Imaging with Wireless Sensor Networks

Transcription

1 Imaging with Wireless Sensor Networks Rob Nowak Waheed Bajwa, Jarvis Haupt, Akbar Sayeed Supported by the NSF

2 What is a Wireless Sensor Network? Comm between army units was crucial Signal towers built on hilltops Wireless comm and coding consisted of smoke signals, fires, flags, cannon fire e.g., during Ming Dynasty a single column of smoke plus a single gun shot would indicate the approach of a hundred enemy soldiers

3 Wireless Sensor Networks Today (or Tomorrow) Goal: Measure, estimate and convey a physical field Ex. Temperature, light, pressure, moisture, vibration, sound, gas concentration, position

4 Wireless Sensors solar cell battery sensor µproc radio GPS module Each node equipped with power source(s) sensor(s) modest computing capabilities radio transmitter/receiver

5 What is a Sensor Network? A network of sensors spatially distributed over - imperial border - forest - Internet - cropland - manufacturing facility - urban environment For monitoring spatially distributed processes - enemy soldiers - fires - spread of computer viruses - temperature, light, moisture - biological and chemical processes

6 What is Information Processing in Sensor Networks? Extracting, Manipulating, and Communicating salient features from raw sensor data and delivering them to a destination Features: - location/magnitude of sources - summary statistics - signals, maps & images - decisions

7 The trade-off Low density network low bandwidth/energy consumption low spatial resolution High density network high bandwidth/energy consumption high spatial resolution

8 But Physical fields are spatially correlated, so information does not grow linearly with network density Knowledge of correlation (e.g., Slepian-Wolf coding) or in-network processing can significantly reduce number of bits that must be transmitted

9 The upshot Basic Trade-off: Field is more accurately characterized with higher density sampling, but data rate increases as density increases Key idea: R(u,v) v u As network density increases, correlation between sensor measurements increases, which reduces communication requirements (more communications, but each is shorter) data rate grows linearly with network density, but information rate grows is sub-linearly

10 Data Compression 458 x 300 pixels - coded using only 37 kb Uncompressed : (458 x 300) x (3 colors x 1 byte) = 412 kb Compression factor 11:1

11 A simple field model Pseudocolor depiction of smooth spatial process moisture or pesticide over cropland chemical distribution in lake or sea biochemical agent in urban environment Assume that field is k-times continuously differentiable i.e., the field is smooth

12 Wireless Sensing wireless sensors at random locations Sensor i makes a noisy measurement Y i of field at its location X i GOAL: Reconstruct f at remote destination How to approximate/model/encode f? estimate f in presence of noise? transmit/reconstruct f at destination?

13 Approximating Smooth Fields Example: Smooth functions can be locally approximated very well by simple polynomial functions

14 Approximating Smooth Fields

15 Approximation and Information Distortion (average squared error):

16 Rate-Distortion Analysis Encoding polyfit parameters of each cell requires a fixed number of bits, thus number bits required is proportional to m log distortion k=3 k=2 k=1 log bits Information content of field is inversely proportional to smoothness (i.e., increasing k)

17 Estimating Fields from Noisy Data 1. Divide into m cells 2. Fit polynomial to noisy sensor data in each cell This accomplishes data compression and denoising What choice of m is best?

18 Noiseless vs. Noisy Sensing Noiseless Sensing Noisy Sensing What choice of m is best?

19 Approximation and Estimation Errors Partition sensor field into m square cells Distortion due to partition-based approximation Distortion due estimating polyfits from noisy data Estimated polynomial fits fluctuate about optimal Taylor approximations due to noise

20 Distortion Analysis Optimal # of cells increases (slowly) with # of sensors Distortion decreases with # of sensors

21 Transmitting Information to Destination wireless comms destination Obtaining optimal field reconstruction at destination can be accomplished in several ways: 1. Transmit raw sensor data to destination 2. Compute/transmit local polynomial fits 3. Imaging wireless sensor ensembles; communications and data-fitting combined in single operation

22 Transmission of Raw Data Transmit raw sensor data to destination using digital comm; destination computes field reconstruction destination There are n sensors, so this requires n transmissions; i.e., the number of bits that must be transmitted is O(n) Communication requirements grow linearly with n

23 In-Network Processing & Communication Partition sensor field into cells; local polyfits are computed in-network using digital communications Nodes in each cell selfconfigure into a wireless network, and cooperatively exchange information to compute polyfit fixed number of parameters computed per cell

24 Out-of-Network Communication Polyfit parameters are transmitted out-of-network to destination via digital communications destination Number of parameters (bits) that must be transmitted to the destination is O(m) = O(n 1/(k+1) ) Communication requirements grow sublinearly with n

25 In-Network Processing & Communication In-network processing and communications drastically reduce resources (bandwidth, power) required to transmit the field information to a remote destination Reduction of out-of-network communication resource demands from O(n) to O(n 1/(k+1) ) BUT overhead of forming wireless cooperative networks in each cell consumes a dominant fraction of the system resources As sensor density increases, we move from a network for sensing to a network for networking!!!

26 Wireless Sensor Ensembles (Waheed Bajwa, Akbar Sayeed and RN 05) An attractive alternative to the conventional sensor network paradigm is a wireless sensor ensemble destination nodes in each cell transmit values via amplitude modulation no cooperative processing or communications required transmissions synchronized in each cell to arrive in-phase processing (averaging) implicitly computed by receive antenna

27 Ensemble Power Gain transmitted signals from one cell A = amplitude of each sensor transmission received signal phase-coherent sum of transmitted signals total transmit power ~ n/m A 2 receive power ~ (n/m A) 2 Beamforming Gain = n/m = number of sensors in each cell

28 Ensemble Beamforming transmitted signals from one cell received signal phase-coherent sum of transmitted signals Phase-coherency beams energy to receive antenna

29 Ensemble Communications and Distortion Let P s denote the transmit power per sensor node Distortion in field reconstruction at destination : approx error (bias) sensor noise variance comm noise variance we want to choose m and P s to minimize distortion

30 Distortion vs. Power minimum distortion when

31 Per Node Power Requirements power requirements decrease as node density increases!

32 Comparison of Three Schemes Power requirements to achieve minimal distortion at destination: 1. Transmit raw sensor data to destination; destination computes field reconstruction power requirements grow linearly with network size! 2. Local polyfits are computed in-network and only the estimated parameters are transmitted out-of-network to destination but requires complicated in-network comms/processing 3. Imaging wireless sensor ensembles; communications and data-fitting combined in single operation only requires (relatively) simple synchronization of nodes

33 Compressible Signals (Known Subspace) Smooth fields like this can be well approximated by truncated series expansions (e.g., Fourier, wavelet, etc.) basis function coefficient Truncated series Approximation error

34 Compressible Signals (Known Subspace) The same approximation, estimation and communications analysis goes through in this more general case Nodes synchronize and weight transmissions according to (known) values of corresponding basis functions desired inner products computed via averaging in receive antenna transmit power per coefficient = constant

35 Proof of Concept : Magnetic Resonance Imaging Sensors = hydrogen atoms Coherent ensemble communications = external EM excitation causes hydrogen atoms to produce coherent externally measurable RF signal proportional to Fourier projection of hydrogen density in brain reconstruction of Rob s brain structure using a wireless sensor ensemble MRI senses spatial distribution of hydrogen atoms in my head

36 Compressible Signals (Unknown Subspace) edge Piecewise smooth fields are compressible, but cannot be well approximated by a simple truncated series (approximating subspace is function-dependent) nonlinear best m-term approximations are required (e.g, wavelet, curvelet) Nonlinear m-term approximation Approximation error

37 Compressive Wireless Sensing (Jarvis Haupt and RN 05) destination sends random seed to sensors Sensors each modulate sensor modifies readings seed by pseudorandom according binary to a local variables attribute and coherently (e.g., location, transmit address) to destination destination After r transmissions, destination has r random projections of sensor readings ANY compressible field can be reconstructed from random projections (Candes & Tao 04, Donoho 04)

38 Noisy Compressive Sampling Theorem (Jarvis Haupt and RN 05) Random projection sampling allows us to use entire ensemble as a coherent beamforming array

39 Distortion vs. Power Distortion Total Power

40 Example: Sparse Signals Suppose f is known to be sparse (m non-zero terms in a certain basis expansion) known subspace & m optimal projections: distortion at receiver: unknown subspace & r > m random projections: distortion at receiver: Random projections are less effective by a factor of r/n ; the fraction of energy they deposit in signal subspace

41 Example: Lowpass vs. Random Projections r low frequency Fourier projections: piecewise smooth field distortion at receiver: r random projections: piecewise smooth field wavelets curvelets distortion at receiver:

42 Conclusions Complexity (entropy) of field grows far more slowly than volume of raw data, as sensor density increases information rate data rate wireless sensor ensembles offer a promising new architecture for dense wireless sensing compressive sampling offers advantages iff target function is very compressible nowak@engr.wisc.edu

43 Papers Matched Source-Channel Communication for Field Estimation in Wireless Sensor Networks, W. Bajwa, A. Sayeed and R. Nowak, IPSN 2005, Los Angeles, CA. Signal Reconstruction from Noisy Random Projections, J. Haupt and R. Nowak, submitted to IEEE Trans. Info. Th., 2005 (short version in Proceedings of 2005 IEEE Statistical Signal Processing Workshop)