RESUMO
Diversity "multiple description" (MD) source coding promises graceful degradation in the presence of a priori unknown number of erased packets in the channel. A simple coding scheme for the case of two packets consists of oversampling the source by a factor of two and delta-sigma quantization. This approach was applied successfully to JPEG-based image coding over a lossy packet network, where the interpolation and splitting into two descriptions are done in the discrete cosine transform (DCT) domain. Moreover, unlike the classical source-channel separation approach - which is designed for a predetermined number of erasures (say, K out of N ), hence its distortion does not improve when the channel behaves better than expected - an MD coding scheme aims to achieve a better reconstruction quality when more or all the N descriptions are received at the decoder side. The extension to a larger number of descriptions, however, suffers from noise amplification whenever the received descriptions form a non-uniform sampling pattern. In this work, we examine inter- and intra-block interpolation methods, and show how noise amplification can be reduced by redesigning the interpolation filter at the encoder. Specifically, for a given total coding rate, we demonstrate that an "irregular" interpolation filter is robust to the pattern of received packets over all ( K out of N ) patterns, with some degradation relative to low-pass (LP) interpolation in the case where all N packets arrived. We provide experimental results comparing LP and irregular interpolation filters, and examine the effect of noise shaping on the trade-off between the central distortion (receiving all packets) and side distortion (receiving K packets).
RESUMO
We draw a random subset of [Formula: see text] rows from a frame with [Formula: see text] rows (vectors) and [Formula: see text] columns (dimensions), where [Formula: see text] and [Formula: see text] are proportional to [Formula: see text] For a variety of important deterministic equiangular tight frames (ETFs) and tight non-ETFs, we consider the distribution of singular values of the [Formula: see text]-subset matrix. We observe that, for large [Formula: see text], they can be precisely described by a known probability distribution-Wachter's MANOVA (multivariate ANOVA) spectral distribution, a phenomenon that was previously known only for two types of random frames. In terms of convergence to this limit, the [Formula: see text]-subset matrix from all of these frames is shown to be empirically indistinguishable from the classical MANOVA (Jacobi) random matrix ensemble. Thus, empirically, the MANOVA ensemble offers a universal description of the spectra of randomly selected [Formula: see text] subframes, even those taken from deterministic frames. The same universality phenomena is shown to hold for notable random frames as well. This description enables exact calculations of properties of solutions for systems of linear equations based on a random choice of [Formula: see text] frame vectors of [Formula: see text] possible vectors and has a variety of implications for erasure coding, compressed sensing, and sparse recovery. When the aspect ratio [Formula: see text] is small, the MANOVA spectrum tends to the well-known Marcenko-Pastur distribution of the singular values of a Gaussian matrix, in agreement with previous work on highly redundant frames. Our results are empirical, but they are exhaustive, precise, and fully reproducible.
