Arithmetic Dynamics

This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic expansions of reals and vectors that have a "dynamical" sense. This means precisely that they (semi-) conjugate a given continuous (or measure-preserving) dynamical system and a symbolic one. The classes of dynamical systems and their codings considered in the paper involve: (1) Beta-expansions, i.e., the radix expansions in non-integer bases; (2) "Rotational" expansions which arise in the problem of encoding of irrational rotations of the circle; (3) Toral expansions which naturally appear in arithmetic symbolic codings of algebraic toral automorphisms (mostly hyperbolic). We study ergodic-theoretic and probabilistic properties of these expansions and their applications. Besides, in some cases we create "redundant" representations (those whose space of "digits" is a priori larger than necessary) and study their combinatorics.Comment: 45 pages in Latex + 3 figures in ep

Fast and Efficient Entropy Coding Architectures for Massive Data Compression

The compression of data is fundamental to alleviating the costs of transmitting and storing massive datasets employed in myriad fields of our society. Most compression systems employ an entropy coder in their coding pipeline to remove the redundancy of coded symbols. The entropy-coding stage needs to be efficient, to yield high compression ratios, and fast, to process large amounts of data rapidly. Despite their widespread use, entropy coders are commonly assessed for some particular scenario or coding system. This work provides a general framework to assess and optimize different entropy coders. First, the paper describes three main families of entropy coders, namely those based on variable-to-variable length codes (V2VLC), arithmetic coding (AC), and tabled asymmetric numeral systems (tANS). Then, a low-complexity architecture for the most representative coder(s) of each family is presented-more precisely, a general version of V2VLC, the MQ, M, and a fixed-length version of AC and two different implementations of tANS. These coders are evaluated under different coding conditions in terms of compression efficiency and computational throughput. The results obtained suggest that V2VLC and tANS achieve the highest compression ratios for most coding rates and that the AC coder that uses fixed-length codewords attains the highest throughput. The experimental evaluation discloses the advantages and shortcomings of each entropy-coding scheme, providing insights that may help to select this stage in forthcoming compression systems

Compressive sensing based image processing and energy-efficient hardware implementation with application to MRI and JPG 2000

In the present age of technology, the buzzwords are low-power, energy-efficient and compact systems. This directly leads to the date processing and hardware techniques employed in the core of these devices. One of the most power-hungry and space-consuming schemes is that of image/video processing, due to its high quality requirements. In current design methodologies, a point has nearly been reached in which physical and physiological effects limit the ability to just encode data faster. These limits have led to research into methods to reduce the amount of acquired data without degrading image quality and increasing the energy consumption. Compressive sensing (CS) has emerged as an efficient signal compression and recovery technique, which can be used to efficiently reduce the data acquisition and processing. It exploits the sparsity of a signal in a transform domain to perform sampling and stable recovery. This is an alternative paradigm to conventional data processing and is robust in nature. Unlike the conventional methods, CS provides an information capturing paradigm with both sampling and compression. It permits signals to be sampled below the Nyquist rate, and still allowing optimal reconstruction of the signal. The required measurements are far less than those of conventional methods, and the process is non-adaptive, making the sampling process faster and universal. In this thesis, CS methods are applied to magnetic resonance imaging (MRI) and JPEG 2000, which are popularly used imaging techniques in clinical applications and image compression, respectively. Over the years, MRI has improved dramatically in both imaging quality and speed. This has further revolutionized the field of diagnostic medicine. However, imaging speed, which is essential to many MRI applications still remains a major challenge. The specific challenge addressed in this work is the use of non-Fourier based complex measurement-based data acquisition. This method provides the possibility of reconstructing high quality MRI data with minimal measurements, due to the high incoherence between the two chosen matrices. Similarly, JPEG2000, though providing a high compression, can be further improved upon by using compressive sampling. In addition, the image quality is also improved. Moreover, having a optimized JPEG 2000 architecture reduces the overall processing, and a faster computation when combined with CS. Considering the requirements, this thesis is presented in two parts. In the first part: (1) A complex Hadamard matrix (CHM) based 2D and 3D MRI data acquisition with recovery using a greedy algorithm is proposed. The CHM measurement matrix is shown to satisfy the necessary condition for CS, known as restricted isometry property (RIP). The sparse recovery is done using compressive sampling matching pursuit (CoSaMP); (2) An optimized matrix and modified CoSaMP is presented, which enhances the MRI performance when compared with the conventional sampling; (3) An energy-efficient, cost-efficient hardware design based on field programmable gate array (FPGA) is proposed, to provide a platform for low-cost MRI processing hardware. At every stage, the design is proven to be superior with other commonly used MRI-CS methods and is comparable with the conventional MRI sampling. In the second part, CS techniques are applied to image processing and is combined with JPEG 2000 coder. While CS can reduce the encoding time, the effect on the overall JPEG 2000 encoder is not very significant due to some complex JPEG 2000 algorithms. One problem encountered is the big-level operations in JPEG 2000 arithmetic encoding (AE), which is completely based on bit-level operations. In this work, this problem is tackled by proposing a two-symbol AE with an efficient FPGA based hardware design. Furthermore, this design is energy-efficient, fast and has lower complexity when compared to conventional JPEG 2000 encoding

Reordering Rows for Better Compression: Beyond the Lexicographic Order

Sorting database tables before compressing them improves the compression rate. Can we do better than the lexicographical order? For minimizing the number of runs in a run-length encoding compression scheme, the best approaches to row-ordering are derived from traveling salesman heuristics, although there is a significant trade-off between running time and compression. A new heuristic, Multiple Lists, which is a variant on Nearest Neighbor that trades off compression for a major running-time speedup, is a good option for very large tables. However, for some compression schemes, it is more important to generate long runs rather than few runs. For this case, another novel heuristic, Vortex, is promising. We find that we can improve run-length encoding up to a factor of 3 whereas we can improve prefix coding by up to 80%: these gains are on top of the gains due to lexicographically sorting the table. We prove that the new row reordering is optimal (within 10%) at minimizing the runs of identical values within columns, in a few cases.Comment: to appear in ACM TOD

Communities and beyond: mesoscopic analysis of a large social network with complementary methods

Community detection methods have so far been tested mostly on small empirical networks and on synthetic benchmarks. Much less is known about their performance on large real-world networks, which nonetheless are a significant target for application. We analyze the performance of three state-of-the-art community detection methods by using them to identify communities in a large social network constructed from mobile phone call records. We find that all methods detect communities that are meaningful in some respects but fall short in others, and that there often is a hierarchical relationship between communities detected by different methods. Our results suggest that community detection methods could be useful in studying the general mesoscale structure of networks, as opposed to only trying to identify dense structures.Comment: 11 pages, 10 figures. V2: typos corrected, one sentence added. V3: revised version, Appendix added. V4: final published versio