M. W. Marcellin and T. R. Fischer, "Trellis coded quantization of memoryless and Gauss--Markov sources," IEEE Trans. Commun., vol. 38, pp. 82--93, Jan. 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....In general, the uncoded scheme can be modified to use a code, C, by changing the uniform scalar quanfizer, F(X ) into a quanfizer which quanfizes X to the nearest codeword in C. To illustrate this procedure we describe a construction using lattices based on trellis codes. As shown in [13] [14], 15] these lattices are superior to the integer lattice both in terms of reducing the quantization distortion and increasing the distance between codewords. We use the same encoding scheme as shown in Figure 3 except that the uniform scalar quantizer is replaced with a trellis quantizer ....

....to the integer lattice both in terms of reducing the quantization distortion and increasing the distance between codewords. We use the same encoding scheme as shown in Figure 3 except that the uniform scalar quantizer is replaced with a trellis quantizer designed for a uniform source as in [14]. The tag bits are then computed and embedded into the least significant bits of the codeword as before t and the resulting codeword is reconstructed to obtain the encoded signal Y . The decoding operation is the same as in the uncoded case except the scalar quantizer is replaced with the trellis ....

M. W. Marcellin and T. R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Transactions on Communications 38, pp. 82-93, January 1990.

....showing improvement with respect to the schemes of [12] The key point in [13] is to decompose the underlying operations into the lower dimensional subspaces. This decomposition avoids the exponential growth of the complexity. Trellis coded quantization (TCQ) introduced by Marcellin and Fischer [14], is based on applying the Ungerboeck notion of set partitioning to the partitions of a scalar quantizer where a trellis structure is used to prune the expanded number of quantization levels down to the desired encoding rate. Entropy constrained TCQ [15,16] is based on using a distance measure ....

....which was introduced in [18] In order to solve the resulting zero one integer program, we approximated it to a simple linear program. The result is a Lagrangian formulation adjoining the distortion and length of codewords. In order to achieve some packing gain, we combine the trellis encoding [14] and the new proposed FEVQ. For the important special case of a source with a monotonically decreasing density, we present a second method with negligible complexity, The rest of article is organized as follows: Section II contains a brief description of the linear program formulation and the ....

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M. W. Marcellin and T. R. Fischer, "Trellis-coded quantization of memoryless and Gauss-Markov sources," IEEE Trans. Commun. vol. 38, pp. 82--93, Nov. 1990.

....presence of long range dependencies of data as, for example, in a bimodal distribution. The goal of this paper is to use the adaptive quantization technique of [4] as a building block in quantization schemes, such as the scalar vector quan tizer (SVq) 5] and the trellis coded quantizer (TCq) [6], which are constructed based on an underlying scalar quantizer (USQ) Our motivation is to demon strate how adaptivity can be added and provide good results for popular quantization techniques such as TCQ and SVQ both of which have useful properties. We will introduce the adaptive scalar vector ....

....scalar quantizer (ECSQ) while quantizing the input vectors at a fixed rate and retaining structural and computational simplicity. The fixed rate approach is attractive to avoid the potential problems of transmitting variable rate quantizer data over channels with error. The popu larity of the TCQ [6] stems from the fact that it can outperform scalar quantizers with an encoding complexity which is still far less than that of vector quan tizers. The paper is organized as follows: In Section 2, we introduce the adaptive scalar quantization scheme used. Then we explain how adaptivity can be ....

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M. Marcellin and T. Fischer, "Trellis coded quantiza- tion of memoryless and Gauss-Markov sources," IEEE Trans. Commun, vol. 38, pp. 82-93, Jan. 1990.

....use an upper bound that is possibly not tight. This yields a R D bound for the Type 1 strategy that might not be representative of its actual performance. 4. Practical Implementation In this section, we develop a practical and feasible compression scheme based on trellis coded quantization (TCQ) [6]. TCQ allows efficient quantization of Gaussian random variables, and its optimization structure is easily adapted for lossy compression of Bernoulli random variables. Although TCQ encodes a sequence of variables one symbol at a time, it achieves compression efficiency by 11 10 00 01 10 00 ....

....I: Magnitude Optimization Stage I assumes that fB i g has been coded lossily to f b B i g, and it attempts to find the optimal trellis coding for fX i g with a desired rate of RX bits per symbol. Our algorithm uses the Ungerboeck trellis and is identical to the Gaussian TCQ algorithm presented in [6]; the only difference in our implementation is the distortion function which is minimized. At each node, we set b Y i = b X i ( b B i ; 1 b B i ) Instead of minimizing the distortion of the Gaussian itself, we choose a trellis path for f b X i g to minimize the distortion of f b Y i g. As in ....

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Michael W. Marcellin and Thomas R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Trans. on Communication, vol. 38, no. 1, January 1990.

....scheme. In general, the uncoded scheme can be modified to use a code, by changing the uniform scalar quanfizer, F(X) into a quantizer which quantizes X to the nearest codeword in . To illustrate this procedure we describe a construction using lattices based on trellis codes. As shown in [13] [14], 15] these lattices are superior to the integer lattice both in terms of reducing the quantization distortion and increasing the distance between codewords. We use the same encoding scheme as shown in Figure 3 except that the uniform scalar quantizer is replaced with a trellis quantizer ....

....to the integer lattice both in terms of reducing the quantization distortion and increasing the distance between codewords. We use the same encoding scheme as shown in Figure 3 except that the uniform scalar quantizer is replaced with a trellis quantizer designed for a uniform source as in [14]. The tag bits are then computed and embedded into the least significant bits of the codeword as before t and the resulting codeword is reconstructed to obtain the encoded signal Y L The decoding operation is the same as in the uncoded case except the scalar quantizer is replaced with the trellis ....

M. W. Marcellin and T. R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Transactions on Communications 38, pp. 82-93, January 1990.

.... encoder without feedback must be used for trellis generation [4] Fortunately, for every convolutional encoder with feedback, there exists a feedback free encoder, for which any given input bit can affect no more that 1 log 2 (N ) outputs, where N is defined as the number of trellis states [5]. A feedback free convolutional encoder produces a similar trellis with the branch bits labeled differently. This construction bounds the diversion from the correct trellis path caused by a bit error and allows the intended path to be recovered. 3. PREDICTION FILTERS The prediction filters used ....

M. W. Marcellin and T. R. Fischer. Trellis coded quantization of memoryless and Gauss-Markov sources. IEEE Transactions on Communications, 38(1):82--93, January 1990.

....data using a finite set of quantized symbols. Notions used in TCQ are similar to that of trellis coded modulation (TCM) widely used today for the design of voice line modems [96] In its simple form, TCQ exploits a codebook of codewords when coding a given source at a bit rate of bits symbol [97]. This is twice the number of codewords needed in a conventional quantization mechanism. It is possible to show that this doubling of number of codewords allows one to obtain nearly all the theoretical gain possible over a B bits symol Lloyd Max quantizers. The codebook is partitioned into ....

M. W. Marcellin and T. R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Trans. Commun., vol. COM-38, pp. 82--93, Jan. 1990.

....the uncoded scheme can be modi ed to use a code, C, by changing the uniform scalar quantizer, F (X n 1 ) into a quantizer which quantizes X n 1 to the nearest codeword in C. To illustrate this procedure we describe a construction using lattices based on trellis codes. As shown in [13] [14], 15] these lattices are superior to the integer lattice both in terms of reducing the quantization distortion and increasing the distance between codewords. We use the same encoding scheme as shown in Figure 3 except that the uniform scalar quantizer is replaced with a trellis quantizer ....

....to the integer lattice both in terms of reducing the quantization distortion and increasing the distance between codewords. We use the same encoding scheme as shown in Figure 3 except that the uniform scalar quantizer is replaced with a trellis quantizer designed for a uniform source as in [14]. The tag bits are then computed and embedded into the least signi cant bits of the codeword as before y and the resulting codeword is reconstructed to obtain the encoded signal Y n 1 . The decoding operation is the same as in the uncoded case except the scalar quantizer is replaced with the ....

M. W. Marcellin and T. R. Fischer, \Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Transactions on Communications 38, pp. 82-93, January 1990.

....The next question in robust image coding is how to choose the quantization scheme. There has been extensive research on quantization, and many schemes have been proposed, including various scalar quantizations and vector quantizations. One recently proposed scheme, trellis coded quantization (TCQ) [7], is shown to be relatively insensitive to the channel errors with moderate complexity. One bit error a ects no more than 1#log # N outputs, where N is the number of trellis states. Therefore TCQ is selected in this research as the quantization scheme and will be discussed in detail in Section ....

....Chen, Z. Sun Signal Processing: Image Communication 14 (1999) 575 584 577 Fig. 2. Four state trellis with four subsets. quantized to #r # #### ### bits sample, and reconstructed as #x( # #### ### . 3. Fixed rate uniform threshold trellis coded quantization Trellis coded quantization [7] is a nite state quantizer that employs a scalar codebook and a Viterbi encoding algorithm and labels the trellis branches with subsets of the codebook. This quantization scheme has low complexity with excellent mean square error (MSE) performance and is also insensitive to channel noise. TCQ ....

M.W. Marcellin, T.R. Fischer, Trellis coded quantization of memoryless and Gauss-Markov sources, IEEE Trans. Commun. 38 (1990) 82}93.

....as possible, while maintaining symmetry among the cosets. In the following, a trellis based partitioning is considered based on convolutional codes and set partitioning rules as in Trellis Coded Modulation (TCM) 11] Note that this is not to be confused with the Trellis Coded Quantization (TCQ) [12] framework: we still use scalar quantization but use trellises to build coset sequences. A bit stream with R bits unit time is used to partition 2 R 1 codevectors taking values in R. The set r is partitioned into 4 subsets (for the sake of clarity) as before. We use Ungerboeck s 4 state trellis ....

M. W. Marcellin and T. R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Trans. on Comm., vol. 38, pp. 82--93, Jan. 1990.

....are straightforward) Let us consider a simple fixed length, 8 level, uniform scalar quantizer with step size Delta = 1:0. Suppose R 1 = R 2 = 2 bits source sample. Let r = fr 0 ; r 1 ; r 7 g be the set of reconstruction levels of the scalar quantizer. Consider the Ungerboeck trellis [13, 14, 11] built on r. We have a finite state machine operating on a bit stream at the rate of 2 bits unit time. It has a rate 2=3 3 It can be noted that this problem is not the same as the quantization followed by coding in the binary space. 5 convolutional encoder, set partitioning rules and a mapping ....

M. W. Marcellin and T. R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Trans. on Comm., vol. 38, pp. 82--93, Jan. 1990.

....such as the Viterbi algorithm. Trellis Coded Quantization Trellis coded quantization, both scalar and vector, improves upon traditional trellis encoded systems by labeling the trellis branches with entire subcodebooks (or subsets ) rather than with individual reproduction levels [345] [344], 166] 167] 522] 343] 478] 514] The primary gain resulting is a reduction in encoder complexity for a given level of performance. As the original trellis encoding systems were motivated by convolutional channel codes with Viterbi decoders, trellis coded quantization was motivated by ....

M. W. Marcellin and T. R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Trans. Comm., vol. 38, pp. 82--93, January 1990.

....has been widely applied to many decoding and estimation applications in communications and signal processing. It can be used to perform maximum likelihood decoding for convolutional codes, or maximum likelihood estimation on Trellis Coded Modulation (TCM) 2] and Trellis Coded Quantization (TCQ) [3, 4]. Trellis Coded Modulation has evolved over the past decade as a combined coding and modulation technique for digital transmission over band limited channels. Trellis Coded Quantization was recently introduced as an effective scheme for encoding memoryless sources with low to moderate complexity. ....

M.W. Marcellin, T.R. Fischer, Trellis Coded Quantization of Memoryless and Gauss-Markov sources, IEEE Trans. Commun., v. 28, pp.92-93, Jan 1990.

....the best path (the maximum likelihood sequence) through a trellis in a dynamic manner through the study of the output sequence of a convolutional encoder received from a noisy channel. Other applications of the Viterbi algorithm are related to communications (TCM) 3] and image compression (TCQ) [4]. In order to establish a notation and briefly introduce the operation of the decoder, it is convenient to start by describing the encoder. A convolutional encoder consists of a shift register with K stages and n binary function generators. We denote as state the content of the K 1 least ....

M.W. Marcellin, T.R. Fischer, Trellis Coded Quantization of Memoryless and Gauss-Markov sources, IEEE Trans. Commun., vol 28, pp.92-93, Jan 1990.

....formal model that permits obtaining a regular and modular design solution optimal for a particular set of area and or speed constraints. 1. INTRODUCTION Trellis Coded Quantization (TCQ) was introduced as an effective scheme for encoding monochrome and color images with low to moderate complexity [1, 2]. For a given encoding rate of R bits per sample, it is possible to obtain nearly all of the theorically possible gain over the R bit per sample Loyd Max quantizer using an encoder with an output alphabet consisting of the output points of the R 1 bit per sample Lloy Max quantizer [2] The ....

.... complexity [1, 2] For a given encoding rate of R bits per sample, it is possible to obtain nearly all of the theorically possible gain over the R bit per sample Loyd Max quantizer using an encoder with an output alphabet consisting of the output points of the R 1 bit per sample Lloy Max quantizer [2]. The fixed rate TCQ applied to images can be improved with entropy constrained schemes and vector quantization. In TCQ systems an expanded codebook is partitioned into subsets, and these subsets are used to label the branches of an appropriate trellis, where a trellis is a transition diagram for ....

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M.W. Marcellin, T.R. Fischer, Trellis Coded Quantization of Memoryless and Gauss-Markov sources, IEEE Trans. Commun., v. 28, pp.92-93, Jan 1990.

....transforming the image with better frequency separation. In order to further speed up the overall algorithm, the factorized nonseparable fast filters, such as the ones proposed in [5] can be used. Trellis coded quantization (TCQ) has shown to be performing quite effectively for image compression [29, 30]. A move from the uniform scalar quantization towards the TCQ might be worthwhile. In order to achieve the objective of lossless compression, the lifting scheme [21] has been used [31] since the main hurdle in lossless compression otherwise is the floating point coefficients generated by the ....

M. W. Marcellin and T. R. Fischer. Trellis Coded Quantization of Memoryless and Gauss-Markov Sources. IEEE Transactions on Communications, 38:82--93, 1990.

....such as the Viterbi algorithm. Trellis Coded Quantization Trellis coded quantization, both scalar and vector, improves upon traditional trellis encoded systems by labeling the trellis branches with entire subcodebooks (or subsets ) rather than with individual reproduction levels [345] [344], 166] 167] 522] 343] 478] 514] The primary gain resulting is a reduction in encoder complexity for a given level of performance. As the original trellis encoding systems were motivated by convolutional channel codes with Viterbi decoders, trellis coded quantization was motivated by ....

M. W. Marcellin and T. R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Trans. Comm., vol. 38, pp. 82--93, January 1990.

....quantizers use a cartesian product reproduction codebook, which often can be rapidly searched. Examples include polar vector quantizers [291, 292, 47, 376, 345, 346, 339, 342, 344] mean removed vector quantizers [18, 19] and shape gain vector quantizers [313, 281] Trellis coded quantizers [249, 247, 123, 124, 248, 324] use a Viterbi algorithm encoder matched to a reproduction codebook with a trellis structure. Hierarchical table lookup vector quantizers provide fixed rate vector quantizers with minimal computational complexity, both lossy encoder and reproduction decoder being accomplished by table lookups [69, ....

....trellis search algorithm such as the Viterbi algorithm. Trellis Coded Quantization Trellis coded quantization, both scalar and vector, improve upon traditional encoded systems by labeling the trellis branches with entire subcodebooks (or subsets ) rather than with individual reproduction levels [249, 247, 123, 124, 369, 248, 324]. The primary gain resulting is a reduction in encoder complexity for a given level of performance. As the original trellis encoding systems were motivated by convolutional channel codes with Viterbi decoders, trellis coded quantization was motivated by Ungerboeck s enormously successful coded ....

M. W. Marcellin and T. R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Trans. on Communications, Vol. 38, pp. 82--93, 1990.

....then quantized and encoded as the compressed image [4] 3 III. PTCQ PTCQ is a combination of Trellis Coded Quantization (TCQ) and predictive coding. It is known that for a memoryless source, TCQ outperforms the optimal scalar quantizer and comes within 0. 21 dB of the distortion rate lower bound [5]. TCQ exploits the duality between modulation for digital communication systems and source coding while following the general principles of Trellis Coded Modulation (TCM) signal expansion described by Ungerboeck [6] 7] When Ungerboeck designed the trellis for TCM, convolutional encoders with ....

.... without feedback can be used to specify the branch selection [8] 9] Fortunately, for every convolutional encoder with feedback, there exists a feedback free encoder, for which any given input bit can affect no more that 1 log 2 (N) outputs, where N is defined as the number of trellis states [5]. A detailed description of the PTCQ algorithm can be found in the seminal PTCQ paper by Marcellin and Fischer [3] IV. Prediction Filters Within the schemes of interest, DPCM and PTCQ, two prediction filters are evaluated: the optimal linear predictor [4] and the nonlinear Ll filter [10] A. ....

M. W. Marcellin and T. R. Fischer. Trellis coded quantization of memoryless and Gauss-Markov sources. IEEE Transactions on Communications, 38(1):82--93, January 1990.

....even further by using vector trellis encoding of the excitation with the goal in mind to simultaneously increase the perceptual quality of the speech and to make it more robust to channel errors, thereby eliminating the need for error correction on the excitation stream. It has been hypothesised [101] that trellis encoding of the excitation is more robust to channel errors as a corrupt bit generates only a short range corruptive effect as it travels through the shift register (state generator) used in reconstructing the excitation sequence. This concept will be explored in more detail below. ....

....7.1 Introduction This chapter deals with characterization of errors in the synthetic speech waveform due to trellis encoding of the residual. Two types of errors are addressed. Firstly the errors caused by channel errors and secondly errors due to the encoding process. Marcellin and Fischer [101] comment that for their trellis source coding scheme, a received error only affects log 2 N 1 output symbols where N is the number of states in the trellis. This comment is no longer valid if a predictor, such as a speech synthesis filter, is driven by the trellis output symbols. In fact the ....

M.W. Marcellin and T.R. Fischer, "Trellis Coded Quantization of Memoryless and Gauss-Markov Sources", IEEE Trans. Comms. Vol 38, No. 1, pp. 82--93, Jan 1990.

....distortion is approximately 1500. This is most likely the result of inadequate motion compensation. This will be further discussed in Section 6. For this reason, subband Y bhl will not be differentially coded. 4 2 Entropy Constrained Trellis Coded Quantization Trellis coded quantization (TCQ) [12] is a source coding technique of low complexity and good distortion rate (meansquared error) performance that is based on ideas from Ungerboeck s trellis coded modulation [13] For a memoryless Gaussian source, Marcellin and Fischer [12] reported that TCQ had comparable or higher performance at ....

.... Trellis Coded Quantization Trellis coded quantization (TCQ) 12] is a source coding technique of low complexity and good distortion rate (meansquared error) performance that is based on ideas from Ungerboeck s trellis coded modulation [13] For a memoryless Gaussian source, Marcellin and Fischer [12] reported that TCQ had comparable or higher performance at integer rates than many coders, including trellis coders and ECSQ. However, for non uniform sources such as video sequences, the output symbols of a TCQ are not equally likely to occur. The desire to remove redundancy to lower the encoding ....

M.W. Marcellin and T.R. Fischer, "Trellis Coded Quantization of Memoryless and Gauss-Markov Sources," IEEE Trans. on Comm., vol. 38, pp. 82-93, Jan. 1990.

....PSNRs as compared to other classification based coding results. Our system has the potential for further improvements if the proposed adaptive scheme is combined with adaptive subband decompositions and or more sophisticated quantizers instead of simple UTQ, e.g. Trellis Coded Quantizer (TCQ) [22], 13] The paper is organized as follows. In Section II we introduce our proposed context based classification technique. An efficient use of forward adaptation to help context based classification is also discussed. In Section III we present techniques to allow on line model parameter estimation ....

M. W. Marcellin and T. R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Trans. Commun., vol. 38, pp. 82--93, Jan. 1990.

....but without their important area growth. 1 INTRODUCTION The Viterbi algorithm [1] is known to be an efficient method for the realization of maximum likelihood decoding of convolutional codes, maximum likelihood estimation of Trellis Coded Modulation (TCM) and Trellis Coded Quantization (TCQ) [2]. The Viterbi algorithm has been widely employed in the communication and image compression fields. For a given data sequence the Viterbi algorithm is used to find the most likely value in a weighted branches graph (trellis) The trellis in all these applications corresponds to the transition ....

M.W. Marcellin, T.R. Fischer, Trellis Coded Quantization of Memoryless and Gauss-Markov sources, IEEE Trans. Commun., v. 28, pp.92-93, Jan 1990.

....D D 0 2 0 2 D D D D 1 3 1 3 S S 0 1 FIGURE 21. TCQ sets and supersets inter and intra band information, and is based on a recursive estimation of the variance of the wavelet coe#cients. Both of them yield excellent coding results. 9. 1 Trellis coded quantization Trellis Coded Quantization (TCQ) [81] is a fast and e#ective method of quantizing random variables. Trellis coding exploits correlations between variables. More interestingly, it can use non rectangular quantizer cells that give it quantization e#ciencies not attainable by scalar quantizers. The central ideas of TCQ grew out of the ....

M. W. Marcellin and T. R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Transactions on Communications, vol. 38, pp. 82--93, January 1990.

....and these dependencies can be quite difficult to analyze understand. To this date, two approaches have been taken around this problem: MD Trellis Coded Quantization. Vaishampayan, Batllo and Calderbank [37] propose to design these two quantizers using the ideas of Trellis Coded Quantization (TCQ) [21]. MD TCQ has been studied by a number of authors [2, 17, 37] and gains of up to 2:2dB over the performance of the MD scalar quantizer have been reported. MD Lattice Vector Quantization. Servetto, Vaishampayan and Sloane [30] give an explicit construction of the index assignments to be used in ....

M. Marcellin and T. Fischer. Trellis Coded Quantization of Memoryless and Gauss-Markov Sources. IEEE Transactions on Communications, 38(1):82--93, 1990.

.... lattice quantizers also occur when pre post filters are incorporated in the coding process (see e.g. 23] or any recent article on the CELP technique for speech coding) Finally, this theorem can be extended to infinite dimensional lattice quantizers, i.e. to trellis coded quantizers (TCQ) [16], 3] having a linear structure. We denote by Q1 the trellis quantizer, and we replace the matrix A by the discrete time invariant shaping filter a = fa n g, n = 0; Sigma1; Sigma2; whose frequency response is A(f) P n a n Delta e Gammaj n2 f , Gamma1=2 f 1=2. The shaped ....

....x) where x is a source sequence, a ( Gamma1) fa ( Gamma1) n g are the coefficients of the inverse filter (i.e. the inverse Fourier transform of 1=A(f ) and denotes the convolution operator. Spectral shaping will also occur when Q1 is in the feedback loop, resembling predictive TCQ [16]. Now, let Z = Z Gamma1 ; Z 0 ; Z 1 : denote the (stationary) quantization noise process of Q1 . As shown in the appendix, in some sense Z is uniformly distributed over the infinite dimensional basic cell of Q1 . Clearly, if the quantization noise spectrum is S z (f ) then after ....

M.W. Marcellin and T.R. Fischer. Trellis coded quantization of memoryless and Gauss-Markov sources. IEEE Trans. Communications, 38:82--93, Jan. 1990.

....to look ahead to find the best available sequence of paths for a sequence of input vectors. This ensures better long run behavior at the expense of added coding delay. A variety of design algorithms for the supercodebook and subcodebooks based on lattices and on clustering have been considered [58, 101, 102]. Hierarchical vector quantization (HVQ) Although the adjective hierarchical has been applied to many variants of VQ (and other things) we here consider the technique of Chang et al. 28] for replacing nearest neighbor computations by a fast table lookup. We illustrate the idea for an 8 D ....

M. W. Marcellin and T. R. Fischer. Trellis coded quantization of memoryless and Gauss-Markov sources. IEEE Trans. Comm., COM-38:82--93, 1990.

....concept of Trellis Coded Quantization (TCQ) The other uses both inter and intra band information, and is based on a recursive estimation of the variance of the wavelet coefficients. Both of them yield excellent coding results. 1.8. 1 Trellis Coded Quantization Trellis Coded Quantization (TCQ) [76] is a fast and effective method of quantizing random variables. Trellis coding exploits correlations between variables. More interestingly, it can use non rectangular quantizer cells that give it quantization efficiencies not attainable by scalar quantizers. TCQ grew out of the ground breaking ....

M. W. Marcellin and T. R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Transactions on Communications, vol. 38, pp. 82--93, Jan. 1990.

.... 2 p k q (p) 6= 0 (12) 2. 2 Trellis Coded Quantization Trellis Coded Quantization (TCQ) is based on the ideas of an expanded signal set and set partioning from coded modulation, and has been shown to be an ecient method with modest complexity for encoding of memoryless sources [1]. As mentioned previously, TCQ is included in JPEG2000 Part II. A trellis is nothing more than a state transition diagram (that takes time into account) for a nite state machine. Trellises are used to study sequences of state transitions, or equivalently, sequences of states. A typical trellis ....

M. W. Marcellin, T. R. Fischer, Trellis coded quantization of memoryless and Gauss-Markov sources, IEEE Transactions on Communications 38 (1990) 82-93.

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M. W. Marcellin, T. R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Trans. Commun., January, 1990.

....(Ericsson Sweden) were also named as coeditors of the standard at that time. Results from round 1 core experiments were selected to modify WTCQ into the first release of the VM (VM 0) 2. 1 The WTCQ Algorithm The basic ingredients of the WTCQ algorithm are: the discrete wavelet transform, TCQ [9, 10] (using step sizes chosen via a Lagrangian rate allocation procedure) and binary arithmetic bitplane coding. The embedding principle [5, 6, 1, 7, 11, 12, 13] asserts the encoded bitstream should be ordered in a way that maximally reduces MSE 3 per bit transmitted. In WTCQ embedding is provided ....

M. W. Marcellin and T. R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Trans. on Commun., vol. 38, no. 1, pp. 82-93, Jan 1990.

....of memoryless Gaussian and Laplacian sources is presented. This work was supported in part by SAIC and by the National Science Foundation under Grant No. 9258374. I INTRODUCTION TCQ has been shown to be an effective technique for quantizing memoryless sources with low to moderate complexity [1]. The Viterbi algorithm [2] is used to pick the allowed trellis path that minimizes the mean square error (MSE) between the input data and output codewords. To improve the high rate performance of TCQ, ECTCQ was developed in [3, 4] and achieves MSE performance near (within about 0.5 dB) the ....

M. W. Marcellin and T. R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Trans. Commun., vol. COM-38, pp. 82--93, Jan. 1990.

....28.47 30.56 33.12 36.55 42.01 sar1 16.37 16.93 17.75 19.14 21.45 26.52 tb64 13.78 15.56 18.56 21.44 25.97 32.47 Table 2: Average first order entropy (across subbands) for the S P [4] encoding algorithm at rate R bits pixel. 3. 2 Zerotree coding using trellis coded quantization Since TCQ is known [12] to provide granular gain over scalar quantization, we investigate an image encoding algorithm using a (near) unitary subband decomposition, an across subband scanning pattern, and uniform threshold TCQ followed by arithmetic coding [13] 3.2.1 Zerotree search pattern An across subband scanning ....

....and the ZTTCQ encoding scheme uses only one uniform quantization stepsize. Subband dependent quantization stepsizes were studied, but experimental results show that this produces negligible PSNR improvement. The TCQ encoding is done using the Viterbi algorithm to search the respective trellis [12], but using the uniform threshold technique described in [3] 3.2.2 Dead zone Figure 3 illustrates the TCQ quantization regions with a deadzone described by the two parameters ff and Delta. Delta is the uniform quantization stepsize used in all subbands, and ff specifies the size of the ....

M. W. Marcellin and T. R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Trans. Comm., vol. COM-38, pp. 82-93, Jan. 1990.

....using separate codebooks for the real and imaginary components of the complex SAR phase history data. The splitting algorithm and the Generalized Lloyd Algorithm [5] were used to generate the VQ codebooks. Trellis coded quantization (TCQ) techniques for data compression have been well documented [6, 7, 8]. TCQ has previously been shown to be highly effective for compressing imagery of all types. In this paper, TCQ techniques are applied to the problem of downlink data rate reduction in SAR systems. We demonstrate that TCQ techniques provide superior SNR performance to BAQ and VQ for SAR phase ....

....and an allowable sequence of codewords. This sequence of codewords can be specified using R bits at each stage of the trellis (one bit to specify the path, and R Gamma 1 bits to select the codeword in the corresponding subset) For further details on TCQ, see the paper by Marcellin and Fischer [6]. 1 D 2 0 D 1 0 D 2 1 D 0 1 D 3 0 D 3 1 D 1 0 D 2 1 D 0 0 D 0 1 D 2 0 D 3 0 D 0 1 D 1 0 D 1 1 D 3 State 1 0 2 3 4 5 6 7 Figure 1: Eight State Trellis. 0 D D D 1 2 3 D l l l l l l l l D D D 0 D 3 2 1 Figure 2: Subset Partitioning for R=2 bits sample. III IMPLEMENTATION OF ....

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M. W. Marcellin and T. R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Trans. Commun., vol. COM-38, pp. 82--93, Jan. 1990.

....such that the codebook contains no more than 2 mr code vectors. In [12] Laroia and Farvardin developed another high performance fixed rate quantizer called trellis based scalar vector quantization (TB SVQ) which combines the scalar vector quantizer (SVQ) with trellis coded quantization (TCQ) [13], 14] 15] The TB SVQ can outperform all other known fixed rate quantizers at reasonable complexity. While SVQ and TB SVQ provide very attractive fixed rate quantizers, they are not compatible with image video compression standards like JPEG and MPEG [16] 17] 18] In general, JPEG ....

M. W. Marcellin and T. R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources", IEEE Transactions on Communications, vol. 38, pp. 82--93, Jan. 1990. December 8, 18

....the wavelet coefficients, and then apply sophisticated entropy coding methods to the quantized coefficients. This paper describes both spectral selection and successive approximation type methods for use with a wavelet TCQ (WTCQ) coder. TCQ has been shown to be an effective method for quantization [10]. It has been used in wavelet based image coding applications, and produced excellent results [5, 11, 12, 7] We investigate the performance of the WTCQ coder with different bit rate allocation and entropy coding techniques. Comparisons with other results from the literature show that our system ....

.... significant bit (i.e. a bit plane coder) similar to those employed in [3, 4] or the hierarchical tree based method used in the Set Partitioning in Hierarchial Trees (SPIHT) coder [6] III Trellis Coded Quantization TCQ has been shown to be an efficient method for encoding memoryless sources [10]. It is based on the ideas of an expanded signal set and set partitioning from coded modulation, and has excellent mean squared error (MSE) performance with modest complexity. For encoding a memoryless source using TCQ a scalar codebook is partitioned into four disjoint subsets D 0 ; D 1 ; D 2 ; D ....

M. W. Marcellin and T. R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources", IEEE Transactions on Communications, vol. 38, pp. 82--93, Jan. 1990.

....: 7 modulo 2 arithmetics) the sequential decoder can be replaced by a Viterbi decoder which is applied to the trellis of the imaginary scrambler. This method of working in a trellis of an imaginary scrambler has simultaneously come up in source coding theory, named trellis coded quantization [28]. All other components of the proposed shaping precoding scheme remain the same; in particular the described advantages are kept. But some additional remarks on the decoder seem to be necessary: For trellis shaping and trellis precoding it is essential to guarantee a continuous path integrity in ....

M.W. Marcellin, T.R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources." IEEE Trans. on Commun. , COM-38, pp. 82--93, Jan. 1990.

....that generate encoded files [8] 10] 2. TRELLIS CODED QUANTIZATION TCQ has been shown to be an effective technique for quantizing memoryless sources with low to moderate This work was supported in part by SAIC and by the National Science Foundation under Grant No. 9258374. complexity [11]. For a given sequence of data to quantize, the Viterbi algorithm [12] is used to pick the allowed trellis path that minimizes the mean square error (MSE) between the input data and output codewords. ECTCQ was developed in [13, 14] and achieves MSE performance near (within about 0.5 dB) the ....

M. W. Marcellin and T. R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Trans. Commun., vol. COM-38, pp. 82--93, Jan. 1990.

No context found.

M. W. Marcellin and T. R. Fischer, "Trellis coded quantization of memoryless and Gauss--Markov sources," IEEE Trans. Commun., vol. 38, pp. 82--93, Jan. 1990.

No context found.

M. W. Marcellin and T. R. Fischer, "Trellis-coded quantization of memoryless and Gauss-Markov sources," IEEE Trans. Commun., vol. 38, pp. 82--93, 1993.

No context found.

M. W. Marcellin and Fischer T. R., "Trellis coded quantization of memoryless and gauss-markov sources," IEEE Trans. on Commun., vol. COM-38, pp. 82--93, Jan. 1990.

No context found.

M. W. Marcellin and T. R. Fischer. Trellis coded quantization of memoryless and gauss-markov sources. IEEE Transactions on Communications, 38(1):82-- 93, January 1990.

No context found.

M.W. Marcellin and T.R. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Trans. Commun., vol. 38, pp. 82-93, Jan. 1990.

No context found.

M. W. Marcellin and T. R. Fischer. Trellis coded quantization of memoryless and gauss-markov sources. IEEE Trans. on Comm., 38(1):82--93, January 1990.

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