Bounds on performance of optimum quantizers

Eliáš, P.

doi:10.1109/tit.1970.1054415

Cited by 25 publications

(7 citation statements)

References 22 publications

(46 reference statements)

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“…Univariate sample K ‐means is encountered when the goal is to divide an N ×1 vector, x , into K groups. Elias (1970) showed that the population partition is optimal when the within‐cluster sums of squares are equal. Based on work in Wong (1982a) on population K ‐means clusters, Wong (1984) derived two double asymptotic theorems ( K →∞ and N →∞, indicating that the length of the cluster intervals approaches zero while the size of each cluster approaches infinity) for univariate sample K ‐means clusters: (a) the sample cluster within‐sums of squares are equal; and (b) the size of the cluster intervals is inversely proportional to the cube root of the underlying density at the midpoints (which are sufficiently close to the mean in large samples) of the intervals.…”

Section: Theoretical Results Concerning K‐meansmentioning

confidence: 99%

“…Univariate K-means Univariate sample K-means is encountered when the goal is to divide an N £ 1 vector, x, into K groups. Elias (1970) showed that the population partition is optimal when the within-cluster sums of squares are equal. Based on work in Wong (1982a) on population K-means clusters, Wong (1984) derived two double asymptotic theorems (K !…”

Section: Theoretical Results Concerning K-meansmentioning

confidence: 99%

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K‐means clustering: A half‐century synthesis

Steinley

2006

Brit J Math & Statis

837

550

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This paper synthesizes the results, methodology, and research conducted concerning the K-means clustering method over the last fifty years. The K-means method is first introduced, various formulations of the minimum variance loss function and alternative loss functions within the same class are outlined, and different methods of choosing the number of clusters and initialization, variable preprocessing, and data reduction schemes are discussed. Theoretic statistical results are provided and various extensions of K-means using different metrics or modifications of the original algorithm are given, leading to a unifying treatment of K-means and some of its extensions. Finally, several future studies are outlined that could enhance the understanding of numerous subtleties affecting the performance of the K-means method.

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Section: Theoretical Results Concerning K‐meansmentioning

confidence: 99%

Section: Theoretical Results Concerning K-meansmentioning

confidence: 99%

K‐means clustering: A half‐century synthesis

Steinley

2006

Brit J Math & Statis

837

550

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show abstract

“…Other realizations of the generalized scalar quantizer, such as quantization with memory (by means of feedback) and quantization with memory and "preview", will be analyzed in Sections 6 and 7, respectively. For a more comprehensive analysis of quantization see, e.g., [45,16,17].…”

Section: Quantizationmentioning

confidence: 99%

Efficient Data Representations for Signal Processing and Control: "Making Most of a Little"

Goodwin

Derpich

Quevedo

2006

2006 Chinese Control Conference

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Abstract:In order that signals can be stored, transmitted or processed it is necessary that they first be converted into digital form. This, in turn, raises the problem of how to digitize data so as to achieve the best trade-off between data load and performance, i.e., "how to make the most out of a little". Two issues are involved in this problem, namely temporal quantization (i.e., sampling) and spatial quantization. These two problems have traditionally been addressed separately. Indeed, there exists substantial literature dealing with the temporal quantization problem, covering both band-limited and non-band-limited signals. The usual underlying paradigm is that of an analysis filter, followed by a sampler, followed by a reconstruction filter. Various parts of this architecture can be optimized once other parts have been specified. On the other hand, spatial quantization has been studied extensively for a given sampling strategy, particularly in the framework of sigma delta conversion. Finally, it is also possible to formulate the joint design problem for sampling and spatial quantization. This typically leads to enhanced performance compared to that achievable by considering the two aspects separately. This paper will survey the general area of sampling and quantization and analyze methods for achieving efficient data representations for signal processing and control applications. We will show how, on the one hand, contemporary control theory can contribute to the design of sampling and quantization systems and, on the other hand, how these systems impact on the performance of modern feedback control systems.

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“…First, the R-D relationship is mainly derived based on the distribution and quantizer that are convenient to calculate [7]. Second, the R-D performance is mostly observed in ideal conditions like extremely high or low bitrate [8], [9]. In contrast, the R-D analysis over the real coding environment, i.e.…”

Section: Introductionmentioning

confidence: 99%

Towards Rate-Distortion analysis of general source distributions: Property and principles

Sun

Duan

Zhang

et al. 2015

2015 IEEE 17th International Workshop on Multimedia Signal Processing (MMSP)

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This paper decouples the complex Rate-Distortion (R-D) analysis problem by inspecting the respective influence of source distribution and quantizer design on R-D performance. First, a universal R-D property is theoretically revealed that, for any source distribution can be expressed as the product of a Scaling Factor (SF) and its remaining part, SF does not affect its derivative R-D function. Second, efficient quantizer design principles are deduced for different source distributions, which can be used as convenient R-D performance classifier and indicator when the dead-zone plus uniform threshold scalar quantizer with nearly-uniform reconstruction quantizer (DZ+UTSQ/NURQ) is applied. These two contributions bring new insight and inspiration towards the R-D analysis of various different sources, being solid infrastructure to benefit various video/image applications.

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Bounds on performance of optimum quantizers

Cited by 25 publications

References 22 publications

K‐means clustering: A half‐century synthesis

K‐means clustering: A half‐century synthesis

Efficient Data Representations for Signal Processing and Control: "Making Most of a Little"

Towards Rate-Distortion analysis of general source distributions: Property and principles

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