Computing and Compressive Sensing in
Wireless Sensor Networks
Zhenzhi Qian, Chu Wang
Department of Electronic Engineering
Shanghai Jiao Tong University, China
1
Outline
Introduction
Definition & Model
Computing in Two-Node Network
Function Computation Rate
Compressive Sensing : A Basic Application
Future Work
2
Introduction
 Wireless Sensor Networks
 Task of sensing the environment
 Task of communicating function values to the sink
node
 Function Types
 Type-sensitive
e.g. Network of temperature sensors
 Type-threshold
e.g. Alarm network
 Aggregate functions under end-to-end flow
 Energy-constrain
 Memory-constrain
 Bandwidth-constrain
3
Introduction
 An alternative solution
 In-network computation
 Perform operations on received data
 A series of Fundament al Issues in In-Network
Computation
 How best to perform distributed computation
 What is the optimal strategy to compute
 Challenges in WSNs Data Gathering
 Global communication cost reduction
 Energy consumption load balancing
4
Outline
Introduction
Definition & Model
Computing in Two-Node Network
Function Computation Rate
Compressive Sensing: A Basic Application
Future Work
5
Definition & Model
 X  {xi | i  1, 2,…} is the set of the measurement data
 f ( x, y ) is the function used for computation
  ij is the Euclidean distance between sensor i and sensor j
 r is the sensor’s transmission range
 Collocated Network(figure.(a))
The network with ij  r ,for all i, j .
 Random Planar Network
The n nodes and the sink
node is i.i.d distributed,
and r ( n) is chosen to
ensure connectivity by
multi-hop communication
6
Outline
Introduction
Definition & Model
Computing in Two-Node Network
Function Computation Rate
Compressive Sensing: A Basic Application
Conclusion & Future Work
7
Computing in Two-Node Network
 The two connecteed processors can exchange bits one at a time
over the link
 When A and B both know the function value f ( x, y ) ,the
communication terminates
 This problem is minimizing computation time given a throughput
constrained link between processors and an input split between
processors
xi  X
A
f ( xi , y j )  Z
yi  Y
B
8
Computing in Two-Node Network
 A general protocol functionality
 Decide which node to transmit
Input : previously transmitted nodes
 Decide the value of the bit to be transmitted
Input : input value + previous transmission
 A naïve protocol
 The communication complexity of function f
log | X |  log | Z | slots
 Optimization ? Lower bound ?
x
A
() is
f ( x, y )
B
9
Computing in Two-Node Network
 The Lower bound of communication complexity:
log | Range( f ) |
 Any two distinct function values must correspond to different
sequences of transmitted bits
f ( x1 , y1 )  f ( x2 , y2 )
f ( x1 , y1 )  f ( x1 , y2 )
A
10
Computing in Two-Node Network
 Protocol : Matrix Representation
A/B
1
2
3
4
1
0
0
0
0
2
0
0
0
1
3
0
0
0
1
4
1
1
0
1
11
Computing in Two-Node Network
There are several ways to derive the lower bound of the
number of the partitions required
 Rank-based: log Rank (C ) 
 fooling set -based: log m
Prove : 1) ONE ROUND
RANK
2) similar to the above
(AT MOST) ½
12
Outline
Introduction
Definition & Model
Computing in Two-Node Network
Function Computation Rate
Compressive Sensing : A Basic Application
Conclusion & Future Work
13
Function Computing Rate
 Scenario of Sensor network Computation
A tree rooted at the collector node
14
Function Computing Rate
 Function types and corresponding results:[kumar]
 Histogram: statistic of node measurements
 Computational rate:
1
O(
)
log n
15
Function Computing Rate
 Function types and corresponding results:
 Type-sensitive:
A symmetric function f () is defined as type-sensitive if
exists some r  (0,1) and integer N, such that for all
n  N and any j  n  rn  , there are two subsets of
{ y j 1 , y j  2 ,..., yn } , {z j 1 , z j  2 ,..., zn } satisfy that:
f ( x1 ,..., x j , z j 1 , z j  2 ,..., zn )  f ( x1 ,..., x j , y j 1 , y j  2 ,..., yn )
Computing
A easy noterate:
:
1
O( )
InAny
a collocated
input of thenetwork:
sensor network
n changes, the sensitive
Examples
of type-sensitive
function
value changes
due to the localfunctions:
small difference.
Computing
rate:
1
Average,median,majority,histogram
O
(
)
In a random planar multihop network:
log n
16
Function Computing Rate
A example
help
understand
threshold

Function to
types
and
corresponding
results:function
Node : Tall,wealthy,handsome
 Type-threshold:
A symmetric function f () is defined as type-threshold if
exists a nonnegative  -vector  , called the threshold
vector , so that
f ( x)  f ' ( ( x ))  f ' (min( ( x), ))
for all x  
 A easy note:
Suppose a protocol of advancing the threshold:
Node: the opposite
When a given sensor measurement is above the threshold,
Theotherwise
function : white
wealth
pulchritude
it is considered by the computation,
it can
be
safely ignored
Thresholds
n

17
Function Computing Rate
 Examples of type-threshold functions:
Maximum, minimum, k-th largest value
 Computing rate in collocated network:
1
O(
)
log n
 Computing rate in random planar multi-hop network:
1
O(
)
log log n
18
Outline
Introduction
Definition & Model
Computing in Two-Node Network
Function Computation Rate
Compressive Sensing : A Basic Application
Conclusion & Future Work
19
Compressive sensing
 Introduction of compressive sensing[2009 mobicom]
 Baseline data collection
 Compressive data gathering
20
Compressive sensing
 Analysis :
 The sink obtains M weighted sums { yi }, i  1, 2,..., M
 Where ij represents the i-th sum round’s corresponding
j-th sensor nodes coefficient. This coefficient is random a
value. But it can be achieved at the sink node by
preserving the series of pseudo random numbers of each
sensor .Meaning the matrix
is saved beforehand.

 N total nodes and M rounds of gathering exists.
21
Compressive sensing
 Data recovery
 Find a particular domain
,and sensor readings
d  [d1 , d 2 ,..., d N ]T is a K-sparse signal in it , thus
x  [ x1 , x2 ,..., xn ] are the coefficients, which given as:

N
d   xi i
i 1
d  x
 The domain is chosen by yourself. Usually the DCT and
wavelet is preferred.
 The compressive sampling theory have:M should satisfy
M  c 2 (, ) K log N
 (, )= N max i , j
1i , j  N
if the K-sparse signal is resconstructable.
22
Compressive sensing
 Data recovery
 We thus summary the conditions for now:
 M sums at the sink node with efficient amount for
reconstruction by the restraints given previously.
 y    x where   and
are known to us.
 As is given in the compressive sensing theory, the
problem is converted to a l1-norm minimization version :
Find min x
satisfying y    x
y
xR N
l1
N

xi
where x l1 
i 1
This can be solved by a linear programming tech[13].
And finally using d  x we can obtain the sensor
readings.
23
Outline
Introduction
Definition & Model
Computing in Two-Node Network
Function Computation Rate
Compressive Sensing: A Basic Application
Future Work
Future Work
Consider mobility in the WSNs
Gossip algorithm
Coding strategy:LDPC
Thank you!
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