Search

We propose a general greedy algorithm for binary de Bruijn sequences, called Generalized Prefer-Opposite Algorithm, and its modifications. By identifying specific feedback functions and initial states, we demonstrate that most previously-known greedy algorithms that generate binary de Bruijn sequences are particular cases of our algorithm. Nanyang Technological University Accepted version Nanyang Technological University Grant M4080456 supports the research carried out by M. F. Ezerman and A. A. Fahreza.

Holographic representations of data encode information in packets of equal importance that enable progressive recovery. The quality of recovered data improves as more and more packets become available. This progressive recovery of the information is independent of the order in which packets become available. Such representations are ideally suited for distributed storage and for the transmission of data packets over networks with unpredictable delays and or erasures. Several methods for holographic representations of signals and images have been proposed over the years and multiple description information theory also deals with such representations. Surprisingly, however, these methods had not been considered in the classical framework of optimal least-squares estimation theory, until very recently. We develop a least-squares approach to the design of holographic representation for stochastic data vectors, relying on the framework widely used in modeling signals and images., Comment: 22 pages with 6 figures

The focus of this work is to show how to combine Zech’s logarithms and each of the cycle joining and cross-join pairing methods to construct binary de Bruijn sequences. Basic implementations are supplied as proofs of concept. The cycles, in the cycle joining method, are typically generated by a linear feedback shift register. We prove a crucial characterization that determining Zech’s logarithms is equivalent to identifying conjugate pairs shared by any two distinct cycles. This speeds up the task of building a connected adjacency subgraph that contains all vertices of the complete adjacency graph. Distinct spanning trees in either graph correspond to cyclically inequivalent de Bruijn sequences. As the cycles are being joined, guided by the conjugate pairs, we track the changes in the feedback function. We show how to produce certificates of existence for spanning trees of certain types to conveniently handle large order cases. The characterization of conjugate pairs via Zech’s logarithms, as positional markings, is then adapted to identify cross-join pairs. A modified m-sequence is initially used, for ease of generation. The process can be repeated on each of the resulting de Bruijn sequences. Most prior constructions in the literature measure the complexity of the corresponding bit-by-bit algorithms. Our approach is different. We aim first to build a connected adjacency subgraph that is certified to contain all of the cycles as vertices. The ingredients are computed just once and concisely stored. Simple strategies are offered to keep the complexities low as the order grows.

Holographic representations of data enable distributed storage with progressive refinement when the stored packets of data are made available in any arbitrary order. In this paper, we propose and test patch-based transform coding holographic sensing of image data. Our proposal is optimized for progressive recovery under random order of retrieval of the stored data. The coding of the image patches relies on the design of distributed projections ensuring best image recovery, in terms of the $\ell_2$ norm, at each retrieval stage. The performance depends only on the number of data packets that has been retrieved thus far. Several possible options to enhance the quality of the recovery while changing the size and number of data packets are discussed and tested. This leads us to examine several interesting bit-allocation and rate-distortion trade offs, highlighted for a set of natural images with ensemble estimated statistical properties. Nanyang Technological University National Research Foundation (NRF) Published version This research is supported by National Research Foundation, Singapore, and Israel Science Foundation under their joint program NRF2015-NRF-ISF001-2597. It is also supported by Nanyang Technological University, Grant Number M4080456.