## Beyond Traditional Transform Coding (1998)

Citations: | 18 - 8 self |

### BibTeX

@TECHREPORT{Goyal98beyondtraditional,

author = {Vivek K Goyal},

title = {Beyond Traditional Transform Coding},

institution = {},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

9231 |
Elements of Information Theory
- Cover, Thomas
- 1990
(Show Context)
Citation Context ...heory directly addresses problems in communication, but it has also had fundamental impact on fields beyond communication including probability, statistics, computation theory, physics, and economics =-=[34]-=-. 3 One of many successes of Shannon was to formulate a concrete but sufficiently flexible abstraction of a communication system. Shannon’s abstraction is shown in Figure 1.1. • The information source... |

7148 |
A mathematical theory of communication
- Shannon
- 1948
(Show Context)
Citation Context ...., yK} ⊂R, y1 <y2 < ···<yK 6 It was with justified confidence that Shannon’s paper, “A Mathematical Theory of Communication,” became “The Mathematical Theory of Communication” in the book with Weaver =-=[173]-=-. 7 The advantages of digital communication are described in many texts; e.g., [115].sCHAPTER 1. INTRODUCTION AND PREVIEW 6 is called the codebook. This can be decomposed into encoding or “quantizatio... |

4457 |
Classification and Regression Trees
- Breiman, Friedman, et al.
- 1984
(Show Context)
Citation Context ...be discussed further in Section 6.3.3, where joint optimization with respect 1 The summary here follows [56]. See [56] and references therein for more details. 2 Decision trees are described fully in =-=[19]-=-. Only cursory familiarity is needed here.sCHAPTER 6. COMPUTATION-OPTIMIZED SOURCE CODING 186 to rate, distortion, and encoding complexity will be considered. Though UVQ is the subject of many analyti... |

4331 |
Computer architecture: A quantitative approach
- Hennessy, Patterson
- 1990
(Show Context)
Citation Context ...2 Cycles/mult 5 2 3 Cycles/divide 19 17 11 Cycles/square root - 30 14 Table 6.1: Summary of relative execution times for arithmetic operations on three floating-point microprocessors. Reproduced from =-=[95]-=-. ψ11 = h1 + h2 − h4 + h6, ψ12 = h4 + h5, ψ21 = h6 + h7, ψ22 = h2 − h3 + h5 − h7. It can be shown that the number of multiplications cannot be reduced below seven. The following two results are used i... |

1774 |
Vector quantization and signal compression
- Gersho, Gray
- 1992
(Show Context)
Citation Context ...ts on the codebook design (using a suboptimal codebook) or by replacing the nearest neighbor encoding rule (using suboptimal encoding) have been proposed. The most important variants are described in =-=[60]-=-, a comprehensive text on VQ. Viewing a source coder as a vector quantizer can lead to valuable insights. Examples of this will be seen in Chapters 2 and 5, where the geometries of partitionings are u... |

1332 |
An algorithm for vector quantizer design
- Linde, Buzo, et al.
- 1980
(Show Context)
Citation Context ...fixed codebook size and entropy-constrained, have been addressed for VQ as well. For the first problem, a design algorithm which generalizes the Lloyd–Max iteration was given by Linde, Buzo, and Gray =-=[124]-=-. It is called the generalized Lloyd, or LBG for the authors, algorithm. An algorithm for entropy-constrained VQ (ECVQ) design was given by Chou, Lookabaugh, and Gray [29]. The main drawback of VQ is ... |

1221 | A universal algorithm for sequential data compression
- Ziv, Lempel
- 1977
(Show Context)
Citation Context ...versal. The ubiquity of universal lossless coding methods is well-documented, the commonplace example being that most general-purpose data compression is done with some variant of Lempel–Ziv encoding =-=[233]-=-. Universal lossy coding is less common in practice and is an active area of research. This chapter presents two algorithms for transform coding of sources that emit independent and identically distri... |

1147 | Matching pursuits with time-frequency dictionaries
- Mallat
- 1993
(Show Context)
Citation Context ...Unless the same jk sequence is used every time, the yk’s are a nonlinear function of the signal. One method for choosing the yk’s and jk’s (described in Section 2.3.1) is called matching pursuit (MP) =-=[131]-=-. A version of this algorithm including quantization is presented in Section 2.3.2. This algorithm, called quantized matching pursuit (QMP), is the natural way to apply MP to source coding. The meldin... |

935 | The JPEG still picture compression standard
- Wallace
- 1992
(Show Context)
Citation Context ...rceptibility of the quantization noise. These step sizes depend on a spectral analysis of the signal. The most common technique for compression of continuous tone images is given by the JPEG standard =-=[198, 199, 153]-=-. A JPEG image coder is typically implemented in the form shown in Figure 1.8. A linear transform, in this case a two-dimensional discrete cosine transform (DCT) [159], is applied to 8 × 8 blocks from... |

922 | P.: Least squares quantization in PCM
- LLOYD
- 1982
(Show Context)
Citation Context ... by decoding to the average value of the cell: D(yi) =E[x | Q(x) =yi]. Except for a few examples with simple probability density functions, optimal quantizer design cannot be done analytically. Lloyd =-=[128]-=- and Max [133] independently suggested that a quantizer be designed iteratively by alternating enforcements of the above conditions. A quantizer designed in this manner is called a Lloyd–Max quantizer... |

873 |
random variables and stochastic processes
- Papoulis
- 1984
(Show Context)
Citation Context ...e proof is to first show that each term of (3.1) has expected value Rx +∆2I/12, has finite variance, and is elementwise uncorrelated with every other term. Applying the Chebyshev law of large numbers =-=[151]-=- then gives the first desired conclusion. The second conclusion follows easily. First note that ˆxk = T T k ˆyk = T T k (yk +(ˆyk − yk)) = xk + T T k (ˆyk − yk). Because of the use of subtractive dith... |

744 |
The Art of Computer Programming, Volume 3: Sorting and Searching
- Knuth
- 1973
(Show Context)
Citation Context ...epresentation of a source with expected length bounded above by R. Denotethe source and the reproduction by x and ˆx, respectively. Then, define the distortion–computation function at rate R by 6 See =-=[108]-=- for the complexity of sorting. 7 The Ω( · ) notation means “grows at least as fast as.” DR(C) = inf Eρ(x, ˆx), (6.3) {A∈A : Ec(A,x)≤C, Eℓ(ˆx)≤R}sCHAPTER 6. COMPUTATION-OPTIMIZED SOURCE CODING 190 whe... |

700 | Vector Quantization
- Gray
- 1984
(Show Context)
Citation Context ... advantage in Chapters 2 and 3. The first use of SDQ was by Roberts [164] to improve the perceptual quality of PCM-encoded images. Subtractive and nonsubtractive dithered quantization are surveyed in =-=[126, 85]-=-. 1.1.3 Series Signal Expansions and Transform Coding Series expansions are the central operations of signal processing. They date back almost two centuries to the work of Fourier [57] on expansions i... |

676 |
Textures: a Photographic Album for Artists and Designers
- Brodatz
- 1965
(Show Context)
Citation Context ...hin the confines of blockwise coding. Results for three images are given in Figure 3.7. The first is a 1024 × 1024 pixel image formed by compositing four uniform texture images from the Brodatz album =-=[20]-=-. The DCT is very effective for uniformly textured regions, but the adaptive coder performs a bit better at moderate rates. This is a somewhat contrived example where information from neighboring bloc... |

628 | MPEG: A Video Compression Standard for Multimedia Applications
- Gall
- 1991
(Show Context)
Citation Context ...ideo coder. the image. The transform coefficients are scalar quantized and jointly entropy coded. Many video coding techniques incorporate coding similar to JPEG still image coding. The MPEG standard =-=[116]-=- and other popular algorithms use a hybrid between predictive coding and DCT coding: pixel values are predicted temporally, and the residual from this prediction is DCT coded. A schematic is shown in ... |

577 |
The wavelet transform, time-frequency localization and signal analysis
- Daubechies
- 1990
(Show Context)
Citation Context ...y Duffin and Schaeffer [44] in the context of non-harmonic Fourier series. Recent interest in frames has been spurred by their utility in analyzing the discretization of continuous wavelet transforms =-=[94, 36, 37]-=- and time-frequency decompositions [137]. The motivation here is to understand quantization effects and efficient representations. Section 2.2.1 begins with definitions and examples of frames. It conc... |

528 |
Near optimum error correcting coding and decoding: Turbo codes
- Berrou, Glavieux
- 1996
(Show Context)
Citation Context .... He showed that with the appropriate choice of inner and outer codes it is possible to have probability of error that decreases exponentially with complexity. The explosive popularity of turbo codes =-=[16]-=- is due to the efficacy of iterative decoding of certain concatenated codes. In particular, it is not just that the codes themselves are good for a given constraint length, but that the complexity ver... |

522 |
Error Control Coding: Fundamentals and Applications
- Lin, Costello
- 1983
(Show Context)
Citation Context ...nformation theory is the subject of many books, notably the text by Cover and Thomas [34]; and the specific aspects of channel coding and rate–distortion theory are explained well by Lin and Costello =-=[123]-=- and Berger [13], respectively. Nevertheless, Shannon’s lucid, original paper remains the best introduction to the field. We are lucky that Shannon had no motivation to submit manuscripts in leastpubl... |

503 | Wavelets and Subband Coding
- Vetterli
- 1995
(Show Context)
Citation Context ...ons in terms of sines and cosines. The classical techniques of Fourier analysis occupy a prominent position in engineering curricula and thus will not be reviewed here. Recommended references include =-=[112, 145, 195]-=-. The signals considered in this thesis will be of the discrete-parameter type and have finite extent; thus, they are finite-dimensional vectors. Fourier analysis applies to such signals, specifically... |

467 |
A mathematical theory of communication, Bell Syst
- Shannon
- 1948
(Show Context)
Citation Context ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 7 Conclusions 203 Publications and Patents 207 Bibliography 210sList of Figures 1.1 Shannon’s communication system abstraction =-=[170]-=- . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Separation of encoding into source and channel coding . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Optimal loading factors for Gauss... |

436 |
Theory and Practice of Recursive Identification
- Ljung, Soderstrom
- 1983
(Show Context)
Citation Context ...well. At issue are the quality of reconstruction that is possible and the efficient computation of good estimates. The classical method for estimating the unknown vector x is least-squares estimation =-=[127, 92]-=-, which attempts to find ˆx such that the ℓ2-norm of the residual sequence yk − ϕT k ˆx is minimized. As in the undithered case, a least-squares estimate may be inconsistent with the bounds on the qua... |

419 |
Internet protocol, version 6 (ipv6) specification
- Deering, Hinden
- 1998
(Show Context)
Citation Context ...TER 5. MULTIPLE DESCRIPTION CODING 103 comes from redundancy spread across packets; redundancy within a packet is not useful. For example, consider a network using Internet Protocol, Version 6 (IPv6) =-=[41]-=-. An IPv6 node is required to handle 576-byte packets without fragmentation, and it is recommended that larger packets be accommodated. 5 Accounting for packet headers, a 576-byte packet may have a pa... |

392 |
Gaussian elimination is not optimal
- Strassen
- 1969
(Show Context)
Citation Context ...ψ11 = x11y11 + x12y21, ψ12 = x11y12 + x12y22, ψ21 = x21y11 + x22y21, ψ22 = x21y12 + x22y22, y21 y22 showing that it can be computed with eight multiplications and four additions. Strassen’s algorithm =-=[181]-=- isa clever rearrangement, with some reuse of intermediate answers, that computes the same result with only seven multiplications: 5 Notation is taken from [207]. h1 = (x12 − x22)(y21 + y22), h2 = (x1... |

386 |
Discrete Cosine Transform: Algorithms, Advantages, Applications
- Rao, Yip
- 1990
(Show Context)
Citation Context ...n matrices, which can be done with nlog2 7 multiplications 9 The latter quantities are Rn(D) points as defined in [13]. 10 This is the “original” DCT first reported in [3] and classified as DCT-II in =-=[159]-=-.sCHAPTER 6. COMPUTATION-OPTIMIZED SOURCE CODING 192 Distortion (MSE/sample) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10 0 D n (R) 10 1 n (block size) no transform no transform KLT DCT D n (R) transform coding 1... |

368 |
Toeplitz forms and their applications
- Grenander, Szegö
- 1958
(Show Context)
Citation Context ...remove redundancy in a similar sense. This is a well-known empirical fact [159] and is also justified for large block sizes by an asymptotic equivalence between frequency-selective unitary transforms =-=[87, 83]-=-. What might be the benefit of leaving redundancy in the signal or introducing redundancy to a signal with independent components? This redundancy can be used to combat channel impairments [89, 21, 17... |

353 |
An Introduction to Numerical Analysis
- Atkinson
- 1989
(Show Context)
Citation Context ...n use of Newton’s method is for computing square roots. To compute the square root of a>0, use f(x) =x2 − a. Then(6.8) becomes xn+1 = 1 � xn + 2 a � , (6.9) and the iteration converges for all x0 �= 0=-=[8]-=-. Newton’s method is used in the implementation of hardware square root instructions [95]. In this case, one can use (6.9) directly or use an alternative form which avoids division: � xn+1 = xn 1+ 1 �... |

336 |
Sparse approximate solutions to linear systems
- Natarajan
- 1995
(Show Context)
Citation Context .... For an unconstrained A, finding such an α is hard: • Given ɛ ∈ R + and L ∈ Z + , determining if there exists α with not more than L nonzero entries such that �Aα − x�2 ≤ ɛ is an NP-complete problem =-=[139, 38]-=-. • Given L ∈ Z + , finding α which minimizes �Aα − x�2 among all vectors with not more than L nonzero entries is NP-hard [38]. Matching pursuit is equivalent to a well-known greedy heuristic for find... |

330 |
Nonlinear Systems Analysis
- Vidyasagar
- 1993
(Show Context)
Citation Context ...The fact that the local exponential stability of the original nonlinear system can be inferred from an eigenvalue condition on the linearized system follows from the continuous differentiability of f =-=[196]-=-. Now evaluate F . Differentiating (4.8) gives ∂2J1 = 2 ∂θℓ ∂θk � � ∂A Yij i�=j (k) ij + A ∂θℓ (k) � ∂Yij ij ∂θℓ = 2 � � ∂A Yij (k) ij + A ∂θℓ (k) ij A(ℓ) � ij . (4.9) i�=j Evaluating (4.9) atΘ=0, Y b... |

329 |
Discrete cosine transform
- Ahmed, Natarajan, et al.
- 1974
(Show Context)
Citation Context ...d entail multiplying pairs of n×n matrices, which can be done with nlog2 7 multiplications 9 The latter quantities are Rn(D) points as defined in [13]. 10 This is the “original” DCT first reported in =-=[3]-=- and classified as DCT-II in [159].sCHAPTER 6. COMPUTATION-OPTIMIZED SOURCE CODING 192 Distortion (MSE/sample) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10 0 D n (R) 10 1 n (block size) no transform no transform ... |

327 |
Introduction to Applied Mathematics
- Strang
- 1986
(Show Context)
Citation Context ... inequalities are elementwise. These inequalities can be combined into � � F ˆx ≤ −F � 1 2 ∆+ˆy 1 2 ∆ − ˆy � . (2.10) The formulation (2.10) shows that ˆx can be determined through linear programming =-=[179]-=-. The feasible set of the linear program is exactly the set of consistent estimates, so an arbitrary cost function can be used. This is summarized in Table 2.1. A linear program always returns a corne... |

313 |
Probability and random processes
- Grimmett, Stirzaker
- 1992
(Show Context)
Citation Context ...ℓ xj ℓ ]=E[(xik )2 (x j k )2 ]=(Rx)ii(Rx)jj +2(Rx) 2 ij . The final computation, which can be viewed as a mixed moment of a jointly Gaussian vector, can be made easily using a characteristic function =-=[88]-=-. Computation of E[xi kxj kxi ℓ˜ej ℓ ]: Let T i k denote the ith row of Tk. To compute the expectation of xi kxj kxi ℓ˜ej ℓ = xi kxj kxi j ℓTℓ eℓ, notethateℓis independent of the rest and has zero mea... |

296 |
Universal coding, information, prediction and estimation
- Rissanen
- 1984
(Show Context)
Citation Context ...imate KLT (which must be communicated to the decoder), and uses this approximate KLT to code the source. This approach will not be pursued here, though one could presumably extend results of Rissanen =-=[163]-=- to show that such a system could be universal. 2 Instead, on-line approaches are considered as described below. The availability of a universal lossless coder is assumed, but, in contrast to [231], w... |

291 |
A class of nonharmonic Fourier series
- Duffin, Schaeffer
- 1952
(Show Context)
Citation Context ... or expected SE. 2.2 Nonadaptive Expansions This section describes frames, which provide a general framework for studying nonadaptive linear transforms. Frames were introduced by Duffin and Schaeffer =-=[44]-=- in the context of non-harmonic Fourier series. Recent interest in frames has been spurred by their utility in analyzing the discretization of continuous wavelet transforms [94, 36, 37] and time-frequ... |

283 | Wavelet transforms that map integers to integers
- CALDERBANK, DAUBECHIES
- 1998
(Show Context)
Citation Context ...sCHAPTER 5. MULTIPLE DESCRIPTION CODING 167 Appendices 5.A Pseudo-linear Discrete Transforms Recently, several researchers have proposed using invertible discrete-domain to discrete-domain transforms =-=[98, 223, 24]-=-. They appear under various names (lossless transforms, integer-to-integer transforms, lifting factorizations) and in various flavors (finite dimensional matrices, or Fourier or wavelet domain operato... |

268 |
Design of multiple description scalar quantizers
- Vaishampayan
- 1993
(Show Context)
Citation Context ...art from the opposite end and devise schemes for multiple description coding of scalars. This approach was pioneered by Vaishampayan, who introduced multiple description scalar quantization (MDSQ) in =-=[188]-=-. Conceptually, MDSQ can be seen as the use of a pair of independent scalar quantizers to give two descriptions of a scalar source sample. As in all multiple description coding, the design challenge i... |

251 |
Coding theorems for a discrete source with a fidelity criterion
- Shannon
- 1958
(Show Context)
Citation Context ... of the source is needed. This is given by rate–distortion theory. The rate–distortion function gives the minimum rate need to approximate a source up to a given distortion. The reader is referred to =-=[170, 172, 13]-=- for details. 1.1.1.1 Source and channel coding The separate characterization of the source and the channel, and a unit of measure to connect them, led to separating encoders into two components: a so... |

245 | Digital Communication
- Lee, Messerschmitt
- 1993
(Show Context)
Citation Context ...“A Mathematical Theory of Communication,” became “The Mathematical Theory of Communication” in the book with Weaver [173]. 7 The advantages of digital communication are described in many texts; e.g., =-=[115]-=-.sCHAPTER 1. INTRODUCTION AND PREVIEW 6 is called the codebook. This can be decomposed into encoding or “quantization” and decoding or “inverse quantization” E : R →{1, 2, ..., K} D : {1, 2, ..., K}→C... |

220 |
Quantizing for Minimum Distortion
- Max
- 1960
(Show Context)
Citation Context ...o the average value of the cell: D(yi) =E[x | Q(x) =yi]. Except for a few examples with simple probability density functions, optimal quantizer design cannot be done analytically. Lloyd [128] and Max =-=[133]-=- independently suggested that a quantizer be designed iteratively by alternating enforcements of the above conditions. A quantizer designed in this manner is called a Lloyd–Max quantizer. The output o... |

217 | Continuous and discrete wavelet transforms
- Heil, Walnut
- 1989
(Show Context)
Citation Context ...y Duffin and Schaeffer [44] in the context of non-harmonic Fourier series. Recent interest in frames has been spurred by their utility in analyzing the discretization of continuous wavelet transforms =-=[94, 36, 37]-=- and time-frequency decompositions [137]. The motivation here is to understand quantization effects and efficient representations. Section 2.2.1 begins with definitions and examples of frames. It conc... |

205 |
Achievable rates for multiple descriptions
- El-Gamal, Cover
- 1982
(Show Context)
Citation Context ...t the September 1979 Shannon Theory Workshop held at the Seven Springs Conference Center, Mount Kisco, New York, the following question was posed by Gersho, Ozarow, Witsenhausen, Wolf, Wyner, and Ziv =-=[48]-=-: 1 If an information source is described by two separate descriptions, what are the concurrent limitations on qualities of these descriptions taken separately and jointly? This problem would come to ... |

200 |
Rate Distortion Theory
- Berger
- 1971
(Show Context)
Citation Context ... of the source is needed. This is given by rate–distortion theory. The rate–distortion function gives the minimum rate need to approximate a source up to a given distortion. The reader is referred to =-=[170, 172, 13]-=- for details. 1.1.1.1 Source and channel coding The separate characterization of the source and the channel, and a unit of measure to connect them, led to separating encoders into two components: a so... |

193 |
Constructing optimal binary decision trees is NP-complete
- Hya, L, et al.
- 1976
(Show Context)
Citation Context ...mapped to the nearest codeword. Still, TSVQ is popular in practice because the performance loss is offset by the lower encoding complexity. Designing an optimal decision tree is an NPcomplete problem =-=[101]-=-, so TSVQ design generally follows one of two heuristic strategies: tree-growing [130, 162] or tree-pruning [29]. Tree-pruning will be discussed further in Section 6.3.3, where joint optimization with... |

189 |
Concatenated Codes
- Forney
- 1966
(Show Context)
Citation Context ...ding complexity increases exponentially with N and the probability of error decreases only subexponentially with the complexity. This subexponential rate inspired Forney’s study of concatenated codes =-=[55]-=-. He showed that with the appropriate choice of inner and outer codes it is possible to have probability of error that decreases exponentially with complexity. The explosive popularity of turbo codes ... |

187 |
Distribution of Eigenvalues for Some Sets of Random Matrices
- Marchenko, Pastur
- 1967
(Show Context)
Citation Context ...ng the eigenvalues of ATA away from zero, where A is a matrix of basis vectors. For random bases generated according to a uniform distribution, the eigenvalues of ATA cannot be bounded away from zero =-=[106, 132]-=-. This negative result is not surprising given that optimally “uniform” frames also fail to give this property. These negative results should not discourage the use of the QF system with small numbers... |

178 | Successive refinement of information
- Equitz, Cover
- 1991
(Show Context)
Citation Context ...(X, Y ) and for the private channels to add refinement information for the individual components. This paper influenced the development of successive refinement and multiresolution coding (see, e.g., =-=[50]-=-) but is not immediately applicable to the multiple description problem. Since the papers on MD coding in the 1980’s did not reference Gray and Wyner, it may be assumed that the later work on this pro... |

145 |
Transmission of information
- Hartley
- 1928
(Show Context)
Citation Context ...on) is provided by Kolmogorov’s ɛ-entropy [109]. This will not be considered here. 5 The use of a logarithmic measure was deduced axiomatically by Shannon and had previously been suggested by Hartley =-=[91]-=-.sCHAPTER 1. INTRODUCTION AND PREVIEW 4 TRANSMITTER SOURCE CODER CHANNEL CODER MESSAGE BITS SIGNAL Figure 1.2: Separation of encoding into source and channel coding. symbols that can reliably be disti... |

142 |
A study of vector quantization for noisy channels
- Farvardin
- 1990
(Show Context)
Citation Context ...t should also be noted that there is a substantial literature on the error-sensitivity of compressed data and more generally on trade-offs between source and channel coding. The reader is referred to =-=[144, 103, 54, 134, 52, 89, 21, 97, 175, 96]-=- for a sampling of the results. Finally, MD coding includes as a special case the more well-known successive refinement or multiresolution coding. The successive refinement problem can also be describ... |

141 |
Topics in Matrix Analysis,” Cambridge Univ
- Horn, Johnson
- 1994
(Show Context)
Citation Context ...n will not be known, but must be estimated or in some sense inferred from {xk} n k=1 . First of all, note that if Xn is known, then a Tn consisting of normalized eigenvectors of Xn solves our problem =-=[99]-=-. A traditional approach would be to construct an estimate � Xn = f({xk} n k=1 )foreachn, andthen use an “off the shelf” method to compute the eigenvectors of � Xn. The difficulty with this is that th... |

122 |
A simple derivation of the coding theorem and some applications
- Gallager
- 1965
(Show Context)
Citation Context ...been been a central research topic in channel coding. 1 Consider a system that uses a block code of length n and maximum likelihood (ML) decoding for communication over a memoryless channel. Gallager =-=[58]-=- showed that, when the code is chosen optimally, the probability of error decreases exponentially with the block length as P (error) ≈ exp(−nE(R)). E(R), called the error exponent, is a function of th... |

119 | Quantized overcomplete expansions in R N : Analysis, synthesis, and algorithms
- Goyal, Vetterli, et al.
- 1998
(Show Context)
Citation Context ... for fine quantization. The dependence on dictionary size and lack of monotonicity indicate complicated geometric factors. Similar experiments with different sources and dictionaries were reported in =-=[66]-=-. As noted earlier, the cells of the partition generated by QMP are convex or the union of two convex cells that share one point. This fact allows the computation of consistent estimates through the m... |