Information Rates of Densely Sampled Gaussian Data: Distributed Vector Quantization and Scalar Quantization with Transforms∗

Motivated by the question of the efficiency of dense sensor networks for sampling and encoding spatial random fields, this paper investigates the rates attainable by several lossy schemes for coding a Gaussian random field to a specified mean-squared error distortion based on sampling at asymptotically large rates. In the first, a densely sampled, spatio-temporal, stationary Gaussian source is distributively encoded. The Berger-Tung upper bound to the distributed rate-distortion function, the Szego asymptotic eigenvalue theorem, and an integral convergence theorem are used to obtain an upper bound, expressed in terms of the source spectral density, to the smallest attainable rate at asymptotically high sampling densities. The bound is tighter than that recently obtained by Kashyap et al. Both indicate that with ideal distributed lossy coding, dense sensor networks can efficiently sense and convey a field, in contrast to the negative result obtained by Marco et al. for encoders based on timeand space-invariant scalar quantization and Slepian-Wolf distributed lossless coding. The second scheme is transform coding, i.e., an orthogonal transform with some family of scalar quantizers to encode the coefficients. A new generalized transform coding analysis, as well as the asymptotic eigenvalue and integral convergence theorems, are used to find the smallest attainable rate at asymptotically high sampling densities in terms of the source spectral density and the operational rate-distortion function of the family of quantizers, which in contrast to previous analyses, need not be convex. The result shows that when a transform is used, scalar quantization need not cause the poor performance found by Marco et al. As a corollary, the final result pursues an approach, originally proposed by Berger, to show that the well known inverse water-pouring formula for the rate-distortion function can be attained at high sampling densities by transform coding with ideal vector quantization to encode the coefficients.