Distributed estimation and coding: A sequential framework based on a side-informed decomposition

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 2, FEBRUARY 2011 759

Distributed Estimation and Coding: A SequentialFramework Based on a Side-Informed Decomposition

Chao Yu, Student Member, IEEE, and Gaurav Sharma, Senior Member, IEEE

Abstract—We propose a sequential framework for the dis-tributed multiple-sensor estimation and coding problem thatdecomposes the problem into a series of side-informed sourcecoding problems and enables construction of good codecs. Ourconstruction relies on a separation result for the simplified sce-nario where information from one sensor is to be sent to theCP that already has information regarding the desired signal.We show that the optimal encoder decoder combination, in thissetting, can be decomposed, without loss in performance, into twosteps: A first preprocessing step to extract relevant informationfrom the indirect observation with consideration of the side in-formation, followed by a second step of side-informed encodingof the preprocessed output. A recursive exploit of the decom-position coupled with side-informed transform coding allows usto construct encoders by reusing scalar Wyner–Ziv codecs. Wedevelop a numerical procedure for obtaining bounds delineatingthe best achievable performance for the proposed sequentialframework and construct and demonstrate a practical codec inthe proposed framework that achieves empirical performanceclose to the bound. Furthermore, we also compare the bounds forthe sequential scheme against bounds for a number of alternateschemes—for some of which codec constructions are obvious andothers for which codec constructions are inobvious. Our resultsshow that, in most cases, our codec exceeds the performancebound of schemes offering obvious constructions. The achievablebound for our sequential framework is also shown to be close to ageneral nonconstructive distributed bound that does not imposethe sequential constraint indicating that the sequential approachmay not cause a significant performance compromise.

Index Terms—Distributed estimation, Gaussian CEO, side-in-formed coding, side-informed estimation, Wyner-Ziv coding.

I. INTRODUCTION

T HE increasing popularity of wireless sensor networksmade up of nodes of devices with sensing, communi-

cation, and computation capabilities has recently motivated

Manuscript received February 17, 2010; revised August 19, 2010; acceptedOctober 17, 2010. Date of publication October 25, 2010; date of currentversion January 12, 2011. The associate editor coordinating the review of thismanuscript and approving it for publication was Prof. Subhrakanti Dey. Thiswork is supported in part by the National Science Foundation under GrantECS-0428157. Part of this work has been presented at the Proceedings of theForty-First Asilomar Conference on Signals, Systems and Computers, PacificGrove, CA, November 4–7, 2007, pp. 681–685.

C. Yu is with the Department of Electrical and Computer Engineering,University of Rochester, Rochester, NY 14627-0126 USA (e-mail:[email protected]; [email protected]; website: http://www.ece.rochester.edu/~chyu/).

G. Sharma is with the Electrical and Computer Engineering Department,the Department of Biostatistics and Computational Biology, and the De-partment of Oncology, University of Rochester, Rochester, NY 14627-0126USA (e-mail: [email protected]; [email protected]; website:http://www.ece.rochester.edu/~gsharma/).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2010.2089622

Fig. 1. � sensors collect observations� � � � �� regarding a desiredsignal �. Each observation � is encoded locally and sent to the CP, where anestimate �� is achieved.

intense research on distributed signal processing techniques [2].In some sensor network applications, it is desirable to directlycollect the observations recorded by the sensor nodes at acentral processor (CP) for the purposes for archival and furtherprocessing. On the other hand, in other application scenarios,the CP is not interested in the raw sensor observations but onlyan aggregated estimate obtained from them. An example ofthis type is a visual sensor network where a CP may wish tocreate a desired view of a scene based on a number of imagescaptured by individual cameras (none of which may be exactlyat the desired viewpoint). In this application, the individualcamera views and the desired view represent lower dimensionalprojections of the underlying scene. Motivated by thisscenario, we consider a setup as illustrated in Fig. 1, where agroup of geographically dispersed nodes (sensors) are deployedto collect observations pertaining to a desired signal, which areindirect and noisy. Each sensor communicates with the CP via ashared rate-constrained channel. Individual sensor observationsare processed and encoded locally before being communicatedto the CP in order to meet the channel rate constraint. TheCP utilizes the received data to compute an estimate of thedesired signal. We consider the design of encoders at thesesensors and the decoder at the CP with a view to minimizingthe mean-square error (MSE) for the estimate.

Our investigation is motivated by the fact that direct encodingof local observations using existing source coding techniquesyields suboptimal performance. The reasons for the failure ofdirect encoding are twofold. First, the observations are indirectand contain irrelevant data and noise. Second, observations col-lected by different sensors are often correlated in a dense net-work, resulting in redundancy among these sensor observations.In this paper, we develop a decomposition and an encoder imple-mentation that addresses these two sources of inefficiency. First,under Gaussian statistics and the MSE distortion metric, we es-tablish that without incurring a rate-distortion penalty, the op-timal encoder can be decomposed into two stages comprising ofa first preprocessing step that extracts relevant information fromlocal observations, referred to as a “side-informed estimation”,followed by side-informed coding of the preprocessed data. The

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processing at the CP is also correspondingly decomposed intotwo steps: a first side-informed decoding step followed by esti-mation of the desired signal using the side information and thereceived data. Next, we consider the general scenario involvingmultiple sensors, and develop a sequential approach that recur-sively exploits our decomposition, each sensor preprocesses andencodes its observation for transmission considering as side-in-formation the information already available at the CP from thesensors preceding it, thereby reducing the distributed estima-tion and coding problem into a sequence of preprocessing andside-informed coding problems.

The decomposition paves the way for the development of con-structive encoders for transmission of indirect observations withside information, by leveraging source coding techniques devel-oped for scenarios where the desired signal is available at theencoder in its pristine form. To demonstrate the utility of this ap-proach, we develop a side-informed codec in a transform codingframework [3], using the conditional Karhunen–Loéve trans-form (KLT) [4], followed by scalar side-informed codes [5], [6],i.e., Wyner–Ziv codes. We also compute an achievable rate-dis-tortion (R-D) bound for our proposed sequential approach anddemonstrate, via simulations, that our codec construction offersperformance close to this R-D bound. The codec’s R-D per-formance in simulations exceeds bounds delineating the bestR-D performance achievable with a number of alternate side-informed and non-side-informed codec constructions, therebyvalidating the proposed methodology. Compared to a more gen-eral distributed achievable R-D bound [7] that does not imposethe sequential constraint, only a small compromise in perfor-mance is seen for the sequential bound, further indicating thatthe overall approach is effective and that the sequential approachmay not incur a significant R-D penalty.

Related work exists in several contexts. For the non-side-in-formed scenario, a separation result analogous to ours wasestablished by Sakrison [8]. This part of our work can beconsidered a generalization of the result of Sakrison. Althoughderived for the case of rate-constrained channels, our decompo-sition is readily applicable for reduced-dimensionality channelsconsidered in [9]–[12]. The optimal linear preprocessing inthese cases can be obtained from the decomposition and theconditional KLT [4]. Our work extends this body of research indistributed estimation to address more realistic rate-constrainedchannels. For rate constrained channels, an achievable ratedistortion bound for the distributed estimation and codingproblem is obtained in [7]. The analysis therein (particularly inSection IV of [7]), arrives at conclusions analogous to, thoughnot identical to, our proposed decomposition. Our analysismethodology and outcome also differ from [6] because of thedifference in our motivation. Particularly, the decomposition atthe decoder side required for our construction is unnecessaryfor computing R-D bounds and therefore missing from [7].

Also related is the problem of distributed estimation usingquantized observations under a rate constraint for estimating un-known deterministic parameters [13], [14] and for random sig-nals [15]–[17]. For these problems, it has been observed thatthe strong statistical dependency among sensor observations,once taken into account, can significantly reduce the data raterequired to communicate the local observation. For this pur-

pose, either a feedback channel from the CP to sensors is used[18] or inter-sensor communications are utilized [15], [16], sothat the sensor can calculate the innovation in its local observa-tion, and only the quantized innovation is sent to the CP, wherethe quantizer enforces the rate constraint. Nested quantizers areused as a simple form of side-informed coding technique to ex-ploit the side information at the CP [17] without feedback ofthe side information. Under the assumption of high rate quan-tization, a separative structure is also suggested in [19] for theasymptotically optimal quantization of noisy sources with sideinformation.

The general distributed estimation problem, without the se-quential constraints that we consider in our framework, is avector version of the Gaussian CEO problem [20]. A completecharacterization of the rate-distortion region for this general sce-nario remains an open problem though several papers have ad-dressed this in the past for the Gaussian CEO problem [21],[22] . Alternate related work [23] establishes that, a succes-sive-refinement scheme can achieve any point in the achievablerate-distortion region of the quadratic Gaussian CEO problem.The proof however, is nonconstructive, relying on random bin-ning, and does not lead to practical coding schemes.

The rest of the paper is organized as follows. Section IIpresents the decomposed structure for the optimal encoder ofa local observation when the CP has side information. Thesequential approach for the multisensor setup is consideredin Section III. We next briefly present in Section IV ourWyner–Ziv codec design. Simulation results are presented inSection V. Sections VI and VII conclude the paper with adiscussion and a summary of the conclusions.

II. OPTIMAL SIDE-INFORMED ESTIMATION UNDER

A RATE CONSTRAINT: A DECOMPOSITION

Fig. 1 illustrates the general setup, where a set of sensorsare deployed to estimate the real-valued random vector .The sensor collects a dimensional ob-servation pertaining to this desired signal . Eachobservation is separately encoded and transmitted to the CP,over a channel that is constrained to a total rate (over all sen-sors) but is otherwise error-free. The CP utilizes the messagesreceived from the sensors to obtain an estimate of . We focuson the scenario where the desired signal and measurements

are jointly Gaussian, and assume the joint statistics ofthese vector random variables are available at the CP and at eachof the sensors. For notational simplicity, we assume all signalsare zero mean, and denote by the cross-co-variance of random vectors and by theautocovariance of vector .

In this section, we consider a simplified setup as illustratedin Fig. 2(a), where an observation needs to be encoded andtransmitted to the CP that already has information avail-able from other sensor nodes and possibly from direct localobservations. The sensor encodes as a messagewhere denotes the output space of the en-coder—defined so as to meet the channel rate constraint. TheCP utilizes the local information and the message that itreceives from the sensor in a “decoder” to obtain an estimate

YU AND SHARMA: DISTRIBUTED ESTIMATION AND CODING: A SEQUENTIAL FRAMEWORK BASED ON A SIDE-INFORMED DECOMPOSITION 761

Fig. 2. Transmission of an indirect observation with side information. (a) Single-stage encoder and decoder. (b) Encoder and decoder are decomposed into side-informed estimation and side-informed coding stages.

of the signal . The resulting MSE distortion for thedecoder is , where represents theexpectation operator, and upper-case symbols represent randomvariables (r.v.s) for which the corresponding lower-case sym-bols are sample realizations, both being bold when these cor-respond to vectors. The MSE depends on the encoder map-ping , and the decoder mapping defined by

. Our overall objective is to constructencoder and decoder mappings for an indirect observationusing as side information, i.e., and , thatminimize the MSE distortion .

In this section, we establish an equivalence that allows us todecompose the above problem of encoding and decoding fromindirect observations into a preprocessing and postprocessingstep combined with the problem of encoding and decoding di-rect observations. The latter problem has been extensively ad-dressed in literature and good codes have been developed forthe scalar observation scenario [5], [6], [24]. Note also that ourobjective is to develop a constructive algorithm by exploitingthe equivalence we obtain, and not the determination of approx-imate or exact rate-distortion bounds.

We begin by considering the distortion . Denoting bythe MMSE estimate of obtained from ,

where [25], we obtain (1),shown at the bottom of the page, where in the final stepwe have used the orthogonality property of the MMSE esti-mators that ensures the prediction erroris orthogonal to any function of the predictor inputs

, hence

and . Observingthat the first term in (1) is a constant independent of theencoder and decoder mappings, we see that the optimal map-pings can be designed to only minimize the second term, viz.

, without any loss ofrate-distortion performance. Now, rewriting as

(2)

where denotes the probability density function (p.d.f)of r.v. , we see that for a fixed decoder mapping , theoptimal encoding for the observation is

(3)

We assume without loss of generality that this encoder is uti-lized at the sensor in the system of Fig. 2(a) and show next thatwe can construct a corresponding partitioned encoder–decoderpair in the structure of Fig. 2(b), where the latter has an expecteddistortion that is no larger than .

We define as the innovation in given, where , and obtain

(4)

Equation (4) relies on the additive property of the MMSE esti-mates for zero-mean jointly Gaussian signals [25, Sec. 11.4]: theMMSE estimate of a signal from two uncorrelated observationscan be obtained as the sum of the two MMSE estimates obtainedfrom each of these two observations, respectively. Equation (4)

(1)


can also be interpreted as a two-stage sequential MMSE estima-tion: an initial estimate is first obtained from , whichis further refined by the innovation contained in .

Under the assumption of joint Gaussian statistics, the MMSEestimator is a linear operator. We denote bythe MMSE estimator of from , where

and . canalso be viewed as a “side-informed” estimator of fromgiven . We further rewrite .Next, we rewrite the objective function in (3) as

(5)

where , denotes the MMSEestimate of from , and

(6)

The decoding rule can be viewed as a side-informeddecoder for , using as side information, induced by thecorresponding side-informed decoding rule of . Conversely,any decoding rule for the (side-informed) decoding

of induces the corresponding decoding rulefor the decoding of which also

satisfies (5). Since (5) establishes that the error criteria for theoptimal encoding rules for these two scenarios, viz. encoding of

with as side information and encoding of with as sideinformation, are identical, we see that the optimum encodingrule for also defines an optimal encoding rule for , bothwith as side information. Equation (6) also defines a decom-posed structure for realizing the encoder–decoder pair whereis obtained at the remote sensor and encoded, and correspond-ingly at the decoder, first the decoder for recovers ,which is then transformed into a decoded value for using (6).

Therefore, for the problem of Fig. 2(a), we can equivalentlyconsider the problem of side-informed encoding of withas side information. Now is trivially a sufficientstatistic of for estimating , thus given the encoding rule,the optimal decoder for utilizing , has exactly the samedistortion as the optimal decoder for utilizing . Defining,

and noting that forwe have , we see that the right-

hand side of (5) can also be written as

(7)

As a result, the optimal encoding and decoding rules for en-coding of with as side information also define optimal en-coding and decoding rules for with as side information.This establishes that for the optimal encoder–decoder instantia-tion, Fig. 2(a) can be decomposed as shown in Fig. 2(b) withoutincurring any rate distortion penalty.

Remark 1: The preprocessing is useful for two reasons.First, the preprocessing successfully extracts relevant infor-mation from . Second, the preprocessing reduces an

indirect encoding problem into a direct encoding problem.Using our decomposition, in (7) is directly computable atthe encoder, and a codec can be designed correspondinglyto minimize . In contrast, is not available atthe encoder, thus we can not readily design a codec to min-imize . Practical code constructions with goodperformance have been demonstrated recently for the directobservation scalar scenario with side information, where thereceiver seeks to reconstruct the one-dimensional observation“itself” [6], [26], [27]. The decomposition presented hereallows us to directly exploit these existing coding schemes forour indirect vector observation situation1.

Note also that the side informed estimation stage can operateon individual observation vectors and does not introduce delay,unlike the side-informed coding codec which requires that theencoding be performed over arbitrarily long blocks of i.i.d. ob-servations for better R-D performance.

Remark 2: Combining the decomposition of Fig. 2(b), withthe Wyner–Ziv result [28], [29] for the Gaussian case, whichstates that the rate-distortion (R-D) function for the side-in-formed coding of with as side information is identical tothe R-D function when is available at both the encoder anddecoder, we can obtain the R-D function for the side-informedestimation as

otherwise(8)

where denotes the R-D boundary for a multivariateGaussian signal with covariance matrix at a distortion value

[30] , and denotes the covariance ma-

trix of , and . Achieving thebound of (8), however, requires infinite length codes.

Remark 3: Mirroring the development of the decompositionin (1), we can readily obtain the following (matrix) decomposi-tion, by utilizing the orthogonality principle:

(9)

Similarly, we can obtain the matrix analog of (5) as

(10)

Equations (9) and (10) indicate that the covariance matrix of theestimation error in Fig. 2(b) is the summation of two parts, thefirst part corresponds to the covariance matrix of the estimationerror when the remote observation is also available at the CP,and the second part corresponds to the covariance matrix of thereconstruction error of the preprocessed signal. We shall findthese versions useful in the sequential estimation problem de-scribed in Section III.

1The decomposition of Fig. 2(b) is in fact a specific instance of the generalside-informed encoder in Fig. 2(a), the former cannot offer an improvementin rate-distortion performance over the latter. Good practical coding methods,however, have not been previously explored for the situation of Fig. 2(a).


Fig. 3. Transmission of an indirect observation without side information.(a) Single-stage encoder and decoder. (b) Encoder is decomposed into anMMSE estimator and an encoder for the estimate.

A. Special Case: No Side Information

If the CP does not have any side information, the system ofFig. 2(a) reduces to the one in Fig. 3(a), a problem studied bySakrison [8]. In this setting, (1) simplifies as

(11)

where denotes the decoding rule for without sideinformation. Equation (11) immediately implies a decomposedstructure for the optimal encoder–decoder chain, as shownin Fig. 3(b): At the encoder, first an MMSE estimateis computed, which is then transmitted to the decoder overthe rate-constrained channel. In this scenario, the innovation

reduces to the observation itself, and the preprocessingreduces into the MMSE estimator .

Thus, Sakrison’s result can be viewed as a special case of theseparation presented here for the case when no side informationis available. We note the Gaussian statistics are not requiredfor the decomposition of Fig. 3(b), and the MMSE estimate

represents a sufficient statistic for estimating.

B. Dimensionality Constrained Channels: A Strong Parallel

The decomposition of Fig. 2(b) is obtained for a rate-con-strained channel under the joint Gaussian assumption. An iden-tical decomposition can also be obtained for a dimensionality-constrained channel [9], [10] where the observationsvector is preprocessed at the encoder via a matrix

, where . The resulting vector is uti-lized, along with the side information , by a linear decoderto obtain for , where is amatrix and is a matrix. In this scenario, the MSEoptimal encoder–decoder pair can be obtained as theconcatenation of the side-informed estimation followed bythe optimal side-informed encoding of this estimate. The latteris obtained via the recently developed conditional KLT [4].

This decomposition result can be obtained by utilizing the or-thogonality property for linear MMSE (LMMSE) estimators bywhich the estimation error is orthogonal to any linear functionof the predictor inputs and does not require the assumption ofjoint Gaussianity. The problem of distributed estimation usingreduced-dimensionality observations has been formulated andaddressed in [9] and [10], where (among other results) the op-timal linear pre and postprocessing matrices were de-rived. We illustrate in Appendix I how these optimal matrices

may also be alternatively obtained by combining our separationresult with the conditional KLT.

III. MULTI-SENSOR SCENARIO: A SEQUENTIAL APPROACH

In this section, we return to the multisensor network ofFig. 1 and consider a sequential approach for distributed es-timation and coding. We assume without loss of generality,that the sensors are indexed such that the CP receives themessage transmitted from sensors in the index order, thusthe information received from the lower-ordered sensors maybe considered as part of the side information for subsequent(higher-ordered) sensors. The estimation and coding at thesesensors can therefore be performed sequentially through arecursive exploit of the decomposition of Fig. 2(b), reducingthe estimation and coding problem into a sequence of side-in-formed estimation and side-informed coding problems. In thissection, we consider the side-informed estimation componentof this framework, deferring discussions of the side-informedencoding to Section IV. In practice, the transmission ordermay be determined by other considerations or may be ar-bitrarily selected. In the latter scenario, we assume that thesensors are ordered in decreasing order of the conditionalmutual information between the current sensor observationand the desired signal given the preceding observations, i.e.,

, for.

A. Sequential Estimation and Coding

For our specific development, we assume all observationsare modeled as linear measurements corrupted by additivenoise , whererepresents a linear observation matrix, and is theadditive noise. At the sensor, denoted by , the noisevectors are assumed to be independent of each otherand of the signal vector , and each of these vectors is as-sumed to be a zero-mean multivariate Gaussian. We denote theallocation of the total rate among the sensors by a vector

, where represents the rate allocated forsensor , and . Rate allocation is described inSection III-B, here we consider the sequential preprocessinggiven an allocation vector.

We next determine the preprocessing matrix at sensor, when the encoded observation from sensors

are available as side information at the CP. We denote by theestimate of obtained by the CP utilizing the messages fromsensors , and by the covariance matrix for the cor-responding estimation error . We will also find it nota-tionally convenient to denote and , the“initial values” of these quantities. The first sensor observation

is preprocessed by , and yields .This preprocessing matrix is obtained from Sakrison’s re-sult in Section II-B since no side information pertaining tois available. Next, is transmitted using rate to the CPwhich obtains an (approximate) MMSE estimate of usingthe coded message that it received from the first sensor. Asindicated earlier, we defer discussion of the specifics of thecoding method to Section IV. To account for the rate restric-tion on the coding, we model the CP’s reconstruction of as


, where denotes the error which we assumeis zero-mean Gaussian and independent of . The separation2

represented by (11) indicates that is also the MMSE estimateof , thus and the estimation error covariance matrix

. Using the matrix decompositionresult in (9) and (10), we obtain , where

(12)

denotes the covariance matrix of which repre-sents the MMSE estimation error when is available at theCP without any distortion. The covariance matrix , and alsocovariance matrices that we require subse-quently, are obtained from the codec as described in Section IV.

Next we consider the processing for the observationat the sensor. We observe that, for the estimation of

, is a sufficient statistic of all the information availableat the CP from sensors . The innovation in canbe represented as , where

is the MMSE estimate of at the CP before isencoded and transmitted. The decomposition of Fig. 2(b) indi-cates that the preprocessing for is , itcan be verified that

(13)

This preprocessed signal is encoded at rateusing a side-informed codec where

is available as side information at the CP. The CP’s reconstruc-tion of via the side-informed decoder is modeled as

, where, as before, we assume is zero-meanGaussian and independent of . Next, the CP refines its es-timate of by using in the second stage of the decompo-sition, represented by a variant of (6),

(14)

The estimation error covariance matrix is updated as, where

(15)

Equation (15) corresponds to a simplified version of the recur-sion in the Kalman filter [25, p. 448]. Now, combining the ini-tialization with the iteration, the recursive processing for the se-quentially received sensor observations at the CP is summarizedin Algorithm 1 , where for clarity of description we have in-cluded the coding and decoding steps which will be described inSection IV. Note that the recursive preprocessing in Algorithm1 only requires the statistical characterization of the codec in

2We note that the separation assumes that the encoder–decoder combinationis optimal and therefore, strictly speaking, holds only asymptotically for codesoperating at the rate distortion bound. The consequence also follows when therate is high and �� can be approximated as uncorrelated with � .

terms of the covariances which, for a given rate allo-cation vector, can be obtained as described in Section III-B.

Algorithm 1: Recursive Estimation and Coding for MultipleSensor Observations

Input: Statistics and a rate vector .Available at all sensors and CP.

1: Initialize: At CP and each Sensor, set ,

2: for to do

3: Sensor records the observation and processes andtransmits as follows:

4:

5:

6: Transmit with rate

7: At CP the reconstruction is updated using informationfrom as follows:

8: The decoder obtains .

9:

10: At CP and update the estimation error covariancematrix:

11: , calculated by (15) and theevaluation of is described in Section III-B.

12: end for

For comparison, we also consider the scenario where eachsensor directly encodes its observation using a KLT codecwithout applying the side-informed estimator and the CP esti-mates the desired signal recursively from the received values.The processing is obtained via a minor modification of Algo-rithm 1, where the sensor observations are transmitteddirectly at each sensor without exploiting side information andthe CP reconstructs an estimate of from these received values.A corresponding test channel provides the required statistics.

B. An Achievable Rate-Distortion Bound and Rate Allocationfor Sequential Estimation and Coding

We develop a numerical algorithm to jointly determine therate allocation among the sensors under the total rate constraint,and an achievable rate-distortion bound under the sequential es-timation and coding framework.

For a prescribed rate allocation , where, an achievable R-D bound can be calculated in a

procedure identical to Algorithm 1 with the actual encoding/de-coding steps omitted and the statistics of reconstruction distor-tion computed under the assumption that the side informa-tion is available at both the encoder and decoder3. Under thisassumption for the channel, the forward test channel model[31, p. 479] specifies the reconstruction distortion ,where and represents the reconstructionof obtained from an optimal encoder–decoder combinationwith rate .

3This is justified by the Wyner–Ziv rate-distortion result for the Gaussiansetting.


Fig. 4. Test channel model for transmitting a vector signal � under rateconstraint.

Denoting , the vector Gaussian test channel(for the sensor) is obtained by using the eigendecom-position , where represents a diagonalmatrix whose entries are the eigenvalues of , arrangedin decreasing order, and is the corresponding matrix ofeigenvectors. The test channel parameters are then obtainedvia the reverse water-filling procedure on the diagonal en-tries in [30]. First, the number of transform channelswith nonzero rate allocation and the corresponding La-grange parameter that defines the per-channel distortion forthese channels is obtained as the solution to the constraints

and , whereand by conven-

tion. The corresponding test channel is then depicted in Fig. 4,where is the submatrix obtained by truncating to itsfirst columns, is a zero mean multivariate Gaussianindependent of , with diagonal covariance matrix where

,) and is a diagonal matrix withdiagonal entries . The covariancematrix of is obtained from the test channel as

(16)

Using the achievable R-D bound for a prescribed rate alloca-tion, we develop an algorithm to allocate the total rate amongthe multiple sensors as summarized in Algorithm 2. All sen-sors are initialized with a 0 bit allocation. Available bits, up tothe total rate , are then allocated by sequentially assigningeach successive bit to the sensor where this additional alloca-tion would result in the largest reduction in distortion. We notethe achievable R-D bound we calculate is limited to the sce-nario where allocated rates are integers, facilitating practical im-plementation. The R-D bound corresponding to the optimizedallocation is then used as a sequential achievable bound forour problem4. For the chosen rate allocation vector, the corre-sponding test channel of Fig. 4 also provides an estimate offor Algorithm 1.

Algorithm 2: Rate Allocation for Sequential DistributedEstimation and Coding

Input: , total rate .

Output: : rate allocation vector (acrosssensors)

1: Initialize , for all

2: for to do

3: for to do

4:

4The achievable bound we define can be shown to be attainable under randomcoding—though practical codecs attaining the corresponding performance havenot been demonstrated yet.

Fig. 5. The decomposition with a side-informed transform codec.

5: if or then

6: , {tentatively assign thebit to sensor }

7: calculate the sequential R-D bound withallocated rate vector using Algorithm 1

8: end if

9: end for

10: , {allocate the bit tothe sensor which results in largest reduction indistortion using this bit.}

11: end for

IV. SIDE-INFORMED CODEC DESIGN

In order to encode the preprocessed sensor observationswithin the corresponding channel rate constraints,

we consider a practical side-informed codec construction ina transform coding framework [3], leveraging prior work onscalar side-informed codes for the direct observation setting [5],[6], [24]. An overview of the codec is illustrated in Fig. 5 usingthe notation of the previous section.

At the sensor node, the preprocessed observation istransformed to obtain a transform vector and cor-respondingly, at the CP, first an estimate is obtained for ,from which, is estimated as . Our transform codeccorresponds to the conditional KLT [4], which ensures that the

transform coefficients forming the vector are conditionallyindependent (given the side information at the CP). Note that theconditional KLT represents the optimal transform coding in thepresence of receiver side information [4] , much like the KLTrepresents the optimal transform coding in the absence of sideinformation [32].5

At the CP, the side information is transformed to obtain. Each of the coefficients in

is then encoded using a scalar Wyner–Ziv codec, where thecoefficient uses the corresponding side information inthe decoding process. The side information is modeled as

, where is assumed to be a zero mean Gaussian in-dependent of with variance ,which corresponds to the conditional variance of the giventhe side information . The available rate at the sensor

5The general optimal vector quantization problem is known to be NP com-plete [33].


Fig. 6. The bit-plane sequential scalar Wyner–Ziv codec obtained by concatenating TCQ with a bit plane sequential Slepian–Wolf code [6].

node is distributed among the coefficients in using the re-verse water-filling algorithm as described in Section III-B toobtain rates where is the rate allocated for coef-ficient . Next, without loss of generality, we consider thisscalar Wyner–Ziv encoding for the coefficient with asthe CP side information. Furthermore for notational simplicity,we drop the subscripts and denote , , , and , respec-tively, by , , , and in the remainder of this description.

The scalar side-informed codec for is implemented as atrellis-coded quantizer (TCQ) [34], followed by a bit-plane se-quential Slepian–Wolf [5], [35] codec,6 which closely follows[6] and represents the current state-of-the-art in scalar GaussianWyner–Ziv coding. Because of space constraints, we includeonly a brief overview here and refer the reader to [6] and ref-erences therein for further details.

The codec operates on a block of i.i.d observationsat the sensor node. First, the samples are quan-

tized using a bit TCQ with representation points andthen a Slepian–Wolf codec is utilized for communicating the bitplanes sequentially where prior bit planes and side informationis utilized for the decoding of each bit plane at the receiver, typi-cally requiring fewer than bits per sample for the transmissionbecause of both the side information and correlations betweenthe bit planes. We describe each of these components next andlater in this Section, we describe how is selected according tothe allocated rate .

For TCQ, we assume a specific configuration7 wherein aquantized representation of the observations, using bitsper sample, is obtained by using an underlying configurationof representation points, which are organized ascosets with points each. The sequence of representationpoints for the samples are constrained so that at each timeinstant, a 2-bit output from a rate 1/2 feedforward binary con-volutional code [36] determines the coset of the representationpoint. As a consequence of this constraint, the coset indexes ofthe representation points can be encoded as the correspondingsequence of input bits for the convolutional encoder using 1 bitper sample, leaving free the remaining bits at each timeinstant for the selection of the representation point closest to theobserved sample, from the points within the coset. At the

6Slepian–Wolf coding refers to side-informed coding for discrete alphabetsources as opposed to side-informed coding for continuous valued sources con-sidered in Wyner–Ziv coding.

7Using alphabet constrained rate-distortion theory, [34] demonstrates that thisconfiguration suffices for obtaining performance close to the R-D bound in theGaussian setting.

encoder, a Viterbi [37], [38] search on the convolutional codetrellis allows efficient determination of the best convolutionalcode input sequence and coset memberidentifiers for the representation of the block of samples, thelatter of which we represent by blocks of bit planesproceeding from most significant bit (MSB) to least significantbit (LSB) as , whererepresents the bit-plane and the bitsindicate the selection of a representation point at time instant ,within the coset determined by the convolutional code outputat time in response to the input sequence .

The TCQ module is coupled with a bit-plane sequentialSlepian–Wolf codec to obtain our overall scalar Wyner–Zivcodec shown in Fig. 6, which, as previously indicated, is closelybased on the construction in [6]. Because incorporates theconvolutional code trellis memory, it is directly transmitted tothe decoder. Given , using the TCQ convolutional encoder,the decoder obtains the sequence of 2-bit(or 4-valued) coset index vectors, where identifies the cosetof representation points used at time . This simplifies theside-informed coding and decoding of remaining bit planesbecause given and the side information , the bit planes1 through at time instant are conditionally inde-pendent of corresponding planes and the side information atother time instants. This allows computation of the conditionallikelihoods required for the Slepian–Wolf decoding in a mem-oryless fashion. To provide rate adaptability, we implementthe individual bit-plane Slepian–Wolf codecs using low-den-sity parity-check accumulate (LDPCA) codes [27], wherethe decoder for the bit-plane utilizes as side informationthe coset index , the side information at the decoder, andprevious bit-planes . The LDPCA coderepresents a rate adaptive version of the LDPC codes [39], [40]that have been used in side-informed coding [26], [27]. Parityinformation is generated by the LDPCA encoder and sent to theCP, where the coset index vector , the parity information andthe side information are used by an iterative message-passingdecoding [41] to estimate the encoded bit values. Appendix IIdescribes how the side information is used in the decodingalgorithm.

The LDPCA encoder provides rate scalability [27] by firsttransmitting a fraction of the computed parity bits, and progres-sively transmitting more bits at the request of the decoder if thedecoding fails. Given the allocated rate-constraint , is firsttransmitted with rate 1 bit per sample, and the bit-plane is


transmitted next, a fractional bit rate is consumed in thisprocess. The codec continues to transmit additional bit-planes

consuming corresponding rates , re-spectively. The bit rate actually consumed is

.Once the decoded bit-planes are available from the LDPCA

Slepian–Wolf decoder, it computes an estimate for . The op-timal estimate can be obtained, by using for each time instant

the conditional mean asthe estimate for . While this nonlinear estimate can be evalu-ated empirically [6], it offers only a minor improvement over asuboptimal linear estimator8 which first uses the TCQ decoderto obtain an estimate for and then usesa linear MMSE estimator to obtain [6], [24]

(17)

where is the variance of the TCQ quan-tizer error, which is observed to be time invariant except for thetrellis initialization phase. is empirically estimated for a

-bit TCQ using a unit variance i.i.d. zero mean Gaussian inputand scaled by the standard deviation of to obtain .

The variance of the estimation error for the LMMSE esti-mator is [25]

(18)

In our codec implementation, for each coefficient, bit-plane, we select this TCQ bit depth

for which is closest to the target rate, where the expression is obtained by considering the R-D

function for a scalar Gaussian with variance and distortiongiven by the right-hand side of (18).

We note this codec introduces delay and requires a binaryfeedback channel.9 The delay constraint can be avoided by usingweaker side-informed codec, e.g., the nested quantizer, at the ex-pense of (practical) rate-distortion performance. The feedbackrequirement can be eliminated, albeit at prohibitive computa-tional cost, by specifically designing irregular LDPC codes foreach individual bit plane customized for the corresponding con-ditional variance.

V. SIMULATION

In order to demonstrate the benefit of the decomposition wepropose, and to investigate the effect of side-informed codecs,we simulate the sequential estimation and coding setup devel-oped in Sections II and III. All sensor observations are modeledas for , where the desired signalis a multivariate Gaussian generated according to a covariancematrix . Several different choices of the measure-ment matrices and the signal covariance matrix

are explored in the simulations. Each of the coefficientsin the vector noise is generated (independently) according

8This is demonstrated in [6] and also independently verified by us.9The binary feedback channel enables rate adaptability by indicating de-

coding failures for the LDPCA code to the sensor, which serves as a signal forthe encoder to send additional parity bits for the scalable LDPCA code.

to the distribution . The value of is obtained ac-cording to a prescribed signal-noise-ratio (SNR) defined as

SNR

SNR values of 10 and 0 dB are utilized in the simulations inorder to illustrate system performance in the high and low SNRregime, respectively.

Our Wyner–Ziv code constructions described in Section IVmake use of 256 state TCQ using the optimized convolutionalcode polynomials listed in [6] that are specified by the pairof polynomials with coefficients (515, 362) in hexadecimalformat. The representation points used in TCQ are chosenas the representation points for a bit Lloyd-Maxquantizer [42].10 For the LDPCA codes, we use the implemen-tation of [27], which allow block lengths that are multiplesof 66. We choose an encoder block length of forcompatibility with this constraint while providing long enoughblocks to allow both the TCQ and the LDPCA codecs tofunction efficiently.

We compare the proposed encoder construction, denoted byP-WZ, against the bounds for achievable performance obtainedin Section III-B for our construction and against bounds delin-eating the best achievable performance for six other alternativescenarios. In total, we have seven scenarios.

1) Sequential preprocessing with side-informedcoding (SEQ-WZ) as described in Section III: Aswe propose, the encoder applies the side-informedestimator as a preprocessing and uses our side-informedcodec. This setting exactly matches our proposedconstruction. The bound derived in Section III-B, whichwe denote (SEQ-BD) represents the best performanceachievable that practical constructions of SEQ-WZ, inparticular, P-WZ, can attain.

2) Preprocessing with conventional transform coding (PTC):The encoder applies the side-informed estimator ofSections II and III as a preprocessing and uses a transformcodec without exploiting the side information. The overallcoding procedure to transmit a vector signal is similaras the transmission of described in Section III-B. Thecodec first applies the orthogonal transform matrixwhich is specified by the eigendecomposition of ,i.e., . The codec next allocates theavailable rate among the coefficients of using thereverse water-filling algorithm. Attainable performancebounds are then obtained in a manner similar to the cal-culation of the achievable rate-distortion bound for theproposed sequential estimation and coding as describedin Section III-B. We denote this bound by PTC-BD andit represents the best attainable performance for practicalconstructions of PTC.

3) Direct side-informed coding (DWZ): Sensor observa-tions are encoded directly using our side-informed codecwithout the preprocessing but with a transform codec that

10In our bit-plane sequential side-informed codec, inter bit-plane correlationsare exploited in the side-informed decoding at the receiver and negligible perfor-mance gain is obtained by using a codebook of representation points optimizedspecifically for TCQ (for instance, with the methods described in [34] and [43]).


uses a KL transform for mirroring the description inPart 2 above. Attainable performance bounds are alsoobtained similarly and denoted by (DWZ-BD).

4) Direct transform coding (DTC): Sensor observations areencoded directly using a conventional transform codecand a corresponding achievable bound (DTC-BD) is com-puted. The direct coding operation for both DTC-BD andDWZ-BD was outlined at the end of Section III-A.

5) Local MMSE estimation with transform coding (ETC-BD): The encoder obtains a local MMSE estimate ,which is sent to the CP. The CP obtains an estimate

. The achievable boundETC-BD for this setting coincides with the decoupledestimate-compress scheme in [7].

6) General distributed estimation without requiring the se-quential constraint (DST): Both preprocessing and side-in-formed coding can be used in DST and since the sequen-tial constraint is not imposed, the performance at least asgood as or better than SEQ-WZ is assured. No practicalcodec constructions are available as yet for DST, though acomputational method for evaluating an achievable bound,which we denote DST-BD, is presented in [7]. We extendthe two-sensor case described in [7] to the multisensor caserelevant to our simulation settings.

7) Joint estimation from all sensor observations (JNT): JNT isinfeasible in practice because it requires all sensor observa-tions at one location. However, for comparison, it is usefulto include the rate-distortion function for the transmissionof the joint MMSE estimate represents anabsolute lower bound of the rate-distortion function for thedistributed estimation problem, which we refer to as thejoint lower bound (JNT-BD).

Both options 1) and 2) operate according to Algorithm. 1, thuslie within the framework of the decomposition that we propose.In all our plots (unless otherwise specified), the abscissa is theaverage bit rate at each sensor and the ordinate is the decoderMSE distortion, expressed in decibels as .All results presented represent an average over ten Monte Carlosimulations. Due to the long block length of our codec, the stan-dard deviation across simulations is quite small and is negligibleon the scale of our plots and therefore not included.

A. Two-Sensor Case

We first simulate the two-sensor scenario as illustrated inFig. 2(a), i.e., . The signal covariance matrix is setas , where, prior to commencing the Monte Carloruns, and the measurement matricesare generated with i.i.d. random coefficients each distributed as

. In this scenario, the information-theoretic rate-distor-tion function is given by (8), and we benchmark our simulationsagainst this theoretical lower bound (RD-BD), which also coin-cides with SEQ-BD, DST-BD, and JNT-BD, thereby reducingus to 5 additional scenarios characterized by bounds in additionto the practical P-WZ scheme.

For our first set of simulations, we set , (for) and obtain the results shown in Fig. 7(a) and (b) for

the low and high noise regime, respectively. Our first obser-vation from these figures is that our practical codec construc-

Fig. 7. Comparison of rate-distortion performances of different encodingschemes for the two-sensor case �� . (a) Low noise regimeSNR � 10 dB. (b) High noise regime, SNR � 0 dB.

tion (P-WZ) does quite well and offers an RD performance ap-proaching the RD-BD except for the very low rate case in thelow noise regime.11 In fact, in the region of interest where thebounds diverge, P-WZ’s empirical performance exceeds eventhe bounds that mark the best attainable performance for all the4 other alternative encoding scenarios.

A further comparison of the bounds is also instructive. Inboth the high and low noise regimes, PTC-BD and ETC-BD arequite close. Furthermore, Fig. 7(a) indicates that in a low noiseregime the distortion corresponding to DTC-BD is larger com-pared to PTC-BD and ETC-BD. PTC-BD slightly outperformsETC-BD. DWZ-BD also significantly outperforms DTC-BD,indicating the effectiveness of Wyner–Ziv coding in the lownoise regime, although the cross-over between PTC-BD andDWZ-BD indicates that at high rates the preprocessing is morehelpful whereas at low rates the side-informed coding is more

11At low rates, the direct transmission of the trellis path encoding bit planeintroduces inefficiency causing this loss in performance [6].


effective. Fig. 7(b) presents results for the high noise regime.In this setting, DTC-BD yields poor performance as comparedto PTC-BD, SEQ-BD and ETC-BD. We observe that PTC-BDhas similar performance as ETC-BD. Comparing Fig. 7(a) and(b), we observe that under the low noise regime, the advantageof Wyner–Ziv coding is more significant, and the degradationof side information at high noise regime also decreases the effi-ciency of side-informed coding.

B. Multi-Sensor Scenario: A Special Case

Next, we consider a special case of the multisensor scenariothat is synthesized to highlight the benefit of the proposed de-composition. The signal of interest and the sensor measure-ments are all derived via linear, potentially noisy, observationsof an 6 1 zero-mean multivariate Gaussian vector whoseelements are i.i.d. with zero mean and unit variance. Specifi-cally, we assume , where ,and the sensor observations are , where themeasurement matrices are12 ,

, and . Thus, wehave , , and for all . In this special case,we allocate the total available rate equally among these sensors.

The rate distortion performance obtained in the simulationsis summarized in Fig. 8(a) and (b) for the low and high noiseregime, respectively. Once again, we see that our P-WZ codecconstruction offers performance close to the correspondingSEQ-BD. Though the gap between P-WZ and SEQ-BD is largerthan the gap in the two sensor case,13 the P-WZ constructionoutperforms all other bounds with the exception of the PTCbound in the high noise regime, where the noise makes theside-information ineffective for coding and local preprocessingcomponent common to PTC and SEQ provides most of thegain. We also observe that the SEQ-BD is close to the DST-BDin both scenarios, despite the absence of the sequential con-straint for DST. This once again demonstrates the effectivenessof the sequential approach. The three best achievable boundsare DST-BD, SEQ-BD, and PTC-BD in that order and theseoutperform other achievable bounds by a significant marginin either high or low noise scenarios. Furthermore, PTC-BDsignificantly outperforms ETC-BD and DTC-BD in both casesby using the preprocessing we propose. DWZ-BD also out-performs DTC-BD in both scenarios, and more significantlyin the low noise regime where the Wyner–Ziv coding is moreeffective.

This special case clearly highlights the effectiveness of theproposed sequential approach, the preprocessing effectively ex-tracts relevant information from the local sensor observation,and the Wyner–Ziv coding explores the side information at theCP to send the preprocessed signal.

12If the measurement model formulation is assumed to be in the frequency do-main, these measurement matrices correspond to sensors with increasing band-width in the order 1, 2, 3, 4 with each of the sensors also covering a band offrequencies that is not of interest for estimation of the vector � desired at theCP.

13With the multiple sensors and channels the low rate inefficiencies of theTCQ codec and the scalable LDPCA codes are encountered more frequently.

Fig. 8. Comparison of rate-distortion performances of different encodingschemes for the multisensor case with the measurement matrices of Section V-B.(a) Low noise regime SNR � 10 dB. (b) High noise regime, SNR � 0 dB.

C. Multi-Sensor Scenario: The General Case

We next simulate a “generic” multisensor scenario. We con-sider , , and for all . In this simula-tion, the autocovariance matrix , where, prior to thecommencing the Monte Carlo runs, and the mea-surement matrices are generated with i.i.d randomcoefficients each distributed as .

The simulation results are illustrated in Fig. 9 and similartrends are observed as in Fig. 8. Among the achievable bounds,in both high and low noise scenarios, DST-BD, SEQ-BD repre-sent the best performance in that order with a small compromisein performance for SEQ-BD over DST-BD. Our practical codecconstruction P-WZ offers performance close to SEQ-BD ex-cept for low rate scenarios. P-WZ also outperforms the achiev-able bounds of other alternative encoding schemes except forlow rate scenarios. We also observe that PTC-BD slightly out-performs ETC-BD, both clearly outperform DTC-BD. Further-more, as expected, in the low noise regime side information


Fig. 9. Comparison of rate-distortion performances of different encodingschemes for the generic multisensor case. (a) Low noise regime SNR � 10 dB.(b) High noise regime, SNR � 0 dB.

is more effective than preprocessing so DWZ-BD outperformsPTC-BD and in the high noise regime the situation is reversed.

VI. DISCUSSION

Our problem setup assumes a distributed setting where thesensors communicate only with the CP and the CP does nottransmit any of the information it receives from the sensors.However, signaling from the CP to the sensors is required toestablish the network transmissions. Specifically, binary feed-back enables rate adaptability for our DSC codes by indicatingwhether the DSC decoding is successful or not. Note that this isconsistent with other DSC schemes [27], [44]. When the blocklength for the Wyner–Ziv codec is large, this rate for feedbacksignaling is negligible, when compared to the rate used for datatransmission at the sensors, e.g., for our LDPCA codec, trans-mitting a block of 6336 bits requires at most 66 (on averagearound 30) bits of binary feedback. The rates consumed in

the feedback change the rate-distortion plots by imperceptibleamounts and are therefore not included in the plots.

Our formulation imposes a sequential constraint on the en-coding and decoding procedures. For the scenarios we com-pared in Figs. 8 and 9, we see interestingly that the achiev-able bounds for the sequential approach (SEQ-BD) and for thegeneral distributed approach (DST-BD) are close. This suggeststhat the sequential approach may in fact be quite effective andmay not levy a significant R-D penalty in a number of scenarios.

The gap between our codec performance and the sequentialR-D bound (SEQ-BD) indicates that further improvements arefeasible within the proposed sequential framework. In partic-ular, optimized irregular LDPC codes would improve upon ourLDPCA code, which provides rate adaptability but levies a mod-erate rate loss [27]. Such an approach is quite inflexible be-cause individually optimized codes are necessary for specificbit planes and side-information statistics. The low rate perfor-mance can also be improved by using trellis coded vector quan-tization (TCVQ). Minor gains are also feasible by enlargingthe number of TCQ states. Each of these approaches furtherincreases computational complexity, which is already signifi-cant for our proposed construction. It is worth noting that in themultisensor scenario, deviations of the practical codec from theideal codec assumptions at a given sensor, create a mismatchwith the modeling assumptions for subsequent sensors whichalso partly explain the greater inefficiency in these scenarios inFigs. 8 and 9.

VII. CONCLUSION

In a sequential framework, the problem of distributed esti-mation and coding under rate constraints reduces to a sequenceof subproblems of side-informed transmission of an indirect(noisy) observation. We show that under jointly Gaussianstatistics and for a MSE metric, the optimal encoder–decoderdesign for indirect observations with side information can bedecomposed into a preprocessing “side-informed” estimationstep followed by “side-informed” coding of the estimate,without any degradation in rate-distortion performance. Thedecomposed structure enables the development of practicalcodec constructions with performance close to the numericalachievable bound for the sequential framework. Furthermore,the achievable bound of our sequential approach is close to theachievable distributed bound without the sequential constraint,implying that the sequential approach, which enables practicalcodec constructions, may not incur significant performanceloss.

APPENDIX IDISTRIBUTED ESTIMATION FROM

DIMENSIONALITY-CONSTRAINED OBSERVATIONS

In this appendix, we consider an alternate instantiation of thedistributed estimation and coding problem of Fig. 1 consideringdimensionality constrained channels. This alternative instanti-ation was previously considered in [9], [10], and [12] and isshown in Fig. 10. Here the rate constraints on the channels arereplaced by dimensionality constraints and the encoders and


Fig. 10. Dimensionality constrained channels. The CP receives reduced-di-mensionality observations and utilizes them in a linear estimator to estimate thedesired signal �.

Fig. 11. (a) Estimation of� from indirect observation� with side information� under dimensionality constraint. (b) A decomposed structure of (a).

decoders are replaced by pre and postprocessing matrices, re-spectively. Let denotes the dimensionality constraint for thechannel connecting and the CP, matrix denotesthe linear processing for , where thus imposes thedimensionality constraint. At the CP, the preprocessed observa-tion is postprocessed by matrix . The finalestimate is obtained by combining the postprocessed outputsas . The optimization of the linear pre andpostprocessing matrices under a MSE distortionmetric can be written as

(19)

We note Gaussian statistics are not required in this setup whenonly linear processing and linear channels are concerned. Wefirst note in (19) that can be obtained as the corre-sponding partitions of the LMMSE estimator once areknown, thus we focus on the optimization of . The ana-lytical solution for the joint optimization of is believedto be intractable [11]. We thus proceed as in prior literature andconsider the optimization of , assuming are known.An approximate solution for the joint optimization ofcan be obtained by using this in an alternating maximizationtechnique [9], where each of is optimized iterativelyuntil convergence. We next consider the optimization of inthe setup of Fig. 11(a) where denotes all the preprocessedobservations available at the CP.

Fig. 12. (a) Estimation of � from single indirect observation � under dimen-sionality constraint. (b) A decomposed structure of (a).

A decomposition similar14 to the one illustrated in Fig. 2(b)applies to the system of Fig. 11(a), and indicates a separativestructure illustrated in Fig. 11(b). The processing for consistsof a first side-informed estimation preprocessing and side-in-formed “coding”, the preprocessing is represented by whichis identical with that in Fig. 2(b), the side-informed “coding”is dimensionality-reduced approximation of with

as the side information. This approximation is obtained bythe conditional KLT [4], which indicates ,

and

where represents the unity matrix,denotes the innovation and represents the matrix of the

eigenvectors of corresponding to the largest eigen-values. can be interpreted as the MMSE estimator of

from . The CP processing is decomposed sim-ilarly, first an approximation of is obtained as

, and then is obtained from by using a variant of (7),i.e., , indicating

Comparing Fig. 11(a) and (b), we conclude

(20)

(21)

(22)

where we substitute and reorganize re-sulting terms. These results are identical with those in [9], how-ever obtained by applying our decomposition and the condi-tional KLT, both of which are also applicable to other channelmodels such as rate-constrained channel for jointly Gaussiansignals considered in our main development.

Fig. 12(a) illustrates the scenario where no side-infor-mation is available (but the channel dimensionality is con-strained to a positive integer ). The decomposition inFig. 3(b) applies and establishes the separative structure inFig. 12(b) [45]. The encoder first obtains the LMMSE estimate

, followed by dimensionality-reducedapproximation of . The latter problem can be reduced to aclassic matrix approximation problem, which is readily ob-tained by the truncated Singular Value Decomposition (SVD)

14The decomposition here is slightly different from the one in Section II butcan be obtained in an analogous fashion by utilizing the orthogonality propertyfor the LMMSE estimator, that the estimation error is orthogonal to any linearfunction of the predictor inputs.


[46]: , . Comparing Fig. 12(a) and(b), we further obtain

(23)

The decomposition of the preprocessing matrix as anLMMSE estimator followed by dimensionality reduction hasbeen observed in [9], [10], and [47] after obtaining the ex-pression of through matrix decomposition or canonicalcorrelation analysis [48]. In contrast, we first establish thisdecomposition and the solution follows in a straightforwardmanner.

APPENDIX IIDECODING WITH SIDE INFORMATION

Our side-informed codec uses the iterative message-passingdecoding algorithm [41] which has been adapted for side-in-formed coding [26]. Here, we briefly summarize how the de-coder inputs are computed and refer the reader to [27] for de-tails of the Wyner–Ziv codec. This decoding algorithm relieson the parity information sent from the encoder, and the initiallog-likelihood ratios (LLR) which are obtained at the decoderfrom the side information.

After the trellis path selector bit-plane is received, weobtain the coset index vector which is used for decodingthe subsequent bit-planes. The decoding of the bit-plane

use already decoded bit-planes, together with and , as side information,

and requires as input the initial LLRs,

(24)

The conditional probabilities in (24) are obtained by marginal-ization. For example,

(25)

We note that the side information is not quantized atthe decoder and the probability term in the summation onthe right-hand side of (25) is obtained by empirical estima-tion. Specifically, the conditional probability mass function

is empirically characterized for the TCQ encoder byassuming time invariance, and the Markov chain relationship

is used to obtain the probabilityin (25). Details can be found in [6].

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers forseveral insightful comments and suggestions that have signifi-cantly improved the paper.

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Chao Yu (S’05) received the B.S. degree inelectronic engineering from Tsinghua University,Beijing, China, in 2004 and the M.S. degree in elec-trical and computer engineering from the Universityof Rochester, Rochester, NY, in 2005.

Since September 2005, he has been workingtowards the Ph.D. degree in the Department of Elec-trical and Computer Engineering at the Universityof Rochester. His research interests lie in the areaof signal and image/video processing, and specif-ically in distributed signal processing, distributed

image/video compression and processing, and resource allocation in (visual)sensor networks.

Mr. Yu is a recipient of the best paper awards at the SPIE Visual Communi-cations and Image Processing (VCIP) Conference, San Jose, CA, in 2009.

Gaurav Sharma (S’90–M’90–SM’00) receivedthe B.E. degree in electronics and communicationengineering from the Indian Institute of TechnologyRoorkee (formerly the University of Roorkee), India,in 1990; the M.E. degree in electrical communicationengineering from the Indian Institute of Science,Bangalore, India, in 1992; and the M.S. degree inapplied mathematics and Ph.D. degree in electricaland computer engineering from North Carolina StateUniversity, Raleigh, in 1995 and 1996, respectively.

From August 1996 through August 2003, he waswith Xerox Research and Technology, Webster, NY, initially as a Member ofResearch Staff and subsequently as a Principal Scientist. From 2008 to 2010,he served as the Director for the Center for Emerging and Innovative Sciences(CEIS), a New York State-funded center for promoting joint university-industryresearch and technology development, which is housed at the University ofRochester. He is currently an Associate Professor at the University of Rochesterin the Department of Electrical and Computer Engineering, the Department ofBiostatistics and Computational Biology, and the Department of Oncology. Heis the Editor of the Color Imaging Handbook (Boca Raton, FL: CRC Press,2003). His research interests include media security and watermarking, colorscience and imaging, genomic signal processing, and image processing for vi-sual sensor networks.

Dr. Sharma is a member of Sigma Xi, Phi Kappa Phi, Pi Mu Epsilon, SPIE,IS&T, and the IEEE Signal Processing Society (SPS) and the IEEE Communi-cations Society. He was the 2007 Chair for the Rochester section of the IEEEand served as the 2003 Chair for the Rochester chapter of the IEEE SPS. He cur-rently serves as the Chair for the IEEE Signal Processing Society’s Image Videoand Multi-Dimensional Signal Processing (IVMSP) technical committee. He ismember of the IEEE SPS’s Information Forensics and Security (IFS) technicalcommittee and an advisory member of the IEEE Standing Committee on In-dustry DSP. He is an Associate Editor of the Journal of Electronic Imaging andin the past has served as an Associate Editor for the IEEE TRANSACTIONS ON

IMAGE PROCESSING and the IEEE TRANSACTIONS ON INFORMATION FORENSICS

AND SECURITY.

Distributed estimation and coding: A sequential framework based on a side-informed decomposition

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