This thesis proposes two constructive methods of approaching the Shannon limit very closely. Interestingly, these two methods operate in opposite regions, one has a block length of one and the other has a block length approaching infinity. The first approach is based on novel memoryless joint source-channel coding schemes. We first show some examples of sources and channels where no coding is optimal for all values of the signal-to-noise ratio (SNR). When the source bandwidth is greater than the channel bandwidth, joint coding schemes based on space-filling curves and other families of curves are proposed. For uniform sources and modulo channels, our coding scheme based on space-filling curves operates within 1.1 dB of Shannon’s rate-distortion bound. For Gaussian sources and additive white Gaussian noise (AWGN) channels, we can achieve within 0.9 dB of the rate-distortion bound. The second scheme is based on low-density parity-check (LDPC) codes. We first demonstrate that we can translate threshold values of an LDPC code between channels accurately using a simple mapping. We develop some models for density evolution

A space-filling curve is a linear traversal of a discrete finite multi-dimensional space. In order that this traversal be useful in many applications, the curve should preserve "locality". We quantify "locality" and bound the locality of multi-dimensional space-filling curves. Classic Hilbert spacefilling curves come close to achieving optimal locality. EDICS: IP 3.1 Corresponding author: Craig Gotsman Dept. of Computer Science Technion, Haifa 32000 Israel Tel: +972-4-294336 Fax: +972-4-294353 Email: gotsman@cs.technion.ac.il # A preliminary version of this work was presented at the IEEE International Conference on Pattern Recognition, Jerusalem, 1994. 1 1

The efficiency of many data structures and algorithms relies on "locality-preserving" indexing schemes for meshes. We concentrate on the case in which the maximal distance between two mesh nodes indexed i and j shall be a slow-growing function of ji jj. We present a new 2-D indexing scheme we call H-indexing , which has superior (possibly optimal) locality in comparison with the well-known Hilbert indexings. H-indexings form a Hamiltonian cycle and we prove that they are optimally locality-preserving among all cyclic indexings. We provide fairly tight lower bounds for indexings without any restriction. Finally, illustrated by investigations concerning 2-D and 3-D Hilbert indexings, we present a framework for mechanizing upper bound proofs for locality.

We present algorithms for rendering realistic images of large terrains and their implementation on a parallel computer for rapid production of terrain animation sequences. By "large" terrains, we mean the use of datasets too large to be contained in RAM, a fact which significantly influences the design of rendering algorithms, and has not been addressed in most other works on this subject. To achieve a good speed without quality degradation, we use a hybrid ray-casting and projection technique, incorporating quadtree subdivision techniques and filter using precomputed bit masks. Hilbert space-filling curves determine the image pixel rendering order, fully exploiting spatial coherence. A parallel version of the algorithm is presented, based on an architecture implemented on a Meiko parallel computer. The architecture is designed to relieve dataflow bottlenecks caused by slow interprocessor communications, and exploit temporal image coherence. Using our parallel system, incorporating 26 processors, we are able to generate a full color terrain image at video resolution, without noticable aliasing artifacts, every 2 seconds, including I/O and communication overheads. The speedup of the system is linear with the number of processors.

A framework for studying texture in general, and for texture mixing in particular, is presented in this paper. The work follows concepts from universal type classes and universal simulation. Based on the well-known Lempel and Ziv (LZ) universal compression scheme, the universal type class of a one dimensional sequence is defined as the set of possible sequences of the same length which produce the same dictionary (or parsing tree) with the classical LZ incremental parsing algorithm. Universal simulation is realized by sampling uniformly from the universal type class, which can be efficiently implemented. Starting with a source texture image, we use universal simulation to synthesize new textures that have, asymptotically, the same statistics as the source texture, yet have as much uncertainty as possible, in the sense that they are sampled from the broadest pool of possible sequences that comply with the statistical constraint. When considering two or more textures, a parsing tree is constructed for each one, and samples from the trees are randomly interleaved according to pre-defined proportions, thus obtaining a mixed texture. As with single texture synthesis, the k-th order statistics of this mixture, for any k, approach, asymptotically, the weighted mixture of the k-th order statistics of each individual texture used in the mixing. We present the underlying principles of universal types, universal simulation, and their extensions and application to mixing two or more textures with pre-defined proportions.