S. P. Lloyd, Least Squares Quantization in PCM," IEEE Transactions on Information Theory, ##, pp. 129-137, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....problems exactly, it is natural to consider approximation, either through polynomial time approximation algorithms, which provide guarantees on the quality of their results, or heuristics, which make no guarantees. One of the most popular heuristics for the k means problem is Lloyd s algorithm [17, 30, 31], which is often called the k means algorithm. Define the neighborhood of a center point to be the set of data points for which this center is the closest. It is easy to prove that any locally minimal solution must be centroidal, meaning that that each center lies at the centroid of its ....

S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, 28:129-- 137, 1982.

....image is wavelet transformed and quadtree coded as discussed in Section 5.2. In the following sections we describe the steps of this algorithm in greater detail. 5.3. 1 Text Block Color Classification Within each text block X , the two main colors are identified with Lloyd Max optimization ([47]) on a two level scalar quantizer. If the two main colors are closer to each other than some threshold, the block is classified as unimodal a block that only contains different shades of the same color and is assigned to be text or non text based on the foreground and background colors of the ....

S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, 28(2):127--135, March 1982.

....the minimal distance between two sampling points. It is difficult to choose this minimal radius: a small radius allows for very non uniform distributions while a too large radius inhibits convergence. In order to overcome the problems of a too small radius McCool and Fiume apply Lloyd s method [7] as a post process. The relaxation technique is applied to the points generated by dart throwing VMV 2001 Stuttgart, Germany, November 21 23, 2001 Figure 1: Illustration of Lloyd s relaxation scheme: a) initial random point set with Voronoi Diagram, b) movement of points during iteration, and c) ....

S. Lloyd. Least Square Quantization in PCM. IEEE Transactions on Information Theory, 28:129--137, 1982. 1.1

....not easy to visualise this property in simulations, because it concerns densities and is thus very sensitive to the observations. Another comment is that the exponent in (6) has no effect on uniform densities. Although different, this result must be related to Lloyd s work on vector quantization [8], who also derives a similar relation between the original density f(x) and the resulting g(x) This last density is however defined in a different way in Lloyd s work (in a few words, g(x) is here reconstructed as a staircase density in which the limits of intervals are defined by the centroids, ....

....by the centroids, and their weights are inversely proportional to their width. The probabilities of intervals are thus supposed equal, and the reconstructed density is shown to be related to f(x) through an exponent as in (6) We insist on the fact that definitions of g(x) are different in [7] and [8], and that further investigation would be necessary to compare these two results. Finally, it must be mentioned that several authors [17, 6] proposed modifications of the algorithm to compensate the exponent in (6) and to obtain discrete densities of centroids closer fromf(x) To our opinion, ....

Lloyd S.P., Least squares quantization in PCM, IEEE Transactions on Information Theory, vol. 1T28, no. 2, pp. 129-149, March 1982.

....In the general case, for the SOM algorithm, the updating concerns not only the winning quantifier, but also its neighbors. The SCL algorithm is in fact the stochastic or on line version of the Forgy algorithm (also called moving centers algorithm, Lloyd s algorithm, LBG) See for example [7] [8], 9] In this version of the algorithm, the quantifiers are randomly initialized. At each step t, the classes Q, C2 . Cn are determined by putting in class C, the data which are closer to q than to any other quantifier qi. Then the mean values of each class is computed and taken as new ....

Lloyd S.P., Least squares quantization in PCM, IEEE Transactions on Information Theory, vol. IT-28, no. 2, pp. 129-149, March 1982.

....idea of our approach is to compute Voronoi diagrams on the MLS surface and add points at vertices of this diagram. Note that the vertices of the Voronoi diagram are exactly those points on the surface with maximum distance to several of the existing points. This idea is related to Lloyd s method [37], i.e techniques using Voronoi diagrams to achieve a certain distribution of points [40] However, computing the Voronoi diagram on the MLS surface is excessive and local approximations are used instead. More specifically, our technique works as follows: In each step, one of the existing points ....

S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, 28:128--137, 1982.

....of the Voronoi region. These two conditions imply that the encoder is optimized for the given decoder by the nearest neighbor condition, and that the decoder is optimized for the given encoder by the centroid condition. A generalization of a least squares quantization method for scalar data [70] to multidimensional data in 1980 was the first vector quantization design technique introduced in the literature, and is known as the generalized Lloyd algorithm (GLA) 69] The GLA is a descent algorithm that finds codebooks with progressively lower average distortions by using iterations of the ....

S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, 28(2):129--136, March 1982.

....idea of our approach is to compute Voronoi diagrams on the MLS surface and add points at vertices of this diagram. Note that the vertices of the Voronoi diagram are exactly those points on the surface with maximum distance to several of the existing points. This idea is related to Lloyd s method [30], i.e techniques using Voronoi diagrams to achieve a certain distribution of points [32] However, computing the Voronoi diagram on the MLS surface is excessive and local approximations are used instead. More specifically, our technique works as follows: In each step, one of the existing points ....

S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, 28:128--137, 1982.

....the minimal distance between two sampling points. It is difficult to choose this minimal radius: a small radius allows for very non uniform distributions while a too large radius inhibits convergence. In order to overcome the problems of a too small radius McCool and Fiume apply Lloyd s method [7] as a post process. The relaxation technique is applied to the points generated by dart throwing VMV 2001 Stuttgart, Germany, November 21 23, 2001 a) b) c) Figure 1: Illustration of Lloyd s relaxation scheme: a) initial random point set with Voronoi Diagram, b) movement of points during ....

S. Lloyd. Least Square Quantization in PCM. IEEE Transactions on Information Theory, 28:129--137, 1982.

....of MLLR is to tie or cluster some Gaussians together in order to force them to share the same matrix A and vector b. 3.1.1 Gaussian Clustering Many algorithms can be used to cluster the Gaussians in order to reduce the number of parameters. One such algorithm is the K means clustering algorithm [9, 15] where one nds the K centers (clusters) which minimize the Euclidean distance between every Gaussian mean and its nearest center (cluster) Note however that this algorithm needs to select beforehand the number K of clusters. Another solution, which has been used in the experiments reported in ....

S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, 28(2):129137, 1982.

....## r# v r# s## ur#iv# p qr## uh## #h #qr## . vrq# s . #### # pryy# #r#t## (#iv##dA Avhyy ##r ##)########)#### f. r v # s#vqvo oovqhy#o oohyr ## ###b# X ) ### X o oorq#i fhpxvt#hq#qr h##A# # bA r#o oohyr ##v####v# #sh ## d# #yro oory# fr. h## # 2.1. Data Analysis Quantization [4] is used in QuantiCubes to determine the set of representative values for an attribute. The crucial issue of accuracy is guaranteed by our new proposal for an adaptable quantization procedure. This adaptable procedure uses an accuracy constraint based in user defined or preset error bounds ....

S.P. Lloyd. Least Squares Quantization in PCM. IEEE Transactions on Information Theory, IT-28:127-135, March, 1982.

....equal to the error introduced by the scalar quantizer q( since, y(n) x(n) p(n) u(n) p(n) d(n) u(n) d(n) e(n) 1.2) where e(n) is the error introduced by the quantizer. The design of a minimum mean squared error (mmse) quantizer based on the pdf of fE n g is addressed in [7] and [8]. For such a quantizer with 2 R levels the asymptotic error variance is given by, e 2 = 2 2 2R d 2 (1.3) where e 2 is the variance of fe(n)g and d 2 is the variance of fd(n)g the prediction error sequence. The constant 2 is dependent upon the pdf of fd(n)g. From the ....

S. P. Lloyd, \Least squares quantization in PCM," IEEE Transactions on Information Theory, vol. 28, pp. 129-137, March 1982.

....advantageous to rewrite compression distortion as the sum of distortions D = P J D J due to quantizing the transform coecients s J = W T J x, where W J is the J th column vector of W . The r JiJ grid mark values that minimize each D J are the reproduction values of a scalar Lloyd quantizer [10] designed for the transform coecients, s J . K J is the number of reproduction values in the quantizer for transform coordinate J . Allocating the log 2 (K) coding bits among the transform coordinates so that we minimize distortion [11] determines the optimal K J s. 3 Local Transform Coder ....

S. Lloyd. Least square optimization in PCM. IEEE Transactions on Information Theory, 28(2):129-137, 1982.

....numerically. They accelerate the task of computing all interparticle forces (the brute force approach takes O#n # # time) through the use of a spatial hierarchy. Our approach uses graphics hardware to find analogous minimum energy configurations. Deussen et al. [1] use Lloyd s method [10] to spread ink dots in a stippling pattern. Round ink dots have no preferred orientation, unlike square Figure 3: Detail from The Betrayal , St. Apollinarie Nuovo, Ravenna, early 6th century. tiles, but otherwise their goals are very similar to ours. Hertzmann s painterly renderings [5] ....

....that all points within a region are closest to its associated site. Figure 4a shows an example. CVDs are voronoi diagrams with the additional property that each site is located at the mass centre (centroid) of its region. Figure 4b shows a CVD. CVDs are easily produced using Lloyd s algorithm [10]: At each iteration, the algorithm moves each site to its region s centroid, then re computes the voronoi diagram. Its convergence properties are only known in one dimension [2] but it seems to work quickly in ab Figure 4: a) Voronoi diagram; b) Centroidal voronoi diagram. ab Figure 5: a) ....

Lloyd, S. Least Square Quantization in PCM. IEEE Transactions on Information Theory 28(1982): 129-137.

....[13] can be used in order to maximize the likelihood of the data given the model. The average log likelihood LL of a sequence X is equal to LL = 1 T T X t=1 log p(x t ) 2. 15) As it is well known that EM is very sensitive to the initialization of the parameters, a simple K means algorithm [32] is usually performed to select a good candidate parameter set before starting the EM procedure. An important property of any parametric model such as a GMM is its capacity [59] which is related to the number of di erent functions (or distributions in the case of GMMs) the model can span using ....

S.P. Lloyd. Least square quantization in PCM. IEEE Transactions on Information Theory, 28(2):129137, 1982. 36

....to write compression distortion as the sum of distortions D = P J D J due to quantizing the transform coefficients s J = W T J x, where W J is the J th column vector of W 3 . The r JiJ grid mark values that minimize each D J are the reproduction values of a scalar Lloyd quantizer [11] designed for the transform coefficients, s J . K J is the number of reproduction values in the quantizer for transform coordinate J . Allocating the log 2 (K) coding bits among the transform coordinates so that we minimize distortion [12] determines the optimal K J s. 3 Local Transform Coder ....

S. Lloyd. Least square optimization in PCM. IEEE Transactions on Information Theory, 28(2):129--137, 1982.

....in practice. Such multi modal distributions are not accurately represented with unimodal densities, so the quantizer reproduction values will not be placed to minimize quantization error. Instead, we must design empirical quantizers using a process such as the Lloyd quantizer design algorithm [13], in which quantizer reproduction values are determined directly from the coecient values. 1.3 Optimal local PCA transform coding In this paper, we present a new algorithm that integrates optimization of the signal space partition, the local transforms, bit allocation and the scalar quantizers. ....

....it with the least distortion. To optimize the transform coder in each region we perform PCA on the vectors assigned to a region to de ne the local transform 4 , allocate the available coding bits among the coecient quantizers to nd the optimal quantizer sizes, and design empirical Lloyd [13] quantizers to code the coecient values with minimal error. To evaluate the compression performance of our algorithm, we performed compression experiments on two types of images, 1) Magnetic Resonance Image (MRI) data and 2) a database of gray scale images of trac moving through street ....

[Article contains additional citation context not shown here]

S. Lloyd. Least square optimization in PCM. IEEE Transactions on Information Theory, 28(2):129-137, 1982.

....not easy to visualise this property in simulations, because it concerns densities and is thus very sensitive to the observations. Another comment is that the exponent in (6) has no effect on uniform densities. Although different, this result must be related to Lloyd s work on vector quantization [8], who also derives a similar relation between the original density f(x) and the resulting g(x) This last density is however defined in a different way in Lloyd s work (in a few words, g(x) is here reconstructed as a staircase density in which the limits of intervals are defined by the centroids, ....

....by the centroids, and their weights are inversely proportional to their width. The probabilities of intervals are thus supposed equal, and the reconstructed density is shown to be related to f(x) through an exponent as in (6) We insist on the fact that definitions of g(x) are different in [7] and [8], and that further investigation would be necessary to compare these two results. Finally, it must be mentioned that several authors [17, 6] proposed modifications of the algorithm to compensate the exponent in (6) and to obtain discrete densities of centroids closer from f(x) To our opinion, ....

Lloyd S.P., Least squares quantization in PCM, IEEE Transactions on Information Theory, vol. IT-28, no. 2, pp. 129-149, March 1982.

....is advantageous to rewrite compression distortion as the sum of distortions D = P J DJ due to quantizing the transform coefficients sJ = W T J x, where WJ is the J th column vector of W . The rJi J grid mark values that minimize each DJ are the reproduction values of a scalar Lloyd quantizer [10] designed for the transform coefficients, sJ . KJ is the number of reproduction values in the quantizer for transform coordinate J . Allocating the log 2 (K) coding bits among the transform coordinates so that we minimize distortion [11] determines the optimal KJ s. 3 Local Transform Coder ....

S. Lloyd. Least square optimization in PCM. IEEE Transactions on Information Theory, 28(2):129--137, 1982.

....for discrete HMM is the K means algorithm. Variants are sometimes known as the Linde Buzo Gray (LBG) algorithm[LBG80] or Lloyd s algorithm. The algorithm was independently discovered by Lloyd in 1957, while working on pattern recognition research but was unpublished. He later published it in 1982[Llo82] The K in the K means refers 50 to the number of clusters used. The discrete LEMS system use K = 256 for each of the three codebooks. The K means algorithm may be described (superscripts imply iteration number) by: Initialization: Choose an arbitrary set of K training vectors, say y 0 k ,k ....

S. P. Lloyd. Least squares quantization in pcm. IEEE Transactions on Information Theory, 28:129--137, March 1982.

....maximum data likelihood. The fitting amounts to estimating shift and width scaling parameters of each density model. From among the candidates, we then choose the fitted density with the highest data likelihood. Our library of model densities defines a library of optimal quantizers. We fit Lloyd [15] quantizers to each of the model densities for a range of quantizer bit rates. The quantizer reproduction values are then automatically scaled and shifted when the densities are fit to the data. Bit Allocation Having established the above quantizer models, we apply Risken s bit allocation ....

S. Lloyd. Least square optimization in pcm. IEEE Transactions on Information Theory, 28(2):129--137, 1982.

....the corresponding representative values for each interval. Even though we are not going to make use of the representative values directly, they also have to be designed together with the decision intervals. A very popular algorithm for designing a quantizer with K intervals is Lloyd s algorithm [22], which is nothing but the K means algorithm for clustering [25, 21] specialized to one dimension: Denote by t n the value of the nth data point along the objective KLT dimension. Start with a given set of intervals [c i ; c i 1 ) for i = 1; K, where c 1 = min n t n and c K 1 = ffl ....

S. P. Lloyd. Least squares quantization in pcm. IEEE Transactions on Information Theory, 28:127--135, March 1982.

....the computationally more convenient canonical ensemble, in which rather than fixing the value of H, we control its average value by a Lagrange multiplier, 1=T . The connection to Statistical Mechanics sketched above was first made by Rose et al.[8] They, however, used a very different cost function[9] one that minimizes the variance within each cluster and hence maps onto a model with glassy behavior. Another important difference between their work and ours is in the way they calculate the equilibrium properties of their model by means of deterministic annealing. This method is unable to ....

S. P. Lloyd, IEEE Transactions on Information Theory 28, 129 (1982).

....the computationally more convenient canonical ensemble, in which rather than fixing the value of H, we control its average value by a Lagrange multiplier, 1=T . The connection to Statistical Mechanics sketched above was first made by Rose et al.[7] They, however, used a very different cost function[8] one that minimizes the variance within each cluster and hence maps onto a model with glassy behavior. Another important difference between their work and ours is in the way they calculate the equilibrium properties of their model by means of deterministic annealing. This method is unable to ....

S. P. Lloyd, IEEE Transactions on Information Theory 28, 129 (1982).

....# then predict # P u#u # ;v#v # Cl;m #u##v# # 0#; else predict # P u#u # ;v#v # Cl;m #u##v# # 0#; Figure 4: The practical algorithm for branch prediction synthesis. deferred to a later paper. There is a considerable body of literature on optimal quantizer design for prediction, see [12] for instance. We restrict the complexity of the constructed predictors as follows. A one register predictor is only allowed to compare the contents of the register with a single literal. If the register value is less than the literal, it predicts the branch predicate to be true (false) else the ....

S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, 28(3):129--137, March 1982.

....Optimal Adaptive K Means [6] was constructed for this purpose and was tested along with the other algorithms, in order to evaluate its performance. Optimal Adaptive K Means is an enhanced version of the neural network implementation of K Means, which is usually referred to as Adaptive K Means [17, 22, 14]. First, a brief description of Adaptive K Means will be given, followed by a presentation of the Optimal Adaptive K Means. 5.1 Adaptive K Means 5.1 Adaptive K Means 9 Weights Input Signals k q y k f( kp k2 w Summing Function Activation Function Output Signal Synaptic w . S ....

S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, IT-28(2):129--137, March 1982.

....if the map dimension does not coincide with the intrinsic dimension of data. Higher approximation flexibility is thus offered by structure free self organizing network models, either with predefined number of units (prototypes) such as Lloyd and MacQueen s K means clustering procedure [9], 10] the neural gas network [11] or, time varying, incremental neural network structure [12] 13] 14] 15] In this paper, we use the dynamic cell structures (DCS) 15] for incremental quantization of the state space which is combined with LLMs. This type of unsupervised neural network ....

S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, 28(2):129--137, 1982.

....are restricted to rth power Euclidean distortions D(x; y) kx Gamma yk r , although the analysis can be generalized to the class of difference distortion measure [46] We denote the codebook size by K. S ff , ff = 1; K are the partitions of d which define the quantizer. Following [33] we define the codebook density function as g K (x) 1 KV (S ff ) if x 2 S ff ; 22) where V (S ff ) is the volume of partition S ff . In the limit of dense quantization levels we expect g K (x) to closely approximate a continuous density function Upsilon(x) 1= KV (S ff ) The total ....

S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, 28(2):129--137, 1982.

....be difficult and there are no known closed form solutions [3] The author may also be reached at his World Wide Web home page at URL: http: www.eecs.umich.edu wkn . The conventional technique for designing a codebook works through a process of iterative refinements of an initial codebook [9]. A brief description is given in Section 2. This technique does not guarantee optimality, it sometimes yield locally optimal codebooks. Genetic algorithms [4, 7] are becoming a widely used and accepted method for very difficult problems [1] There has been some recent work in the application of ....

....the performance between the genetic and conventional algorithms in Section 4. Finally, the last section concludes with a summary of our work and directions for future research. 2. Generalized Lloyd Algorithm A widely used technique for codebook design is the Generalized Lloyd Algorithm (GLA) [9] as Step 0 Given: An input sequence X and an initial codebook Y . Step 1 Map each vector x 2 X into a vector in Y that yields the minimum distortion. Step 2 Compute total distortion for all x 2 X. Step 3 If total distortion does not improve beyond a threshold, quit. Step 4 Update the ....

S. P. Lloyd. Least Squares Quantization in PCM. IEEE Transactions on Information Theory , Vol. 28, No. 2, pp. 127--135, March 1982.

....small, e.g. at 5 percent, and grows linearly if too many clusters are formed. Background pixel padding, 0 padding, is done to the right and bottom to ensure the same size in (1) To create better templates we invoke a Vector Quantization algorithm [4] based on the Generalized Lloyd Algorithm [8, 9], which basically shifts the cluster centers in a direction of lower distortion which is the sum of all the differences in equation (1) A subset of the cluster centers which were obtained from a sample image is shown in Figure 1. The symbolic image of the sample image and the associated error map ....

S. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, 28:127-- 135, 1984.

....where the mean square distortion d( x i ; Q( x i ) k x i Gamma Q( x i )k 2 . This problem is known to be difficult and there are no known closed form solutions [4] The conventional technique for designing a codebook works through a process of iterative refinements of an initial codebook [10]. A brief description is given in Section 7. This technique does not guarantee optimality, it sometimes yield locally optimal codebooks. In Sections 7 to 9, we present a genetic algorithm to approximate a solution to the optimal codebook design problem and evaluate its performance. 7 Generalized ....

....yield locally optimal codebooks. In Sections 7 to 9, we present a genetic algorithm to approximate a solution to the optimal codebook design problem and evaluate its performance. 7 Generalized Lloyd Algorithm A widely used technique for codebook design is the Generalized Lloyd Algorithm (GLA) [10] as shown in Figure 3. The algorithm begins with a set of input vectors and an initial codebook. For each input vector, a codeword from the codebook is chosen that yields the minimum distortion. If the sum of distortions from all input vectors does not improve beyond some threshold, the algorithm ....

S. P. Lloyd. Least Squares Quantization in PCM. IEEE Transactions on Information Theory, Vol. 28, No. 2, pp. 127--135, March 1982.

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S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, IT-28(2):129--136, Mar. 1982.

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S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, IT-28(2):129--136, Mar. 1982.

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S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, IT-28(2):129--136, Mar. 1982.

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S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, IT-28(2):129--136, Mar. 1982.

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S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, IT-28(2):129--136, Mar. 1982.

....noise. The function of the filter is to estimate X k given a sequence of the quantised versions of the continuous measurements Y k . Much of the previous research in quantisation aims to reduce the distortion between the original and quantised signals under some minimum mean squared error criteria [5, 6, 13], where the principal measure of performance is the error (or function of) between the input and output of the quantiser. Other authors have considered quantisation where the quantised data is used to form a test of hypotheses for the purposes of signal detection [1, 3, 8] Our approach differs ....

....to significant improvement over the uniform case, particularly when oe 1. However, it was surprising to find that KL maximisation did not lead to better filter performance. While KL maximisation sled to reduction of quantisation noise (consistent with traditional non uniform quantisation schemes [5, 6]) the resulting filtering error remains comparable with errors obtained from filters constructed using uniform quantisation. This may be understood by examining the resultant quantisation levels (see Fig. 5 7) It can be seen that for small oe, minimising conditional entropy places levels very ....

S. P. Lloyd. Least squares quantisation in PCM. IEEE Transactions on Information Theory, IT-28(2):129-- 137, Mar. 1982.

....and identically distributed (i.i.d. noise. The function of the filter is to estimate X k given quantised versions of Y k . Much of previous work dealing with quantisation aims to reduce the distortion between the original and quantised signals under some minimum mean squared error criteria [2, 3, 6]. In contrast, the primary focus here is to minimise the probability of error of a nonlinear filter with quantised observations as inputs. Some of the questions we address are: 1) Is there an optimal strategy for choosing the quantisation levels which minimises the filtering error (2) How does ....

S. P. Lloyd. Least squares quantisation in PCM. IEEE Transactions on Information Theory, IT-28(2):129-- 137, Mar. 1982.

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S. P. Lloyd, Least Squares Quantization in PCM," IEEE Transactions on Information Theory, ##, pp. 129-137, 1982.

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S. P. Lloyd. Least square quantization in PCM. IEEE Transaction on Information Theory, 28(2):129--137, March 1982.

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S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, IT-28(2):129--136, Mar. 1982.

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Lloyd,S. Least square quantization in PCM. IEEE Transactions on Information Theory, IT-28(2):129--137, March 1982.

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S. P. Lloyd. Least squares quantization in pcm. IEEE Transactions on Information Theory, 28(2):129--137, March 1982.

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S. P. Lloyd. Least squares quantization in pcm. IEEE Transactions on Information Theory, 28:127--135, March 1982.

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S. P. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, 28:129-137, 1982.

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S. Lloyd. Least square quantization in PCM. IEEE Transactions on Information Theory, 28:129--137, 1982.

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