Source Localizationon a budget

Volkan Cevher

volkan@rice.eduRice University

Petros Rich Anna Martin Lance

Localization Problem

• Goal: Localize targetsby fusing measurementsfrom a network of sensors

[Cevher, Duarte, Baraniuk; EUSIPCO 2007| Model and Zibulevsky; SP 2006| Cevher et al.; ICASSP 2006| Malioutov, Cetin, and Willsky; IEEE TSP, 2005| Chen et al.; Proc. of IEEE 2003]

Localization Problem

• Goal: Localize targetsby fusing measurementsfrom a network of sensors

– collect time signal data– communicate signals across

the network– solve an optimization

problem

Digital Revolution

• Goal: Localize targetsby fusing measurementsfrom a network of sensors

– collect time signal data– communicate signals across

the network– solve an optimization

problem

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Digital Data Acquisition

Foundation: Shannon/Nyquist sampling theorem

time space

“if you sample densely enough (at the Nyquist rate), you can perfectly reconstruct the original analog data”

Major Trends in Sensing

higher resolution / denser sampling

large numbers of sensors

increasing # of modalities / mobility

• Goal: Localize targetsby fusing measurementsfrom a network of sensors

– collect time signal data requires potentially

high-rate (Nyquist)sampling

– communicate signalsacross the network potentially large

communicationburden

– solve an optimizationproblem e.g., MLE

Need compression

Problems of the Current Paradigm

Approaches

• Do nothing / Ignore

be content with the existing approaches

– generalizes well

– robust

Approaches

• Finite Rate of Innovation

Sketching / Streaming

Compressive Sensing

[Vetterli, Marziliano, Blu; Blu, Dragotti, Vetterli, Marziliano, Coulot; Gilbert, Indyk, Strauss, Cormode, Muthukrishnan; Donoho; Candes, Romberg, Tao; Candes, Tao]

Approaches

• Finite Rate of Innovation

Sketching / Streaming

Compressive Sensing

PARSITY

Agenda

• A short review of compressive sensing

• Localization via dimensionality reduction

– Experimental results

• A broader view of localization

• Conclusions

A Short Review of Compressive Sensing

Theory

Compressive Sensing 101

• Goal: Recover a sparse orcompressible signal from measurements

• Problem: Randomprojection not full rank

• Solution: Exploit the sparsity / compressibilitygeometry of acquired signal

• Goal: Recover a sparse orcompressible signal from measurements

• Problem: Randomprojection not full rankbut satisfies Restricted Isometry Property (RIP)

• Solution: Exploit the sparsity / compressibility geometry of acquired signal

– iid Gaussian– iid Bernoulli– …

Compressive Sensing 101

• Goal: Recover a sparse orcompressible signal from measurements

• Problem: Randomprojection not full rankbut satisfies Restricted Isometry Property (RIP)

• Solution: Exploit the sparsity / compressibility geometry of acquired signal

via convex optimization or greedy algorithm

– iid Gaussian– iid Bernoulli– …

Compressive Sensing 101

• Sparse signal: only K out of N coordinates nonzero

– model: union of K-dimensional subspacesaligned w/ coordinate axes

Concise Signal Structure

sorted index

• Sparse signal: only K out of N coordinates nonzero

– model: union of K-dimensional subspaces

• Compressible signal: sorted coordinates decay rapidly to zero

well-approximated by a K-sparse signal(simply by thresholding)

sorted index

Concise Signal Structure

power-lawdecay

Restricted Isometry Property (RIP)• Preserve the structure of sparse/compressible signals

• RIP of order 2K implies: for all K-sparse x1 and x2

K-planes

Restricted Isometry Property (RIP)• Preserve the structure of sparse/compressible signals

• Random subGaussian (iid Gaussian, Bernoulli) matrix has the RIP with high probability if

K-planes

Recovery Algorithms

• Goal:given

recover

• and convex optimization formulations– basis pursuit, Dantzig selector, Lasso, …

• Greedy algorithms– orthogonal matching pursuit,

iterative thresholding (IT), compressive sensing matching pursuit (CoSaMP)

– at their core: iterative sparse approximation

Performance of Recovery

• Using methods, IT, CoSaMP

• Sparse signals

– noise-free measurements: exact recovery – noisy measurements: stable recovery

• Compressible signals

– recovery as good as K-sparse approximation

CS recoveryerror

(METRIC)

signal K-termapprox error

noise

Universality

• Random measurements can be used for signals sparse in any basis

Universality

• Random measurements can be used for signals sparse in any basis

Universality

• Random measurements can be used for signals sparse in any basis

sparsecoefficient

vector

nonzeroentries

Signal recovery is not always required.

ELVIS:

Enhanced Localization via Incoherence and Sparsity

(Back to Localization)

An Important Detail

• Solve two entangled problems for localization

– estimate source locations

– estimate source signals

Today

• Instead, solve one localization problem

– estimate source locations by exploiting random projections of

observed signals– estimate source signals

ELVIS

• Instead, solve one localization problem

– estimate source locations by exploiting random projections of

observed signals– estimate source signals

• Bayesian model order selection & MAP estimation in a decentralized sparse approximation framework that leverages

– source sparsity

– incoherence of sources

– spatial sparsity of sources

[VC, Boufounos, Baraniuk, Gilbert, Strauss; IPSN’09]

Problem Setup

• Discretize space into a localization grid withN grid points

– fixes localization resolution

– P sensors do not have tobe on grid points

Localization as Sparse Approximation

localizationgrid

actual sensormeasurements true

targetlocation

local localizationdictionary

Multiple Targets

localizationgrid

actual sensormeasurements 2 true

targetlocations

Local Dictionaries

• Sample sparse / compressible signal using CS

– Fourier sampling [Gilbert, Strauss]

• Calculate at sensor i using measurements

– for all grid positionsn=1,…,N:

assume that target is at grid position n

– for all sensors j=1,…,P: use Green’s function to

estimate signal sensor j would measure if target was at position n

Valid Dictionaries • S.A. works when columns of

are mutual incoherent

• True when target signal has fast-decaying autocorrelation

• Extends to multiple targets with small cross-correlation

Typical Correlation FunctionsToyota Prius

Isuzu Rodeo

Chevy Camaro

ACF: Toyota Prius

ACF: Isuzu Rodeo

ACF: Chevy Camaro

CCF: Rodeo vs. Prius

CCF: Rodeo vs. Camaro

CCF: Camaro vs. Prius

An Important Issue

localizationgrid

actual sensormeasurements

Need to sendacross

the network

Enter Compressive Sensing

• Sparse localization vector <>acquire and transmit compressive measurementsof the actual observations without losing information

So Far…

• Use random projections of observed signals two ways:

– create local sensor dictionaries that sparsify source locations

– create intersensor communication messages

(K targets on N-dim grid)

populatedusing recovered

signalsrandom iid

ELVIS Highlights

• Use random projections of observed signals two ways:

– create local sensor dictionaries that sparsify source locations

sample at source sparsity– create intersensor communication

messagescommunicate at spatial sparsityrobust to (i) quantization (1-bit quantization–paper)

(ii) packet drops

No Signal Reconstruction

ELVIS Dictionary

ELVIS Highlights

• Use random projections of observed signals two ways:

– create local sensor dictionaries that sparsify source locations

sample at source sparsity– create intersensor communication

messagescommunicate at spatial sparsityrobust to (i) quantization (1-bit quantization–paper)

(ii) packet drops

• Provable greedy estimation for ELVIS dictionaries

Bearing pursuit – computationally efficient reconstruction

No Signal Reconstruction

Experiments

Field Data Results5 vehicle convoy

>100 × sub-Nyquist

Sensing System Problems

• Common theme so far…

sensors > representations > metrics > “do our best”

• Purpose of deployment

– Multi-objective: sensing, lifetime, connectivity, coverage,

reliability, etc…

Competition among Objectives

• Common theme so far…

sensors > representations > metrics > “do our best”

• Purpose of deployment

– Multi-objective: sensing, lifetime, connectivity, coverage,

reliability, etc…

• Limited resources > conflicts in objectives

Diversity of Objectives

• Multiple objectives

– localization area– lifetime time– connectivity probability– coverage area– reliability probability

• Unifying framework

– utility

Optimality

• Pareto efficiency

– Economics / optimization literature

• Pareto Frontier

– a fundamental limit for achievable utilities

Pareto Frontiers for Localization

• Mathematical framework for multi objective design best sensors portfolio

• Elements of the design

– constant budget > optimization polytope

– sensor dictionary

– random deployment

– communications

[VC, Kaplan; IPSN’09, TOSN’09]

Pareto Frontiers for Localization

• Mathematical framework for multi objective design

• Elements of the design

– constant budget

– sensor dictionary > measurement type (bearing or range) and error, sensor reliability,

field-of- view, sensing range, and mobility

… …

$10 $30 $200 $1M $5M

Pareto Frontiers for Localization

• Mathematical framework for multi objective design

• Elements of the design

– constant budget

– sensor dictionary

– random deployment > optimize expected / worst case

utilities– communications

Pareto Frontiers for Localization

• Mathematical framework for multi objective design

• Elements of the design

– constant budget

– sensor dictionary

– random deployment

– communications > bearing or range

Statement of Results – 1

• Theory to predict the localization performancewith management

– signals > performance characterizations

– key idea:

duality among sensors <>existence of a reference sensing system

• Provable diminishing returns

• Optimal heterogeneity

– sparse solutions

bounded by # of objectives

– key theorems:

concentration of resources

dominating sensor pairs

• Solution algorithms

– integer optimization

Statement of Results – 2

Conclusions

• CS

– sensing via dimensionality reduction

• ELVIS

– source localization via dimensionality reduction

– provable and efficient recovery via bearing pursuit

• Current work

– clock synchronization

– sensor position errors via linear filtering

• Pareto Frontiers w/ ELVIS: reactive systems

Questions?

Volkan Cevher / volkan@rice.edu