Reduced-Dimension Linear Transform Coding of Correlated Signals in Networks

A model, called the linear transform network (LTN), is proposed to analyze the compression and estimation of correlated signals transmitted over directed acyclic graphs (DAGs). An LTN is a DAG network with multiple source and receiver nodes. Source nodes transmit subspace projections of random correlated signals by applying reduced-dimension linear transforms. The subspace projections are linearly processed by multiple relays and routed to intended receivers. Each receiver applies a linear estimator to approximate a subset of the sources with minimum mean squared error (MSE) distortion. The model is extended to include noisy networks with power constraints on transmitters. A key task is to compute all local compression matrices and linear estimators in the network to minimize end-to-end distortion. The non-convex problem is solved iteratively within an optimization framework using constrained quadratic programs (QPs). The proposed algorithm recovers as special cases the regular and distributed Karhunen-Loeve transforms (KLTs). Cut-set lower bounds on the distortion region of multi-source, multi-receiver networks are given for linear coding based on convex relaxations. Cut-set lower bounds are also given for any coding strategy based on information theory. The distortion region and compression-estimation tradeoffs are illustrated for different communication demands (e.g. multiple unicast), and graph structures.Comment: 33 pages, 7 figures, To appear in IEEE Transactions on Signal Processin

Computation in Multicast Networks: Function Alignment and Converse Theorems

The classical problem in network coding theory considers communication over multicast networks. Multiple transmitters send independent messages to multiple receivers which decode the same set of messages. In this work, computation over multicast networks is considered: each receiver decodes an identical function of the original messages. For a countably infinite class of two-transmitter two-receiver single-hop linear deterministic networks, the computing capacity is characterized for a linear function (modulo-2 sum) of Bernoulli sources. Inspired by the geometric concept of interference alignment in networks, a new achievable coding scheme called function alignment is introduced. A new converse theorem is established that is tighter than cut-set based and genie-aided bounds. Computation (vs. communication) over multicast networks requires additional analysis to account for multiple receivers sharing a network's computational resources. We also develop a network decomposition theorem which identifies elementary parallel subnetworks that can constitute an original network without loss of optimality. The decomposition theorem provides a conceptually-simpler algebraic proof of achievability that generalizes to $L$-transmitter $L$-receiver networks.Comment: to appear in the IEEE Transactions on Information Theor

Matching and compressing sequences of visual hulls

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2004.Includes bibliographical references (p. 61-63).In this thesis, we implement the polyhedral visual hull (PVH) algorithm in a modular software system to reconstruct 3D meshes from 2D images and camera poses. We also introduce the new idea of visual hull graphs. For data, using an eight camera synchronous system after multi-camera calibration, we collect video sequences to study the pose and motion of people. For efficiency in VH processing, we compress 2D input contours to reduce te number of triangles in the output mesh and demonstrate how subdivision surfaces smoothly approximate the irregular output mesh in 3D. After generating sequences of visual hulls from source video, to define a visual hull graph, we use a simple distance metric for pose by calculating Chamfer distances between 2D shape contours. At each frame of our graph, we store a view independent 3D pose and calculate the transition probability to any other frame based on similarity of pose. To test our approach, we synthesize new realistic motion by walking through cycles in the graph. Our results are new videos of arbitrary length and viewing direction based on a sample source video.by Naveen Goela.M.Eng

Modern Low-Complexity Capacity-Achieving Codes for Network Communication

Communication over unreliable, interfering networks is one of the current challenges inengineering. For point-to-point channels, Shannon established capacity results in 1948, and it took more than forty years to find coded systems approaching the capacity limit with feasible complexity. Significant research efforts have gone into extending Shannon's capacity results to networks with many partial successes. By contrast, the development of low-complexity codes for networks has received limited attention to date. The focus of this thesis is the design of capacity-achieving network codes realizable by modern signal processing circuits. For classes of networks, the following codes have been invented on the foundation of algebraic structure and probability theory: i ) Broadcast codes which achieve multi-user rates on the capacity boundary of several types of broadcast channels. The codes utilize Arýkan's polarization theory of random variables, providing insight into information-theoretic concepts such as random binning, superposition coding, and Marton's construction. Reproducible experiments over block lengths n = 512, 1024, 2048 corroborate the theory; ii ) A network code which achieves the computing capacities of a countably infinite class of simple noiseless interfering networks. The code separates a network into irreducible parallel sub-networks and applies a new vector-space function alignment scheme inspired by the concept of interference alignment for channel communications. New bounds are developed to tighten the standardcut-set bound for multi-casting functions. As an additional example of low-complexity codes, reduced-dimension linear transforms and convex optimization methods are proposed for the lossy transmission of correlated sources across noisy networks. Surprisingly, simple un-coded or one-shot strategies achieve a performance which is exactly optimal in certain networks, or close to optimal in the low signal-to-noise regime relevant for sensor networks

Distributed Karhunen-Loeve Transform With Nested Subspaces

A network in which sensors observe a common Gaussian source is analyzed. Using a fixed linear transform, each sensor compresses its high-dimensional observation into a low-dimensional representation. The latter is provided to a central decoder that reconstructs the source according to a mean squared error (MSE) distortion metric. The Distributed Karhunen-Loeve Transform (d-KLT) has been shown to provide a (locally) optimal linear solution for compression at each sensor. While the d-KLT achieves the lowest distortion linear reconstruction known, it does not maintain a nested subspace structure. In the case of ideal links to the decoder, this paper presents transforms that maintain nested subspaces, allowing the decoder to approximate a delay-limited source in an online fashion according to a desired sensor schedule. A distortion envelope for one distributed transform with nested subspace properties (d-nested-KLT) is provided. In the case of i.i.d. noise to the decoder, under assumptions of power allocation over subspaces, it is also possible to achieve nested subspaces utilizing correlations between sensors' observations. Results are applicable for data access over networks, and online information processing in sensor networks

Linear Compressive Networks

A linear compressive network (LCN) is defined as a graph of sensors in which each encoding sensor compresses incoming jointly Gaussian random signals and transmits (potentially) low-dimensional linear projections to neighbors over a noisy uncoded channel. Each sensor has a maximum power to allocate over signal subspaces. The networks of focus are acyclic, directed graphs with multiple sources and multiple destinations. LCN pathways lead to decoding leaf nodes that estimate linear functions of the original high dimensional sources by minimizing a mean squared error (MSE) distortion cost function. An iterative Optimization of local compressive matrices for all graph nodes is developed using an optimal quadratically constrained quadratic program (QCQP) step. The performance of the optimization is marked by power-compression-distortion spectra, with converse bounds based on cut-set arguments. Examples include single layer and multi-layer (e.g. p-layer tree cascades, butterfly) networks. The LCN is a generalization of the Karhunen-Loeve Transform to noisy multi-layer networks, and extends previous approaches for point-to-point and distributed compression-estimation of Gaussian signals. The framework relates to network coding in the noiseless case, and uncoded transmission in the noisy case

Network Coding with Computation Alignment

Abstract—Determining the capacity of multi-receiver networks with arbitrary message demands is an open problem in the network coding literature. In this paper, we consider a multisource, multi-receiver symmetric deterministic network model parameterized by channel coefficients (inspired by wireless network flow) in which the receivers compute a sum of the symbols generated at the sources. Scalar and vector linear coding strategies are analyzed. It is shown that computation alignment over finite field vector spaces is necessary to achieve the computation capacities in the network. To aid in the construction of coding strategies, network equivalence theorems are established for the decomposition of deterministic models into elementary sub-networks. The linear coding capacity for computation is characterized for all channel parameters considered in the model for a countably infinite class of networks. The constructive coding schemes introduced herein for a specific class of networks provide an optimistic viewpoint for the application of structured codes in network communication. Index Terms—Vector linear network coding, computation capacity, computation alignment, structured codes. I