In electrical engineering and computer science, Lloyd's algorithm, also known as Voronoi iteration or relaxation, is an algorithm named after Stuart P. Lloyd for finding evenly spaced sets of points in subsets of Euclidean spaces and partitions of these subsets into well-shaped and uniformly sized convex cells. Like the closely related k-means clustering algorithm, it repeatedly finds the centroid of each set in the partition and then re-partitions the input according to which of these centroids is closest. In this setting, the mean operation is an integral over a region of space, and the nearest centroid operation results in Voronoi diagrams. Although the algorithm may be applied most directly to the Euclidean plane, similar algorithms may also be applied to higher-dimensional spaces or to spaces with other non-Euclidean metrics. Lloyd's algorithm can be used to construct close approximations to centroidal Voronoi tessellations of the input, which can be used for quantization, dithering, and stippling. Other applications of Lloyd's algorithm include smoothing of triangle meshes in the finite element method.
History The algorithm was first proposed by Stuart P. Lloyd of Bell Labs in 1957 as a technique for pulse-code modulation. Lloyd's work became widely circulated but remained unpublished until 1982. A similar algorithm was developed independently by Joel Max and published in 1960, which is why the algorithm is sometimes referred as the Lloyd-Max algorithm.
Algorithm description Lloyd's algorithm starts by an initial placement of some number k of point sites in the input domain. In mesh-smoothing applications, these would be the vertices of the mesh to be smoothed; in other applications they may be placed at random or by intersecting a uniform triangular mesh of the appropriate size with the input domain. It then repeatedly executes the following relaxation step:
The Voronoi diagram of the k sites is computed. Each cell of the Voronoi diagram is integrated, and the centroid is computed. Each site is then moved to the centroid of its Voronoi cell.
Integration and centroid computation Because Voronoi diagram construction algorithms can be highly non-trivial, especially for inputs of dimension higher than two, the steps of calculating this diagram and finding the exact centroids of its cells may be replaced by an approximation.
Approximation A common simplification is to employ a suitable discretization of space like a fine pixel-grid, e.g. the texture buffer in graphics hardware. Cells are materialized as pixels, labeled with their corresponding site-ID. A cell's new center is approximated by averaging the positions of all pixels assigned with the same label. Alternatively, Monte Carlo methods may be used, in which random sample points are generated according to some fixed underlying probability distribution, assigned to the closest site, and averaged to approximate the centroid for each site.
Exact computation Although embedding in other spaces is also possible, this elaboration assumes Euclidean space using the L2 norm and discusses the two most relevant scenarios, which are two, and respectively three dimensions. Since a Voronoi cell is of convex shape and always encloses its site, there exist trivial decompositions into easy integratable simplices:
In two dimensions, the edges of the polygonal cell are connected with its site, creating an umbrella-shaped set of triangles. In three dimensions, the cell is enclosed by several planar polygons which have to be triangulated first: Compute a center for the polygon face, e.g. the average of all its vertices. Connecting the vertices of a polygon face with its center gives a planar umbrella-shaped triangulation. Trivially, a set of tetrahedra is obtained by connecting triangles of the cell's hull with the cell's site. Integration of a cell and computation of its centroid (center of mass) is now given as a weighted combination of its simplices' centroids (in the following called c i {\textstyle \mathbf {c} _{i}} ).
Two dimensions: For a triangle the centroid can be easily computed, e.g. using cartesian coordinates. Weighting computes as simplex-to-cell area ratios. Three dimensions: The centroid of a tetrahedron is found as the intersection of three bisector planes and can be expressed as a matrix-vector product. Weighting computes as simplex-to-cell volume ratios. For a 2D cell with n triangular simplices and an accumulated area A C = ∑ i = 0 n a i {\textstyle A_{C}=\sum _{i=0}^{n}a_{i}} (where a i {\textstyle a_{i}} is the area of a triangle simplex), the new cell centroid computes as:
C = 1 A C ∑ i = 0 n c i a i {\displaystyle C={\frac {1}{A_{C}}}\sum _{i=0}^{n}\mathbf {c} _{i}a_{i}}
Analogously, for a 3D cell with a volume of V C = ∑ i = 0 n v i {\textstyle V_{C}=\sum _{i=0}^{n}v_{i}} (where v i {\textstyle v_{i}} is the volume of a tetrahedron simplex), the centroid computes as:
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