Information Rates of Densely Sampled Data: Distributed Vector Quantization and Scalar Quantization With Transforms for Gaussian Sources
This paper establishes rates attainable by several lossy schemes for coding a continuous parameter source to a specified mean-squared-error distortion based on sampling at asymptotically large rates. First, a densely sampled, spatiotemporal, stationary Gaussian source is distributively encoded. The Berger-Tung bound to the distributed rate-distortion function and three convergence theorems are used to obtain an upper bound, expressed in terms of the source spectral density, to the smallest attainable rate at asymptotically large sampling rates. The bound is tighter than that recently obtained by Kashyap 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 for encoders based on scalar quantization and Slepian-Wolf distributed lossless coding. The second scheme is transform coding with scalar coefficient quantization. A new generalized transform coding analysis, as well as the aforementioned convergence theorems, is used to find the smallest attainable rate at asymptotically large sampling rates 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 As a corollary, the final result pursues an approach, originally proposed by Berger, to show that the inverse water-pouring formula for the rate-distortion function can be attained at high sampling rates by transform coding with ideal vector quantization to encode the coefficients. Also established in the paper are relations between operational rate-distortion and distortion-rate functions for a continuous parameter source and those for the discrete parameter source that results from sampling.