The emerging field of signal processing on graph plays a more and more important role in processing signals and information related to networks. Existing works have shown that under certain conditions a smooth graph signal can be uniquely reconstruct
ed from its decimation, i.e., data associated with a subset of vertices. However, in some potential applications (e.g., sensor networks with clustering structure), the obtained data may be a combination of signals associated with several vertices, rather than the decimation. In this paper, we propose a new concept of local measurement, which is a generalization of decimation. Using the local measurements, a local-set-based method named iterative local measurement reconstruction (ILMR) is proposed to reconstruct bandlimited graph signals. It is proved that ILMR can reconstruct the original signal perfectly under certain conditions. The performance of ILMR against noise is theoretically analyzed. The optimal choice of local weights and a greedy algorithm of local set partition are given in the sense of minimizing the expected reconstruction error. Compared with decimation, the proposed local measurement sampling and reconstruction scheme is more robust in noise existing scenarios.
The Hales numbered $n$-dimensional hypercube and the corresponding adjacency matrix exhibit interesting recursive structures in $n$. These structures lead to a very simple proof of the well-known bandwidth formula for hypercube, whose proof was thoug
ht to be surprisingly difficult. A related problem called hypercube antibandwidth, for which Harper proposed an algorithm, is also reexamined in the light of the above recursive structures, and a close form solution is found.
We present a joint source-channel multiple description (JSC-MD) framework for resource-constrained network communications (e.g., sensor networks), in which one or many deprived encoders communicate a Markov source against bit errors and erasure error
s to many heterogeneous decoders, some powerful and some deprived. To keep the encoder complexity at minimum, the source is coded into K descriptions by a simple multiple description quantizer (MDQ) with neither entropy nor channel coding. The code diversity of MDQ and the path diversity of the network are exploited by decoders to correct transmission errors and improve coding efficiency. A key design objective is resource scalability: powerful nodes in the network can perform JSC-MD distributed estimation/decoding under the criteria of maximum a posteriori probability (MAP) or minimum mean-square error (MMSE), while primitive nodes resort to simpler MD decoding, all working with the same MDQ code. The application of JSC-MD to distributed estimation of hidden Markov models in a sensor network is demonstrated. The proposed JSC-MD MAP estimator is an algorithm of the longest path in a weighted directed acyclic graph, while the JSC-MD MMSE decoder is an extension of the well-known forward-backward algorithm to multiple descriptions. Both algorithms simultaneously exploit the source memory, the redundancy of the fixed-rate MDQ, and the inter-description correlations. They outperform the existing hard-decision MDQ decoders by large margins (up to 8dB). For Gaussian Markov sources, the complexity of JSC-MD distributed MAP sequence estimation can be made as low as that of typical single description Viterbi-type algorithms.
