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## Centroidal Voronoi tessellations: Applications and algorithms (1999)

Venue: | SIAM Review |

Citations: | 379 - 39 self |

### Citations

13852 |
Computers and Intractability; A Guide to the Theory of NP‐Completeness
- Garey, Johnson
- 1990
(Show Context)
Citation Context ...imple arithmetic mean by the more general notion of the mass centroid given in section 1.1. For discussions of complexity issues involved in k-clustering as well as related clustering algorithms, see =-=[12, 30, 32, 31, 48]-=- and the references cited therein. Additional references on clustering analysis are also provided in later sections. 2.4. Finite Difference Schemes Having Optimal Truncation Errors. For the sake of ea... |

2968 | Some methods for classification and analysis of multivariate observations
- MacQueen
(Show Context)
Citation Context ...L VORONOI TESSELLATIONS 653 Many results involve properties of centroidal Voronoi tessellations as the number of sample points becomes large, i.e., as m →∞. The convergence of the energy was shown in =-=[40]-=-. The convergence of the centroids, i.e., the cluster centers, was proved in [52] for the Euclidean metric under certain uniqueness assumptions. The asymptotic distribution of cluster centers was cons... |

2752 |
Algorithms for Clustering Data
- Jain, Dubes
- 1981
(Show Context)
Citation Context ...y∈V z∈V ∗ y∈V where the sums extend over the points belonging to V, and V ∗ can be taken to be V or it can be a larger set like R N . In the statistical and vector quantization literature (see, e.g., =-=[15, 19, 34]-=-) discrete centroidal Voronoi tessellations are often related to optimal k-means clusters and Voronoi regions and centroids are referred to as clusters and cluster centers, respectively. More discussi... |

2207 |
Clustering algorithms
- Hartigan
- 1995
(Show Context)
Citation Context ...y∈V z∈V ∗ y∈V where the sums extend over the points belonging to V, and V ∗ can be taken to be V or it can be a larger set like R N . In the statistical and vector quantization literature (see, e.g., =-=[15, 19, 34]-=-) discrete centroidal Voronoi tessellations are often related to optimal k-means clusters and Voronoi regions and centroids are referred to as clusters and cluster centers, respectively. More discussi... |

2101 |
Vector Quantization and Signal Compression
- Gersho, Gray
- 1991
(Show Context)
Citation Context ...y∈V z∈V ∗ y∈V where the sums extend over the points belonging to V, and V ∗ can be taken to be V or it can be a larger set like R N . In the statistical and vector quantization literature (see, e.g., =-=[15, 19, 34]-=-) discrete centroidal Voronoi tessellations are often related to optimal k-means clusters and Voronoi regions and centroids are referred to as clusters and cluster centers, respectively. More discussi... |

1663 |
Combinatorial Optimization: Algorithms and Complexity
- Papadimitriou, Steiglitz
- 1982
(Show Context)
Citation Context ... clusters can be solved in O(mkN+1 ) steps. This bound is very pessimistic for large k. However, it is also known that, in general, the problem of finding the optimal clustering is an NP-hard problem =-=[12, 13, 48]-=-. Other results on the k-means method and its relation to the optimality conditions of a related mathematical programming problem can be found in [55]. 5.2. Lloyd’s Method. Next, we discuss a method t... |

1621 |
An Algorithm for Vector Quantization Design
- Linde, Buzo, et al.
- 1980
(Show Context)
Citation Context ...efined by yρ(y) dy ρ(y) dy , Vi(Z) = Voronoi region for zi, i=1,...,k. T =(T1, T2,...,Tk) T . Clearly, centroidal Voronoi tessellations are fixed points of T(Z). 5.3. Variations on Lloyd’s Method. In =-=[38]-=- (see also [15]), a probabilistic variation of Lloyd’s method is proposed. The method is essentially a continuation method for the global optimization problem in the spirit of simulated annealing. The... |

1341 | Least Square Quantization in PCM
- Lloyd
- 1982
(Show Context)
Citation Context ... we will not discuss termination procedures since they are very much dependent on the specific application.) This method, at least in the electrical engineering literature, is known as Lloyd’s method =-=[39]-=-. (A second method was also proposed by Lloyd for one-dimensional problems; it is similar to a shooting approach, i.e., one guesses the leftmost mass center, then determines the end-point of the leftm... |

1076 |
Random Number Generation and Quasi-Monte Carlo Methods
- Niederreiter
- 1992
(Show Context)
Citation Context ...in the literature. For instance, the classical Gaussian integration rules [7] and the quasi–Monte Carlo rules based on number-theoretic methods all share optimal properties in some sense. We refer to =-=[46]-=- for a comprehensive treatment of the latter subject; see also [26, 49, 57] for some recent developments. In general, one can consider optimal quadrature rules over a given function space and for spec... |

315 | Exploratory projection pursuit
- Friedman
- 1987
(Show Context)
Citation Context ...n. For many practical problems, it is desirable to visualize the resulting centroidal Voronoi diagrams or the optimal clusters by some means. Various techniques, such as the projection pursuit method =-=[11, 18, 29, 58]-=-, have been studied in the statistics literature. They can be used to characterize the centroidal Voronoi diagrams via lower dimensional approximations. 3.4. Connections between the Discrete and Conti... |

251 |
Asymptotically optimal block quantization
- Gersho
- 1979
(Show Context)
Citation Context ...s equally distributed in the Voronoi intervals and the sizes of the Voronoi intervals are inversely proportional to the one-third power of the underlying density at the midpoints of the intervals. In =-=[14]-=-, an important conjecture is made which states that asymptotically, for the optimal centroidal Voronoi tessellation, all Voronoi regions are approximately congruent to the same basic cell that depends... |

195 |
classification EM algorithm for clustering and two stochastic versions
- Celeux, Govaert, et al.
- 1992
(Show Context)
Citation Context ...to the finite-dimensional656 QIANG DU, VANCE FABER, AND MAX GUNZBURGER version of the algorithm. Some deterministic and stochastic variants of the k-means algorithm in the discrete case are given in =-=[4]-=-. 4.2. A Random 2-Clustering Algorithm for the Discrete Case. For variancebased k-clustering analysis of a finite-dimensional set of points W = {yℓ} M ℓ=1 belonging to R N , another type of probabilis... |

183 | When are quasi-Monte Carlo algorithms efficient for high dimensional integrals
- Sloan, Wo´zniakowski
- 1998
(Show Context)
Citation Context ... rules [7] and the quasi–Monte Carlo rules based on number-theoretic methods all share optimal properties in some sense. We refer to [46] for a comprehensive treatment of the latter subject; see also =-=[26, 49, 57]-=- for some recent developments. In general, one can consider optimal quadrature rules over a given function space and for specific types of function evaluations. The appearance of the centroidal Vorono... |

164 |
K-means-type algorithms: A generalized convergence theorem and characterization of local optimality
- Selim, Ismail
- 1984
(Show Context)
Citation Context ...uster center zi∗ to be the mean of the points belonging to the corresponding cluster. An efficient implementation of this algorithm for large data sets is given in [21]; see also [10]. It is shown in =-=[55]-=- that, for the quadratic metric, the energy converges to a local minimum value. Other deterministic algorithms for determining the cluster centers are given in [25, 62, 64]. These involve recursive cl... |

103 |
Stong consistency of k-means clustering
- Pollard
- 1981
(Show Context)
Citation Context ... tessellations as the number of sample points becomes large, i.e., as m →∞. The convergence of the energy was shown in [40]. The convergence of the centroids, i.e., the cluster centers, was proved in =-=[52]-=- for the Euclidean metric under certain uniqueness assumptions. The asymptotic distribution of cluster centers was considered in [53], where it was shown that the cluster centers, suitably normalized,... |

100 | Applications of weighted voronoi diagrams and randomization to variance-based k-clustering
- Inaba, Katoh, et al.
- 1994
(Show Context)
Citation Context ...k-clustering analysis of a finite-dimensional set of points W = {yℓ} M ℓ=1 belonging to R N , another type of probabilistic algorithm, which we refer to as the m-sample algorithm, has been studied in =-=[24, 30, 31, 32]-=-. The 2-clustering version is given as follows: 0. Sample an initial subset T of m points from W , e.g., by using a Monte Carlo method. 1. For every linearly separable 2-clustering (T1,T2) ofT , • com... |

75 |
Concrete and Abstract Voronoi Diagrams
- Klein
- 1989
(Show Context)
Citation Context ...ons of boundaries with nonempty interiors get assigned to specific Voronoi regions. Such a partition, in general, only guarantees that the Voronoi regions are star shaped (with respect to the metric) =-=[37]-=-. Note that if one instead considers (1.8) Vi = { x ∈ Ω | d(x, zi) ≤ d(x, zj), j =1,...,k, j = i } , then the regions may overlap but they remain convex. A generalized definition of the centroid z∗ ... |

61 |
A central limit theorem for k-means clustering
- Pollard
- 1982
(Show Context)
Citation Context ...ergence of the centroids, i.e., the cluster centers, was proved in [52] for the Euclidean metric under certain uniqueness assumptions. The asymptotic distribution of cluster centers was considered in =-=[53]-=-, where it was shown that the cluster centers, suitably normalized, have an asymptotic normal distribution. These results generalize those of [20]. Generalization to separable metric spaces was given ... |

59 |
Direct discretization of planar div-curl problems
- Nicolaides
- 1992
(Show Context)
Citation Context ...f the dual Voronoi– Delaunay tessellations. Unlike the finite difference scheme discussed above, unknowns can be associated with edges of either or both the Voronoi polygons or the Delaunay triangles =-=[45]-=-. Here, it is possible to use centroidal Voronoi grids to obtain schemes for which the L2-error in the approximate solution is of order h2 and for which the error is only of order h for general grids.... |

55 |
Computational Geometry--An Introduction
- Preparata, Shamos
- 1985
(Show Context)
Citation Context ...veys on the subject and comprehensive lists of references to the literature; see also [16]. In the same spirit, centroidal Voronoi diagrams also play a central role in clustering analysis; see, e.g., =-=[19, 34, 54]-=-. Clustering, as a tool to analyze similarities and dissimilarities between different objects, is fundamental and is used in various fields of statistical analysis, pattern recognition, learning theor... |

48 |
The complexity of the generalized Lloyd-Max problem
- Garey, Johnson, et al.
- 1982
(Show Context)
Citation Context ... clusters can be solved in O(mkN+1 ) steps. This bound is very pessimistic for large k. However, it is also known that, in general, the problem of finding the optimal clustering is an NP-hard problem =-=[12, 13, 48]-=-. Other results on the k-means method and its relation to the optimality conditions of a related mathematical programming problem can be found in [55]. 5.2. Lloyd’s Method. Next, we discuss a method t... |

42 |
Clustering Algorithms based on Minimum and Maximum Spanning Trees
- Asano, Bhattacharya, et al.
- 1988
(Show Context)
Citation Context ...ering is a centroidal Voronoi diagram if we use the above variance-based criteria. In computational geometry, criteria such as the diameter or radius of the subsets have been well studied; see, e.g., =-=[2]-=-. Other criteria have also been proposed, e.g., L1 based [25] and variance based [62]. In some applications of statistical analysis, only the clusters are of interest; the cluster centers are not impo... |

42 |
On polynomial-based projection indices for exploratory projection pursuit
- Hall
- 1989
(Show Context)
Citation Context ...n. For many practical problems, it is desirable to visualize the resulting centroidal Voronoi diagrams or the optimal clusters by some means. Various techniques, such as the projection pursuit method =-=[11, 18, 29, 58]-=-, have been studied in the statistics literature. They can be used to characterize the centroidal Voronoi diagrams via lower dimensional approximations. 3.4. Connections between the Discrete and Conti... |

40 |
Description of cellular patterns by Dirichlet domains: The two-dimensional case
- Honda
- 1978
(Show Context)
Citation Context ...ally monolayered or columnar, whose geometrical shapes are polygonal. In many cases, they can be identified with a Voronoi tessellation and, indeed, with a centroidal Voronoi tessellation; see, e.g., =-=[27, 28]-=- for examples. Likewise, centroidal Voronoi tessellations can be used to model how cells are reshaped when they divide or are removed from a tissue. Here we consider one example provided in [27, 28], ... |

39 | Optimal adaptive k-means algorithm with dynamic adjustment of learning rate - Chinrungrueng, Sequin - 1995 |

38 |
The Hexagon Theorem
- Newman
- 1982
(Show Context)
Citation Context ...al Voronoi tessellation, all Voronoi regions are approximately congruent to the same basic cell that depends only on the dimension. The basic cell is shown to be the regular hexagon in two dimensions =-=[44]-=-, but the conjecture remains open for three and higher dimensions. The equidistribution of energy principle, however, can be established based on Gersho’s conjecture [14, 17]. 6.4.2. Linear Convergenc... |

26 |
A fast Voronoi diagram algorithm with applications to geographical optimization problems
- Iri, Murota, et al.
- 1984
(Show Context)
Citation Context ...nts are fixed points of the Lloyd map T(Z). See section 6.2 for a further discussion. For the discrete case, gradient methods, including steepest descent methods with line searches, are considered in =-=[33]-=-. The relation between centroidal Voronoi tessellations, e.g., optimal clustering, and stationary points of associated functions is discussed in [55]. 5.5. Newton-Like Methods. One may naturally compu... |

25 |
Convergence of vector quantizers with applications to optimal quantization
- Abaya, Wise
- 1984
(Show Context)
Citation Context ..., D(Q, F ) = Ed(X, Q(X)). The mathematical formulation of the average distortion is similar to that of the energy functional given in (3.1) for the continuous case and (3.5) for the discrete case. In =-=[1]-=-, several convergence and continuity properties for sequences of vector quantizers or block source codes with a fidelity criterion are developed. Conditions under which convergence of a sequence of qu... |

25 | Faster evaluation of multidimensional integrals
- Papageorgiou, Traub
- 1997
(Show Context)
Citation Context ... rules [7] and the quasi–Monte Carlo rules based on number-theoretic methods all share optimal properties in some sense. We refer to [46] for a comprehensive treatment of the latter subject; see also =-=[26, 49, 57]-=- for some recent developments. In general, one can consider optimal quadrature rules over a given function space and for specific types of function evaluations. The appearance of the centroidal Vorono... |

25 |
An algorithm for multidimensional data clustering
- Wan, Wong, et al.
- 1988
(Show Context)
Citation Context ...n computational geometry, criteria such as the diameter or radius of the subsets have been well studied; see, e.g., [2]. Other criteria have also been proposed, e.g., L1 based [25] and variance based =-=[62]-=-. In some applications of statistical analysis, only the clusters are of interest; the cluster centers are not important themselves. In other applications, e.g., the color image compression problem, t... |

20 |
Geometrical models for cells in tissues
- Honda
- 1983
(Show Context)
Citation Context ...ally monolayered or columnar, whose geometrical shapes are polygonal. In many cases, they can be identified with a Voronoi tessellation and, indeed, with a centroidal Voronoi tessellation; see, e.g., =-=[27, 28]-=- for examples. Likewise, centroidal Voronoi tessellations can be used to model how cells are reshaped when they divide or are removed from a tissue. Here we consider one example provided in [27, 28], ... |

17 |
Hexagonal territories
- Barlow
- 1974
(Show Context)
Citation Context ...oms by spitting sand away from the pit centers toward their neighbors. For a high enough density of fish, this reciprocal spitting results in sand parapets that are visible territorial boundaries. In =-=[3]-=-, the results of a controlled experiment were given. Fish were introduced into a large outdoor pool with a uniform sandy bottom. After the fish had established their territories, i.e., after the final... |

17 |
OptiSim: an extended dissimilarity selection method for finding diverse representative subsets
- Clark
- 1997
(Show Context)
Citation Context ...g to some criteria that distinguishes between them. This is commonly referred to as k-clustering analysis. For example, in combinatorial chemistry, k-clustering analysis is used in compound selection =-=[6]-=-, where the similarity criteria may be related to the compound components as well as their structure. Clustering analysis provides a selection of a finite collection of templates that well represent, ... |

17 |
Locally optimal block quantizer design
- Gray, Kieffer, et al.
- 1980
(Show Context)
Citation Context .... The image compression example we discussed earlier belongs to this subject as well. We refer to [15, 17] for surveys on the subject and comprehensive lists of references to the literature; see also =-=[16]-=-. In the same spirit, centroidal Voronoi diagrams also play a central role in clustering analysis; see, e.g., [19, 34, 54]. Clustering, as a tool to analyze similarities and dissimilarities between di... |

14 |
Uniqueness of locally optimal quantizer for log-concave density and convex error function
- Kieffer
- 1983
(Show Context)
Citation Context ...M(G(Z (n) )) −1 . 6.3. Convergence of Lloyd’s Method for a Class of One-Dimensional Problems. Lloyd’s method has been shown to be convergent locally for a class of onedimensional problems; see, e.g., =-=[36]-=-. Here, we present a simple proof for a result of this type. Without loss of generality, we assume that Ω = [0, 1]. Let the density function ρ be smooth and strictly positive. In addition, ρ is assume... |

13 | Efficient algorithms for variance-based kclustering
- Hasegawa, Imai, et al.
- 1993
(Show Context)
Citation Context ... different color. Our task is to approximate the picture by replacing the color at each pixel by one from a smaller set of colors {zi} k i=1 , where each zi ∈ W . For related discussions, we refer to =-=[24, 25, 32]-=-. In particular, a discussion of different clustering algorithms in image compression applications can be found in [24]. We give a concrete example. Suppose the picture is composed of 106 pixels and t... |

13 |
On the calculation of the most probable values of frequency constants, for data arranged according to equidistant divisions of a scale
- Sheppard
(Show Context)
Citation Context ...m of finding the centroidal Voronoi diagram. A similar and even earlier example is the question of round-off, i.e., representing real numbers by the closest integers. This problem has been studied in =-=[56]-=- and the answer is obviously a simple example of one-dimensional centroidal Voronoi diagrams. The representation of a given quantity with less information is often referred to as quantization and it i... |

12 |
Geometrical models of territory. I. Models for synchronous and asynchronous settlement of territories
- Tanemura, Hasegawa
- 1980
(Show Context)
Citation Context ...are seen to be polygonal and, in [27, 59], it was shown that they are very closely approximated by a Voronoi tessellation. A behavioral model for how the fish establish their territories was given in =-=[22, 23, 60]-=-. When the fish enter a region, they first randomly select the centers of their breeding pits, i.e., the locations at which they will spit sand. Their desire to place the pit centers as far away as po... |

9 | The asymptotic efficiency of randomized nets for quadrature
- Hong
- 1999
(Show Context)
Citation Context ... rules [7] and the quasi–Monte Carlo rules based on number-theoretic methods all share optimal properties in some sense. We refer to [46] for a comprehensive treatment of the latter subject; see also =-=[26, 49, 57]-=- for some recent developments. In general, one can consider optimal quadrature rules over a given function space and for specific types of function evaluations. The appearance of the centroidal Vorono... |

8 |
On the design of an optimal quantizer
- Trushkin
- 1993
(Show Context)
Citation Context ...e” distributions yield “close” optimal quantizers. These properties, however, have not been shown to be valid in all cases. Another generalization, in the form of a two-stage algorithm, is studied in =-=[61]-=-. For that algorithm, the first stage of Lloyd’s iteration is applied so as to reduce the value of error functionals such as k∑ ∫ ρ(y)f(d(y, zi)) dy i=1 Vi and its discrete analogs. At the second stag... |

7 |
An efficient Vector Quantizer Providing Globally Optimal Solutions
- Moller, Galicki, et al.
- 1998
(Show Context)
Citation Context ...quency regions or in regions with uncorrelated data; polynomial approximations are used in smooth regions or in regions with highly correlated data. A number of other generalizations are discussed in =-=[15, 43]-=-. 5.4. Descent or Gradient Methods. One can construct general iterative procedures for determining the centroidal Voronoi tessellations by following a descent search algorithm of the type Z (n+1) = Z ... |

7 |
Approximation of a tessellation of the plane by a Voronoi diagram
- Suzuki, Iri
- 1986
(Show Context)
Citation Context ...ding pits were established, the parapets separating the territories were photographed. In Figure 2.2, the resulting photograph from [3] is reproduced. The territories are seen to be polygonal and, in =-=[27, 59]-=-, it was shown that they are very closely approximated by a Voronoi tessellation. A behavioral model for how the fish establish their territories was given in [22, 23, 60]. When the fish enter a regio... |

7 |
A fast k-means type clustering algorithm
- Wu, Witten
- 1985
(Show Context)
Citation Context ...[21]; see also [10]. It is shown in [55] that, for the quadratic metric, the energy converges to a local minimum value. Other deterministic algorithms for determining the cluster centers are given in =-=[25, 62, 64]-=-. These involve recursive clustering into hyperboxes having faces perpendicular to the coordinate axes.CENTROIDAL VORONOI TESSELLATIONS 657 If m is finite, then obviously the clustering problem can b... |

6 |
On the pattern of space division by territories
- Hasegawa, Tanemura
- 1976
(Show Context)
Citation Context ...are seen to be polygonal and, in [27, 59], it was shown that they are very closely approximated by a Voronoi tessellation. A behavioral model for how the fish establish their territories was given in =-=[22, 23, 60]-=-. When the fish enter a region, they first randomly select the centers of their breeding pits, i.e., the locations at which they will spit sand. Their desire to place the pit centers as far away as po... |

5 |
Experimental results of randomized clustering algorithms
- Inaba, Imai, et al.
- 1996
(Show Context)
Citation Context ... different color. Our task is to approximate the picture by replacing the color at each pixel by one from a smaller set of colors {zi} k i=1 , where each zi ∈ W . For related discussions, we refer to =-=[24, 25, 32]-=-. In particular, a discussion of different clustering algorithms in image compression applications can be found in [24]. We give a concrete example. Suppose the picture is composed of 106 pixels and t... |

3 |
Color image quantization frame buffer display
- Heckert
- 1982
(Show Context)
Citation Context ...sed to obtain the subset with the maximum variance to be divided in the subsequent step. Extensive numerical tests were performed in [32] along with comparisons to other methods such as those used in =-=[25, 62]-=-. Suggestions on directly sampling k-clusters, without using the top-down binary partition technique, are also proposed. 5. Deterministic Approaches to Determining Centroidal Voronoi Tessellations. In... |

3 |
Dynamic Grid Adaptation and Grid Quality
- McRae, Laflin
- 1999
(Show Context)
Citation Context ... equations is another potentially important application of centroidal Voronoi tessellations. Indeed, a Lloyd-like iteration is already in use in existing adaptive grid generation methods; see [9] and =-=[41]-=-. In these methods, a functional such as (2.1) is defined where the density function depends, e.g., on first or second derivatives or differences of a computed solution. The new position of the grid p... |

3 |
Asymptotics of k-Means clustering based on projection pursuit
- Stute, Zhu
- 1995
(Show Context)
Citation Context ...n. For many practical problems, it is desirable to visualize the resulting centroidal Voronoi diagrams or the optimal clusters by some means. Various techniques, such as the projection pursuit method =-=[11, 18, 29, 58]-=-, have been studied in the statistics literature. They can be used to characterize the centroidal Voronoi diagrams via lower dimensional approximations. 3.4. Connections between the Discrete and Conti... |

2 |
Adaptive grid generation, Comput
- Eiseman
- 1987
(Show Context)
Citation Context ...erential equations is another potentially important application of centroidal Voronoi tessellations. Indeed, a Lloyd-like iteration is already in use in existing adaptive grid generation methods; see =-=[9]-=- and [41]. In these methods, a functional such as (2.1) is defined where the density function depends, e.g., on first or second derivatives or differences of a computed solution. The new position of t... |

2 |
Spatial patterns of territories
- Hasegawa, Tanemura
- 1980
(Show Context)
Citation Context ...are seen to be polygonal and, in [27, 59], it was shown that they are very closely approximated by a Voronoi tessellation. A behavioral model for how the fish establish their territories was given in =-=[22, 23, 60]-=-. When the fish enter a region, they first randomly select the centers of their breeding pits, i.e., the locations at which they will spit sand. Their desire to place the pit centers as far away as po... |

2 |
Bootstrapping K-Means Clustering
- JHUN
- 1990
(Show Context)
Citation Context ... be the class of all minimizing sets A∗ = {a∗ 1,...,a∗ k }. T he almost sure convergence Wk(Pn) → Wk(P )asPn converges to P weakly is proved in [51]. Variations of the k-means methods are proposed in =-=[35]-=- using bootstrapping techniques. Advantages of the bootstrap methods are discussed and the performance of bootstrap confidence sets is compared with that of the Monte Carlo confidence sets discussed i... |

2 |
Strong consistency of k-means clustering criterion in separable metric spaces
- Parna
- 1986
(Show Context)
Citation Context ...here it was shown that the cluster centers, suitably normalized, have an asymptotic normal distribution. These results generalize those of [20]. Generalization to separable metric spaces was given in =-=[50]-=-. In [63], some large-sample properties of the k-means clusters (as the number of clusters k approaches ∞ with the total sample size) are obtained. In one dimension, it is established that the sample ... |

1 |
Polynomial approximation and vector quantization—A region-based integration
- Denatale, Desoli, et al.
- 1995
(Show Context)
Citation Context ...sented for computing the optimal parameters with a known precision, using a generalization of Lloyd’s method. In image-data compression applications, region-based algorithms have also been studied in =-=[8]-=-. Voronoi diagram–based vector quantization techniques are used in highfrequency regions or in regions with uncorrelated data; polynomial approximations are used in smooth regions or in regions with h... |

1 |
A remark on Algorithm AS136: k-means clustering algorithm
- England, Benyon
- 1981
(Show Context)
Citation Context ...; 3. recompute the cluster center zi∗ to be the mean of the points belonging to the corresponding cluster. An efficient implementation of this algorithm for large data sets is given in [21]; see also =-=[10]-=-. It is shown in [55] that, for the quadratic metric, the energy converges to a local minimum value. Other deterministic algorithms for determining the cluster centers are given in [25, 62, 64]. These... |

1 |
Asymptotic distribution for clustering centers
- Hartigan
- 1978
(Show Context)
Citation Context ...tic distribution of cluster centers was considered in [53], where it was shown that the cluster centers, suitably normalized, have an asymptotic normal distribution. These results generalize those of =-=[20]-=-. Generalization to separable metric spaces was given in [50]. In [63], some large-sample properties of the k-means clusters (as the number of clusters k approaches ∞ with the total sample size) are o... |

1 |
Projection pursuit, with discussion, Ann
- Huber
- 1985
(Show Context)
Citation Context |

1 |
Geometric k-clustering with applications
- Imai, Inaba
- 1995
(Show Context)
Citation Context ... ∫ ∫ (1.10) V ρ(y)f(d(z ∗ , y)) dy = inf z∈V ∗ V ρ(y)f(d(z, y)) dy for some function f such that f(d(z∗ − y)) is convex in y. The notion of Voronoi regions may be extended to weighted Voronoi regions =-=[30]-=- and to more abstract spaces, and the generators can also be lines, areas, or other more abstract objects [47]. In addition, the metrics need not be induced by a norm. 2. Applications. In this section... |

1 | Vector quantization of images using the l∞ distortion measure
- Mathews
- 1997
(Show Context)
Citation Context ...or given k, the best strategy is given by the centroidal Voronoi diagrams in the L 1 -norm. The use of non-Euclidean metrics also has been considered in the vector quantization literature; see, e.g., =-=[42]-=-. 3. Some Results about Centroidal Voronoi Tessellations and Their Minimization Properties. 3.1. Results Involving Centroidal Voronoi Tessellations as Minimizers. For the sake of completeness, we prov... |

1 |
On the stability of k-means clustering in metric spaces
- Parna
- 1988
(Show Context)
Citation Context ... in step 3 by using the more general definition of the mass center. A proof of the weak convergence of the energy functional was also given in [40]. Extensions to separable metric spaces are given in =-=[51]-=-. Let P be a probability measure on the separable metric space (T,d). Define the energy functional (clustering criterion) by ∫ W (A, P )= min 1≤i≤k φ(d(x, ai))P (dx) , where A = {a1,...,ak} ⊂T is the ... |

1 |
Asymptotic properties of univariate sample k-means clusters
- Wong
- 1984
(Show Context)
Citation Context ...as shown that the cluster centers, suitably normalized, have an asymptotic normal distribution. These results generalize those of [20]. Generalization to separable metric spaces was given in [50]. In =-=[63]-=-, some large-sample properties of the k-means clusters (as the number of clusters k approaches ∞ with the total sample size) are obtained. In one dimension, it is established that the sample k-means c... |