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Example [1]: Measuring, conveying and reproducing a temperature field Region G -3YGO,2XGF      Data Correlation$A Motivation for Compression Spatial Correlation & Temporal Correlation Similar to image compression in a distributed fashion Correlation Structure Model A stationary random process Y(x, t); Example: One-dimensional Gaussian random field [2] where measures the intensity of correlation; scaling factor of temporal correlation; 1/2: Gauss-Markov field model; 1: squared distance model; la%a%  h; Information Fidelity$Fidelity Measure - distortion Original Field: X, Reconstructed Field X Distortion d(X, X) - difference between two field values; Example: Mean-Square Error E[(X - X)2] Distortion Sources [1, 2, 3] Finite number of sensors - interpolation error, spatial distortion; Conversion of Continuous-magnitude samples to a discrete format; Lossy Coding for discretized samples; Delay - time distortion in real-time applications; ; ,P = 8Information Fidelity (Cont )$Total distortion [1] ( ) is the value at position in the original (reconstructed) field; when sensors are dense, the distortion due to interpolation error is negligible (i.e., satisfying Nyquist sampling theorem), i.e., total distortion is approximately determined by the data processing at each sensor (if time distortion is not considered). 4poCompression via Coding$EQuestion What is the minimal data rate for the given distortion constraint D? Rate-Distortion Function [4] where the minimization is over all conditional distributions for which the joint distribution satisfies the expected distortion constraint and the average mutual information between and is  F E,8>Compression via Coding (Cont )  $Quantization + Entropy Coding Rate-Distortion Theorem provides a lower bound for the achievable information rate; -  ideal encoder A more practical way where Ik is the index of quantization cell in which X(k) lies. Entropy coding is lossless to quantized data; The size of quantization cell should satisfy distortion requirement;As 6Y6 V KLossless Coding$Three types of lossless coding schemes for data aggregation in wireless sensor networks [1] Independent encoding and decoding; Conditional encoding and decoding; Distributed source coding (DSC) - Slepian-Wolf [4]; *\|\| O.Lossless Coding (Cont )$ Independent encoding and decoding Each sensor encodes its quantization index independently; Simplest, but no compression gain (blind to the correlation); Number of generated bits from all sensors to be sent to the sink which is the same for any routing structure.6"." .Lossless Coding (Cont )$Conditional encoding and decoding Each sensor encodes its local quantization index conditioned on the received side information (indices) from its descendants in the routing tree; Compression gain depends on the routing structure; Explicit communication among nodes; Partially takes advantage of the correlation structure; Number of generated bits from all sensors to be sent to the sinkB"g" .Lossless Coding (Cont )$_Distributed Source Coding - Slepian-Wolf [4,5] Theorem [4]: Let be jointly ergodic sources with distribution . Then the set of rate vectors achievable for distributed source coding with separate encoders and a common decoder is defined by for all where and Example: 2-source case L/-/0=& .Lossless Coding (Cont )$Distributed Source Coding - Slepian-Wolf [4,5] Each sensor encodes its index without knowing other sensor indices, but with the assumption that the decoder will know other sensor indices at the time of decoding; Full knowledge of the correlation structure; Number of generated bits from all sensors to be sent to the sink No redundancy in generated traffic! is independent of the choice of the routing structure and High complexity in encoding; distributed implementation? @/&|/MAggregation with Compression$Based on different coding schemes, there are two types of aggregation (routing) strategies: Aggregation with Distributed Source Coding (Slepian-Wolf) [5]; Aggregation with Explicit Communication (EC) [6, 7]; Routing Driven Compression (RDC); Compression Driven Routing (CDR);@\tD\tD* Aggregation with DSC$ If sensor nodes have perfect knowledge about their correlation, they can encode/compress data so as to avoid transmitting redundant information; then each source can send its encoded data to the sink along the shortest path without the need for intermediate aggregation. *+ Aggregation with EC$&Routing Driven Compression (RDC) [6]: The sensors do not have any knowledge about their correlation and send data along the shortest paths to the sink while allowing for OPPORTUNISTIC aggregation wherever the paths overlap. Compression is not the objective, only an opportunity. ^&"'8,Aggregation with EC$Compression Driven Routing (CDR) [6]: The sensors have no knowledge of the correlations but the data is aggregated close to the sources and initially routed so as to allow for maximum possible aggregation at each hop. Eventually, the collected data is sent to the sink along the shortest possible path. Compression is the objective, but CDR in [6] is only an extreme case. No Optimality in transmission structure! @&#Q'R An Example  $Objective Minimize the total transmission cost of transporting the information collected by the sources, to the sink. Optimization (Joint) Transmission (routing) structure from sources to the sink; (Information) Rate allocation at each source; Strategies DSC (Slepian-Wolf); Explicit Communication (EC); Assumptions Distortion is handled by quantization; Snap-shot aggregation (No temporal correlation is considered);  nk 1 j lh 0  f-Problem Formulation [5]$ NGraph G=(V,E), |V|=N+1; Sources Vs={1,2,& ,N}; Sink {N+1}; Edge e=(i,j) with weight we; where Ri is the net traffic (i.e., rate) generated at source i and di is x\L" 3   > ,b;0Strategy 1 - DSC$Corollary 1 - Optimality of the shortest path tree (SPT) for the single-sink data aggregation problem [5]: When there is a single sink in the data aggregation problem and Slepian-Wolf coding is used, the SPT is optimal, in terms of minimizing the total flow cost, for any given rate allocation. The joint optimization problem is separated First, optimizing the transmission structure by building SPT; Second, optimizing the rate allocation for the given SPT;l-xW-x10Strategy 1 - DSC (Cont )$When the SPT is found, assume path weights from sources to the sink are ordered as For a linear cost function F(R), the optimal rate allocation is i.e., most of load to nodes close to the sink, small rates to nodes at extremity of the network. NP7S50Strategy 1 - DSC (Cont )$IDifficulties for distributed implementation The order of the path weights is required as a-priori knowledge to allocate rate (i.e., Global knowledge of SPT); Global knowledge of the correlation structure is required for the node to calculate conditional entropy; Approximated Slepian-Wolf coding [7] Assumption - correlation decays fast with distance; Algorithm - locally order the path weights: Find SPT; For each node i: find in the neighborhood N(i) the set Ci of nodes that are closer to the sink, on the SPT, than node i; Coding rate Ri = H(Xi | Ci) for its quantization index; ,%`,-%`B 4    bK!60Strategy 1 - DSC (Cont )$Performance loss of approximated Slepian-Wolf [7] Gaussian random field (squared distance model); Area size: 100*100; 50 nodes, uniform dis. Intensity of correlation c = 0.001(high) ~ 0.01(low)d2F2x, c97Strategy 2 - EC$+Conditional encoding is used; Rate allocation and transmission structure selection are NOT separable; SPT is not necessary to be optimal in transmission structure; Example: link weight = 1; Shortest Path Tree (SPT, left); Traveling Salesman Path (TSP, right); if r < 0.5R, TSP is better than SPT; V | |=.Strategy 2 - EC (Cont )$-Problem formulation: Find: the spanning tree ST = {T, L} with T (non-terminal nodes) and L (leaves), T U L = V; Such that where, a simple correlation model is assumed: the rate at leaves is R and is r at non-terminal nodes; correlation coefficient is NP-complete for general [7][ +[   +:.Strategy 2 - EC (Cont )$Approximation Algorithms [7] SPT: reference scheme; Greedy algorithm: start from an initial subtree only with the sink; then, successively, add the node who causes the least cost increment to the existing subtree. Leaves deletion algorithm: start from SPT; check possible cost improvement by making leaf nodes change their parent to some other leaf node in their neighborhood; Balanced SPT/TSP: builds SPT up to a radius away from the root and then builds TSP starting from the leaves of SPT in their respective sub-regions; *@.Strategy 2 - EC (Cont )$-Performance of Approximation Algorithms [7] *,,,AComparison: DSC vs. EC$ One-dimensional case *D<Comparison: DSC vs. EC (Cont )$,@Cost ratio [5]: If entropy rate > 0 If entropy rate = 0 and *11IConclusion & Future Work$CA fundamental tradeoff is information fidelity and compression gain; For the given distortion, different coding schemes combined with routing strategies can achieve different gains; The joint optimization for routing and rate allocation (coding) is generally difficult; DSC (Slepian-Wolf) is elegant in separation of the joint optimization problem, but its distributed implementation is still under investigation [8, 9]; Aggregation with EC is practical, but good approximation algorithms are needed for finding the optimal transmission structure for general correlated sources;DD)SConclusion & Future Work$VMultiple-Sink case is more complicated: [5] shows that the problem to find the optimal transmission structure is NP-complete even with Slepian-Wolf coding; Multiple rate problem at the single node - increasing coding complexity; Lossy compression More interesting optimization problems (e.g., sensor density, placement, etc.) )(R%(14(%d      ,\bJ References $h [1] D. Neuhoff,  Field-Gathering Sensor Networks, Distributed Encoding and Oversmapling , Canadian workshop on Information Theory, May 2003. [2] R. Cristescu and M. Vetterli,  On the Optimal Density for Real-time Data Gathering of Spatio-Temporal Processes in Sensor Networks , in the Proc. of ACM IPSN 05, 2005. [3] A. Scaglione and S. Servetto,  On the Interdependence of Routing and Data Compression in Multi-hop Sensor Networks , to appear in ACM/Kluwer Journel on Mobile Networks and Applications (MONET), also see the short version in MobiCom 2002. [4] T. Cover and J. Thomas, Elements of Information Theory, John Wiley and Sons, Inc., 1991. [5] R. Cristescu, B. Beferull-Lozano and M. Vetterli,  Networked Slepian-Wolf: Theory, Algorithms and Scaling Laws , to appear in IEEE Trans. on Information Theory, 2005. [6] S. Pattem, B. Krishnamachari and R. Govindan,  The Impact of Spatial Correlation on Routing with Compression in Wireless Sensor Networks , in the Proc. of ACM IPSN 04, Apr 2004. [7] R. Cristescu, B. Beferull-Lozano and M. Vetterli,  On Network Correlated Data Gathering , in the Proc. of IEEE Infocom 04, 2004. [8] S. Pradham, J. Kusuma and K. Ramchandran,  Distributed Compression in a Dense Microsensor Network , IEEE Signal Processing Magazine, pp.51 - 60, Mar 2002. [9] Aaron, A., Girod, B.: Compression with Side Information Using Turbo Codes, in Proc.IEEE Data Compression Conference (DCC 02), Snowbird, UT, pp. 252-261, Apr 2002. Z' >  m z /H< <  9/  jKk  i e"   0" % +"=9T   Thank You!  U!Backup$1Optimal SPT radius for Balanced SPT/TSP algorithm22  ` ^WN̙f` ^WNff3` 999MMM` ^WNf` ff33` ff33` ^WN̙f>?" dd@(?zdd@ y x@" `  n?" dd@   @@``@n?" dd@  @@``PR   @ ` ` p>> (    s *  `  T Click to edit Master title style! !   c $<  @  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     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