Vivek K. Goyal

Quantized overcomplete expansions in IR/sup N/: analysis, synthesis, and algorithms

Coefficient quantization has peculiar qualitative effects on representations of vectors in IR with respect to overcomplete sets of vectors. These effects are investigated in two settings: frame expansions (representations obtained by forming inner products with each element of the set) and matching pursuit expansions (approximations obtained by greedily forming linear combinations). In both cases, based on the concept of consistency, it is shown that traditional linear reconstruction methods are suboptimal, and better consistent reconstruction algorithms are given. The proposed consistent reconstruction algorithms were in each case implemented, and experimental results are included. For frame expansions, results are proven to bound distortion as a function of frame redundancy r and quantization step size for linear, consistent, and optimal reconstruction methods. Taken together, these suggest that optimal reconstruction methods will yield O(1/r/sup 2/) mean-squared error (MSE), and that consistency is sufficient to insure this asymptotic behavior. A result on the asymptotic tightness of random frames is also proven. Applicability of quantized matching pursuit to lossy vector compression is explored. Experiments demonstrate the likelihood that a linear reconstruction is inconsistent, the MSE reduction obtained with a nonlinear (consistent) reconstruction algorithm, and generally competitive performance at low bit rates.

Generalized multiple description coding with correlating transforms

Multiple description (MD) coding is source coding in which several descriptions of the source are produced such that various reconstruction qualities are obtained from different subsets of the descriptions. Unlike multiresolution or layered source coding, there is no hierarchy of descriptions; thus, MD coding is suitable for packet erasure channels or networks without priority provisions. Generalizing work by Orchard, Wang, Vaishampayan and Reibman (see Proc IEEE Int. Conf. Image Processing, vol.I, Santa Barbara, CA, p.608-11, 1997), a transform-based approach is developed for producing M descriptions of an N-tuple source, M/spl les/N. The descriptions are sets of transform coefficients, and the transform coefficients of different descriptions are correlated so that missing coefficients can be estimated. Several transform optimization results are presented for memoryless Gaussian sources, including a complete solution of the N=2, M=2 case with arbitrary weighting of the descriptions. The technique is effective only when independent components of the source have differing variances. Numerical studies show that this method performs well at low redundancies, as compared to uniform MD scalar quantization.

Multiple description coding with many channels

An achievable region for the L-channel multiple description coding problem is presented. This region generalizes two-channel results of El Gamal and Cover (1982) and of Zhang and Berger (1987). It further generalizes three-channel results of Gray and Wyner (1974) and of Zhang and Berger. A source that is successively refinable on chains is shown to be successively refinable on trees. A new outer bound on the rate-distortion (RD) region for memoryless Gaussian sources with mean squared error distortion is also derived. The achievable region meets this outer bound for certain symmetric cases.

Multiple description transform coding of images

Generalized multiple description coding (GMDC) is source coding for multiple channels such that a decoder which receives an arbitrary subset of the channels may produce a useful reconstruction. This paper reports on applications of two recently proposed methods for GMDC to image coding. The first produces statistically correlated streams such that lost streams can be estimated from the received data. The second uses quantized frame expansions and hence is conceptually similar to block channel coding, except it is done prior to quantization.

Multiple description transform coding: robustness to erasures using tight frame expansions

We propose the use of a tight frame expansion for generalized multiple description coding. This method is conceptually very similar to using a block channel code, except it is done on continuous-valued data prior to quantization. A sketch of the distortion analysis is given.

Quantized frame expansions as source-channel codes for erasure channels

Quantized frame expansions are proposed as a method for generalized multiple description coding, where each quantized coefficient is a description. Whereas previous investigations have revealed the robustness of frame expansions to additive noise and quantization, this represents a new application of frame expansions. The performance of a system based on quantized frame expansions is compared to that of a system with a conventional block channel code. The new system performs well when the number of lost descriptions (erasures on an erasure channel) is hard to predict.

Optimal multiple description transform coding of Gaussian vectors

Multiple description coding (MDC) is source coding for multiple channels such that a decoder which receives an arbitrary subset of the channels may produce a useful reconstruction. Orchard et al. (1997) proposed a transform coding method for MDC of pairs of independent Gaussian random variables. This paper provides a general framework which extends multiple description transform coding (MDTC) to any number of variables and expands the set of transforms which are considered. Analysis of the general case is provided, which can be used to numerically design optimal MDTC systems. The case of two variables sent over two channels is analytically optimized in the most general setting where channel failures need not have equal probability or be independent. It is shown that when channel failures are equally probable and independent, the transforms used in Orchard et al. are in the optimal set, but many other choices are possible. A cascade structure is presented which facilitates low-complexity design, coding, and decoding for a system with a large number of variables.

Transform coding with backward adaptive updates

The Karhunen-Loeve transform (KLT) is optimal for transform coding of a Gaussian source. This is established for all scale-invariant quantizers, generalizing previous results. A backward adaptive technique for combating the data dependence of the KLT is proposed and analyzed. When the adapted transform converges to a KLT, the scheme is universal among transform coders. A variety of convergence results are proven.

Recursive consistent estimation with bounded noise

Estimation problems with bounded, uniformly distributed noise arise naturally in reconstruction problems from over complete linear expansions with subtractive dithered quantization. We present a simple recursive algorithm for such bounded-noise estimation problems. The mean-square error (MSE) of the algorithm is almost O(1/n/sup 2/), where n is the number of samples. This rate is faster than the O(1/n) MSE obtained by standard recursive least squares estimation and is optimal to within a constant factor.

Robust Predictive Quantization: Analysis and Design Via Convex Optimization

Predictive quantization is a simple and effective method for encoding slowly-varying signals that is widely used in speech and audio coding. It has been known qualitatively that leaving correlation in the encoded samples can lead to improved estimation at the decoder when encoded samples are subject to erasure. However, performance estimation in this case has required Monte Carlo simulation. Provided here is a novel method for efficiently computing the mean-squared error performance of a predictive quantization system with erasures via a convex optimization with linear matrix inequality constraints. The method is based on jump linear system modeling and applies to any autoregressive moving average (ARMA) signal source and any erasure channel described by an aperiodic and irreducible Markov chain. In addition to this quantification for a given encoder filter, a method is presented to design the encoder filter to minimize the reconstruction error. Optimization of the encoder filter is a nonconvex problem, but we are able to parameterize with a single scalar a set of encoder filters that yield low MSE. The design method reduces the prediction gain in the filter, leaving the redundancy in the signal for robustness. This illuminates the basic tradeoff between compression and robustness.