R. M. Gray, Source coding theory, Boston: Kluwer Academic Publishers, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....it is believed that we will be able to quantize the components of y more efficiently. In general, this process results in a non uniform distribution of energy between the transform coefficients. The KL transform is optimum in the sense that it completely decorrelates the vector components [1]. This transform is based on eigenvector decomposition and is data dependent. 2 The Structure In this work, we are interested in a linear transformation that optimally decorrelates the vector components in a causal manner. We are particularly interested in a causal form since it can be easily ....

....in E[xk] akjE[xixj] j=l 2 k N, 1 i k (20) and subsequently, akl ] e,ll :e,12 . ak2 :e,21 :e,22 . Lak k lJ ahk 1 1 ahk 1 2 ahk 1 k 1 1 rxi21 ,2 k N, 21) grx, i where rx,ij = xixj] 1 i,j N. Considering the Orthogonality Principle [1] [6] Equation (17) also result in the minimization of the transform coefficients power in Equation (16) or the maximization of the corresponding gains (see next section) We can observe that the transform coefficients y are, in fact, the linear prediction errors when all the components of ....

A. Gresho and R. M. Gray, Vector Quantization and Signal Compression, Boston: Kluwer Academic Publishers, 1992.

....presented in Table 4.1 for two choices of A. This gain is defined as G i =10log 10 E[l i (n) E[lr i (n) dB (4.6) From Table 4.1, it is seen that by employing SLP, the energy of the signal to be encoded is reduced by4:76 dB on the average. However, the LSF parameters of the See reference [12] for details. mid frequency range can be predicted more efficiently as compared to those of the low and high frequency regions. It is also observed that employing VLP provides an additional prediction gain of 0:1 dB. Therefore, we select the first order VLP as the prediction matrix to be used. ....

A. Gresho and R. M. Gray, Vector Quantization and Signal Compression, Boston: Kluwer Academic Publishers, 1992.

....in Figure 5.1 and is characterized by multiple encoders arranged sequentially so that the quantization error signal from each stage is the input to the next stage. In source coding, this structure is used to reduce the encoder complexity and storage requirements for vector quantization (VQ) [75]. For example, a multistage VQ (MSVQ) consisting of m stages with codebook size N i at stage i, has search complexity (assuming greedy stage by stage encoding) and storage equal to P m i=1 N i instead of Q m i=1 N i (for an equivalent full search vector quantizer) The reproduction vector is ....

....Segment into Packets used by decoder Unused packets after the first erasure 00011010 Transmit over packet loss channel Entropy Decoder 1011 001 001 000 Truncated Quantization Indices Figure 5.10 Single stage encoding for an image. distortion of the recovered value using n bits is [75] D n = 1 12 2 2n : 5.41) In the high rate case, early truncation leads to significant distortion compared to the recovered value using the full index. This is not surprising since only a small portion of the bits can be used in the reconstruction. Coding multiple stages, or descriptions, ....

A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Boston: Kluwer Academic Publishers, 1992.

....partitioning a data set arises in two different forms dependent on the data representation as vectorial or proximity data. A partitioning approach known as central clustering derives a set of reference or prototype vectors which quantize a set of vectorial data with minimal quantization error [6, 7]. Data compression is achieved by transmission and storage of the indices of reference vectors rather than the original data vectors. The second approach to data clustering, referred to as pairwise data clustering [8] partitions a set of data into clusters in which the data are indirectly ....

A. Gersho and R. M. Gray, Vector Quantization and Signal Processing. Boston: Kluwer Academic Publisher, 1992.

....singularity. For the purposes of the sphere such coordinate singularities are not significant. However, the point is made here since a general manifold may not be so easily described by a single geodesic coordinate chart. A common method of direct manifold representation is vector quantization [26]. As depicted in Figure 6, a small region of the manifold is represented by a single 13 Representative for each Partition Figure 6. Vector quantization point. These regions are chosen (or learned using a self organizing map [44] to satisfy some criterion (e.g. Voronai regions satisfy an ....

A. Gersho and R. Gray, Vector Quantization and Signal Compression, Boston: Kluwer Academic Publishers, 1992.

.... algorithm is SPIHT [1] an extension of Shapiro s Embedded Zerotree Wavelet method [3] These two new algorithms are 2 a significant breakthrough in lossy image compression in that they give substantially higher compression ratios than prior techniques including JPEG [4] vector quantization [5], and the discrete wavelet transform [6] combined with quantization. In addition, the algorithms allow for progressive transmission [7] meaning coarse approximations of an image can be reconstructed quickly from beginning parts of the bit stream) require no training, and are of low computational ....

A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Boston: Kluwer Academic Publishers, 1992.

.... is to reduce the number of bits required to represent the source signal while ensuring that the distortion between the original and reconstructed signals remains below an acceptable threshold (in a dual framework, the aim is to minimize the distortion while limiting the number of bits required) [23]. The compressibility of a signal is often described in terms of the redundancy inherent in the signal. While the goal of data compression is to eliminate the redundancy in the signal, the presence of delay and or complexity limits and inadequate knowledge of the source statistics hinder the ....

A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Boston: Kluwer Academic Publishers, 1992.

....assumption. A sequence of optimal bit allocations is produced with monotonically decreasing bit rates. The key to the algorithm is that in Step 2, all of the slope calculations are performed ahead of time. We use modified notation from Westerink, Biemond, and Boekee [1] and from Gersho and Gray [7]. Here, our S(1; 1) which is the of the pruning algorithm, is the reciprocal of their s(1; 1) S i (j; j 0 1) is just the magnitude of the slope of the QF of class i between rates j 0 1 and j bits. Let B i be the number of bits allocated to class i. Again, there are M classes and a maximum of q ....

A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Boston: Kluwer Academic Publishers, 1990.

.... is an extension of Shapiro s Embedded Zerotree Wavelet (EZW) algorithm [3] These two new algorithms are a significant breakthrough in lossy image compression in that they give substantially higher compression ratios than prior lossy compression techniques including JPEG [4] vector quantization [5], and the discrete wavelet transform [6] combined with quantization. In addition, the algorithms allow for progressive transmission [7] meaning coarse approximations of an image can be reconstructed quickly from beginning parts of the bitstream) require no training, and are of low computational ....

A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Boston: Kluwer Academic Publishers, 1992.

....partitioning a data set arises in two different forms dependent on the data representation as vectorial or proximity data. A partitioning approach known as central clustering derives a set of reference or prototype vectors which quantize a set of vectorial data with minimal quantization error [6] [7]. Data compression is achieved by transmission and storage of the indices of reference vectors rather than the original data vectors. The second approach to data clustering, referred to as pairwise data clustering [8] partitions a set of data into clusters in which the data are indirectly ....

A. Gersho and R.M. Gray, Vector Quantization and Signal Processing. Boston: Kluwer Academic Publisher, 1992.

.... the minimax proximity of the filter s frequency response to an ideal brick wall filter [3, 4, 5, 6] or the smoothness of the scaling and wavelet functions as measured by regularity [7] The former quality offers promising theoretical results for subband coding if the frequency responses are ideal [8], while the latter is consistent with the smoothness typically observed in natural images and the intuitively desirable objective of retaining that smoothness after quantization. Of course, the relative merit of other filter characteristics has been debated, including orthonormality, coding gain, ....

A. Gersho and R.M. Gray, Vector Quantization and Signal Compression. Boston: Kluwer Academic Publishers, 1992.

.... is an extension of Shapiro s Embedded Zerotree Wavelet al..gorithm [2] These two new algorithms are a significant breakthrough in lossy image compression in that they give substantially higher compression ratios than prior lossy compression techniques including JPEG [3] vector quantization [4], and the discrete wavelet transform [5] combined with quantization. In addition, the algorithms allow for progressive transmission [6] meaning coarse approximations of an image can be reconstructed quickly from beginning parts of the bitstream) require no training, and are of low computational ....

A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Boston: Kluwer Academic Publishers, 1992.

....Q 0 for a source random variable X 2 R d is Delta S = E kX Gamma Q 0 (X)k 2 = X i2I Z R i kx Gamma y i k 2 d (x) 7) where is the probability distribution of the input X. Necessary conditions for the optimality of a vector quantizer using the mean squared distortion are (see [18] for example) the Centroid Condition: y i = E [XjX 2 R i ] 8i 2 I (8) and the Nearest Neighbor Condition: R i = fx 2 R d : kx Gamma y i k kx Gamma y j k 8j 2 I n figg 8i 2 I: 9) Locally optimal vector quantizers satisfying both necessary conditions (8) and (9) can be obtained using the ....

.... Condition: y i = E [XjX 2 R i ] 8i 2 I (8) and the Nearest Neighbor Condition: R i = fx 2 R d : kx Gamma y i k kx Gamma y j k 8j 2 I n figg 8i 2 I: 9) Locally optimal vector quantizers satisfying both necessary conditions (8) and (9) can be obtained using the Generalized Lloyd Algorithm [18]. The high resolution (i.e. large R S ) behavior of Delta S for optimal quantization of a bounded source is described by Zador s formula, which is stated below in a convenient form. Lemma 2 (Zador[3] The minimum mean squared error of a rate R S vector quantizer is asymptotically (as R S 1) ....

A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Boston: Kluwer Academic Publishers, 1992.

.... Gamma1) ff 3 = Gamma:5; Gamma:5) and shown in Figure 4. If the source is uniform over the support region, then the natural binary assignment is again optimum. 2 Comments Direct sum quantizers are often implemented in a multistage or successive approximation fashion ( 14] p. 667, 18] [19] p. 194 ff. 451 ff, and [20] In the scalar case, assuming jff L Gamma1 j : jff 0 j 0, a source sample x is first quantized into the closer of Gammaff 0 and ff 0 . Next the error of this first stage is quantized into the closer of Gammaff 1 and ff 1 , and so on until an approximation ....

A. Gersho and R.M. Gray, Vector Quantization and Signal Compression. Boston: Kluwer Academic Publishers, 1992.

....quantization, and lossless coding. A variety of systems and algorithms for compression are described to provide context, but the method chosen for the current study is a compromise among a variety of considerations. The algorithm used was predictive pruned treestructured vector quantization [14, 15, 16, 17]. This technique involves fast encoding and decoding, and provides additional advantages such as simple progressive transmission and potential incorporation of other signal processing techniques such as classification [18, 19] The algorithm does not perform a signal decomposition such as a DCT or ....

....It is a standard exercise in information theory to demonstrate that any uniquely decodable channel codebook can be made into a channel codebook with the same codeword lengths that also satisfies the prefix condition, and hence no essential generality is lost by the assumption. See, for example, [22, 15]. The decoder fi : f0; 1g C is a mapping from the space of finite length binary sequences onto a set C j ffi(w) w 2 Wg called the reproduction codebook , with members called reproduction codewords or templates. The members of C are chosen from a reproduction alphabet A which typically, ....

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A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Boston: Kluwer Academic Publishers, 1992.

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R. M. Gray, Source coding theory, Boston: Kluwer Academic Publishers, 1990.

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R. M. Gray, Source coding theory, Boston: Kluwer Academic Publishers, 1990.

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