption of smoothing is reasonable and (2) by being able to provide analyses that are both flexible and robust. Many different algorithms are used in smoothing. Smoothing may be distinguished from the related and partially overlapping concept of curve fitting in the following ways: * curve fitting often involves the use of an explicit function for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygon Mesh
In 3D computer graphics and solid modeling, a polygon mesh is a collection of , s and s that defines the shape of a polyhedral object. The faces usually consist of triangles (triangle mesh), quadrilaterals (quads), or other simple convex polygons ( n-gons), since this simplifies rendering, but may also be more generally composed of concave polygons, or even polygons with holes. The study of polygon meshes is a large sub-field of computer graphics (specifically 3D computer graphics) and geometric modeling. Different representations of polygon meshes are used for different applications and goals. The variety of operations performed on meshes may include: Boolean logic ( Constructive solid geometry), smoothing, simplification, and many others. Algorithms also exist for ray tracing, collision detection, and rigid-body dynamics with polygon meshes. If the mesh's edges are rendered instead of the faces, then the model becomes a wireframe model. Volumetric meshes are distinct f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marc Alexa
Marc Alexa is a professor of computer science at TU Berlin working in the fields of computer graphics, geometric modeling and geometry processing. Life Alexa studied computer science at TU Darmstadt, receiving a Diplom in 1997 and a PhD in 2002. After his graduation, he spent time as a postdoctoral researcher with Greg Turk at Georgia Tech, returning the same year to become assistant professor at TU Darmstadt. In 2005, he became an associate professor for computer graphics at TU Berlin, transitioning to the full professorship in 2010. He conducted research at Caltech, Carnegie Mellon University, Disney Research, ETH Zurich, and the University of Toronto. From 2018 to 2021, he was the editor-in-chief of ACM Transactions on Graphics. Awards Alexa received numerous best paper awards at conferences, in particular the Symposium on Geometry Processing. Other noteworthy distinctions: *2022: ERC Advanced Grant *2018: Fellow of the Eurographics Association *2014: Eurographics Outs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the nabla operator), or \Delta. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian of a function at a point measures by how much the average value of over small spheres or balls centered at deviates from . The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that densit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tutte Embedding
In graph drawing and geometric graph theory, a Tutte embedding or barycentric embedding of a simple, 3-vertex-connected, planar graph is a crossing-free straight-line embedding with the properties that the outer face is a convex polygon and that each interior vertex is at the average (or barycenter) of its neighbors' positions. If the outer polygon is fixed, this condition on the interior vertices determines their position uniquely as the solution to a system of linear equations. Solving the equations geometrically produces a planar embedding. Tutte's spring theorem, proven by , states that this unique solution is always crossing-free, and more strongly that every face of the resulting planar embedding is convex. It is called the spring theorem because such an embedding can be found as the equilibrium position for a system of springs representing the edges of the graph. Example Let ''G'' be the graph of a cube, and (selecting one of its quadrilateral faces as the outer face) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mesh Generation
Mesh generation is the practice of creating a mesh, a subdivision of a continuous geometric space into discrete geometric and topological cells. Often these cells form a simplicial complex. Usually the cells partition the geometric input domain. Mesh cells are used as discrete local approximations of the larger domain. Meshes are created by computer algorithms, often with human guidance through a GUI , depending on the complexity of the domain and the type of mesh desired. A typical goal is to create a mesh that accurately captures the input domain geometry, with high-quality (well-shaped) cells, and without so many cells as to make subsequent calculations intractable. The mesh should also be fine (have small elements) in areas that are important for the subsequent calculations. Meshes are used for rendering to a computer screen and for physical simulation such as finite element analysis or computational fluid dynamics. Meshes are composed of simple cells like triangles because ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
