## Bennett's Integral for Vector Quantizers (1995)

Venue: | IEEE Trans. Inform. Theory |

Citations: | 32 - 6 self |

### BibTeX

@ARTICLE{Na95bennett'sintegral,

author = {Sangsin Na and David L. Neuhoff},

title = {Bennett's Integral for Vector Quantizers},

journal = {IEEE Trans. Inform. Theory},

year = {1995},

volume = {41},

pages = {886--900}

}

### OpenURL

### Abstract

This paper extends Bennett's integral from scalar to vector quantizers, giving a simple formula that expresses the rth-power distortion of a many-point vector quantizer in terms of the number of points, point density function, inertial profile and the distribution of the source. The inertial profile specifies the normalized moment of inertia of quantization cells as a function of location. The extension is formulated in terms of a sequence of quantizers whose point density and inertial profile approach known functions as the number of points increases. Precise conditions are given for the convergence of distortion (suitably normalized) to Bennett's integral. Previous extensions did not include the inertial profile and, consequently, provided only bounds or applied only to quantizers with congruent cells, such as lattice and optimal quantizers. The new version of Bennett's integral provides a framework for the analysis of suboptimal structured vector quantizers. It is shown how the loss...

### Citations

338 | Digital Coding of Waveforms - Jayant, Noll - 1984 |

199 |
Asymptotically optimal block quantization
- Gersho
- 1979
(Show Context)
Citation Context ...Appendix. For scalar quantizers, Bennett's integral was extended to rth-power distortion by Algazi [6]. It was extended to vector quantizers with congruent cells (as in a lattice quantizer) by Gersho =-=[5]-=-. Yamada et. al. [7] gave a Bennett-like lower bound to distortion that applies to all vector quantizers and difference distortion measures. Bucklew [8,9] extended Bennett's integral to companders in ... |

120 |
Asymptotic quantization error of continuous signals and the quantization dimension
- Zador
- 1982
(Show Context)
Citation Context ...least distortion of any k-dimensional quantizer with N points is D *sr,k,N @ M *sr,k 1 N r/k �� �� �� �� �� �� ��sp(x) k/(k+r)sdxs(k+r)/k , (2.3) 5 which is the well known =-=Zador-Gersho formula. Zador [15,16]-=- derived the form of (2.3); Gersho [5] recognized that the multiplicative constant is M *sr,k . See also [10]. Although as just shown, the vector version of Bennett's integral enables a direct derivat... |

104 |
Asymptotically efficient quantizing
- Gish, Pierce
- 1968
(Show Context)
Citation Context ... original cell were split into cells of different sizes with points in their middle, then even though the macroscopic point density is l(x), Bennett's integral would not correctly predict distortion. =-=(3)-=- Hypothesis (iv), requiring UACI-p, may be viewed as demanding that the cells shrink rapidly over a large enough region of the support of p. From (3.3) and conditions (1) and (2) of Appendix A we see ... |

71 |
Spectra of quantized signals
- Bennett
- 1948
(Show Context)
Citation Context ...N �� S is||x-y i || r p(x) dx , (1.1) where ||x-y|| = ( ��i=1 ks(x i -y i ) 2 ) 1/2 denotes Euclidean distance and p(x) is the (k-dimensional) probability density of X. The pioneering work of =-=Bennett [1] sho-=-wed that the mean-squared error (r=2) of a scalar quantizer (k=1) with many small cells (N large) and with each y i in the center of its cell may be accurately approximated as D(S,C) @ 1 12N 2 �� ... |

50 |
High-resolution quantization theory and the vector quantizer advantage
- Lookabaugh, Gray
- 1989
(Show Context)
Citation Context ...ntizers We now address the frequently asked question: Why do vector quantizers outperform scalar quantizers for stationary, memoryless sources? The best answer to date is given by Lookabough and Gray =-=[25]-=-. By examining the point density and cell shape losses of kth power quantizers, we are able to add further insight. To do so, consider a stationary, memoryless source {X k } with first-order density p... |

50 |
Measure and Integral: An Introduction to Real Analysis
- Wheeden, Zygmund
- 1977
(Show Context)
Citation Context ...he volume of points not contained in any of these sets is zero. Finally, a sequence of functions {f N (x)} is said to be uniformly absolutely continuously integrable with respect to a density p (c.f. =-=[29], p. 192, [30], p. 247) if for a-=-ll e > 0 there exists d > 0 such that �� �� �� �� �� �� �� Fsf N (x) p(x) dxse for every N and every set F with probability P(F) = ��Fsp(x) dxsd. As an abbreviation, we... |

41 |
Multidimensional asymptotic quantization theory with rth power distortion measures
- Bucklew, Wise
- 1982
(Show Context)
Citation Context ...fference distortion measures. Bucklew [8,9] extended Bennett's integral to companders in higher dimensions. All these results were heuristically formulated and derived. Subsequently, Bucklew and Wise =-=[10]-=- gave a rigorous formulation and proof of (1.2) for scalar quantizers. Although they stated their result for companders, when translated to arbitrary quantizers, their approach was to show that if a s... |

38 |
Development and Evaluation of Procedures for Quantizing Multivariate Distributions
- Zador
- 1963
(Show Context)
Citation Context ...least distortion of any k-dimensional quantizer with N points is D *sr,k,N @ M *sr,k 1 N r/k �� �� �� �� �� �� ��sp(x) k/(k+r)sdxs(k+r)/k , (2.3) 5 which is the well known =-=Zador-Gersho formula. Zador [15,16]-=- derived the form of (2.3); Gersho [5] recognized that the multiplicative constant is M *sr,k . See also [10]. Although as just shown, the vector version of Bennett's integral enables a direct derivat... |

32 |
Asymptotic Performance of Block Quantizers with Difference Distortion Measures
- Yamada, Tazaki, et al.
- 1980
(Show Context)
Citation Context ... quantizers, Bennett's integral was extended to rth-power distortion by Algazi [6]. It was extended to vector quantizers with congruent cells (as in a lattice quantizer) by Gersho [5]. Yamada et. al. =-=[7]-=- gave a Bennett-like lower bound to distortion that applies to all vector quantizers and difference distortion measures. Bucklew [8,9] extended Bennett's integral to companders in higher dimensions. A... |

20 |
Introduction to the Theory of Integration
- Hildebrandt
- 1963
(Show Context)
Citation Context ...oints not contained in any of these sets is zero. Finally, a sequence of functions {f N (x)} is said to be uniformly absolutely continuously integrable with respect to a density p (c.f. [29], p. 192, =-=[30], p. 247) if for all e > 0 there-=- exists d > 0 such that �� �� �� �� �� �� �� Fsf N (x) p(x) dxse for every N and every set F with probability P(F) = ��Fsp(x) dxsd. As an abbreviation, we will say {f N... |

19 | On Optimum Quantization - Wood - 1969 |

15 |
A simple class of asymptotically optimal quantizers
- Cambanis, Gerr
- 1983
(Show Context)
Citation Context ...ching a given l(x), then under certain technical conditions on l and the source density p, N r D(S N ,C N ) �� 1 12 �� 1 l(x) r p(x) dx as N ��s. (1.3) A similar result was given by Camban=-=is and Gerr [11]-=-. For vector quantizers, Bucklew [12] showed that under certain technical conditions, a form of Bennett's integral holds, whenever the sequence of quantizers is asymptotically optimum for some other s... |

15 |
Two Results on the Asymptotic Performance of Quantizers
- Bucklew
- 1984
(Show Context)
Citation Context ...n technical conditions on l and the source density p, N r D(S N ,C N ) �� 1 12 �� 1 l(x) r p(x) dx as N ��s. (1.3) A similar result was given by Cambanis and Gerr [11]. For vector quantize=-=rs, Bucklew [12]-=- showed that under certain technical conditions, a form of Bennett's integral holds, whenever the sequence of quantizers is asymptotically optimum for some other source density. Over the years, other ... |

14 |
Useful approximations to optimum quantization
- Algazi
- 1966
(Show Context)
Citation Context ...thesis (i) implies that the cell volumes are shrinking as 1/N. Unfortunately, we were unable to prove theorem without hypothesis (iii). Nor were we able to show that the other hypotheses imply (iii). =-=(6)-=- Although our proof of the theorem relies on the piecewise continuity of the source density, there might be a proof that would not rely on this assumption. For example, it is shown in [31, Theorem 3.2... |

13 | Quantizing Distortion in Pulse-Count Modulation with Nonuniform Spacing of Levels - Panter, Dite - 1951 |

13 | Quantizing for minimum distortion - Roe |

11 |
Companding and random quantization in several dimensions
- Bucklew
- 1981
(Show Context)
Citation Context ...ent cells (as in a lattice quantizer) by Gersho [5]. Yamada et. al. [7] gave a Bennett-like lower bound to distortion that applies to all vector quantizers and difference distortion measures. Bucklew =-=[8,9]-=- extended Bennett's integral to companders in higher dimensions. All these results were heuristically formulated and derived. Subsequently, Bucklew and Wise [10] gave a rigorous formulation and proof ... |

7 |
On the performance of tree structured vector quantizers
- Neuhoff, Lee
- 1991
(Show Context)
Citation Context ...ular structure, then one can use Bennett's integral to quantify the resulting loss of performance. Examples of qualitative and quantitative loss analyses include those of tree-structured quantization =-=[20]-=-, two-stage quantization [21,22], scalar quantization (Subsection C below) and transform coding (Subsection D below). In the following we show how to separately identify the losses due to the point de... |

6 |
Least squares quantization in PCM," Bell Laboratories Memorandum 1957, also published in
- Lloyd
- 1982
(Show Context)
Citation Context ...n the values of l N (x) and m N (x) for an x within both S N,i and the support are influenced by the portion of S N,i outside the support. For example, if S N,i has infinite volume, then l N (x) = 0. =-=(2)-=- Although hypothesis (ii) is a reasonable way to specify that quantizers have, approximately, model inertial profile m, the following example shows there are other ways to specify such and have Bennet... |

6 |
A note on optimal multidimensional companders
- Bucklew
- 1983
(Show Context)
Citation Context ...ent cells (as in a lattice quantizer) by Gersho [5]. Yamada et. al. [7] gave a Bennett-like lower bound to distortion that applies to all vector quantizers and difference distortion measures. Bucklew =-=[8,9]-=- extended Bennett's integral to companders in higher dimensions. All these results were heuristically formulated and derived. Subsequently, Bucklew and Wise [10] gave a rigorous formulation and proof ... |

5 |
Bounds on performance of optimum quantizers
- Elias
- 1970
(Show Context)
Citation Context ...er the years, other work has focused on deriving the asymptotically best performance of quantizers without explicit use of Bennett's integral [13-19]. A thorough summary of early work is contained in =-=[19]-=-. A complete extension of Bennett's integral to vector quantizers must take into account the shape of the quantization cells and, indeed, the possibility that in two or more dimensions the cells may h... |

4 | On the quantization of finite-dimensional messages - Schutzenberger - 1958 |

4 |
Conditionally corrected two-stage vector quantization
- Lee, Neuhoff
- 1990
(Show Context)
Citation Context ...use Bennett's integral to quantify the resulting loss of performance. Examples of qualitative and quantitative loss analyses include those of tree-structured quantization [20], two-stage quantization =-=[21,22]-=-, scalar quantization (Subsection C below) and transform coding (Subsection D below). In the following we show how to separately identify the losses due to the point density and the inertial profile. ... |

4 |
Block quantization of correlated Gaussian variables
- Huang, Schultheiss
- 1963
(Show Context)
Citation Context ...ix T (the "transform") before quantization by a product quantizer. 7 (The inverse transform is applied at the decoder.) Although the performance of such codes is well understood in the high-=-=rate case [28]-=-, we obtain additional insight by reexamining them from the point density, cell shape viewpoint. As usual in high-rate analyses of transform coding, we restrict attention to stationary Gaussian source... |

2 |
Cell-conditioned two-stage vector quantization of speech
- Lee, Neuhoff, et al.
- 1991
(Show Context)
Citation Context ...use Bennett's integral to quantify the resulting loss of performance. Examples of qualitative and quantitative loss analyses include those of tree-structured quantization [20], two-stage quantization =-=[21,22]-=-, scalar quantization (Subsection C below) and transform coding (Subsection D below). In the following we show how to separately identify the losses due to the point density and the inertial profile. ... |

2 |
Theory of Functions of a Real Variable, translated by L
- Natanson
- 1955
(Show Context)
Citation Context ...ssuch that |f N (x)|sK for all N and all x. 2. There exists a function g such that ��sg(x) p(x) dxs, and |f N (x)|sg(x) for all N, x. 3. The integrals ��sf N (x) 2 p(x) dx are uniformly bounde=-=d (c.f. [32], vo-=-l. 1, p. 159). 4. The f N 's are uniform integrable (c.f. [33], p. 139); i.e., for any e > 0 there exists a > 0 such that �� |f N (x)|>asf N (x) p(x) dxse for all N. Actually, uniform integrabilit... |

1 |
Rate Distortion Theory, p. 173
- Berger
(Show Context)
Citation Context ...ely, hypothesis (iv) may be replaced by the condition that the e N (x)'s (or the functions N 1/k d N (x)) are uniformly bounded from above by some function whose integral with respect to p is finite. =-=(4)-=- Much of the complexity of the proof derives from the fact that the source density is allowed to be unbounded and to have unbounded support. However, we emphasize that the UACI-p hypothesis is needed,... |

1 |
On the performance of structured quantizers
- Neuhoff
- 1988
(Show Context)
Citation Context ...okabough and Gray [25] called it the space filling loss 4 . It has been conjectured that a cube has the smallest n.m.i. of any polytope with 2k faces (or, at least, of any such tessellating polytope) =-=[26]-=-. In this light, one may view L cu to be a consequence of the fact that the cells of a kth-power quantizer have fewer faces than those of an optimum quantizer. The oblongitis loss is the ratio of the ... |

1 | Source Coding of Composite Sources with Segmental Fidelity Criteria - Na - 1989 |

1 |
k cubic L cu oblongitis L * ob point dens. L * pt shape L * ob L * pt total L cu L
- Cohn, Theory, et al.
- 1980
(Show Context)
Citation Context ...tion g such that ��sg(x) p(x) dxs, and |f N (x)|sg(x) for all N, x. 3. The integrals ��sf N (x) 2 p(x) dx are uniformly bounded (c.f. [32], vol. 1, p. 159). 4. The f N 's are uniform integrabl=-=e (c.f. [33], p.-=- 139); i.e., for any e > 0 there exists a > 0 such that �� |f N (x)|>asf N (x) p(x) dxse for all N. Actually, uniform integrability is equivalent to UACI-p plus uniform boundedness of the integral... |