Universal lossless source coding with the Burrows Wheeler transform

The Burrows Wheeler transform (1994) is a reversible sequence transformation used in a variety of practical lossless source-coding algorithms. In each, the BWT is followed by a lossless source code that attempts to exploit the natural ordering of the BWT coefficients. BWT-based compression schemes are widely touted as low-complexity algorithms giving lossless coding rates better than those of the Ziv-Lempel codes (commonly known as LZ'77 and LZ'78) and almost as good as those achieved by prediction by partial matching (PPM) algorithms. To date, the coding performance claims have been made primarily on the basis of experimental results. This work gives a theoretical evaluation of BWT-based coding. The main results of this theoretical evaluation include: (1) statistical characterizations of the BWT output on both finite strings and sequences of length n → ∞, (2) a variety of very simple new techniques for BWT-based lossless source coding, and (3) proofs of the universality and bounds on the rates of convergence of both new and existing BWT-based codes for finite-memory and stationary ergodic sources. The end result is a theoretical justification and validation of the experimentally derived conclusions: BWT-based lossless source codes achieve universal lossless coding performance that converges to the optimal coding performance more quickly than the rate of convergence observed in Ziv-Lempel style codes and, for some BWT-based codes, within a constant factor of the optimal rate of convergence for finite-memory source

Scanning and Sequential Decision Making for Multi-Dimensional Data - Part I: the Noiseless Case

We investigate the problem of scanning and prediction ("scandiction", for short) of multidimensional data arrays. This problem arises in several aspects of image and video processing, such as predictive coding, for example, where an image is compressed by coding the error sequence resulting from scandicting it. Thus, it is natural to ask what is the optimal method to scan and predict a given image, what is the resulting minimum prediction loss, and whether there exist specific scandiction schemes which are universal in some sense. Specifically, we investigate the following problems: First, modeling the data array as a random field, we wish to examine whether there exists a scandiction scheme which is independent of the field's distribution, yet asymptotically achieves the same performance as if this distribution was known. This question is answered in the affirmative for the set of all spatially stationary random fields and under mild conditions on the loss function. We then discuss the scenario where a non-optimal scanning order is used, yet accompanied by an optimal predictor, and derive bounds on the excess loss compared to optimal scanning and prediction. This paper is the first part of a two-part paper on sequential decision making for multi-dimensional data. It deals with clean, noiseless data arrays. The second part deals with noisy data arrays, namely, with the case where the decision maker observes only a noisy version of the data, yet it is judged with respect to the original, clean data.Comment: 46 pages, 2 figures. Revised version: title changed, section 1 revised, section 3.1 added, a few minor/technical corrections mad

Results on the Redundancy of Universal Compression for Finite-Length Sequences

In this paper, we investigate the redundancy of universal coding schemes on smooth parametric sources in the finite-length regime. We derive an upper bound on the probability of the event that a sequence of length $n$, chosen using Jeffreys' prior from the family of parametric sources with $d$ unknown parameters, is compressed with a redundancy smaller than $(1-\epsilon)\frac{d}{2}\log n$ for any $\epsilon>0$. Our results also confirm that for large enough $n$ and $d$, the average minimax redundancy provides a good estimate for the redundancy of most sources. Our result may be used to evaluate the performance of universal source coding schemes on finite-length sequences. Additionally, we precisely characterize the minimax redundancy for two--stage codes. We demonstrate that the two--stage assumption incurs a negligible redundancy especially when the number of source parameters is large. Finally, we show that the redundancy is significant in the compression of small sequences.Comment: accepted in the 2011 IEEE International Symposium on Information Theory (ISIT 2011

Lossy compression of discrete sources via Viterbi algorithm

We present a new lossy compressor for discrete-valued sources. For coding a sequence $x^n$, the encoder starts by assigning a certain cost to each possible reconstruction sequence. It then finds the one that minimizes this cost and describes it losslessly to the decoder via a universal lossless compressor. The cost of each sequence is a linear combination of its distance from the sequence $x^n$ and a linear function of its $k^{\rm th}$ order empirical distribution. The structure of the cost function allows the encoder to employ the Viterbi algorithm to recover the minimizer of the cost. We identify a choice of the coefficients comprising the linear function of the empirical distribution used in the cost function which ensures that the algorithm universally achieves the optimum rate-distortion performance of any stationary ergodic source in the limit of large $n$, provided that $k$ diverges as $o(\log n)$. Iterative techniques for approximating the coefficients, which alleviate the computational burden of finding the optimal coefficients, are proposed and studied.Comment: 26 pages, 6 figures, Submitted to IEEE Transactions on Information Theor

A Universal Scheme for Wyner–Ziv Coding of Discrete Sources

We consider the Wyner–Ziv (WZ) problem of lossy compression where the decompressor observes a noisy version of the source, whose statistics are unknown. A new family of WZ coding algorithms is proposed and their universal optimality is proven. Compression consists of sliding-window processing followed by Lempel–Ziv (LZ) compression, while the decompressor is based on a modification of the discrete universal denoiser (DUDE) algorithm to take advantage of side information. The new algorithms not only universally attain the fundamental limits, but also suggest a paradigm for practical WZ coding. The effectiveness of our approach is illustrated with experiments on binary images, and English text using a low complexity algorithm motivated by our class of universally optimal WZ codes