16 IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 10, NO. 1 ...nehorai/paper/TNB_2011.pdf · Microsphere...

16 IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 10, NO. 1, MARCH 2011

Statistical Design of Position-EncodedMicrosphere Arrays

Pinaki Sarder�, Student Member, IEEE, and Arye Nehorai, Fellow, IEEE

Abstract—We propose a microsphere array device with micro-spheres having controllable positions for error-free target identi-fication. We conduct a statistical design analysis to select the op-timal distance between the microspheres as well as the optimaltemperature. Our design simplifies the imaging and ensures a de-sired statistical performance for a given sensor cost. Specifically,we compute the posterior Cramér–Rao bound on the errors in es-timating the unknown target concentrations. We use this perfor-mance bound to compute the optimal design variables. We discussboth uniform and sparse concentration levels of targets, and re-place the unknown imaging parameters with their maximum like-lihood estimates. We illustrate our design concept using numer-ical examples. The proposed microarray has high sensitivity, effi-cient packing, and guaranteed imaging performance. It simplifiesthe imaging analysis significantly by identifying targets based onthe known positions of the microspheres. Potential applications in-clude molecular recognition, specificity of targeting molecules, pro-tein–protein dimerization, high throughput screening assays forenzyme inhibitors, drug discovery, and gene sequencing.

Index Terms—Maximum likelihood estimation, microspherearray, optimal statistical design, position-encoding, posteriorCramér–Rao bound.

I. INTRODUCTION

M ICROARRAY devices are used to measure concentra-tions of targets such as mRNAs, proteins, antibodies,

and cells [1]. Conventional microarrays used in medicine aretwo-dimensional (2-D). They employ spots of specific shapespositioned in predefined locations and conjugated in surfacewith molecular probes to capture targets [2]. Thus, they haveposition encoding which avoids target identification errors.

Recently, a microsphere array technology has been developed[1], [3], [4]. The main advantages of these microarrays over 2-Dare their directional binding capability, higher sensitivity, andhigher surface-to-volume ratio that offers faster reaction. Theirmicrospheres are conjugated in surface with molecular probesto capture targets, and contain quantum-dot (QD) barcodes toidentify the captured targets. Optical reporters (e.g., fluorescent

Manuscript received February 19, 2010; revised September 11, 2010;accepted December 19, 2010. Date of publication February 22, 2011; date ofcurrent version April 27, 2011. This work was supported by the Departmentof Defense under the Air Force Office of Scientific Research MURI GrantFA9550-05-1-0443, and NSF Grants CCF-1014908 and CCF-0963742. As-terisk indicates corresponding author.

*P. Sarder is with the Department of Biostatistics, Harvard School of PublicHealth, Boston, MA 02115, USA (e-mail: [email protected]).

A. Nehorai is with the Department of Electrical and Systems Engineering,Washington University in St. Louis, St. Louis, MO 63130, USA (e-mail: [email protected]).

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

Digital Object Identifier 10.1109/TNB.2010.2103570

Fig. 1. Schematic of an existing microsphere array, where the microspheresare randomly placed on a substrate. The colors indicate quantum-dot barcodes,allowing to identify targets captured by microspheres. For clarity, we representthe microspheres here without their dedicated molecular probes.

dyes, QDs, nanospheres) are employed to quantify the targetconcentrations [5], [6]. Their imaging is performed using a flu-orescence microscope and an image sensor.

In existing microsphere arrays, the microspheres are typicallyrandomly placed on a substrate [1], [3], [4], see Fig. 1. Suchrandom placement of the microspheres renders their packinginefficient. It also hampers the quality of the imaging in areaswhere the microspheres are closely clustered, and it makes theautomatic imaging analysis difficult. Additionally, the existingmicrosphere arrays are prone to errors in identifying targets, dueto noise in the measured QD barcode spectra.

To overcome these drawbacks, we propose a new compactmicrosphere array layout with determinate microsphere posi-tions. These microspheres are thus position encoded, similar tothe spots in 2-D microarrays, thus identifying targets without er-rors through position encoding and simplifying significantly thedata processing. We surround the microspheres (captured withtargets) with nanospheres embedded with QDs. We use theseQD lights to quantify the target concentrations. We develop astatistical design approach to select the minimal distance be-tween the microspheres for a desired performance in imagingthe proposed microarray and achieve an efficient microspherepacking. We also compute the optimal operating temperatureof the image sensor fitting this performance, considering thatthe cost of such sensors varies proportionally with their coolingrequirements. Thus, our proposed design ensures a desired sta-tistical imaging performance for a given image sensor cost. Thefeasibility of implementing the proposed microarray layout withthe position-encoded microspheres is being demonstrated in aparallel research effort.

We first construct a statistical measurement model, assumingthe imaging is space-variant and employing the classicalthree-dimensional (3-D) point-spread function (PSF) proposedin [7]. Here we consider that the target distributions on the

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SARDER AND NEHORAI: STATISTICAL DESIGN OF POSITION-ENCODED MICROSPHERE ARRAYS 17

microspheres are either uniform or sparse. The uniform dis-tributions correspond to cases where the target concentrationis high, and/or the time period of the sensing is sufficientlylong. The sparse distributions correspond to low target concen-trations and/or short sensing durations. We assume the targetconcentrations are unknown, with known prior distributions,and the noise is additive Gaussian. We optimize the designby computing the sum of the posterior Cramér–Rao bounds(PCRBs) [8] on the errors in estimating the target concentra-tions. In this performance measure, we substitute the maximumlikelihood (ML) estimates of the photon-conversion factor ofthe image sensor and its background noise variance. We usethis estimated performance measure to compute the optimaldistance and temperature.

Some of the key advantages of the proposed microarrayover existing microsphere arrays are efficient packing, highsensitivity, simplified imaging, and guaranteed accuracy inestimating the target concentrations, as we will discuss inmore detail in Sections II and IV. Our microarray eliminatestarget identification errors because of the known positions ofthe microspheres. The proposed microarray can also simplifysignificantly the simultaneous screening of targets.

The paper is organized as follows. In Section II, we describethe configuration and imaging of the proposed microarray. InSection III, we compute the PCRB on the errors in estimatingthe target concentrations in imaging the proposed device. InSection IV, we present our method to compute the optimal dis-tance and temperature. In Section V, we present our estimationmethod. In Section VI, we show our results obtained in the nu-merical examples, and we conclude in Section VII.

II. POSITION-ENCODED MICROARRAY DEVICE

We discuss the configuration of the proposed microarray, itsimage-acquisition procedure, and its image analysis advantagescompared to the existing microsphere arrays.

A. Sensing Device Configuration

Fig. 2(a) illustrates a schematic diagram of our proposed po-sition-encoded compact microsphere array device. We assumethat all the microsphere centers are positioned in a plane parallelto the plane. Here we place the microspheres in a uniform2-D grid in controllable positions. For simplicity, we representthem without their dedicated receptors. The microspheres aremade of polystyrene and are around 5 m in diameter. Foreach microsphere, we encode specific receptors (antibodymolecules) to detect a target of interest. Thus, we identify eachtarget without errors from each microsphere location. We termthis property as position encoding for microsphere arrays. Bycoding different microspheres with corresponding receptors,we are able to identify multiple targets simultaneously withouterrors.

To optimally design the layout of the proposed device, wecompute the minimal distance between the microspheresto estimate the target concentrations with a desired accuracy.This optimal design maximizes the microsphere packing in theproposed microarray.

To detect and quantify the targets, we use nanospheres ( 100nm in diameter) embedded with identical QDs and conjugated

Fig. 2. (a) Schematic of a position-encoded microsphere array, where the mi-crospheres are separated by an optimal distance. (b) A target molecule capturedon a microsphere.

with receptors. The nanospheres allow label-free targeting (tar-gets do not contain any optical reporter, and thus their structuresand chemical properties remain unchanged), on-off signaling,and enhance the detection sensitivity [1]. The targets are cap-tured by the microspheres on one side, and are tagged by thenanospheres on their other side, see Fig. 2(b).

Thus, the main differences (mentioned so far) between theconfigurations of the proposed and existing microsphere arraysare the proposed position encoding and optimal selection ofthe minimal distance between the microspheres to estimate thetarget concentrations with a desired accuracy. Also, the posi-tion encoding offers higher sensitivity. Namely, in existing mi-crosphere arrays two or more microspheres often come in closeproximity of each other, and hence the receptors in that close-proximity region are unable to capture targets. This reduces thesensitivity of the existing microsphere arrays. In contrast, themicrospheres in our proposed microarray do not come close toeach other as their microsphere positions are controllable, andhence the sensitivity of our proposed microarrays is higher.

B. Preparing and Collecting Data

To physically prepare the data, we propose to follow the pro-cedure for the microsphere array in [1], except for identifyingthe targets. Namely, a microfluid stream with the targets ispassed through the sensors, and then a cocktail of nanospheresis released periodically [1]. The targets bind to the intendedmicrosphere surfaces on one side and to the nanospheres on theother side (Fig. 2(b), showing one target and one nanosphere asan example) [1]. All nanosphere QDs emit light upon excitationby UV light, and produce a source of light in the form of aspherical shell around each microsphere. The levels of theshell lights quantify the target concentrations. We identify the


targets using the known positions of the microspheres. Thisis in contrast to other approaches [1], where the targets areidentified by the colors of QD barcodes in the microspheres,creating possible errors.

To collect the data, we propose to follow again the procedurein [1]. Namely, to image the target-captured specimen, a fluores-cence microscope is focused at different depth planes of the en-semble, parallel to the plane of the target-free device shownin Fig. 2(a). This produces a series of 2-D cross-section imagesof lights emitted by the nanosphere QDs, see [9]. Thus, eachcross-section image of the spherical shell light formed around amicrosphere forms the image of a ring.

To capture the images, a CCD or CMOS image sensor withhigh quantum efficiency [10], i.e., with high sensitivity, is em-ployed. Examples of such sensors are those produced by WatecInc. or Micron Inc. [11], [12]. Sensors produced by these com-panies have high sensitivity, but require temperature coolingusing external electronics to reduce the background noise. Thecost of such sensors proportionally varies with their cooling re-quirements. Thus, we propose to select the optimal operatingtemperature of the image sensor as a trade-off between minimalcooling versus maximal estimation accuracy, using our statis-tical performance results as a function of the distance betweenthe microspheres and temperature in their image sensing.

C. Image Analysis Comparison With Existing MicrosphereArrays

Analyzing the images of the proposed microarrays should besignificantly simpler and more accurate than in existing micro-sphere arrays, where the random microsphere placement oftencauses some imaged microspheres to cluster [4], [13]. Also, thenumber of the segments in these imaged micropsheres in ex-isting microsphere arrays is not known a priori. Furthermore,their QD barcode spectra for the target identification are noisy[4]. Imaging such randomly placed microspheres requires com-plex segmentation and estimation of their number. Identificationof targets from the noisy QD light barcodes in the existing mi-crosphere arrays requires computations and is prone to errors. Incontrast, our proposed microarrays do not require such compu-tations for segmentation or target identification, and has no sucherrors. This is useful, for example, for simultaneous imaging ofmultiple targets.

III. PERFORMANCE ANALYSIS

We present a statistical performance analysis for estimatingthe target concentrations from our proposed device. We first de-scribe the measurement model for the fluorescence microscopyimaging of the proposed device with targets captured on mi-crospheres and with nanosphere QD lights. We then derive theperformance bounds on the errors in estimating the target con-centrations, for the statistical design.

A. Measurement Model

The measurement at the image sensor output, in fluorescencemicroscopy imaging of a QD illuminating object, is (see [13])

(1)

where , , and; , , and denote the numbers of

measurement voxels; is the unknown random parametervector in imaging; is the microscope output;

is a zero-mean Gaussian noise with variance, and is the photon-conversion factor of the image

sensor [14], [15]; models the interference due to thephoton counting process in the image sensor, and is indepen-dent from voxel to voxel; models the backgroundnoise, which is a zero-mean Gaussian noise with variance[13]; is due to the thermal noise1 of the image sensor[16], is independently and identically distributed (iid) fromvoxel to voxel, and is statistically independent with .Thus, is Gaussian distributed with mean andvariance , independent from voxel to voxel [13].In this paper, we assume that the image sensor output is freeof constant offset [16]. We also assume and are known.Otherwise, we estimate them using images captured from atraining experiment, see Section V.

Assuming a space-variant microscopy, the fluorescence mi-croscope output is given by (see [7])

(2)where is the fluorescence microscope PSF fora point source at a depth in the QD illuminating object

.We group the measurements into a vector form

(3)

where , , , and are —dimensional vectorswhose

components are , , , and , respectively;with and similarly for

and .1) Object Model (Nanosphere QD Intensity Profile of Two

Neighboring Microspheres): For the statistical design, we com-pute the error in estimating the target concentrations of twoneighboring microspheres as a function of their distance and thetemperature in their image sensing. Recall from Section II-Bthat the target concentrations on the microspheres are propor-tional to the intensity levels of the spherical-shell lights sur-rounding them. Consider two such shells and

with unknown parameters and , respec-tively, corresponding to the concentrations of the targets sur-rounding two neighboring microspheres with a distance apart.We model the object as

(4)

with unknown parameters . Below we considertwo different models to define , where the microspheres

1Note that the background noise considered in this paper is an approximation,as there exist other types of background noise, e.g., electronic noise, readoutnoise, and quantization noise [16]. In principle, these latter types of backgroundnoise could be avoided using external control. However, in any case the thermalnoise is the dominant component; it depends on the sensor material and increaseswith the sensor operating temperature [17].


Fig. 3. Left: Schematic of cross-section depicting target molecules captured (a) fully or (b) partially by a microsphere, and sandwiched by nanospheres, see the(a) full-shell and (b) sparse-shell models for � ��. Right: Ideal cross-section ring intensity image of the resulting (a) full shell or (b) sparse shell, associated withthe nanosphere QD lights. We schematize the left- and right-column figures of (a) and (b) without consistent scaling.

are either fully or partially covered with the targets. The fullshell corresponds to cases where the target concentration ishigh, and/or the time period of the sensing is sufficiently long.The sparse shell corresponds to low target concentrationsand/or short sensing durations.

• Full-shell model for : For a microsphere fully cov-ered with the captured target molecules, the emitted nanosphereQD lights completely surround the microsphere, and result in aspherical shell source with known radii and as follows:

ifotherwise,

(5)

where is the unknown intensity level which is constant in theshell and indexes two neighboring microspheres;see Fig. 3(a) (right) where the color level signifies the targetconcentration [5].

We define the prior distribution of the unknown parameterfor the shell using a uniform distribution

(6)

where is a uniform random variable, distributedfrom zero to a known maximum value [18]. We assumethat the prior distributions of for are statisticallyindependent of each other. We adopt a uniform distribution priorfor , since no additional information other than the maximumvalue of the target-concentration level is available in general.

• Sparse-shell model for : Here we consider a sparsemodel to describe the nanosphere QD light intensity profile forcases where the microspheres are surrounded only partially withtargets. In such cases, the target molecules are likely to be at-tached to each microsphere without fully covering it, and hencethe resulting QD intensity profile is sparse

ifotherwise,

(7)

where is the unknown intensity level which is sparsein each measured voxel of the shell, and indexes twoneighboring microspheres; see Fig. 3(b) (right) where the colorlevel in the figure signifies the target concentration [5]. For the

shell , we assume that the total number ofvoxels, where the measurements are captured, is . We denotethe values of at these voxels are , whichwe stack in an dimensional vector .

We define the prior distribution of the unknown parameterfor the shell at its measured voxel

using an exponential distribution

(8)

where is an exponential random variable, with a scaleparameter [18]. We assume that the prior distributions offor and are statistically indepen-dent of each other. Here the exponential distribution prior im-poses a sparsity in , and the parameter of this distributioninversely controls the sparsity level of . Namely, a very smallvalue of in (8) restricts the value of to be very close to zero,and thus constrains to be sparse. Note that a good knowledgeof is important for solving the corresponding sparse parameterestimation problem using the prior model (8). If not known, onecan attempt to estimate this parameter from the measurementsin some way, and thus the resulting sparse parameter estimationmethod becomes completely free of user parameters [19], [20].Note also that a Laplace distribution prior is widely used in theliterature to define the prior distribution of sparse parameters,for solving maximum a posteriori (MAP) estimation problems[21]. However in our work, since is positive, we define theprior distribution of this parameter using an exponential distri-bution prior instead of a Laplace distribution prior. Intuitively,the probability density function (pdf) along the positive axis ofa Laplace distribution with zero location parameter value andthe pdf of an exponential distribution are similar.

2) PSF Model: The fluorescence microscope typically dis-torts the 3-D object image [9], [22]–[24]. We describe this dis-


tortion using the classical model in [7], which allows us com-pute the known PSF using the microscope imaging parametersfollowing the manufacturer’s specification. This model is

(9)

where is the Bessel function of the first kind, the mi-croscope numerical aperture, the normalized radius in theback focal plane, the lens magnification, the QD emis-sion wavelength. Further,

(10)

is the optical path difference function. Moreover, and arethe refractive indexes of the immersion oil and the specimen,respectively, and is the depth at which the point source is lo-cated in the object.

B. Posterior Cramér–Rao Bound

We compute the PCRB on the error in estimating the un-known parameters of (1) to optimize the design. We employPCRB instead of CRB, as PCRB permits us to use realistic priorknowledge of the target-concentration levels for the proposeddesign. Namely, PCRB allows us to exploit the positivity infor-mation of the light-intensity level for the full-shell model (5) orthe sparse-shell model (7), to constrain the target-concentrationlevel from zero to a known maximum value for the full-shellmodel (5), and to exploit the sparsity information of the target-concentration level for the sparse-shell model (7). Below, wefirst briefly discuss the concept of the PCRB. We then introducethe joint likelihood of the measurement and unknown param-eters. After that we present the expressions of the elements ofthe (Fisher) information matrix, which we use to compute thePCRB.

1) PCRB: Let represents a vector of the measured data,be an dimensional unknown random

parameter to be estimated, be the joint probabilitydensity of the pair , and is an estimate of , which isa function of . The PCRB on the estimation error has the form

(11)

where denotes the statistical expectation with respect tothe joint pdf and is the (Fisher) informationmatrix with the elements

(12)

provided that the derivatives and in (11)and (12) exist. The inequality in (11) means that the difference

is a positive semidefinite matrix. We compute thePCRBs on the errors in estimating the unknown random pa-

rameters in corresponding to the diagonal elements of[8], [25].

2) Joint Likelihood Function: The joint likelihood functionof the measurement and the unknown random parameterusing (1) is

(13)

where is the conditional pdf of given andis the marginal pdf of .

• Expression of :

(14)where is the covariance matrix of . The expression of isgiven by

(15)

where denotes a diagonal matrix formed by the ele-ments of and is an identity matrix of dimension .

• Expression of for the full-shell model (5)

(16)

where is the prior pdf of the unknown parameter forthe shell . Recall from Section III-A1 that fol-lows a uniform distribution prior with a range from 0 to ,see (6). Also note with is the unknownrandom parameter to be estimated for the full-shell case.

• Expression of for the sparse-shell model (7)

(17)

where is the prior pdf of the unknown parameterfor the shell at its measured voxel

. Recall from Section III-A1 thatfollows an exponential distribution prior with a scale parameter

, see (8). Also notewith

is the unknown random parameter to be estimated for thesparse-shell case.

3) Information Matrix: We derive the elements of the(Fisher) information matrix using (12) for computing thePCRBs on the error in estimating the unknown random param-eters in , see Section III-B1. Below we present the expressionsof these elements for both the object models.

• (Fisher) information matrix for the full-shell model: Herewe consider that the object model in (4) is formed usingthe full-shell model in (5). Recall that the unknown random pa-rameter for the statistical design using the full-shell model is

, see Sections III-A1 and III-B2.We define

ifotherwise

(18)


and

ifotherwise.

(19)Using (18) and (19), we further define

(20)

The expressions of the elements of the 2 2 symmetric matrixusing (20) are

(21)

where we compute with respect to the pdf in (16)using the Monte-Carlo integration estimation technique [26],see Appendix I.

• (Fisher) information matrix for the sparse-shell model: Herewe consider that the object model in (4) is formed usingthe sparse-shell model in (7). Recall that the unknown randomparameter for the statistical design using the sparse-shell modelis , see Sections III-A1 and III-B2.

We assume that the measured voxel of , that correspondsto the element of , is

, where , , and. Using this assumption, we define

ifotherwise.

(22)We follow similar assumption and definition correspondingto the each element of . We then redefine for thesparse-shell case for by insertingfrom (22) in (20).

The expressions of the elements of the symmetric matrixusing (20) are

(23)

where we compute with respect to the pdf in (17)using the Monte-Carlo integration estimation technique [26],see Appendix I.

4) Comment: The expression of in (21) or (23) involvescomputing for .Here the second derivative of with respect to doesnot exist at the boundary points of the prior pdf . How-ever, the integral here with respect to , in computing thestatistical expectation, is zero for almost surely at the boundarypoints of . This is because the probability measure ofthe prior pdfs at each of their boundary point is zero, as the priorpdfs are continuous in our analysis. Thus, we arbitrarily include

or exclude the boundary points of the prior pdfs in the compu-tation, and assume that the second derivative of withrespect to exists for almost surely with probability one on theset of points where the prior pdfs are nonzero [27].

IV. STATISTICAL DESIGN

We present our statistical design method for selecting theoptimal (minimal) distance between the microspheres as wellas the optimal operating temperature in their image sensing.We first present the performance measure for the design as afunction of distance and temperature. We then present a least-squares (LS) estimation algorithm for automatically selectingthe minimal distance from the performance measure at a giventemperature [28]. We thereafter discuss how we select the op-timal operating temperature.

A. Performance Measure

We define the performance measure in estimating the targetconcentrations as the sum of the PCRBs on the errors in esti-mating the target concentrations. Namely, we define the perfor-mance measure as

(24)

where “tr” is the matrix trace operation and[29]–[31]s. We compute this measure as a function of the designvariables, i.e., the distance between the microspheres and theoperating temperature of the image sensor. From our discus-sion so far, it is evident that is a function of , see Section III.Below we discuss the relationship between this measure with .

The performance measure is a function of the noise level ,see (1), which in turn is a function of . Thus, is a function of

. Specifically,

(25)

where is a constant, is the known bandgap of the image-sensor material, and is the known Boltzmann constant [17].Here we assume is known; otherwise, we estimate it usingimages captured from a training experiment, see Section V. Wefurther assume that is constant for a given image sensor ma-terial, although varies with in reality. In this paper, weconsider the image sensor material is Silicon (Si), and the rela-tionship between with for Si (see, e.g., [32]) is

(26)

where the dependent second term is negligible for the tem-perature range that we use in the numerical examples presentedin Sections VI-C and VI-D to illustrate the concept of our pro-posed design. Hence, we consider is constant and its valueto be 1.15 in this paper. Note that one should replace in (26)and consider its temperature dependency based on the choice ofthe image sensor material of interest.

B. Minimal Distance Selection

We compute the minimal distance by analyzing as afunction of the distance between the microspheres at a giventemperature, to obtain a desired error in estimating the target


Fig. 4. (a) Schematic diagram of ��. (b) Graph example of the proposedparametric model � �� that represents �� shape. Similar to ��, this graphfirst decreases as � increases, and it then starts to flatten from � .

concentrations. We conduct an LS estimation to automaticallyselect the minimal distance. Below we first discuss our mo-tivation to conduct the estimation for the minimal distanceselection, and we then discuss the corresponding analysisdetails. Here we use to denote as a function of .

1) Motivation: Intuitively, as we increase the distance be-tween the microspheres, the light signals from their nanosphereQDs do not interfere with each other. Thus, flattens, seeFig. 4(a), and the error in estimating the target concentrationsis essentially due to the background noise in each microspherelocation individually. In other words, the errors between the mi-cropheres are decoupled, and the PCRB matrix should be blockdiagonal. Thus, we could automatically estimate the minimaldistance from corresponding to the distance at which sucha decoupling occurs.

To estimate at what distance starts to flatten, we firstreplace in the ML estimates of and . (See in Section V adiscussion on the ML estimation.) We denote this estimatedas . We then fit with a parametric curve, that modelsthe shape of as a function of , using an LS estimation. TheLS estimate of the distance at which starts to flatten shouldbe the minimal distance estimate.

2) Parametric Shape Model of : We propose a para-metric curve to model the shape of ; see Fig. 4(a) whichessentially resembles the shape of . This model is given by

(27)

where , , , and are the unknown parameters, andis an indicator function given by

fotherwise.

(28)

Similar to , here in (27) first decreases as increases,and it then starts to flatten from ; see Fig. 4(b) for anillustrative example.

3) Minimal Distance Estimation Using Least-Squares: Weestimate using an LS estimation method. Namely, we firstcompute at increasing values of at

, and we then fit these computed values with computedat . The relationship between and in amatrix-vector form is given by

(29)

where ,, and is the error vector.

We rewrite (29) further as

(30)

where is an dimensional matrix with rowas , ,

and .The least-squares estimates of the unknown parameters (see,

e.g., [28]) are

(31)

where argmax stands for the argument of the maximum, i.e.,the value of the given argument for which the value of theexpression attains its maximum value, and is theprojection matrix on the column space of [28], given as

(32)

We select the minimal distance as

(33)

C. Optimal Operating Temperature Selection

We select the optimal operating temperature by ana-lyzing as a function of the temperature in image sensing,to obtain a desired accuracy in estimating the target concen-trations. Namely, we select the that ensures a desiredperformance through for all possible distances between themicrospheres; see more details in Sections VI-C3 and VI-D3.The ability to select the optimal operating temperature usingthe performance analysis is critical for employing less expen-sive sensors, while attaining a desired estimation accuracy,see Sections II-B, VI-C3, and VI-D3. Specifically, we choosethe optimal operating temperature as a trade off between less


cooling (i.e., reducing the device cost) versus higher estimationaccuracy.

V. ESTIMATING AND BY OPTIMIZING THE EXISTING

MICROSPHERE ARRAY IMAGES

In this section, we estimate and , which we use for thestatistical design, using a training experiment that we conductwith the existing microsphere array layout. Namely, we imageusing the desired image sensor the lights generated by the QDsembedded in number of target-free microspheres placed ran-domly on a substrate [6]. We estimate using a method-of-mo-ments (MoM) estimation method [28] from each microsphereimage, and estimate from the noise-only section of the cap-tured image. The estimate of from one microsphere image tothe other varies in general, see Section VI-A. Hence, we use alarge number of microsphere images, estimate from each ofthem, and substitute the statistical median of these estimates toreplace for the statistical design described in Section IV. (Wediscuss in Section VI-A2 our motivation of using the statisticalmedian of the estimates instead of their statistical mean forthe design.) Below we first describe the measurement model forfluorescence microscopy imaging of a target-free microsphereembedded with QDs. We then present our proposed analysis toestimate and .

A. Measurement Model

Here we employ the measurement model (1), assuming thatthe object with unknown parameter is the QDlight intensity profile of a single microsphere, and assuming alsothat and are the other unknown parameters. We rewrite themeasurement model as

(34)

where , , and; and ( to ).

We make similar assumptions for and .• Object Model (Microsphere QD Intensity Profile): We

model this intensity profile using a parametric sphere of con-stant intensity level per voxel [13]. Namely, we define

ifotherwise,

(35)

where denotes the unknown average intensity level which isconstant in the sphere, , , and are the unknown centerlocation parameters, and is the known radius of the micro-sphere. We denote the unknown parameter vector of the objectby .

For simplicity, we assume a constant intensity level at everymicrosphere voxel. Intuitively, this assumption is justified be-cause the QDs are typically tightly and uniformly packed in-side each microsphere [6]; they produce light at nm resolution,whereas the microscope measurement is done at m resolution.Note that more complex models here could be used to obtainmore realistic results tailored to specific applications.

B. Estimation

In this part, we present our proposed procedure for estimatingand , using the captured image from a training experiment,

see Section VI-A. Below, we first propose an MoM estimationmethod [28] for estimating in (34) from each microsphereimage. This estimation needs the estimates of and . There-fore, we present next how we estimate . We then briefly re-view our parametric ML estimation method [13] for estimatingthe object parameter in (34) from each microsphere image.

1) Estimating : The reciprocal photon-conversion factorfor fluorescence microscopy is determined by several phys-

ical parameters, such as the integration time and the quantumefficiency of the detector [23], which are unknown in our re-search. Hence, we estimate in (34) using an MoM estimationmethod [28] from each microsphere image. This estimate is (seeAppendix II) given by

(36)

where is the estimate of the object parameter from the corre-sponding microsphere image, and is the estimate of the back-ground noise variance in the captured image, see Section V-B2.We denote the estimates of from the number of microsphereimages as . We substitute the statistical medianof these estimates to replace for the statistical design analysisdescribed in Section IV; see also Section VI-A for more details.

2) Estimating : We estimate from the noise-only sec-tion of the captured image. Recall that the background noise

in the captured-image is a zero-mean Gaussiannoise with variance , and is iid from voxel to voxel, seeSection III-A. Recall also that is related to following(25). We thus estimate first from the noise-only section ofthe captured image using the classical ML estimation methoddiscussed in [33, Ch. 6]. We estimate then as

(37)

where is the estimate of and is the temperature at whichthe image is captured in the training experiment. Note that it ispossible here to use sufficient number of measurement samples,and to ensure the estimate of is consistent [28].

3) Estimating : We estimate from each microsphereimage. Here we assume a large , since we employ an imagesensor with high sensitivity, see Section II-B. We also assumethe contribution of is negligible in (34), sincethe QD light imaging is a high signal-to-noise ratio (SNR)imaging [5]. Thus, estimating is essentially equivalent tofitting to the available measurementof a single microsphere at each voxel of the measurement.Therefore, we approximate (34) as follows:

(38)


Considering, and defining, we rewrite (38) as

(39)

With these assumptions and notations, we group the measure-ments into a vector form

(40)

where , , and are —dimensional vectorswhose

components are , , and respectively. The log-likelihood function for estimating using (40) is given by

(41)

where denotes the Euclidean vector-norm operation.The ML estimate of the parameters (see, e.g., [34]) is

(42)

where is the projection matrix on the column space of[28], given as

(43)

We then denote the estimate of as , which we usefor estimating in Section V-B1.

VI. RESULTS

We present our results for statistically designing the proposedposition-encoded microsphere array. Recall that our statisticaldesign analysis uses the values of the imaging parametersand . We estimate them from fluorescence microscopy im-ages of number of target-free microspheres placed randomlyon the substrate of the existing microsphere array layout, seeSections IV and V. Thus, we first present our results in esti-mating and from these microsphere images. Following thispresentation, we provide a brief discussion on validation of ourmeasurement model. We then present two numerical examplesto illustrate the concept of our proposed statistical design usingthe full- and sparse-shell models.

For the purpose of the illustration only, we consider in thispaper a Zeiss Axioscope 2 Mot+ fluorescence microscope [35]with an Axiocam MRm monochrome camera [36], to image themicrosphere QD lights. However, our proposed statistical de-sign analysis is general, and can be applicable in imaging theproposed microarray using any fluorescence microscope andany CCD or CMOS image sensor. In particular, one can em-ploy inexpensive image sensors produced by Watec Inc. [11] orMicron Inc. [12], which require temperature cooling in imaging,see Section IV-C.

A. Estimating and

In this part, we present our results in estimating and fromfluorescence microscopy images of number of target-free mi-crospheres placed randomly on a substrate. We first present theimaging experiment, and we then present our estimation results.

Fig. 5. (a) Focal-plane quantum-dot intensity image of all the microspheres.(b) Histograms of the estimated � from the individual microsphere images.

1) Imaging Experiment: We randomly placed target-free mi-crospheres on a polydimethylsiloxane substrate. These micro-spheres are made of polystyrene, and have a refractive index of1.334. They are QD-embedded, and are 5 m in diameter. Theycontain cadmium selenium sulphide QDs measuring 6 nm indiameter [6]. The QDs were excited at wavelengths lower than500 nm using blue/UV lights [6].

To image the microsphere QD lights, we employed a 10objective with the numerical aperture of the microscope as1.3, and used water as an immersion medium for the objective.We imaged the QD emission in 535 nm wavelength at

C. We captured the 3-D image with a resolution ofm along the -direction, and m/pixel

along the lateral direction.We show the focal-plane intensity image of all the micro-

spheres in Fig. 5(a). This image in Fig. 5(a) illustrates opticalcross-talk (as mentioned in Section I) in the locations where themicrosphere images bind in clusters. Naturally, the microsphereimages are optically indistinguishable in these locations.

2) Results: In Fig. 5(b), we present a histogram of theestimates that we obtain from the image shown in Fig. 5(a).We find 65 microsphere images appear as individual objectsin Fig. 5(a). We manually segment these images, and estimate

from each of them. We observe that the estimates vary inFig. 5(b), and also note a few outliers in the histogram of their


estimates. The presence of these outliers motivates us to use thestatistical median of the estimates for the statistical design,which we compute as 305.21. We also compute the statisticalmedian of the estimates as 0.0053. (Recall that is the av-erage QD intensity level of a microsphere, see (35).) We furthercompute from the noise-only section of theimage in Fig. 5(a), and we then compute using(37) and using , see Section IV-A. We substitutethe values of the estimated and the median of the estimated

to replace and in our proposed statistical design in thenext two subsections. Moreover, we use the median of the esti-mated to decide the value of in the numerical examplebelow for the statistical design using the full-shell models; seeSection VI-C for more details.

B. Validation of Our Measurement Model

In this part, we briefly discuss a validation of ourmeasurement model (1). Note that we rewrite this modelin (34) to estimate some of its parameters. The statistical modelincorporated in this measurement model is a combination ofGaussian approximated Poisson noise model and Gaussianbackground noise model. This statistical model is alreadyvalidated and/or well documented in [9] and [13]–[17], seeSection III-A. Similarly, the photon-conversion factor of theimage sensor used in this measurement model is also wellestablished in the literature in [14] and [15], see Section III-A.The authors in [37] further validate the performance of ML esti-mation of the photon-conversion factor from a statistical modelsimilar to this measurement model, using experimental andnumerical computations and theoretical performance analysis.

The full spherical model (5) or (35) for object and estima-tion of such object parameters from our measurement model arevalidated in our earlier work in [13], using experimental and nu-merical computations and theoretical performance analysis. Thesparse spherical object model (7) is a more general form of thefull spherical object model, and hence such model is more real-istic. For a known microscope PSF, estimation of such model pa-rameters from our measurement model is similar to performinga linear regression analysis for sparse parameter vectors, andsuch analysis is already validated in [20]. We also note that theuniform prior model (6) used to describe the intensity level ofthe full spherical shell model is a noninformative prior model,and thus this model is preferable when no other informative as-sumption for the prior on the intensity level is available, see [28].Moreover, the exponential prior model (8) used to describe theintensity level of the sparse spherical shell model is validatedby the authors in [21] in their study on sparse regression. Fur-thermore, the classical model (9) for microscope PSF employedby our measurement model is proved by the authors in [7] usingphysical analysis.

We therefore justify the models incorporated in our work, andmove to our numerical examples to illustrate the concept of thestatistical design.

C. Example 1: Statistical Design for the Full-Shell Case

In this example, we illustrate the concept of our proposedstatistical design of the position-encoded microsphere ar-rays for the full-shell models. Here we use the shell radii

m and m in (5) for protein targets ofdiameter 250 nm. We compute these radii by considering therespective sizes of the microspheres, nanospheres, and bio-re-ceptors (e.g., IgG antibody), which are 5 m, 100–200 nm,and 10–12 nm in diameter, respectively. We also use the orderof in (6) is similar as the median of the estimates inSection VI-A2. Note that the choice of is not so criticalhere, as we show below in Section VI-C2 that the statisticaldesign is robust with respect to .

1) Effect of Microspheres’ Distance on Performance: InFig. 6(b), we present the effect of the microspheres’ dis-tance on the statistical imaging performance. Here we use

and C. We observe that the estimatedperformance measure first decreases as increases, andit then flattens. This result is similar with what we intuitivelypredict in Section IV-B1 on the shape of as a functionof . We estimate using the proposed LS estimation method(see Section IV-B3) the distance at which starts to flatten.Recall that we define this distance as the minimal distancebetween the microspheres in our proposed statistical designanalysis, see Section IV-B1. In this example, we estimate theminimal distance to be 17 m.

2) Effect of Maximum Light Level on Design: In Fig. 6(b),we present the effect of the maximum microsphere lightlevel on the statistical design performance. Here weuse , 0.00525, and 0.0052, and C.We qualitatively observe that the minimal distance does notchange with varying . This result suggests that the min-imal distance is robust with respect to the maximum possibletarget-concentration level.

3) Effect of Temperature on Performance: In Fig. 6(c), wepresent the effect of the imaging temperature on the statisticaldesign performance. Here we use , and considerthat the microspheres are very close to each other with a distanceof 13 m. We observe that the performance degrades with highertemperature at a fixed distance. This result is useful to selectthe optimal operating temperature of the image sensor for thedesired performance in imaging; see also Section IV-C.

4) Distance and Temperature Effects on Performance: InFig. 6(d), we present the effects of the microspheres’ distanceand the imaging temperature on the statistical design perfor-mance. Here we use . We qualitatively observethat the statistical design performance is more sensitive on thetemperature in imaging than the distance between the micro-spheres, for the full-shell models.

D. Example 2: Statistical Design for the Sparse-Shell Case

We illustrate the concept of our proposed statistical designof the position-encoded microsphere arrays for the sparse-shellmodels. Here we use similar values for the shell radii andin (7) as we use in Example 1, and consider protein targets ofdiameter 250 nm. We also use and in (8) for moresparsity and for less sparsity, respectively. Note that the choiceof is not so critical here, as we show below in Section VI-D2that the statistical design is robust with respect to .

1) Effect of Microspheres’ Distance on Performance: InFig. 7(a), we present the effect of the microspheres’ distanceon the statistical imaging performance. Here we use


Fig. 6. Design results for the full-shell models, see Section VI-C. (a) Minimal distance is 17 �m. (b) Design at 0 C for varying � . (c) Design at � � �� m.(d) Performance as a function of temperature and distance.

and C. We observe that the estimated performancemeasure first decreases as increases, and it then flattens.This result is similar with what we obtain for the full-shellmodels in Example 1. We then compute the minimal distanceto be 11 m following the same procedure that we employ inExample 1.

2) Effect of Sparsity on Design: In Fig. 7(b), we present theeffect of the sparsity on the statistical design performance. Herewe use and , and C. We qualita-tively observe that the minimal distance does not change withvarying . This result suggests that the minimal distance is ro-bust with respect to the sparsity level.

3) Effect of Temperature on Performance: In Fig. 7(c), wepresent the effect of the imaging temperature on the statisticaldesign performance. Here we use , and consider similarto Example 1 that the microspheres are very close to each otherwith a distance of 7.5 m. In this setup, we obtain a similar resultthat we obtain for the full-shell models in Example 1. Namely,we observe that the performance degrades with higher tempera-ture at a fixed distance. Thus, similar to Example 1, we find theresult here is useful to select the optimal operating temperature

of the image sensor for the desired performance in imaging; seealso Section IV-C.

4) Distance and Temperature Effects on Performance: InFig. 7(d), we present the effects of the microspheres’ distanceand the imaging temperature on the statistical design perfor-mance. Here we use . We qualitatively observe that thestatistical design performance degrades with higher temperatureand/or with closer distance between the microspheres.

E. Validation of the Parameters Used for Statistical Design inExamples 1 and 2

We justify our selection of parameters that we use for ourstatistical design in Examples 1 and 2. The selections of shellradii and microscope PSF parameters do not require any formof validation, as we assume their values are known in the design.We thus consider realistic sizes of microspheres, nanospheres,protein targets, and bio-receptors to compute the shell radii, seeSection VI-C, and we use typical experimental convention tochoose microscope PSF parameters, see Section VI-A1. Simi-larly, the design variables and do not require any form ofvalidation, as they are user chosen parameters.


Fig. 7. Design results for the sparse-shell models, see Section VI-D. (a) Minimal distance is 11 �m. (b) Design at�� C with � � �� and �. (c) Designat � � �� m. (d) Performance as a function of temperature and distance.

In our statistical design, we use the estimates for the truevalues of the order of the prior parameter of the full-shell object model, the constant factor of in (25), andthe photon-conversion factor Recall from Section VI-A thatwe obtain these estimates from images of randomly depositedmicrospheres using our measurement model. Using these es-timates for the true parameter values thus validates our mea-surement model for the design. We note that our design is ro-bust with respect to the prior parameters for both the full-shelland sparse-shell object models, see Sections VI-C2 and VI-D2.Therefore, it is possible to use arbitrary values for these priorparameters in our analysis, which also do not require any formof validation.

VII. CONCLUSION

In this paper, we proposed a new approach to designingmicroarrays with position-encoded microspheres and discussedthe potential advantages of such devices. The main advan-tages of our proposed microarray over existing microspherearrays are more efficient packing, higher sensitivity, simplified

imaging, and guaranteed accuracy in estimating the targetconcentrations. Potential applications of this device includemolecular recognition, specificity assessment of targetingmolecules, protein–protein dimerization, high throughputscreening of assays for enzyme inhibitors, drug discovery, andgene sequencing.

To position the microspheres in our proposed microarray,we developed an optimal statistical design analysis consid-ering imaging the microarray using a fluorescence microscopeand an image sensor. In designing our microarray layout,we first derived posterior Cramér–Rao bounds on the errorsin estimating the target concentrations for uniform or sparsetarget profiles. We then showed quantitatively the effects of themicrosphere distance and operating temperature on the imagingperformance. Using these statistical performance analysis re-sults, we computed the optimal (minimal) distance between themicrospheres at a given temperature and the optimal operatingtemperature of the image sensor. The minimal microspheredistance guarantees a desired level of statistical accuracy inimaging the proposed microarray with efficient microsphere


packing. The optimal sensor temperature acts as a tradeoffbetween the sensor cost and our desired statistical performance.

To compute the optimal variables in our statistical design, wesubstituted the maximum likelihood estimates for the true valuesof the photon-conversion factor of the image sensor and its back-ground noise variance. We generated these estimates using im-ages of multiple target-free microspheres, that were embeddedwith quantum-dots and placed randomly on a substrate. To ob-tain the estimate of the photon-conversion factor, we used amethod-of-moments (MoM) estimation. In performing this es-timation, we replaced the quantum-dot intensity levels and lo-cations of the imaged microspheres with their maximum likeli-hood estimates.

Our collaborators are currently implementing our proposedmicroarray with position-encoded microspheres using the min-imal separation distance between them. In our future work, toextend our design for sparse-shell models at a low signal-to-noise ratio, we plan to derive tighter bounds to analyze the statis-tical performance of imaging the position-encoded microarrays.We also plan to develop algorithms to estimate target concentra-tions from the imaging data generated by our device—the focusof which will be sparse target profiles. For uniform target pro-files, the concentration estimation algorithm will be similar towhat we developed in [13], and here the only difference will bethat we will need to use the full-shell model (5) as the unknownobject model.

APPENDIX IMONTE-CARLO INTEGRATION ESTIMATION

We present the Monte-Carlo integration estimation technique[26] for single parameter cases, which can be easily extended formultiparameter cases. Suppose we wish to compute integrals ofthe form

(44)

where is a function of the variable and is a pdf.We generate a large number of samples of fromand compute the Monte-Carlo estimate of the integral (44)

(45)

The accuracy of depends on , but is independent of(i.e., the dimensionality of the integral (44)). By the law of largenumbers,

(46)

APPENDIX IIPROOF OF EQUATION (36)

For each microsphere image segment, we have from (34)

(47)

In (47), replacing by its corresponding estimate and aver-aging both sides over all the voxels of the microsphere imagesegment of interest, we obtain (36).

REFERENCES

[1] A. Mathur, “Image analysis of ultra-high density, multiplexed, mi-crosphere-based assays,” Ph.D. dissertation, Northwestern Univ.,Evanston, IL, 2006.

[2] P. Sarder, A. Nehorai, P. H. Davis, and S. Stanley, “Estimating genesignals from noisy microarray images,” IEEE Trans. NanoBiosci., vol.7, no. 2, pp. 142–153, Jun. 2008.

[3] W. Xu, K. Sur, H. Zeng, A. Feinerman, D. Kelso, and J. B. Ketterson,“A microfluidic approach to assembling ordered microsphere arrays,”J. Micromech. Microeng., vol. 18, p. 075027, 2008.

[4] K. D. Bake and D. R. Walt, “Multiplexed spectroscopic detections,”Annu. Rev. Anal. Chem., vol. 1, pp. 15–47, 2008.

[5] M. Han, X. Gao, J. Z. Su, and S. Nie, “Quantum-dot-tagged microbeadsfor multiplexed optical coding of biomolecules,” Nat. Biotechnol., vol.19, pp. 631–635, 2001.

[6] Crystalplex Corporation. Pittsburgh, PA [Online]. Available: http://www.crystalplex.com

[7] S. F. Gibson and F. Lanni, “Experimental test of an analytical model ofaberration in an oil-immersion objective lens used in three-dimensionallight microscopy,” J. Opt. Soc. Amer., vol. 8, pp. 1601–1612, 1991.

[8] H. L. van Trees, Detection, Estimation and Modulation Theory. NewYork: Wiley, 1968.

[9] P. Sarder and A. Nehorai, “Deconvolution methods of 3D fluorescencemicroscopy images: An overview,” IEEE Signal Process. Mag., vol. 23,pp. 32–45, 2006.

[10] M. Christenson, “The application of scientific grade CCD cameras tobiological imaging,” in Imaging Neurons: A Laboratory Manual ch. 6,R. Yuste, F. Lanni, and A. Konnerth, Eds. Cold Spring Harbor, NY,Cold Spring Harbor Lab. Press, pp. 23–32, 2000.

[11] Watec [Online]. Available: http://www.wateccameras.com/index.php[12] Micron MT9M product sheet [Online]. Available: http://www.aptina.

com/products/image_sensors/[13] P. Sarder and A. Nehorai, “Estimating locations of quantum-dot-en-

coded microparticles from ultra-high density 3D microarrays,” IEEETrans. NanoBiosci., vol. 7, no. 4, pp. 284–297, Dec. 2008.

[14] D. L. Snyder and M. I. Miller, Random Point Processes in Time andSpace, 2nd ed. New York: Springer-Verlag, 1991.

[15] T. M. Jovin and D. J. Arndt-Jovin, “Luminescence digital imaging mi-croscopy,” Annu. Rev. Biophys. Biophys. Chem., vol. 18, no. 1, pp.271–308, Jun. 1989.

[16] G. M. P. van Kempen, “Image restoration in fluorescence microscopy,”Ph. D. dissertation, Delft Tech. Univ., Delft, The Netherlands, 1999.

[17] J. Ohta, Smart CMOS Image Sensors and Applications. Boca Raton,FL: CRC, 2007.

[18] A. Papoulis and S. U. Pillai, Probability, Random Variables, and Sto-chastic Processes, 4th ed. New York: McGraw-Hill, 2001.

[19] E. I. George and D. P. Foster, “Calibration and empirical bayes variableselection,” Biometrica, vol. 87, p. 731747, 2000.

[20] E. G. Larsson and Y. Selén, “Linear regression with a sparse parametervector,” IEEE Trans. Signal Process., vol. 55, no. 2, pp. 451–460, Feb.2007.

[21] S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEETrans. Signal Process., vol. 56, no. 6, pp. 2346–2356, Jun. 2008.

[22] J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-di-mensional imaging by deconvolution microscopy,” Methods Enzymol.,vol. 19, no. 3, pp. 373–385, Nov. 1999.

[23] P. J. Verveer, “Computational and optical methods for improving res-olution and signal quality in fluorescence microscopy,” Ph. D. disser-tation, Delft Tech. Univ., Delft, The Netherlands, 1998.

[24] H. T. M. van der Voort and K. C. Strasters, “Restoration of confocalimages for quantitative image analysis,” J. Microsc., vol. 178, pp.165–181, 1995.

[25] P. Tichavsky, C. H. Muravchik, and A. Nehorai, “PosteriorCramér–Rao bounds for discrete-time nonlinear filtering,” IEEETrans. Signal Process., vol. 46, no. 5, pp. 1386–1396, May 1998.

[26] G. Casella, Monte Carlo Statistical Methods, 2nd ed. New York:Springer-Verlag, 2004.

[27] Y. S. Chow and H. Teicher, Probability Theory: Independence, Inter-changeability, Martingales, 3rd ed. New York: Springer, 2003.

[28] S. M. Kay, Fundamentals of Statistical Signal Processing, EstimationTheory. Englewood Cliffs, NJ: Prentice-Hall, 1993.

[29] F. Pukelsheim, “Information increasing orderings in experimental de-sign theory,” Int. Stats. Rev., vol. 55, pp. 203–219, 1987.


[30] F. Pukelsheim, Optimal Design of Experiments, 1st ed. New York:Wiley, 1993.

[31] K. Shah and B. Sinha, Theory of Optimal Designs. New York:Springer-Verlag, 1989.

[32] B. V. V. Zeghbroeck, Principles of Semiconductor Devices and Het-erojunctions. Upper Saddle River, NJ: Prentice-Hall, 2008.

[33] R. V. Hogg and A. T. Craig, Introduction to Mathematical Statistics,5th ed. Engelwood Cliffs, NJ: Prentice-Hall, 1995.

[34] B. Porat, Digital Processing of Random Signals: Theory andMethods. Englewood Cliffs, NJ: Prentice-Hall, 1994.

[35] Axioskop 2 plus / Axioskop 2 FS plus [Online]. Avail-able: http://www.zeiss.com/C1256D18002CC306/0/974174FE5C0B97FBC1256D5900335191/$file/40-083_e.pdf

[36] Microscopia [Online]. Available: http://www.smt.zeiss.com/0625690B000338EF/Contents-Frame/4AE544FB791F7753C1256BF80043CCB2

[37] F. Aguet, D. V. D. Ville, and M. Unser, “A maximum-likelihood for-malism for sub-resolution axial localization of fluorescent nanoparti-cles,” Opt. Express, vol. 13, pp. 10503–10522, 2005.

Pinaki Sarder (S’03) received the B.Tech degreein electrical engineering from the Indian Institute ofTechnology, Kanpur, in 2003, and the M.Sc. and Ph.D.degrees in electrical engineering from WashingtonUniversity in St. Louis, MO (WUSTL) in 2010.

He is currently a Postdoctoral Fellow in thedepartment of Biostatistics at Harvard School ofPublic Health, Boston, MA. His research interestsare mainly in computational biology, statisticalsignal processing, and biomedical imaging.

Dr. Sarder was a recipient of the Imaging SciencesPathway program fellowship for graduate students at WUSTL starting from Jan-uary 2007 to December 2007.

Arye Nehorai (S’80-M’83-SM’90-F’94) receivedthe B.Sc. and M.Sc. degrees from the Technion, Is-rael, and the Ph.D. degree from Stanford University,Stanford, CA.

He was a faculty member at Yale University andthe University of Illinois at Chicago. In 2001 he wasnamed University Scholar of the University of Illi-nois. He is currently the Eugene and Martha LohmanProfessor and Chair of the Preston M. Green Depart-ment of Electrical and Systems Engineering at Wash-ington University in St. Louis, MO (WUSTL). He

serves as the Director of the Center for Sensor Signal and Information Pro-cessing at WUSTL.

Dr. Nehorai has been a Fellow of the Royal Statistical Society since 1996. Hereceived the 2006 IEEE SPS Technical Achievement Award and the 2010 IEEESPS Meritorious Service Award. He was elected Distinguished Lecturer of theIEEE SPS for the term 2004 to 2005. He was Corecipient of the IEEE SPS1989 Senior Award for Best Paper, Coauthor of the 2003 Young Author BestPaper Award, and Corecipient of the 2004 Magazine Paper Award. He servedas Editor-in-Chief of the IEEE TRANSACTIONS ON SIGNAL PROCESSING during2000–2002. In 2003–2005 he was Vice President (Publications) of the IEEESignal Processing Society (SPS), Chair of the Publications Board, and memberof the Executive Committee of this Society. He was the Founding Editor of thespecial columns on Leadership Reflections in the IEEE SIGNAL PROCESSING

MAGAZINE from 2003 to 2006. He has been the Principal Investigator of severalmultiuniversity grants including the Multidisciplinary University Research Ini-tiative (MURI) project entitled Adaptive Waveform Diversity for Full SpectralDominance.

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