arXiv subjects

Ralf Koetter

Publications and source records attributed to Ralf Koetter.

At least 19 recordsLinked to original sources

arXiv2013

A Factorized Variational Technique for Phase Unwrapping in Markov Random Fields

Some types of medical and topographic imaging device produce images in which the pixel values are "phase-wrapped", i.e. measured modulus a known scalar. Phase unwrapping can be viewed as the problem of inferring the number of shifts between each and every pair of neighboring pixels, subject to an a priori preference for smooth surfaces, and subject to a zero curl constraint, which requires that the shifts must sum to 0 around every loop. We formulate phase unwrapping as a mean field inference problem in a Markov network, where the prior favors the zero curl constraint. We compare our mean field technique with the least squares method on a synthetic 100x100 image, and give results on a 512x512 synthetic aperture radar image from Sandia National Laboratories.

cs.CV↗

arXiv2010

A Theory of Network Equivalence, Parts I and II

A family of equivalence tools for bounding network capacities is introduced. Part I treats networks of point-to-point channels. The main result is roughly as follows. Given a network of noisy, independent, memoryless point-to-point channels, a collection of communication demands can be met on the given network if and only if it can be met on another network where each noisy channel is replaced by a noiseless bit pipe with throughput equal to the noisy channel capacity. This result was known previously for the case of a single-source multicast demand. The result given here treats general demands -- including, for example, multiple unicast demands -- and applies even when the achievable rate region for the corresponding demands is unknown in the noiseless network. In part II, definitions of upper and lower bounding channel models for general channels are introduced. By these definitions, a collection of communication demands can be met on a network of independent channels if it can be met on a network where each channel is replaced by its lower bounding model andonly if it can be met on a network where each channel is replaced by its upper bounding model. This work derives general conditions under which a network of noiseless bit pipes is an upper or lower bounding model for a multiterminal channel. Example upper and lower bounding models for broadcast, multiple access, and interference channels are given. It is then shown that bounding the difference between the upper and lower bounding models for a given channel yields bounds on the accuracy of network capacity bounds derived using those models. By bounding the capacity of a network of independent noisy channels by the network coding capacity of a network of noiseless bit pipes, this approach represents one step towards the goal of building computational tools for bounding network capacities.

cs.IT↗

arXiv2010

The Re-Encoding Transformation in Algebraic List-Decoding of Reed-Solomon Codes

The main computational steps in algebraic soft-decoding, as well as Sudan-type list-decoding, of Reed-Solomon codes are bivariate polynomial interpolation and factorization. We introduce a computational technique, based upon re-encoding and coordinate transformation, that significantly reduces the complexity of the bivariate interpolation procedure. This re-encoding and coordinate transformation converts the original interpolation problem into another reduced interpolation problem, which is orders of magnitude smaller than the original one. A rigorous proof is presented to show that the two interpolation problems are indeed equivalent. An efficient factorization procedure that applies directly to the reduced interpolation problem is also given.

cs.IT↗

arXiv2009

An Algebraic Watchdog for Wireless Network Coding

In this paper, we propose a scheme, called the "algebraic watchdog" for wireless network coding, in which nodes can detect malicious behaviors probabilistically, police their downstream neighbors locally using overheard messages, and, thus, provide a secure global "self-checking network". Unlike traditional Byzantine detection protocols which are receiver-based, this protocol gives the senders an active role in checking the node downstream. This work is inspired by Marti et. al.'s watchdog-pathrater, which attempts to detect and mitigate the effects of routing misbehavior. As the first building block of a such system, we focus on a two-hop network. We present a graphical model to understand the inference process nodes execute to police their downstream neighbors; as well as to compute, analyze, and approximate the probabilities of misdetection and false detection. In addition, we present an algebraic analysis of the performance using an hypothesis testing framework, that provides exact formulae for probabilities of false detection and misdetection.

cs.NI↗

arXiv2009

On Counteracting Byzantine Attacks in Network Coded Peer-to-Peer Networks

Random linear network coding can be used in peer-to-peer networks to increase the efficiency of content distribution and distributed storage. However, these systems are particularly susceptible to Byzantine attacks. We quantify the impact of Byzantine attacks on the coded system by evaluating the probability that a receiver node fails to correctly recover a file. We show that even for a small probability of attack, the system fails with overwhelming probability. We then propose a novel signature scheme that allows packet-level Byzantine detection. This scheme allows one-hop containment of the contamination, and saves bandwidth by allowing nodes to detect and drop the contaminated packets. We compare the net cost of our signature scheme with various other Byzantine schemes, and show that when the probability of Byzantine attacks is high, our scheme is the most bandwidth efficient.

cs.NI↗

arXiv2008

Network Coding in a Multicast Switch

The problem of serving multicast flows in a crossbar switch is considered. Intra-flow linear network coding is shown to achieve a larger rate region than the case without coding. A traffic pattern is presented which is achievable with coding but requires a switch speedup when coding is not allowed. The rate region with coding can be characterized in a simple graph-theoretic manner, in terms of the stable set polytope of the "enhanced conflict graph". No such graph-theoretic characterization is known for the case of fanout splitting without coding. The minimum speedup needed to achieve 100% throughput with coding is shown to be upper bounded by the imperfection ratio of the enhanced conflict graph. When applied to KxN switches with unicasts and broadcasts only, this gives a bound of min{(2K-1)/K,2N/(N+1)} on the speedup. This shows that speedup, which is usually implemented in hardware, can often be substituted by network coding, which can be done in software. Computing an offline schedule (using prior knowledge of the flow rates) is reduced to fractional weighted graph coloring. A graph-theoretic online scheduling algorithm (using only queue occupancy information) is also proposed, that stabilizes the queues for all rates within the rate region.

cs.NI↗

arXiv2008

Coding for Errors and Erasures in Random Network Coding

The problem of error-control in random linear network coding is considered. A ``noncoherent'' or ``channel oblivious'' model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic. Motivated by the property that linear network coding is vector-space preserving, information transmission is modelled as the injection into the network of a basis for a vector space $V$ and the collection by the receiver of a basis for a vector space $U$. A metric on the projective geometry associated with the packet space is introduced, and it is shown that a minimum distance decoder for this metric achieves correct decoding if the dimension of the space $V \cap U$ is sufficiently large. If the dimension of each codeword is restricted to a fixed integer, the code forms a subset of a finite-field Grassmannian, or, equivalently, a subset of the vertices of the corresponding Grassmann graph. Sphere-packing and sphere-covering bounds as well as a generalization of the Singleton bound are provided for such codes. Finally, a Reed-Solomon-like code construction, related to Gabidulin's construction of maximum rank-distance codes, is described and a Sudan-style ``list-1'' minimum distance decoding algorithm is provided.

cs.IT↗

arXiv2007

A Low Complexity Algorithm and Architecture for Systematic Encoding of Hermitian Codes

We present an algorithm for systematic encoding of Hermitian codes. For a Hermitian code defined over GF(q^2), the proposed algorithm achieves a run time complexity of O(q^2) and is suitable for VLSI implementation. The encoder architecture uses as main blocks q varying-rate Reed-Solomon encoders and achieves a space complexity of O(q^2) in terms of finite field multipliers and memory elements.

cs.IT↗

arXiv2007

Network Coding over a Noisy Relay : a Belief Propagation Approach

In recent years, network coding has been investigated as a method to obtain improvements in wireless networks. A typical assumption of previous work is that relay nodes performing network coding can decode the messages from sources perfectly. On a simple relay network, we design a scheme to obtain network coding gain even when the relay node cannot perfectly decode its received messages. In our scheme, the operation at the relay node resembles message passing in belief propagation, sending the logarithm likelihood ratio (LLR) of the network coded message to the destination. Simulation results demonstrate the gain obtained over different channel conditions. The goal of this paper is not to give a theoretical result, but to point to possible interaction of network coding with user cooperation in noisy scenario. The extrinsic information transfer (EXIT) chart is shown to be a useful engineering tool to analyze the performance of joint channel coding and network coding in the network.

cs.IT↗

arXiv2007

On Coding for Reliable Communication over Packet Networks

We present a capacity-achieving coding scheme for unicast or multicast over lossy packet networks. In the scheme, intermediate nodes perform additional coding yet do not decode nor even wait for a block of packets before sending out coded packets. Rather, whenever they have a transmission opportunity, they send out coded packets formed from random linear combinations of previously received packets. All coding and decoding operations have polynomial complexity. We show that the scheme is capacity-achieving as long as packets received on a link arrive according to a process that has an average rate. Thus, packet losses on a link may exhibit correlation in time or with losses on other links. In the special case of Poisson traffic with i.i.d. losses, we give error exponents that quantify the rate of decay of the probability of error with coding delay. Our analysis of the scheme shows that it is not only capacity-achieving, but that the propagation of packets carrying "innovative" information follows the propagation of jobs through a queueing network, and therefore fluid flow models yield good approximations. We consider networks with both lossy point-to-point and broadcast links, allowing us to model both wireline and wireless packet networks.

cs.IT↗

arXiv2006

Network Coding in a Multicast Switch

We consider the problem of serving multicast flows in a crossbar switch. We show that linear network coding across packets of a flow can sustain traffic patterns that cannot be served if network coding were not allowed. Thus, network coding leads to a larger rate region in a multicast crossbar switch. We demonstrate a traffic pattern which requires a switch speedup if coding is not allowed, whereas, with coding the speedup requirement is eliminated completely. In addition to throughput benefits, coding simplifies the characterization of the rate region. We give a graph-theoretic characterization of the rate region with fanout splitting and intra-flow coding, in terms of the stable set polytope of the 'enhanced conflict graph' of the traffic pattern. Such a formulation is not known in the case of fanout splitting without coding. We show that computing the offline schedule (i.e. using prior knowledge of the flow arrival rates) can be reduced to certain graph coloring problems. Finally, we propose online algorithms (i.e. using only the current queue occupancy information) for multicast scheduling based on our graph-theoretic formulation. In particular, we show that a maximum weighted stable set algorithm stabilizes the queues for all rates within the rate region.

cs.NI↗

arXiv2006

On Separation, Randomness and Linearity for Network Codes over Finite Fields

We examine the issue of separation and code design for networks that operate over finite fields. We demonstrate that source-channel (or source-network) separation holds for several canonical network examples like the noisy multiple access channel and the erasure degraded broadcast channel, when the whole network operates over a common finite field. This robustness of separation is predicated on the fact that noise and inputs are independent, and we examine the failure of separation when noise is dependent on inputs in multiple access channels. Our approach is based on the sufficiency of linear codes. Using a simple and unifying framework, we not only re-establish with economy the optimality of linear codes for single-transmitter, single-receiver channels and for Slepian-Wolf source coding, but also establish the optimality of linear codes for multiple access and for erasure degraded broadcast channels. The linearity allows us to obtain simple optimal code constructions and to study capacity regions of the noisy multiple access and the degraded broadcast channel. The linearity of both source and network coding blurs the delineation between source and network codes. While our results point to the fact that separation of source coding and channel coding is optimal in some canonical networks, we show that decomposing networks into canonical subnetworks may not be effective. Thus, we argue that it may be the lack of decomposability of a network into canonical network modules, rather than the lack of separation between source and channel coding, that presents major challenges for coding over networks.

cs.IT↗

arXiv2006

On the Block Error Probability of LP Decoding of LDPC Codes

In his thesis, Wiberg showed the existence of thresholds for families of regular low-density parity-check codes under min-sum algorithm decoding. He also derived analytic bounds on these thresholds. In this paper, we formulate similar results for linear programming decoding of regular low-density parity-check codes.

cs.IT↗

arXiv2006

Bounds on the Threshold of Linear Programming Decoding

Whereas many results are known about thresholds for ensembles of low-density parity-check codes under message-passing iterative decoding, this is not the case for linear programming decoding. Towards closing this knowledge gap, this paper presents some bounds on the thresholds of low-density parity-check code ensembles under linear programming decoding.

cs.IT↗

arXiv2006

Towards Low-Complexity Linear-Programming Decoding

We consider linear-programming (LP) decoding of low-density parity-check (LDPC) codes. While it is clear that one can use any general-purpose LP solver to solve the LP that appears in the decoding problem, we argue in this paper that the LP at hand is equipped with a lot of structure that one should take advantage of. Towards this goal, we study the dual LP and show how coordinate-ascent methods lead to very simple update rules that are tightly connected to the min-sum algorithm. Moreover, replacing minima in the formula of the dual LP with soft-minima one obtains update rules that are tightly connected to the sum-product algorithm. This shows that LP solvers with complexity similar to the min-sum algorithm and the sum-product algorithm are feasible. Finally, we also discuss some sub-gradient-based methods.

cs.IT↗

arXiv2006

Minimum-Cost Multicast over Coded Packet Networks

We consider the problem of establishing minimum-cost multicast connections over coded packet networks, i.e. packet networks where the contents of outgoing packets are arbitrary, causal functions of the contents of received packets. We consider both wireline and wireless packet networks as well as both static multicast (where membership of the multicast group remains constant for the duration of the connection) and dynamic multicast (where membership of the multicast group changes in time, with nodes joining and leaving the group). For static multicast, we reduce the problem to a polynomial-time solvable optimization problem, and we present decentralized algorithms for solving it. These algorithms, when coupled with existing decentralized schemes for constructing network codes, yield a fully decentralized approach for achieving minimum-cost multicast. By contrast, establishing minimum-cost static multicast connections over routed packet networks is a very difficult problem even using centralized computation, except in the special cases of unicast and broadcast connections. For dynamic multicast, we reduce the problem to a dynamic programming problem and apply the theory of dynamic programming to suggest how it may be solved.

cs.IT↗

arXiv2005

Graph-Cover Decoding and Finite-Length Analysis of Message-Passing Iterative Decoding of LDPC Codes

The goal of the present paper is the derivation of a framework for the finite-length analysis of message-passing iterative decoding of low-density parity-check codes. To this end we introduce the concept of graph-cover decoding. Whereas in maximum-likelihood decoding all codewords in a code are competing to be the best explanation of the received vector, under graph-cover decoding all codewords in all finite covers of a Tanner graph representation of the code are competing to be the best explanation. We are interested in graph-cover decoding because it is a theoretical tool that can be used to show connections between linear programming decoding and message-passing iterative decoding. Namely, on the one hand it turns out that graph-cover decoding is essentially equivalent to linear programming decoding. On the other hand, because iterative, locally operating decoding algorithms like message-passing iterative decoding cannot distinguish the underlying Tanner graph from any covering graph, graph-cover decoding can serve as a model to explain the behavior of message-passing iterative decoding. Understanding the behavior of graph-cover decoding is tantamount to understanding the so-called fundamental polytope. Therefore, we give some characterizations of this polytope and explain its relation to earlier concepts that were introduced to understand the behavior of message-passing iterative decoding for finite-length codes.

cs.IT↗

arXiv2005

Characterizations of Pseudo-Codewords of LDPC Codes

An important property of high-performance, low complexity codes is the existence of highly efficient algorithms for their decoding. Many of the most efficient, recent graph-based algorithms, e.g. message passing algorithms and decoding based on linear programming, crucially depend on the efficient representation of a code in a graphical model. In order to understand the performance of these algorithms, we argue for the characterization of codes in terms of a so called fundamental cone in Euclidean space which is a function of a given parity check matrix of a code, rather than of the code itself. We give a number of properties of this fundamental cone derived from its connection to unramified covers of the graphical models on which the decoding algorithms operate. For the class of cycle codes, these developments naturally lead to a characterization of the fundamental polytope as the Newton polytope of the Hashimoto edge zeta function of the underlying graph.

cs.IT↗