Uniform Scalar Quantization Research Articles

Optimal rate allocation for entropy-coded uniform scalar quantization of dependent sources in nonbinary hypothesis testing

We propose a closed-form rate allocation scheme (RAS) for entropy-coded uniform scalar quantization of dependent sources in classification problems. The proposed RAS is applicable to nonbinary classification with piecewise monotonic unquantized Bayes decision boundaries. The RAS is also extended to joint compression and classification.

On the Uniform Quantization of a Class of Sparse Sources

We consider the uniform scalar quantization of a class of mixed distributed memoryless sources, namely sources having a Bernoulli Generalized Gaussian (BGG) distribution. Both for low and high resolutions, asymptotic expressions of the distortion for a pth-order moment error measure, and close approximations of the entropy are provided for these sources. Operational rate-distortion functions at high bit rate and their slope factors at low bit rate are derived. The dependence of these results on p and the distribution parameters as well as the relation to the Shannon optimal rate-distortion bound are then discussed. The application of these results to transform coding in two simple cases is finally highlighted.

A Bound on the MSE of Oversampled Dithered Quantization With Feedback

We analyze the behavior of the mean squared error (MSE) achievable by oversampled, uniform scalar quantization using feedback, pre- and post-filters of unrestricted order, when encoding wide-sense stationary discrete-time random sources having (possibly) unbounded support. Our results are based upon the use of subtractively dithered uniform scalar quantizers. We consider the number of quantization levels, N, to be given and fixed, which lends itself to fixed-rate encoding, and focus on the cases in which N is insufficient to avoid overload. In order to guarantee the stability of the closed-loop, we consider the use of a clipper before the scalar quantizer. Our results are valid for zero-mean sources having independent innovations whose moments satisfy some mild requirements, which are met by infinite-support distributions such as Gaussian and Laplacian. We show that, for fixed N, the MSE can be made to decay with the oversampling ratio lambda as O(e-c 0 lambda 1/3) when lambda tends to infinity, where c 0 [0.5(N-1)]2/3. We note that the latter bound is asymptotic in lambda but not in N, and that it includes clipping errors.

An Efficient Spectrum Sensing Scheme for Cognitive Radio

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The paper combines distributed source coding and compressive sampling for efficient spectrum estimation. Two or more cognitive radios sample the spectrum compressively and independently compress their observations using multiterminal source coding. A central hub collects compressed streams from these radios before performing joint multiterminal source decoding followed by iterative signal reconstruction. Simulation results are provided for two radios performing practical multiterminal source coding with uniform scalar quantization and systematic turbo codes for Slepian-Wolf coding and error protection. </para>

A Low-Complexity Universal Scheme for Rate-Constrained Distributed Regression Using a Wireless Sensor Network

We propose a scheme for rate-constrained distributed nonparametric regression using a wireless sensor network. The scheme is universal across a wide range of sensor noise models, including unbounded and nonadditive noise; it has low complexity, requiring simple operations such as uniform scalar quantization with dither and message passing between neighboring nodes in the network, and attains minimax optimality for regression functions in common smoothness classes. We present theoretical results on the tradeoff between the compression rate, communication complexity of encoding, and the MSE and demonstrate empirical performance of the scheme using simulations.

On the Operational Rate-Distortion Performance of Uniform Scalar Quantization-Based Wyner–Ziv Coding of Laplace–Markov Sources

Wyner-Ziv (WZ) coding has recently been proposed as a low encoding complexity alternative to traditional DPCM coding for compression of sources with memory, in particular, in applications like multimedia compression. The viability of this alternative approach clearly depends on the compression performance of WZ coding compared to that of DPCM coding. In an attempt to understand the performance gap between WZ coding and DPCM coding, this paper studies the operational rate-distortion performance of WZ coding, using uniform scalar quantization followed by perfect Slepian-Wolf coding, for compression of a Laplace-Markov (LM) source. It is shown that at low rates or for weakly correlated LM sources, WZ coding is indeed a competitive alternative to DPCM coding. However, at high rates the performance gap becomes non-negligible for strongly correlated LM sources. In order to reduce the gap at high rates, a hybrid approach that combines DPCM coding and WZ coding is further investigated. It is shown that the hybrid approach is indeed competitive to DPCM coding at all rates even for strongly correlated LM sources.

Lloyd–Max's Algorithm Implementation in Speech Coding Algorithm Based on Forward Adaptive Technique

In this paper a detail analysis of speech coding algorithm based on forward adaptive technique is carried out. We consider an algorithm that works on frame-by-frame basis, where a frame consists of a certain number of speech samples. Buffering frame-by-frame an estimation of the gain defined as squared root of the frame variance is enabled. The information about the gain (side information) and the code book of a nonadaptive quantizer, which is designed for the unit variance case of the input signal, are further used when designing an adaptive quantizer. In such a way better quantizer adaptation to the varying input statistics is provided. Observe that the goal of this paper is to investigate the preference that for the wide range of variance change could be achieved when implementing in the forward adaptive speech coding algorithm, the recently developed effective method for the Lloyd-Max's algorithm initialization, which provides optimal Lloyd-Max's quantizer performances for the unit variance case of the input signal. We destine to consider the speech coding algorithm based on forward adaptive technique since the backward adaptation provides SQNR (signal to quantization noise ratio) within 1 dB of the forward adaptation. We provide theoretical and experimental results (performances of our algorithm) which are compared with the optimal results. Additionally, we discuss the performances of speech coding schemes designed according to G. 711 standard and we point out the benefits that can be achieved by using our algorithm. Finally, in order to find better solution for implementation of the proposed algorithm in practice we consider the performances of our algorithm when log-uniform as well as uniform scalar quantizer are used for gain quantizing.

Uniform and Non-Uniform Optimum Scalar Quantizers Performances: A Comparative Study

The aim of this research is to investigate source coding, the representation of information source output by finite R bits/symbol. The performance of optimum quantisers subject to an entropy constraint has been studied. The definitive work in this area is best summarised by Shannon' s source coding theorem, that is, a source with entropy H can be encoded with arbitrarily small error probability at any rate R (bits/source output) as long as R>H. Conversely, If R<H the error probability will be driven away from zero, independent of the complexity of the encoder and the decoder employed. In this context, the main objective of engineers is however to design the optimum code. Unfortunately, the rate­distortion theorem does not provide the recipe for such a design. T he t heorem d oes, h owever, p rovide t he t heoretical l imit s o t hat w e know how close we are to the optimum. The full understanding of the theorem also helps in setting the direction to achieve such an optimum. In this research, we have investigated the performances of two practical scalar quantisers, i.e., a Lloyd­Max quantiser and the uniformly defined one and also a well­known entropy coding scheme, i.e., Huffman coding against their theoretically attainable optimum performance due to Shannon' s limit R. I t ha s be en s hown that our uniformly defined quantiser could demonstrate superior performance. The performance improvements, in fact, are more noticeable at higher bit rates.

Wavelet-based compression algorithm for still omnidirectional 3d integral images

Three dimensional (3D) integral imaging is a method that allows the display of full colour images with continuous parallax within a wide viewing zone. Due to the significant quantity of data required to represent a captured 3D integral image with high resolution, image compression becomes mandatory for the storage and transmission of integral images. This paper investigates the use of 2D discrete wavelet transform (2D-DWT) for the compression of omnidirectional 3D integral images (OII). The method requires the extraction of different viewpoint images from the integral image. A single viewpoint image is constructed by extracting one pixel from each microlens, then each viewpoint image is decomposed using a 2D-DWT. The resulting array of coefficients contains several frequency bands. The lower frequency bands of the viewpoint images are assembled and compressed using a 3D discrete cosine transform (3D-DCT) followed by Huffman coding. Whereas, the remaining higher bands are fed directly into a quantisation process followed by arithmetic coding. Simulations are performed on a set of several grey level 3D OII using a uniform scalar quantizer with deadzone. It was found that the algorithm achieves better rate-distortion performance and reconstructs the images with much better image quality at very low bit rates than previously reported 3D-DCT-based scheme.

The right-hemispheric auditory cortex in humans is sensitive to degraded speech sounds

We investigated how degraded speech sounds activate the auditory cortices of the left and right hemisphere. To degrade the stimuli, we introduce uniform scalar quantization, a controlled and replicable manipulation, not used before, in cognitive neuroscience. Three Finnish vowels (/a/, /e/ and /u/) were used as stimuli for 10 participants in magnetoencephalography registrations. Compared with the original vowel sounds, the degraded sounds increased the amplitude of the right-hemispheric N1m without affecting the latency whereas the amplitude and latency of the N1m in the left hemisphere remained unaffected. Although the participants were able to identify the stimuli correctly, the increased degradation led to increased reaction times which correlated positively with the N1m amplitude. Thus, the auditory cortex of right hemisphere might be particularly involved in processing degraded speech and possibly compensates for the poor signal quality by increasing its activity.

Quantization Methods for Equal Gain Transmission With Finite Rate Feedback

We consider the design and analysis of quantizers for equal gain transmission (EGT) systems with finite rate feedback-based communication in flat-fading multiple input single output (MISO) systems. EGT is a beamforming technique that maximizes the MISO channel capacity when there is an equal power-per-antenna constraint at the transmitter, and requires the feedback of t-1 phase angles, when there are t antennas at the transmitter. In this paper, we contrast two popular approaches for quantizing the phase angles: vector quantization (VQ) and scalar quantization (SQ). On the VQ side, using the capacity loss with respect to EGT with perfect channel information at transmitter as performance metric, we develop a criterion for designing the beamforming codebook for quantized EGT (Q-EGT). We also propose an iterative algorithm based on the well-known generalized Lloyd algorithm, for computing the beamforming vector codebook. On the analytical side, we study the performance of Q-EGT and derive closed-form expressions for the performance in terms of capacity loss and outage probability in the case of i.i.d. Rayleigh flat-fading channels. On the SQ side, assuming uniform scalar quantization and i.i.d. Rayleigh flat-fading channels, we derive the high-resolution performance of quantized EGT and contrast the performance with that of VQ. We find that although both VQ and SQ achieve the same rate of convergence (to the capacity with perfect feedback) as the number of feedback bits B increases, there exists a fixed gap between the two

Joint thresholding and quantizer selection for compression of medical ultrasound images in the wavelet domain

This paper introduces a simple and efficient technique for compression of medical ultrasound (US) images in the wavelet domain. The statistics of subband wavelet coefficients are modelled using the generalized Gaussian distribution (GGD). By exploiting these statistics, a uniform scalar quantizer is designed which adapts very well to the changing statistics of the signal across various subbands and scales. To increase the quantization performance, a threshold is chosen adaptively to zero-out the insignificant wavelet coefficients in the detail subbands before quantization. A distinctive feature of the proposed technique is that it unifies the two approaches to image adaptive coding: rate-distortion (R-D) optimized quantizer selection and R-D optimal thresholding, in order to increase the compression performance of the coder. The operational R-D criterion used for joint optimization is derived in the minimum description length (MDL) framework. The experimental results show that the joint R-D optimization leads to significant improvement in the compression performance of the proposed coder, named JTQ-WV, over the best state-of-the-art image coder, SPIHT. For example, the coding of US images at 0.25 bpp by JTQ-WV yields a PSNR gain of 1.0 dB over the benchmark SPIHT.

Scalable compression of audio and other signals

Disclosed are scalable quantizers for audio and other signals characterized by a non-uniform, perception-based distortion metric, that operate in a common companded domain which includes both the base-layer and one or more enhancement-layers. The common companded domain is designed to permit use of the same unweighted MSE metric for optimal quantization parameter selection in multiple layers, exploiting the statistical dependence of the enhancement-layer signal on the quantization parameters used in the preceding layer (52). One embodiment features an asymptotically optimal entropy coded uniform scalar quantizer. Another embodiment (52, 54) is an improved bit rate scalable multi-layer Advanced Audio Coder (AAC) which extends the scalability of the asymptotically optimal entropy coded uniform scalar quantizer to systems with non-uniform base-layer quantization, selecting the enhancement-layer quantization methodology to be used in a particular band based on the preceding layer quantization coefficients.

The Validity of the Additive Noise Model for Uniform Scalar Quantizers

A uniform scalar quantizer with small step size, large support, and midpoint reconstruction levels is frequently modeled as adding orthogonal noise to the quantizer input. This paper rigorously demonstrates the asymptotic validity of this model when the input probability density function (pdf) is continuous and satisfies several other mild conditions. Specifically, as step size decreases, the correlation between input and quantization error becomes negligible relative to the mean-squared error (MSE). The model is even valid when the input density is discontinuous at the origin, but discontinuities elsewhere can prevent the correlation from being negligible. Though this invalidates the additive model, an asymptotic formula for the correlation is found in terms of the step size and the heights and positions of the discontinuities. For a finite support input density, such as uniform, it is shown that the support of the uniform quantizer can be matched to that of the density in ways that make the correlation approach a variety of limits. The derivations in this paper are based on an analysis of the asymptotic convergence of cell centroids to cell midpoints. This convergence is fast enough that the centroids and midpoints induce the same asymptotic MSE, but not fast enough to induce the same correlations.

Size of the Dictionary in Matching Pursuit Algorithm

The matching pursuit algorithm has been successfully applied in many areas such as data compression and pattern recognition. The performance of matching pursuit is closely related to the selection of the dictionary. In this paper, we propose an algorithm to estimate the optimal dictionary distribution ratio and discuss the decay of the norm of residual signal in matching pursuit when the coefficients are quantized by a uniform scalar quantizer. It is proposed that if the approximation error E and the dimension of the space N are given, the optimal size of the dictionary and the optimal quantizer step should be obtained by minimizing the number of bits required to store the matching pursuit representation of any signal in the space to satisfy the error bound.

Suboptimality of the Karhunen–LoÈve Transform for Transform Coding

We examine the performance of the Karhunen-Loeve transform (KLT) for transform coding applications. The KLT has long been viewed as the best available block transform for a system that orthogonally transforms a vector source, scalar quantizes the components of the transformed vector using optimal bit allocation, and then inverse transforms the vector. This paper treats fixed-rate and variable-rate transform codes of non-Gaussian sources. The fixed-rate approach uses an optimal fixed-rate scalar quantizer to describe the transform coefficients; the variable-rate approach uses a uniform scalar quantizer followed by an optimal entropy code, and each quantized component is encoded separately. Earlier work shows that for the variable-rate case there exist sources on which the KLT is not unique and the optimal quantization and coding stage matched to a "worst" KLT yields performance as much as 1.5 dB worse than the optimal quantization and coding stage matched to a "best" KLT. In this paper, we strengthen that result to show that in both the fixed-rate and the variable-rate coding frameworks there exist sources for which the performance penalty for using a "worst" KLT can be made arbitrarily large. Further, we demonstrate in both frameworks that there exist sources for which even a best KLT gives suboptimal performance. Finally, we show that even for vector sources where the KLT yields independent coefficients, the KLT can be suboptimal for fixed-rate coding.

Asymptotic Analysis of Optimal Uniform Two-Dimensional

The simple and complete asymptotical analysis is given for a uniform two-dimensional quantizer for Laplace source with the respect to the mean-square error. The significant gain over the optimum uniform scalar quantization is obtained (about 6.8 dB for the rate of 8 bits per sample). The obtained gain using rectangular cells can even be compared to the boundary gain in highdimensional space. Ill. 1, bibl. 6 (in English; summaries in Lithuanian, Russian, English).

Image retrieval using color histograms generated by Gauss mixture vector quantization

Image retrieval based on color histograms requires quantization of a color space. Uniform scalar quantization of each color channel is a popular method for the reduction of histogram dimensionality. With this method, however, no spatial information among pixels is considered in constructing the histograms. Vector quantization (VQ) provides a simple and effective means for exploiting spatial information by clustering groups of pixels. We propose the use of Gauss mixture vector quantization (GMVQ) as a quantization method for color histogram generation. GMVQ is known to be robust for quantizer mismatch, which motivates its use in making color histograms for both the query image and the images in the database. Results show that the histograms made by GMVQ with a penalized log-likelihood (LL) distortion yield better retrieval performance for color images than the conventional methods of uniform quantization and VQ with squared error distortion.

Wavelet-based fixed and embedded L-infinite-constrained image coding

∞A new wavelet-based L∞-constrained fixed and embedded coding technique is proposed. The embedded bit stream can be truncated for any desired distortion bound at a corresponding bit rate, so that the target upper bound on the elements of the reconstruction error signal is guaranteed. The original image can also be coded up to a fixed a priori user-defined distortion bound, ranging up to lossless coding. A lifting-based wavelet decorrelating transform is employed on the original image, and exact relations are established between spatial and wavelet domain distortions. The wavelet coefficients are quantized by symmetric uniform quantizers for fixed-distortion coding and by families of embedded uniform deadzone scalar quantizers for embedded coding. The quantized coefficients are finally losslessly encoded using a quadtree-based coding algorithm. Any floating-point lifting-based wavelet transform can be used, and a few of the popular wavelet transforms included in the JPEG2000 verification model are worked out as examples. We compare other L∞-constrained coding schemes and show that our proposed coder offers in addition a fully embedded L∞-oriented bit stream. We illustrate also that the proposed coder retains the same capabilities as the state-of-the-art embedded wavelet-based codecs, while providing superior compression results and embeddedness with respect to the L distortion measure.

Effects of natural images on the detectability of simple and compound wavelet subband quantization distortions.

Quantization of the coefficients within a discrete wavelet transform subband gives rise to distortions in the reconstructed image that are localized in spatial frequency and orientation and are spatially correlated with the image. We investigated the detectability of these distortions: Contrast thresholds were measured for both simple and compound distortions presented in the unmasked paradigm and against two natural-image maskers. Simple and compound distortions were generated through uniform scalar quantization of one or two subbands. Unmasked detection thresholds for simple distortions yielded contrast sensitivity functions similar to those reported for 1-octave Gabor patches. Detection thresholds for simple distortions presented against two natural-image backgrounds revealed that thresholds were elevated across the frequency range of 1.15-18.4 cycles per degree with the greatest elevation for low-frequency distortions. Unmasked thresholds for compound distortions revealed relative sensitivities of 1.1-1.2, suggesting that summation of responses to wavelet distortions is similar to summation of responses to gratings. Masked thresholds for compound distortions revealed relative sensitivities of 1.5-1.7, suggesting greater summation when distortions are masked by natural images.