DISTRIBUTED MULTISCALE DATA ANALYSIS AND PROCESSING FOR SENSORNETWORKS

Raymond Wagner, Shriram Sarvotham, Hyeokho Choi, Richard Baraniuk

Rice UniversityDepartment of Electrical and Computer Engineering

Houston TX, USA

ABSTRACT

While multiresolution data analysis, processing, and compressionhold considerable promise for sensor network applications, progresshas been confounded by two factors. First, typical sensor data areirregularly spaced, which is incompatible with standard wavelettechniques. Second, the communication overhead of multiscalealgorithms can become prohibitive. In this paper, we take a firststep in addressing both shortcomings by introducing two new dis-tributed multiresolution transforms. Our irregularly sampled Haarwavelet pyramid and telescoping Haar orthonormal wavelet ba-sis provide efficient piecewise-constant approximations of sensordata. We illustrate with examples from distributed data compres-sion and in-network wavelet de-noising.

1. INTRODUCTION

In the recent call-to-arms found in [1], the authors emphasize thatspatially irregular data sampling is an inescapable reality whenconsidering real-world sensor network deployments. Using the ex-ample of compression of a sensor network measurement field fortransmission to a single, external sink via a tree-like routing struc-ture, they emphasize that traditional regular 2-D signal processingschemes simply do not translate to the irregular setting. Observingthat much of sensor network signal processing literature assumesregularity of sensor placement, the authors motivate the need fornewer, irregularity-tolerant solutions for the sensor network set-ting.

To that end, we propose what is to our knowledge the firstdistributed, irregular wavelet transform for sensor networks. Em-ploying successive piecewise constant approximations inspired bythe Haar wavelet in the regular setting, our transform integratesseamlessly with existing hierarchical routing schemes to align itsdata flow with efficient communication paths in the network. Wepresent two variants on our technique and outline exactly how toimplement each in a real sensor network setting. Finally, using ourtransform, we specify protocols for distributed solutions to boththe data compression and transmission problem discussed aboveand the problem of de-noising sensor measurements within the net-work.

A good deal of prior work has addressed wavelet-based pro-cessing in sensor networks, but none has yet to resolve the diffi-culties of working with irregularly-spaced data while appreciatingthe need to minimize communication overhead to reduce network

This work was supported by NSF, AFOSR, ONR, andthe Texas Instruments Leadership University Program. Email:{rwagner,shri,choi,richb}@rice.edu. Web: dsp.rice.edu.

power consumption. Dimensions [2, 3] proposes an in-networkmultiscale wavelet transform and hierarchical coefficient routingwith its wavRoute protocol but assumes sensor measurements lieon a square grid. Similarly, Wisden [4] also assumes a square gridand application of regular wavelets for compression purposes, asdo [5] and [6]. Finally, Fractional Cascading [7] assigns to a sen-sor a view of the entire network which decays in resolution as thenetwork becomes more distant from the sensor. While this ap-proach does not assume regular sensor placement and is useful foranswering measurement range queries, it does not have the broadapplicability of a wavelet transform to a variety of signal process-ing applications.

In specifying our transform algorithms, we assume only thateach sensor knows its location and shares this information with adata sink and that an efficient multiscale routing hierarchy is al-ready in place. A number of algorithms for sensor localization ad-dress the former problem (see [8]), and both wavRoute mentionedabove and COMPASS [9] provide examples of the latter. COM-PASS, in particular, creates a hierarchical routing scheme by clus-tering nodes, electing clusterheads, and iteratively clustering theclusterheads. Under this paradigm, local communication withinclusters and up and down the routing hierarchy are less expensivethan communication across cluster boundaries. Adopting the rout-ing hierarchy as our multiscale decomposition hierarchy allows usto tailor data flows to match the economics of the routing protocol.

In Section 2 we review challenges of irregular data samplingand motivate the need for wavelet transforms tailored to irregulardata. Section 3 provides the mathematical foundations of the twoproposed wavelet transforms, beginning with the multiscale data-flow model induced by the routing topology. The details of the twotransforms, one a tight frame pyramid and the other a complete or-thonormal basis, follow. Section 4 addresses details of implement-ing these transforms, and Section 5 provides protocols for both dis-tributed data compression/ transmission and distributed de-noisingof sensor measurements. Section 6 demonstrates the aforemen-tioned applications in a simulated sensor network setting, and Sec-tion 7 concludes with a discussion of future directions for suchirregular wavelet transforms.

2. CHALLENGES OF IRREGULAR DATA SAMPLING

A large body of elegant wavelet theory suited to regularly-spaceddata has emerged in recent decades, but application of waveletanalysis to irregularly spaced samples is a relatively new endeavor.The Fourier mathematics underlying regular wavelets no longerapply in the regular setting, and second-generation wavelet the-

Fig. 1. Voronoi polygons (solid) bounding a field of sensors con-nected by edges in a Delaunay triangulation (dashed).

ory such as the lifting scheme [10] has risen to replace traditionaltechniques. Extending these ideas to two dimensions proves anever greater challenge. The pyramid-based approach of [11] andthe non-redundant technique given in [12] represent some of thefirst sophisticated attempts at tackling this problem. Unfortunately,such approaches are typically intended to operate in a centralizedfashion on an entire dataset, so as such they are not directly appli-cable to the sensor network problem. For example, they typicallyassume complete freedom to construct their own multiscale anal-ysis hierarchies, and as stated in Section 1, communication effi-ciency suggests that transforms must direct their flows along rout-ing hierarchies formed with respect to physical constraints such assensor placement and wireless channel quality.

Rather than attempting to directly apply these ideas, we takeinspiration from the approaches contained therein. Varying sensordensity poses one of the key challenges in computing an irregulartransform, as measurements from nodes in more dense areas of thegraph will appear to have greater “importance” than those fromless dense regions. To counter this problem, we adopt the tech-nique in [12] of assigning weight to sensor measurements basedon the area of a Voronoi polygon [13] around the sensor. Figure 1illustrates a set of Voronoi tiles drawn as solid lines around a set ofsensor locations. The figure also shows a Delaunay triangulationof the sensor locations drawn in dashed lines. Voronoi tesselationsare in fact geometric duels of Delaunay triangulations, and a sensorcan compute the area of its surrounding Voronoi polygon merelyby knowing the edges for which it is an endpoint in the Delaunaytriangulation. Delaunay triangulations can be computed in a dis-tributed fashion within the sensor network, as will be discussed inSection 4.

3. MULTISCALE TRANSFORMS WITH IRREGULARSAMPLES

We now present the mathematical details behind our proposed trans-forms. First, we discuss the interplay between network routing andmultiscale data decomposition. Then, we delve into the details be-hind our two proposed transforms, one a tight-frame Haar pyramidand the other an orthonormal basis Haar “telescoping” transform.

3.1. Hierarchical Routing and Data Decomposition

In Section 1 we introduced the idea of matching the transform hi-erarchy to the routing hierarchy. Such a hierarchical routing topol-ogy will typically consist of clusters of sensors, one of which isdesignated as the clusterhead. Figure 2(a) describes the scenario

m1

m2 m3m4

m5

m6

ks1

k ,jd1

(a)

kd1

k

l s1

dk d

3

ks3

l d1

k d

ks2

ks

2

1

4

1l

1k

2k

3k

4k

(b)

Fig. 2. Data flow (a) within a single clusterk1 and (b) from clus-terheads ink1 throughk3 to the superclusterl1 head in clusterk4.

in a typical 6-node cluster, labeledk1. Under the assumed routingeconomics, local communications within a cluster are relativelycheap, so each sensorm1 throughm5 sends it measurement tothe clusterheadm6 for processing. Using the setm of measure-ments, the clusterhead computes two quantities. Ascaling coeffi-cientsk1

describes some average behavior over the cluster, whilea setdk1

of wavelet coefficientsencode deviations of the measure-ments from the average value.

The clusterhead stores wavelet coefficientsdk1, but the scaling

coefficientsk1flow up to the next level of the transform/routing

hierarchy, as show in Figure 2(b). Along with clusterk1 from Fig-ure 2(a), clustersk2 throughk4 are grouped together to form asuperclusterl1, whose clusterhead coincides with that of clusterk4. Scaling coefficientssk1

throughsk3are passed to clusterk4,

which combines them with its ownsk4as the input to the next level

of multiresolution analysis. A scaling coefficientsl1 for the supercluster is generated along with a setdl1 of wavelet coefficients.Note that the node acting as the super clusterhead performs dou-ble duty. As the clusterhead of clusterk4, it stores the setdk4

oflevel-k wavelet coefficients, and as the clusterhead for the super-clusterl1, it maintains the setdl1 of level-l wavelet coefficients.The scaling coefficientsl1 for the supercluster is then passed upthe hierarchy to participate in subsequent analysis. The iterativeprocess terminates when one supercluster spans the entire networkand a scaling coefficient for a single root node is computed. Thisroot scaling value, which describes the average behavior over thewhole network, and the entire set of wavelet coefficients at eachlevel represent the completed multiresolution analysis of the sen-sor field. Note that, under this model, the hierarchical waveletdecomposition matchesexactlywith the routing topology, group-ing data arbitrarily as dictated by routing concerns. As such, ourproposed techniques are extraordinarily flexible in the realm of ir-regular wavelet transforms.

∆ 1 ∆ 2 ∆ 3 ∆ 4 ∆ 5

∆ 2

∆ 1∆ 4

∆ 3

m1

m2

m3

m4m5

∆ 5

m2

m3

m4

m5m1

Fig. 3. Conceptual mapping of piecewise-constant function over2-D intervals of area∆i to piecewise-constant 1-D function overintervals of width∆i

Given this transform model, we can present the details of ourtwo proposed transforms. Both are based on piecewise-constantapproximations as seen in the well-known Haar wavelet in the reg-ular setting by the. We adopt a similar piecewise-constant modelfor the data gathered in the network, assuming that measurementshold constant over the Voronoi polygons surrounding each sen-sor, as described in Section 2. Section 3.2 details a pyramid-liketransform whose analysis and synthesis functions are remarkablyeasy to implement but lead to a slightly redundant representation,computing 1 scaling andN wavelet coefficients for a cluster ofNsensors. Section 3.3 presents a more advanced transform whichemploys a complete orthonormal basis expansion of sensor mea-surements, achieving a non-redundant coefficient representation(N coefficients for anN -sensor cluster). Providing a sparser rep-resentation of the sensor field, this transform nonetheless carriesa slightly higher cost in terms of computational cost and ease ofimplementation.

3.2. Tight-Frame Haar Pyramid

The first transform, originally introduced in [14], represents anextraordinarily intuitive and easy-to-implement decomposition ofsensor data into wavelet and scaling coefficients. Harkening backto the pyramid image coders which preceded modern wavelet coders[15], the transform passes a smoothed average to the next level ofdecomposition via the scaling coefficients and leaves behind asmany detail coefficientsd as there are sensors in a cluster — inessence, a redundant transform with the slightest possible redun-dancy.

We now describe how to calculate the transform coefficientsgiven anN -length vectorm of measurements for a cluster ofNsensors. We assume that each sensor in the cluster knows it’s sup-port size as defined by the Voronoi polygon surrounding the sensor.These areas,{∆1, ∆2, . . . , ∆N}, can be computed in a distributedfashion, as detailed in Section 4. Given the set of areas and mea-surements, we can conceptually map the 2-D transform problem to1-D as shown in Figure 3, where sensor measurements are takento be constant over intervals whose lengths correspond to sensors’Voronoi region areas. Note that this mapping is purely for illus-trative purposes and that all subsequent integrations in fact involve

∆ 1 ∆ 2 ∆ 3

s

w1

w2

w3

Fig. 4. Tight frame analysis functions for a cluster of 5 nodes,including one scaling functionS and 3 wavelet functionsW1

throughW3

piecewise-constant functions over 2-D regions.Given the set{∆j}Nj=1 of areas, we form the(N + 1) × N

matrix K, given by

K =

k0 k0 k0 · · · k0

k′

1 k1 k1 · · · k1

k2 k′

2 k2 · · · k2

k3 k3 k′

3 · · · k3

......

......

kN kN kN · · · k′

N

, (1)

where the total area over the cluster is given as∆tot =∑N

i=1 ∆i.The bulk of the entries ofK are defined as

ki =

1√

∆tot

, i = 0

√∆i

∆tot, i > 0

(2)

while the first sub-diagonal entries are given by

k′

i = −ki(∆tot −∆i)

∆i

, i > 0. (3)

Taken together, the entries ofK define the basis functions givenin Figure 4. The first row ofK represents the constant-valuedscaling function, while each(i, j)th element in the remaining rowsgives the value of wavelet functioni − 1 over the Voronoi cellsurrounding sensorj

Given the matrixK describing the set of basis functions, allthat is left to do is simply multiply against the sensor measure-ments and integrate over the set of support areas. Defining a anN -by-N matrix∆ as a diagonal matrix of the areas{∆j}Nj=1, weform the analysis matrix as

TA = K∆. (4)

Using the analysis matrix, we compute the(N + 1)× 1 vector oftransform coefficientsc as

c = TAm, (5)

where the first element of cis the scaling coefficients and the re-mainingN elements give the wavelet detail coefficientsd. Defin-ing the synthesis matrix as

TS = ∆−1

TTA, (6)

we recover the set of measurements from the coefficients as

m = TSc. (7)

As stated previously, the analysis and synthesis functions forma tight frame. In fact, they form a special class of tight frameknown as aParseval tight frame[16]. Given this membership,we conclude that, even though the frame is redundant, we have anequivalence between energy in the signal domain and energy in thecoefficient domain:

E =N∑

i=0

c2i , (8)

whereE is calculated by squaring the piecewise-continuous sig-nal given in Figure 3 and integrating it over its support regions.This equivalence implies that, were we to selectively discardq

wavelet coefficients, the energyE of the resultant signal would bemaximized by discarding the smallestq coefficients — exactly theprocedure we turn to for the compression application discussed inSection 5.1. Moreover, given properties of the transform, we canbound the difference in energies between the reconstructed andoriginal signals as

q∑

i=1

c2i ≤ (E − E) ≤ 2(

q∑

i=1

c2i ), (9)

ensuring that the approximated signal will not blow up when theinverse transform is applied — a useful sanity check when workingwith tight frames, whose redundancy can induce problems in suchsituations. For a complete proof of the transform’s Parseval tightframe properties and error bounds, we refer the reader to [17].

Clearly, the tight frame pyramid transform is extremely easy toimplement. Given a set{∆j}Nj=1 of sensor support areas, comput-ing the wavelet decomposition amounts to multiplying the sensormeasurements againstTA, which depends entirely on the{∆j}Nj=1.Unfortunately, this does entail suffering redundancy in the coeffi-cient representation — a problem we can rectify with a second,more computationally intensive piecewise-constant transform.

3.3. Orthonormal Basis Haar Telescope

Clearly, a non-redundant, distributed wavelet transform for sen-sor networks is superior from a data-representation standpoint. Inthis section we present a technique which appeals to classic Haarwavelets in the 1-D setting and forms a set of basis functions whichby construction are orthonormal and span the piecewise-constantmeasurements in a cluster of sensors. Figure 5(a) recalls the reg-ular Haar scaling and wavelet functions in 1-D. Applied to datawhich is piecewise constant over the intervals[0, 0.5] and[0.5, 1],the scaling function merely sums measurements in the two regionswhile the wavelet function differences those measurements. Thetwo functions are each unit norm, mutually orthogonal (their prod-uct integrates to zero), and span the entire interval (any function

0.5 0.5

1

1

1

w

s

h1

h2

h0

∆

∆ 1

1 2∆

2∆

(a) (b)

Fig. 5. (a) Regular, 1-D Haar wavelet (W ) and scaling (S) ba-sis functions and (b) corresponding matched pair telescope basisfunctions.

which is piecewise constant over[0, 0.5] and[0.5, 1] can be formedas a linear combination of the Haar scaling and wavelet functions).

By grouping sensors in our irregular setting into pairs, we canperform a decomposition mirroring that of the classic Haar case,though we must take into account the support sizes of our sensormeasurements when doing so. Appealing to the 1-D visualizationof Section 3, a conceptual mapping of sensor measurements to 1-Dintervals places paired sensors in adjacent positions on the line sothat pairwise basis functions such as those shown in Figure 5(b)apply over the pair of intervals. Again we stress that this mappingis merely for visualization purposes — the pertinent informationis contained in adjacent sensor pairings, which we describe how togenerate later in the section.

To each pair of sensors, we apply the set of scaling and wavelet

coefficients given in the figure, whereh1 =√

∆2

∆1(∆1+∆2), h2 =

−√

∆1

∆2(∆1+∆2), andh1 =

√1

∆1+∆2. Thus, for a given cluster,

we have that

s = (m1∆1 + m2∆2)√

1∆1+∆2

d = m1∆1

√∆2

∆1(∆1+∆2)+−m2∆2

√∆1

∆2(∆1+∆2).

(10)

Using theh0, h1, andh2 expressions, it is easily verified that thepair of basis functions are mutually orthogonal, have unit-norm,and span the space of functions constant over the∆1 and∆2 in-tervals.

Pairing measurements to generate wavelet and scaling coef-ficients according to Equation 10 gives us roughlyN/2 scalingcoefficients over the cluster ofN sensors. Recall, though, that tofit into the model of Figure 2(a) we must describe the whole clus-ter with a single scaling value for transport to the next layer ofthe hierarchy. To do so, we merely iterate the pairing and trans-form process, operating now on scaling coefficients whose sup-ports spans those of the pairs from which they were derived. Thisprocess effectively turns the arbitrary,(N − 1)–to–1 single-depthtree provided by the routing hierarchy into avirtual binary treeasshown in Figure 6. We refer to this structure as atelescope treesince it “stretches” the cluster into a deeper tree. Note that the tree

Fig. 6. Generating a virtual telescoping tree from the routing hier-archy cluster specification.

∆ 1 ∆ 2 ∆ 3 ∆ 4

∆ 1 ∆ 2 ∆ 3 ∆ 4∆ 1 ∆ 2 ∆ 3 ∆ 4

w1

w2

w3

s

∆ 1 ∆ 2 ∆ 3 ∆ 4

Fig. 7. Comparing a pair of disjoint wavelet functionsW1 andW2

with the wavelet functionW3 and scaling functionS spanning thecombined support at the next level of telescope hierarchy.

is, in fact, virtual, as the entire transform is calculated by the clus-terhead given the measurements and supports for all sensors in thecluster, as described in the introduction to Section 3.

Orthonormality of the basis functions holds over levels of thetelescope tree, as illustrated in Figure 7.W1 andW2 in the fig-ure describe wavelet functions in separate pairs over the intervals{∆1, ∆2} and{∆3, ∆4} on the same level of the tree. Clearly,these functions are orthogonal given their disjoint support inter-vals. W3 andS describe the wavelet and scaling functions at thenext level of the hierarchy, which pairs the scaling coefficient over{∆1, ∆2} with that over{∆3, ∆4}. It is not difficult to see thatbothW1 andW2 at the lower level of the hierarchy are orthogonalto the next-level functions. By construction, all wavelet functionsintegrate to zero against a constant value. AsW1 andW2 bothreside in regions spanned by constants inW3 andS, their innerproducts with those functions are zero, ensuring orthonormalityof the complete set of wavelet functions and the scaling functionemployed in the telescoped decomposition.

Given pairings that form the binary tree in Figure 6 and re-peated application of Equation 10 to the pairs, we can form a setof N − 1 wavelets and a scaling coefficient for a cluster ofN sen-sors. Moreover, as we have a complete orthonormal basis acrossthe cluster, the relation in Equation 8 and approximation energyerror is fixed at

E − E =

q∑

i=1

c2i , (11)

wherei again indexesq smallest-magnitude wavelet coefficients,which have been set to zero prior to reconstruction.

Computational complexity in this telescoping transform clearlyresides in the pairing scheme used. We appeal to a techniqueknown as Perfect Matching [18], which pairs points so that someobjective function is minimized. As clusters may have an oddnumber of nodes, as illustrated in Figure 6, care is needed to spec-ify the non-paired node, which retains its value and support intothe next round of matching. We discuss the details of matching, aswell as a variety of other practical concerns, in Section 4.

4. IMPLEMENTATION DETAILS

Given both the tight-frame pyramid algorithm described in Section3.2 and the orthonormal basis algorithm of Section 3.3, we nowhave two methods for computing piecewise-constant distributedwavelet transforms for sensor networks. In this section we discussimplementation issues of the two transforms, distinguishing theformer from the latter. But before addressing their differences, webegin with a common requirement of both.

As discussed in Section 2, the reality of irregular samplingrequires that we appropriately weight sensor measurements to ac-count for non-uniform sensor densities. This motivates us to calcu-late the area of the Voronoi polygon surrounding each sensor, giv-ing a useful measure of the support-size of each sensor value. Asthe transform progresses up levels in the hierarchy, support areasare merely aggregated and applied to the scaling coefficients whichsummarize finer-level measurements. Clearly, all higher-level areainformation can flow up the hierarchy, leading to efficient aggre-gation. Thus, the real challenge becomes efficiently computing thebase-level Voroni cell areas.

Fortunately, efficient, distributed algorithms already exist forcomputing a Delaunay triangulation of sensors. Recall from Sec-tion 2 that the Delaunay triangulation is the geometric dual of theVoronoi polygon. Given the set of Delaunay edges to which itbelongs, a sensor can easily compute the area of its Voronoi cellby simple geometric construction. A method for computing thetriangulation of ann-node network with anO(n lg n) commu-nication cost is given in [19], which constructs a subset of theVoronoi graph having all but the largest edges in the triangulation.Edges not found by the distributed algorithm can conceivably besent from outside the network, as the network user is assumed toknow all sensor locations and can easily identify these edges. Asarea calculation need happen only once, this extra communicationoverhead should have a small amortized cost.

Once we know support sizes, we can implement each trans-form as detailed in Algorithms 1 and 2, which respectively pro-vide pseudocode for the pyramid and telescope transforms withina cluster. Note the disparity in complexity between the two. Thepyramid transform merely computes the analysis matrix specifiedby Equation 4 given the set of support areas and multiplies thematrix against the value set for the cluster, passing the resultantscaling coefficient and aggregate area up the hierarchy.

The telescope transform, which yields a superior, non-redundantwavelet expansion, requires a bit more computation. Each level ofthe telescope tree within a cluster requires identification of valuepairs. To compute these pairings, we appeal to Perfect Matching

Algorithm 1 PyramidXfm(Values, Areas)1: form analysis matrixTA (Eqn. 4) using Areas2: Coefficients← TAV alues3: RETURN Coefficients(1), sum(Areas)

Algorithm 2 TelescopeXfm(Values, Areas)1: if size of Values is 1then2: RETURN Values, Areas3: end if4: newValues← ∅, newAreas← ∅5: Pairs← perfectMatch(Areas)6: for each pair∈ Pairsdo7: compute pair scaling/wavelet coefficients (Eqn. 10)8: insert scaling coefficient in newValues9: insert sum of pair areas in newAreas

10: end for11: if number of values is oddthen12: insert leftover value in newValues13: insert leftover area in newAreas14: end if15: TelescopeXfm(newValues, newAreas)

as described in [18]. The Perfect Matching algorithm computesthe lowest-cost match between points in a set given a cost func-tion. For the telescoping tree, we choose distance between pairedsensors as our minimizing metric and constrain chosen edges tocorrespond to edges in the Delaunay triangulation (guaranteeingmerged cells share a common boundary). As Voronoi cells mergeinto super-cells, we define the distance between the super-cells tobe the minimum distance between centers of their member Voronoipolygons. This distance-based matching requirement insures thatdifference coefficients remain as small as possible by exploitingthe similarity in spatially proximate measurements from smoothfields.

Though the Perfect Matching requirement does add computa-tional and implementation complexity to the transform computedat a cluster, fast algorithms exist for computing the match [20].Additionally, the telescope tree within a cluster need only be com-puted once and stored, provided nodes remain stationary and donot drop out of the network. Such an approach also opens thedoor to computing all pairings outside the network and flowingthe match information down the hierarchy to clusters. Such an ap-proach would entail far more communication, but the amortizedcost could potentially be tolerable when spread over many subse-quent transform operations.

Now that we have completely described our proposed trans-forms and their implementation issues, we proceed in Section 5to outline how transform coefficients can be applied to a pair ofpertinent signal processing problems.

5. APPLICATION OF TRANSFORM DATA

Transmitting a compressed version of the entire measurement fieldto an exterior sink represents an expected sensor network task, assuggested in [1]. Removing noise in a distributed fashion fromsensor measurements polluted by additive IID processes will alsofind much use in sensor networks, especially as a pre-processingstep for more advanced distributed algorithms. We describe belowalgorithms for doing so, given wavelet coefficients calculated as

Algorithm 3 xmtCompressed(ClsthdID,Threshold)1: CandidateCoeffs← Coefficients(ClsthdID)2: Coeffs2xmt← ∅, ID2xmt← ∅3: for each coefficient∈ CandidateCoeffsdo4: if (magnitude of coefficient)< Thresholdthen5: insert coefficient into Coeffs2xmt, node ID into ID2xmt6: end if7: end for8: send Coeffs2xmt and ID2xmt up the hierarchy to the root9: for each node in clusterdo

10: xmtCompressed(nodeID,Threshold)11: end for

described above.

5.1. Distributed Compression and Transmission

Sections 3.2 and 3.3 appeal to Equation 8 to motivate thresholdingof wavelet coefficients as an optimal procedure for approximatingthe function they represent. As reconstruction error energy equalsthat of discarded wavelet coefficients, discarding smallest coeffi-cients first leads to an optimal approximation, given that only aspecified number of coefficients can be retained.

Thus, given a magnitude threshold below which wavelet coef-ficients are presumed insignificant, compression and transmissionof the measurement field proceeds as described in Algorithm 3,which is called on the root node.

Given a threshold, a clusterhead sends up the hierarchy sig-nificant wavelet coefficients along with node identifiers indicatingwhich nodes in the transform generated the coefficients. It thenpasses the threshold down the hierarchy to child nodes in the clus-ter.

5.2. Distributed Measurement De-Noising

Noise in sensor measurements manifests itself as small wavelet co-efficients. Therefore, de-noising proceeds much as compression,where wavelet coefficients are compared against a threshold. Animportant difference arises, though. IID sensor noise applies onlyto individual sensor measurements and not to their entire supportareas. As wavelet coefficients are derived with respect to continu-ous functions over the plane, we must tailor the threshold to eachcoefficient.

We inject an estimateσn of the noise variance into the net-work. Considering the telescoped transform and a single cluster,we fashion a threshold for each wavelet coefficient in the telescopetree as follows. Where{∆i}ui=1 and{∆i}vi=u+1 describe thevbase-level Voronoi cells aggregated into the positive and negativesupport regions of the functions given in Figure 5(b), the thresh-old t for the wavelet coefficient generated from those basis pairs iscomputed as

t = σn

√√√√h21

u∑

i=1

∆2i + h2

2

v∑

j=u+1

∆2j (12)

Note that this requires knowledge at each transform level of thebase-level Voronoi cells spanned by the transform function sup-ports. Such information can be computed in a single bottom-uppass along the routing hierarchy and stored for future reference.

0 50 100 150 200 25010

−10

10−8

10−6

10−4

10−2

100

Coeffs retained

Nor

mal

ized

err

or e

nerg

y

Haar PyramidHaar TelescopeDauechies

Fig. 8. Comparison of Haar Pyramid and Haar Telescope to D-2,D-4, D-6, and D-8 wavelets in a regular setting.

Now, given that appropriate thresholds can be applied to eachwavelet coefficient, de-noising proceeds as follows. Starting at thehierarchy root, we threshold wavelet coefficients, compute inversetransforms, and pass reconstructed values down to child clustersas scaling values for subsequent inverse transforms. This processiterates top-down until the base-level values are recovered. Thesethen represent de-noised versions of original sensor measurements.

6. EXAMPLES

We now proceed to demonstrate the effectiveness of our proposedtransforms on a variety of simulated sensor measurement fields.We show results for both compression and de-noising, but first weprovide a comparison of our technique, operating in the regularsetting, to a standard regular wavelet transform.

6.1. Comparison to Wavelets on Regular Grids

As a brief sanity check, we compare the performance of our pro-posed transforms in the regular setting to transforms using regu-lar Daubechies-2, Daubechies-4, Daubechies-6, and Daubechies-8wavelets. For all transforms, we plot reconstruction error versusnumbers of coefficients retained with increasing fidelity. For thissetting, the sensor network consists of a regular square grid of 256sensors. We allow sensors to cluster randomly and evaluate theresultant reconstructions against those given by the Daubechieswavelets operating on the entire square grid without clustering.Figure 8 illustrates the relative performances. Though both theHaar Pyramid and Haar Telescope transforms are out-performedby the Daubechies wavelets (as expected), their performance isclearly comparable. Thus, the transforms are capable of perform-ing well in both the regular and irregular domains — a claim theDaubechies wavelets cannot make.

6.2. Compression and Transmission of Network Data

Now, we move back into the irregular domain for an examina-tion of the relative performances of the Haar Pyramid and HaarTelescope transforms. Figures 9 (a), (b), and (c) give the threesimulated sensor measurement fields used in the study. The first

consists of a piecewise-planar field with discontinuities in its leftand right corners, the second of a smooth quadratic, and the thirdof a noisy quadratic with a discontinuity. Relative performances ofthe Haar Pyramid and Telescope transforms are given in Figures 9(d), (e), and (f), with the non-redundant Haar Telescope outper-forming the easier-to-implement Haar Pyramid, as expected. Fi-nally Figures 9 (g), (h), and (i) show the fields reconstructed fromthe Haar Telescope function by retaining 250 of the 2,500 coeffi-cients. Clearly, major features of the fields are retained even in thepresence of such heavy compression

6.3. In-Network De-noising of Measurements

To evaluate the de-noising operation, we took samples from thenoisy quadratic function of Figure 9(c). Over 60 trials we realizeda 5.48 dB improvement in signal-to-noise ratio using soft thresh-olding.

7. CONCLUSION

In conclusion, we have successfully demonstrated novel irregularwavelet transforms which are ideally suited to the setting of sen-sor networks. Providing transforms which are both implementableand practically valuable, we have demonstrated their utility in theapplications of distributed data compression and distributed de-noising. To extend this work, we intend to pursue higher-orderapproximations, enabling superior data compression.

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Fig. 9. Sample piecewise-constant (with a discontinuity) (a), quadratic (b), and noisy quadratic (with a discontinuity) (c) fields;approximation-error versus coefficient count curves (d),(e),(f); representative reconstructed fields (g), (h), (i)

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