Multiresolution Quantization Research Articles

Asymptotically Scale-Invariant Multi-Resolution Quantization

A multi-resolution quantizer is a sequence of quantizers where the output of a coarser quantizer can be deduced from the output of a finer quantizer. In this paper, we propose an asymptotically scale-invariant multi-resolution quantizer, which performs uniformly across any choice of average quantization step, when the length of the range of input numbers is large. Scale invariance is especially useful in worst case or adversarial settings, ensuring that the performance of the quantizer would not be affected greatly by small changes of storage or error requirements. We also show that the proposed quantizer achieves a tradeoff between rate and error that is arbitrarily close to the optimum.

Designing Optimal Multiresolution Quantizers with Error Detecting Codes

This paper investigates the design of optimal multiresolution vector quantizers for broadcast channels with cyclic redundancy checks (CRC). Given a CRC-coded broadcast system with multiresolution vector quantization (MRVQ), a closed-form formula for the weighted end-to-end distortion (EED) is first derived under random index assignment. The closed-form expression is then further utilized to identify necessary optimality conditions to minimize the EED, from which an iterative algorithm is proposed for quantization design. Experiments conducted under both the point-to-point and broadcast channels demonstrate that for a wide range of channel error probability, inclusion of CRC significantly reduces the EED without sacrificing bandwidth. Further analyses are conducted to determine the best tradeoff between bits allocated for source quantization and CRC error detection.

Design of Fixed and Adaptive Companding Quantizer with Variable-Length Codeword for Memoryless Gaussian Source

The problem we address in this paper is the design of a quantizer that in comparison to the classical fixed-rate scalar quantizers provides more sophisticated bit rate reduction while restricting the class of quantizers to be scalar. We propose a switched variable-length code (VLC) optimal companding quantizer composed of two optimal companding scalar quantizers, the inner and the outer one, both designed for the memoryless Gaussian source of unit variance. Quantizers composing the proposed quantizer have a different codebook sizes and a different compressor functions. Particularly, we assume a smaller size of the inner quantizer's codebook in order to provide assignment of the shorter codewords to the high probability low amplitude speech samples belonging to the support region of the inner quantizer. We study the influence of codebook size of the inner and the outer quantizer on the Signal to Quantization Noise Ratio (SQNR). In such a manner the conclusion of the proposed quantizer significance in speech compression is distinctly shown in the paper. For the proposed quantizer model and its forward adaptive version the SQNR robustness analysis in a wide variance range is also presented in the paper. It is shown that our multi-resolution quantizer can satisfy G.712 Recommendation for high-quality quantization at the bit rate of 6.3 bit/sample achieving the compression of 1.7 bit/sample over the G.711 quantizer.

Improvement in Spectral Properties of Quantization Noise of Harmonic Signal Using Multiresolution Quantization

Multiresolution quantization (MQ) is used in modern equipment designed for wideband spectrum measurements with high dynamic range. Such time-domain measurement systems significantly reduce measurement time. Only basic theoretical analyses of the possible influences of MQ on the fidelity of the measured spectrum have been performed until today. Therefore, the real contribution of the method to the quality of measurement should be clarified. In this paper, a new model of quantization noise is proposed for MQ. It was derived from a model of power spectral density previously used for uniform quantization. A system with two different quantization steps used for the quantization of harmonic input signal is considered. The model suitability for such conditions is proved by simulation and confronted with experimental results. Multiresolution measurements were acquired using a dual-channel data acquisition card and personal-computer postprocessing.

Multiresolution, perceptual and vector quantization based video codec

This paper presents a novel Multiresolution, Perceptual and Vector Quantization (MPVQ) based video coding scheme. In the intra-frame mode of operation, a wavelet transform is applied to the input frame and decorrelates it into its frequency subbands. The coefficients in each detail subband are pixel quantized using a uniform quantization factor divided by the perceptual weighting factor of that subband. The quantized coefficients are finally coded using a quadtree-coding algorithm. Perceptual weights are specifically calculated for the centre of each detail subband. In the inter-frame mode of operation, a Displaced Frame Difference (DFD) is first generated using an overlapped block motion estimation/compensation technique. A wavelet transform is then applied on the DFD and converts it into its frequency subbands. The detail subbands are finally vector quantized using an Adaptive Vector Quantization (AVQ) scheme. To evaluate the performance of the proposed codec, the proposed codec and the adaptive subband vector quantization coding scheme (ASVQ), which has been shown to outperform H.263 at all bitrates, were applied to six test sequences. Experimental results indicate that the proposed codec outperforms the ASVQ subjectively and objectively at all bit rates.

Monotonicity-Based Fast Algorithms for MAP Estimation of Markov Sequences Over Noisy Channels

In this correspondence, we study algorithmic approach to solving the problem of maximum a posteriori (MAP) estimation of Markov sequences transmitted over noisy channels, which is also known as the MAP decoding problem. For the class of memoryless binary channels that produce independent substitution and erasure errors, the MAP sequence estimation problem can be formulated and solved as one of the longest path in a weighted directed acyclic graph. But for algorithm efficiency, we transform the graph problem to one of matrix search. If the underlying matrix is totally monotone, then the complexity of MAP sequence estimation can be greatly reduced. We give a sufficient condition for the matrix induced by MAP sequence estimation to be totally monotone, which is indeed the case if the input sequence is Gaussian Markov. Under this condition, the complexity of MAP decoding can be reduced from O(N/sup 2/M) to O(NM), where N is the size of source alphabet and M is the length of input sequence. Furthermore, for Markov sequences of fixed-length code we propose a block parsing strategy to reduce the complexity of MAP sequence estimation to O(M+N/sup 2/M/logM) or to O(M+NM/logM), depending on if the total monotonicity holds. Another significance of this correspondence lies in the applicability of the presented algorithmic approach, which has been thoroughly studied in computer science literature, to many other discrete optimization problems encountered in both source and channel coding, ranging from optimal multiresolution and multiple-description quantizer design, to context quantization for minimum conditional entropy, and to optimal packetization with uneven error protection.

Algorithms for optimal multi-resolution quantization

Multi-resolution quantization is a way of constructing a progressively refinable description of a discrete random variable. The underlying discrete optimization problem is to minimize an expected distortion over all refinement levels weighted by the probability or importance of the descriptions of different resolutions. This research is motivated by the application of multimedia communications via variable-rate channels. We propose an O( rN 2) time and O( N 2log N) space algorithm to design a minimum-distortion quantizer of r levels for a random variable drawn from an alphabet of size N. It is shown that for a very large class of distortion measures the objective function of optimal multi-resolution quantization satisfies the convex Monge property. The efficiency of the proposed algorithm hinges on the convex Monge property. But our algorithm is simpler (even though of the same asymptotic complexity) than the well-known SMAWK fast matrix search technique, which is the best existing solution to the quantization problem. For exponential random variables our approach leads to a solution of even lower complexity: O( rN) time and O( Nlog N) space, which outperforms all the known algorithms for optimal quantization in both multi- and single-resolution cases. We also generalize the multi-resolution quantization problem to a graph problem, for which our algorithm offers an efficient solution.

Universal multiresolution source codes

A multiresolution source code is a single code giving an embedded source description that can be read at a variety of rates and thereby yields reproductions at a variety of resolutions. The resolution of a source reproduction here refers to the accuracy with which it approximates the original source. Thus, a reproduction with low distortion is a "high-resolution" reproduction while a reproduction with high distortion is a "low-resolution" reproduction. This paper treats the generalization of universal lossy source coding from single-resolution source codes to multiresolution source codes. Results described in this work include new definitions for weakly minimax universal, strongly minimax universal, and weighted universal sequences of fixed- and variable-rate multiresolution source codes that extend the corresponding notions from lossless coding and (single-resolution) quantization to multiresolution quantizers. A variety of universal multiresolution source coding results follow, including necessary and sufficient conditions for the existence of universal multiresolution codes, rate of convergence bounds for universal multiresolution coding performance to the theoretical bound, and a new multiresolution approach to two-stage universal source coding.