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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 5, MAY 2009 1839

A Biologically Inspired Compound-Eye DetectorArray—Part I: Modeling and Fundamental Limits

Zhi Liu, Student Member, IEEE, Arye Nehorai, Fellow, IEEE, and Eytan Paldi, Member, IEEE

Abstract—This is the first part of a two-part paper. In thispaper, we propose a detector array for detecting and localizingsources that emit particles including photons, neutrons, or chargedparticles. The array consists of multiple “eyelets.” Each eyelethas a conical module with a lens on its top and an inner subarraycontaining multiple particle detectors. The array configurationis inspired by and generalizes the biological compound eye: it isspherically shaped and has a larger number of detectors in eachindividual eyelet. Potential applications of this biomimetic arrayinclude artificial vision in medicine (e.g., artificial eyes for theblind) or robotics (e.g., for industry or space missions), astronomyand astrophysics, security (e.g., for radioactive materials), andparticle communications. In this part, we assume Poisson distribu-tion for each detector’s measurement within the observation timewindow. Then we construct a general parametric model for the de-tection rate of the Poisson-distributed measurements illustrated bya circular Gaussian lens-shaping function (LSF) approximation,which is commonly used in optical and biological disciplines. Toillustrate how this “prototype” model fits practical cases, we applyit to an example of localizing a candle from 20 miles away andestimating the parameters under this circumstance. In addition,we also discuss the hardware setup and performance measure ofthe proposed array, as well as its fundamental constraints. Part Iforms the theoretical basis for Part II, in which we analyze theperformance of the array, both analytically and numerically.

Index Terms—Artificial compound eye, biologically inspired,biomimetics, eyelet, fundamental limits, lens, mean-square an-gular error, modeling, particle communication, particle detectorarray, particle source localization, weighted direction estimator.

I. INTRODUCTION

I N Part I of our paper, we propose a novel detector arrayfor detecting and localizing particle-emitting sources. The

array consists of multiple “eyelets,” each having a conicalmodule with a lens on its top and an inner subarray containinga large number of small particle detectors (for different typesof particle detectors, see e.g., [1]–[4]). The eyelets can be used

Manuscript received October 27, 2007; accepted November 26, 2008. Firstpublished February 06, 2009; current version published April 15, 2009. Theassociate editor coordinating the review of this manuscript and approving it forpublication was Prof. Mounir Ghogho. This work was supported by the AFOSRGrant FA9550-05-1-0018 and the National Science Foundation Grants CCR-0330342 and CCF-0630734.

Z. Liu is with OSIsoft, Inc., San Leandro, CA 94577 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]).

E. Paldi is with the Department of Mathematics, Israel Institute of Tech-nology, Technion City, Haifa 32000, Israel (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/TSP.2009.2014699

to detect uncharged particles such as photons and neutrons. Forthe case of visible-light sources, this array configuration is ageneralization of the biological compound eye, being spheri-cally shaped and having a larger number of detectors in eachindividual eyelet. Compared with most existing optical sensorarrays (e.g., planar CCD or CMOS sensor arrays in digitalcameras), the proposed sensor array has a larger field of view(because of its spherical shape) and higher sensitivity to thesource direction (because of multiple eyelets). In other words,the proposed array combines the advantages of the biologicalcompound eye (large field of view) and of human eyes (highspatial resolution). Moreover, the size of the spherical array iscompact, compared with other distributed sensor arrays (e.g.,distributed antenna arrays for localizing EM sources).

This work extends our previous work on directional detectorarrays, in which we introduced several detector arrays, com-puted their performance bounds for source localization [5], andproposed a detection algorithm to determine the presence of afar-field particle source [6]. The current extensions include theintroduction of multiple eyelets, lenses, and subarrays. Com-pared with the arrays with projection response to the sourcedirection in [5] and [6], the proposed array with lenses on topof subarrays possesses a sharper response function and onethat is adjustable, and hence the angular resolution is increasedmarkedly. In this two-part paper, we extend our earlier resultsin [7], where we briefly introduced the configuration of theproposed array, and in [8], where we analyzed the performanceof the array numerically. We expand our work in [7] and [8], bygeneralizing the mathematical model and adding new analyticaland numerical results.

A. Motivation and Applications

This work is motivated by the need for sensitive particle sen-sors in applications such as artificial vision, which can be impor-tant in robotics (see, e.g., [9]) or medicine (e.g., artificial eyesfor the blind); security, where the detection of radioactive mate-rials is needed (e.g., in verifying compliance with nuclear non-proliferation treaties); and astronomy assisted (e.g., estimationof orientation and angular velocity with respect to the stars forsatellites and space vehicles). Potential applications also include“particle communication.” Different types of particles in poten-tial applications are listed in Table I. Notably, an artificial com-pound eye that mimics a biological compound eye anatomicallyas well as functionally was recently built [10]. This achievementindicates that the array configuration proposed in this paper isnot merely a “thought experiment” but is a potentially practicalone that may have wide impact in the above areas.

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1840 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 5, MAY 2009

TABLE IPARTICLES IN DIFFERENT APPLICATIONS

The choice of particles and detectors depends on the specific application.

Charged particles are subject to scattering.

Fig. 1. (a) Dragonfly’s compound eye; (b) anatomical cross section of a compound eye and an ommatidium; (c) optics of an apposition compound eye; and(d) optics of a superposition compound eye.

Remark 1.1: Although we used visible photons in the anal-ysis of this paper, the proposed array and the mathematical/practical model apply to all the types of particles listed inTable I.

B. Outline of This Part

This part focuses mainly on modeling the array’s behaviorin localizing particle sources, both statistically and physically.First, we assume a Poisson distribution for each detector’smeasurement within the observation time window. Then, weconstruct a general parametric model for the detection rate ofthe Poisson-distributed measurements illustrated by a circularGaussian lens-shaping function (LSF) approximation, which iswidely used in optical and biological disciplines. To illustratehow this “prototype” model fits in practical cases, we apply itto an example of localizing a candle from 20 miles away andestimate the parameters under this circumstance. In addition,we also discuss the hardware setup of the proposed array andthe fundamental constraints on the array variables. Finally, weintroduce a performance measure of the array and analyze itsfundamental limitations.

Part I forms the analytical basis for Part II, in which we willanalytically and numerically analyze the performance of thearray. Part I is organized as follows: in Section II, we briefly de-scribe the optical properties of different types of compound eyesand compare them with the proposed array; in Section III andSection IV, we introduce the general mathematical measure-ment model and the statistical measurement model for the array,respectively; in Section V, we present an example to show howthe model can be used in practice; in Section VI, we discuss thehardware setup and fundamental limits of the array variables; inSection VII, we introduce the array’s performance measure andanalyze its fundamental bounds; in Section VIII, we summarizethis work.

II. BIOLOGICAL COMPOUND EYES AND THE PROPOSED ARRAY

We briefly describe the optical properties of biological com-pound eyes and introduce the configuration of the proposedarray.

A. Biological Compound Eyes

A compound eye is a visual organ found in certain arthropods(some insects and crustaceans), as shown in Fig. 1(a). It usuallycontains multiple “ommatidia,” each consisting of a lens and arhabdom; see Fig. 1(b). In contrast to the single-aperture eyes ofvertebrates, compound eyes have no central lens or retina. Froman optical viewpoint, compound eyes can be classified into twofundamental categories [11]: apposition and superposition.

As the commonest type and possibly the ancestor of theothers, an apposition eye has its rhabdoms receiving light onlyfrom their “own” corneal facets [12]. This eye consists of anarray of up to 30 000 individual ommatidia, each containinga lens on the top with a cluster of (usually seven or eight)photoreceptor cells inside, which form a single photosensitivelight-guiding rod, or rhabdom [13]. In other words, the omma-tidia work as optically isolated units in these eyes, as shown inFig. 1(c). In these eyes, each ommatidium collects light rayscoming from a small part of the object and forms a small imageof that part on its distal tip. Combination of all these smallimages creates a complete image of the whole object.

In contrast, superposition eyes have the characteristic that theommatidia are not optically isolated. The term “superposition”refers to the fact that contributions from many optical elementsare superimposed in the image plane [11]. In these eyes, eachrhabdom senses light from a large number of corneal lenses,and each facet forms an image that extends over many rhab-doms [see Fig. 1(d)]. The superposition mechanism combinesthe small images of each rhabdom into a single image of theobject.


LIU et al.: A BIOLOGICALLY INSPIRED COMPOUND-EYE DETECTOR ARRAY—PART I 1841

Fig. 2. (a) Cross section of a segment of the proposed array and (b) schematic description of each eyelet, with a lens on top of a conical module and an insidesubarray of detectors.

TABLE IITERMINOLOGY USED FOR THE BIOLOGICAL COMPOUND EYE AND THE PROPOSED ARRAY

Compared with single-aperture eyes, compound eyes havea much wider field-of-view (FOV), better capability to detectmoving objects, and higher sensitivity to light intensity. More-over, a compound eye may perform the function of multiple ar-rays simultaneously, each having different spectral responses,polarization sensitivity, integration times, or adaption organisms[12]. However, because of the tiny aperture size of each corneallens and small number of detectors in each ommatidium, thediffraction effect of light limits the spatial resolution to about 1,yielding an image roughly 100 times coarser than that of humaneyes.

B. The Proposed Array

The proposed array is spherical and contains multiple opti-cally isolated eyelets [see Fig. 2(a)] inspired by an appositioncompound eye. The reasons for choosing the apposition com-pound eye as the “template” include its simplicity of structureand popularity in nature. As shown in Fig. 2(b), each eyelet hasa conical module with a lens on top and a subarray inside con-taining a large number of small particle detectors. Between thedetectors and the edge of the conical module, there is a “de-tector-free margin” whose purpose is to make negligible the dis-tortion of the lens-shaping function due to scattering of particlesnear the cone’s edge. Under the focusing effect of the lens, theincident particles form an intensity distribution on the subarray,inducing a random measurement at each detector. These mea-surements contain information about the source direction, whichcan be extracted using statistical methods.

We configure the proposed array with two major modifica-tions on the apposition compound eye: i) providing the proposedarray with a rad FOV by a complete spherical shape (in prac-tice the FOV may be less if mounting is considered), and ii) al-lowing a larger number of detectors in each subarray. Both mod-ifications, as shown in the analysis below, remarkably improve

the array performance. To further understand the similarities be-tween the biological compound eye and the proposed array, wecompare the terminology used for both structures in Table II.

III. GENERAL MATHEMATICAL MEASUREMENT MODEL

We consider several remote point sources whose intensitiesin the array vicinity are assumed to be homogeneous. The mea-surements of the array are particle counts of all detectors withinan observation time window containing information about thesource direction of interest. In order to analyze the array perfor-mance, we construct a model for these measurements.

A. Concepts, Notations, Assumptions, and Transfer FunctionDerivation

In Appendix D, we introduce several definitions, notations,and assumptions that are needed for the derivation of the transferfunction for the general model. Based on those definitions, wederive the general model of the proposed array below.

B. Orthonormal Basis Expansion for the Array TransferFunction

Since the transfer function for each detector is(from a physical consideration) arbitrarily closely approximatedthrough its truncation by a threshold energy, we may assumethat it vanishes outside some finite energy band . Let

, be any given complete system of orthonormalbasis functions on , the transfer function can be ex-panded according to this basis with corresponding coefficients

and converge in to the transfer function.In this sense, the dependencies of the transfer function onand are separated. The coefficients play the phys-ical role of the detector cross section for the function .Since this expansion holds for any given complete orthonormalsystem of basis functions, it is of importance to find an optimal



orthonormal basis functions for the transfer function. We findthis optimal basis functions using a (continuous) version of sin-gular value decomposition (SVD) for integral kernels.

C. Optimization of the Orthonormal Basis Functions and LPGFunctions Approximation

The optimal basis functions problem is to find an orthonormalbasis function such that the truncation of the transfer functionexpansion after any given number of terms should give the best

approximation to the transfer function over the energy-direc-tion space (EDS). The construction of the optimal basis and itsproof are very similar to the discrete kernel (i.e., matrix) case.So we give below only the construction.

Define the integral kernel

where US denotes the unit sphere. It is easy to verify that thekernel is well defined over the square , nonnegative,symmetric and sufficiently smooth.

This kernel induces a compact integral operator fromto itself which is self-adjoint and has a complete

orthonormal eigensystem of sufficiently smooth eigenfunc-tions. The corresponding eigenfunctions arrangedaccording to decreasing eigenvalues are the desired optimalbasis functions. It is known1 (from classical theory of Fredholmintegral equations) that each eigenvalue has a finite dimensionaleigenspace (spanned by finitely many eigenfunctions) and eachsimple eigenvalue uniquely determine its eigenfunction (i.e., itsone-dimensional eigenspace). Moreover, it follows that for theoptimal basis functions, the corresponding coefficientsof the transfer function are orthogonal as well (over the US).

If the transfer function is strictly positive (in fact, ifis strictly positive), a continuous analog of the Perron–Frobe-nius theorem2 (for nonnegative matrices) holds for the kernel

as well—implying that the first eigenfunction (themost important one) is uniquely defined and its correspondingcoefficient is positive over the US. Since the coefficients

are orthogonal over the US, it follows that for allthe coefficients must change signs over the US. Thus, thefirst coefficient is the only one that is positive over theUS! This implies the importance of this coefficient which givesthe best rank-1 approximation of by . Thisrank-1 approximation may serve to define the associate direc-tion for each detector (independently of ). If from somephysical reason, the dependence of the transfer function onis almost uninfluenced by the direction , then the approxima-tion of the transfer function by a rank-1 kernel is justified.

For a rank-1 kernel approximation, it follows that the fre-quency dependence indicates the detector detection ef-ficiency with respect to energy (hence imply the effective band-width ) whereas gives the “lens shaping function”for a point source (as clearly seen by applying the transfer func-tion to the “spherical delta function” representing the directionaldependence for point-like source intensity).

1See, e.g., [14].2See, e.g., [15].

Finally, as explained in Appendix E, we approximate locallyeach peak of by a Gaussian function (whose associatedcovariance is determined by the inverse of the Hessian of

) and multiply it by a spherical polynomial to forma polynomial-Gaussian (PG function) approximant. By takinga linear combination of such PG functions (LPG functions), wecan approximate any transfer function arbitrarily closely.

Based on Remark D.8, the general modeling for point-likeparticle sources and rank- approximation for the transfer func-tion, takes the form

where denotes the detection rate for the th detector.is the solid angle of the cone of associated directions for the

th detector, are the above cross section coefficients(of the transfer function with respect to the basis function

) for the th detector determined for the direction ofthe th particle source, is the th coefficient (with respect tothe basis function ) for the particle intensity for theth particle source, and are the resulting total noisy detection

rate for the th detector. Note that the coefficientsare approximated as LPG functions of the direction of the thparticle source.

This general modeling has the following specifica-tions: a) rank- approximation of the transfer function by

; b) rank-1 approximation by ;c) LPG approximation of —especially for multifocaltransfer function; d) PG approximation of ; and e) circularGaussian approximation of and uniform approximationfor —this is the simplest or “prototype model” forcircular Gaussian transfer function.

IV. STATISTICAL MEASUREMENT MODEL

We consider a single remote point-like source whose intensityin the array vicinity is assumed to be homogeneous and time in-variant.3 The measurements of the array are particle counts of alldetectors within the observation time window, containing infor-mation for the source direction of interest. In order to analyzethe array’s performance, first we construct a model for thesemeasurements.

Let be the number of eyelets in the array and be thenumber of detectors in each eyelet on the subarray’s detectionsurface (DS), which is defined as a smooth surface tightly en-veloping the detector subarray’s physical surface.4 To obtainisotropic performance (i.e., the array has identical performancein localizing sources over all directions), all eyelets (includinglenses, subarrays, and detectors therein) are assumed to haveidentical physical and geometrical properties. In addition, thelocations of the eyelets are assumed to be approximately uni-formly distributed over the spherical surface.

3The time invariance of the intensity is not really needed and assumed hereonly for simplicity.

4For simplicity, we assume the DS to be planar. Note that the analysis pre-sented below applies also to nonplanar DS cases, for example, the roughly spher-ical retina in human eyes.



Fig. 3. (a) Reference frames� and � and (b) right-handed rotation by � w.r.t.the � axis.

A. Reference Frames

We construct reference frames for the modeling:the array’s frame “ ” and the th eyelet’s frame “ ”

, as shown in Fig. 3(a). For an arbitraryvector in frame , let be its representation in frame .

In frame , we choose the origin at the center ofthe array. Denote the coordinate system as . Let

be the unit vector alongthe source direction, where and are the azimuth and eleva-tion angles, respectively. Denote by as the unit vector alongthe symmetry axis of the th eyelet (or the conical module)pointing outward, with and being its azimuth and ele-vation angles. From the isotropic assignment of the (so-called“balanced design”) eyelets, we have .

Reference frame , with its coordinate system denoted as, results from two consecutive right-handed rotations

[see Fig. 3(b)] from frame : first rotate frame with respectto the axis by angle ; then rotate the resulting frame withrespect to its axis by angle . It is clear that the axis inframe coincides with ; i.e., . Following therotation rules of coordinate systems in (see, e.g., [16]), it iseasy to show that for any vector , or ,where is the rotation matrix from frame to frame whoseexpression is given as , where

and

B. Poisson Measurement Model

From the nature of the particle-counting process, we assumethe following Poisson probability mass function (PMF) modelfor each detector’s particle count:

(1)

where is the stochastic measurement of the th detectorin the th eyelet (or the th detector; ,

) within time window , and is the particlecounting rate of that detector. Obviously, the determination of

is the foremost issue in the practical modeling, since it sta-tistically characterizes the detector’s measurement.

From a signal processing viewpoint, the detection rate ofthe th detector can be naturally expressed as

s (2)

where s and are, respectively, the expected particlecounting rate of the th detector coming from the source(s)and the noise. We consider two categories of noise: the count ofparticles coming from noise sources outside the array and theerroneous particle counts due to inner measurement (e.g., elec-tronic or thermal) errors. From the remote and time-invariantpoint-source assumptions, we assume the source-particle inten-sity coming from different directions is homogeneous over thearray’s surface and time invariant; similarly, the noise intensityis also assumed5 to be isotropic, homogenous, and time invariantthroughout the array.

As a specific model based on Section III and (2), we considerthe simplest rank-1 PG basis function model of

(3)

where is the solid angle of the cone of associated directionsof the th detector, is the th eyelet’s detection cross sec-tion area with respect to , is the exciting source particles in-tensity at the array surface (in s m ) ignoring the sensor’spresence, is the angular density of the noise particles intensityat the array surface6 (in s m rad ), and de-notes the circular Gaussian LSF:

(4)

In (4), is the associated source direction of the th de-tector and is the angular standard deviation of the GaussianLSF (in radians), which depends on the lens surface shape andoptical properties. Comparing (2) and (3), we have s

and , indicatinga good match between the mathematical and practical inter-pretations of . The vector of unknown parameters is then

.In Sections V and VI, we will discuss how to calculate the

parameters and variables defined above on a practical basis, howthey are associated with physical configurations of the array, andwhat their fundamental limitations are.

Remark 4.1: Gaussian-shaped point-spread functions (PSFs)for lens systems are widely used in the optical discipline. Exam-ples include optical lenses [17], scanning laser ophthalmoscopes[18], stellar photometry [19], and plasma lenses [20]. Moreover,Gaussian functions are commonly used to describe the angularsensitivity function (ASF) of a retina cell in the compound-eyeliterature (see, e.g., [21]).

5This assumption is made only for simplicity.6We assume � � � � � , where � and � correspond to the noise

coming from outside the array (e.g., noisy particle-emitting sources) and thoseinside the array (e.g., thermal or electronic noises), respectively.



Fig. 4. Candle and the detector array.

TABLE IIILIST OF VARIABLES USED IN NUMERICAL EXAMPLES

Remark 4.2: Model (3) has the property that all its parameterscan be experimentally calibrated (i.e., externally observable).Hence, we may consider it as a “minimal model.”

Remark 4.3: The th detector’s associated source directionis defined as the unit vector direction of the point-like source

that maximizes the detector’s particle counting rate. In otherwords, source particles coming from should focus on thedetection surface of the th detector. The method used to de-termine is given in Section V.

Remark 4.4: Note that .In other words, while the Gaussian LSF describes a distribu-tion of detection rate with respect to centered at , equiva-lently, it also defines a “focusing distribution” of centeredat : detectors with associated directions closer to source direc-tion have higher detection rates; and describes the width ofthe Gaussian LSF. In addition, the circular Gaussian LSF (4) issimple, in the sense that it is linear in in the exponent and in-variant to the chosen reference frame.

V. A PROTOTYPE MODEL AND ITS PARAMETERS AND

As discussed, model (3) is particularly applicable for the pro-posed compound-eye array. Its parameters, such as and , aredetermined according to practical applications. As a practicalexample (which will be used as the numerical example in thesecond part of the paper), we consider the problem of local-izing a candle from 20 miles away on a clear night, assuming nomoonlight but regular starlight, as shown in Fig. 4. This is theweakest light source that a normal naked human eye can detect[22]. We treat both the signal source (the candle) and the noise

sources (the stars) according to the theory of black-body radia-tion [23]. In Appendix A, we estimate the attenuated values ofparameters and in this environment as

s m

and

s m rad

VI. HARDWARE SETUP AND FUNDAMENTAL

LIMITATIONS OF THE ARRAY

In this section, we discuss the hardware setup of the arrayby defining its essential configurational variables. We also con-sider the fundamental limitations of these variables, which willconstrain the configuration of any real-world implementation ofsuch an array.

A. Hardware Setup

In Table III, we list the configuration variables of the hard-ware setup of the array, which are also shown schematically inFig. 5. In the following, we associate the directions in model(4), with the array’s physical variables.

We determine the associated source direction of the th de-tector in the th eyelet’s reference frame , with all involvedvariables shown in Fig. 5. Define the radial direction of the

th detector as a unit vector along the line joining the



Fig. 5. Determination of �� in the �th eyelet’s reference frame.

spherical array center and the center of the th detectorand pointing outward from the spherical array; and as a unitvector in the (outward) direction of the symmetry axis of the theyelet. Let points s and be the center of the subarray DSand the center of mass of the lens, respectively. Define asthe angle between and , and define as the angle be-tween and .

Because of the focusing effect of the lens, the associated di-rection generally differs from the radial direction ofthe th detector (if s). By Snell’s law (see, e.g.,[24]), we have i) 7; ii) and lie on bothside of ; and iii) , , and are on the same plane. Wethen define the angular expansion factor of the th detector as

. Note that in general may be differentfor different and , but when the conical module of each eyeletis narrow (i.e., ) we assume it to be a constant for alland , denoted by . Similar to , completely depends onthe lens properties: shape of inner and outer surfaces, refractivefactors, etc.

By the definition of and the above observations ii) andiii), we can determine in frame by

(5)

where. Then .

B. Limitations of Array Variables

We discuss in Theorem 6.1 the fundamental limitations onseveral of the array variables as defined above.

Theorem 6.1: The following fundamental limitations withminimal particle wavelength apply to array variables ,

, and :

(6a)

(6b)

7This is valid since the lens refractive index is always larger than that of thearray’s environment.

(6c)

where is the first positive zero of the Bessel function .Proof: See Appendix B.

VII. PERFORMANCE MEASURE AND ITS

FUNDAMENTAL BOUNDS

For analyzing the array’s performance in localizing particlesources, a natural candidate for the performance measure couldbe the Cramér–Rao bound (CRB) matrix of the unknown param-eter vector . However, a meaningful scalar quantity is more ap-propriate than a matrix measure since scalar quantities are easierto compare. In [25], we defined the mean-square angular error(MSAE) of any source direction estimator as

, where is the angular error between and its estimate. Clearly, depends on the array configuration,

the estimation algorithm, the value of unknown parameters ,and the length of observation time window . Another (DOAindependent) performance measure is the (uniform)average of over the US.

Moreover, we defined in [25] (see also [26]) another quantity:

(7)

where and are the (scalar) Cramér–Raobounds for the azimuth and elevation angles, respectively.We showed that for any regular model and estimator,

, whereis the number of snapshots, which is proportional to andin this case. Thus, serves as an asymptotic

lower bound on .It follows that is i) independent of the estima-

tion algorithm, ii) an asymptotically tight bound attainable bysome second-order efficient regular estimators (usually the max-imum-likelihood estimator (MLE); see, e.g., [27]), and iii) in-variant to the choice of reference coordinate frame, since theinformation content in the data is invariant under rotation. Con-sidering the above desirable properties, we use asa performance measure showing the best asymptotic accuracyfor each specific array in source-direction estimation. We will



derive the for the proposed array in the second partof this paper [28].

A. Universal Bound 1:

Now that we know that indicates the optimalperformance of a specific array configuration, we can now askthe following question: What is the “best” performance of anarbitrary array in principle? We try to answer this questionby proposing a universal lower bound on MSAE, denoted by

, for an arbitrary array inside a sphere with radius, regardless of the array’s hardware or software configura-

tion; i.e., , array configurations. For thisuniversal lower bound, we assume that neither outside noise(from noise particle sources) nor inside electronic/thermal noiseis present. We need also the following (reasonable) assumption.

Assumption 7.1: “Practical” assumption: It is impossible toestimate by any combination of hardware and software the DOAof incoming particles during a certain time window better thenthe inherent MSAE of the (stochastic) direction of their totalmomentum. (Note that their total momentum is stochastic be-cause of the quantum-mechanical uncertainty principle.)

The following theorem follows from derivations based on theuncertainty principle of quantum mechanics (see, e.g., [29] and[30]).

Theorem 7.1: Under Assumption 7.1, a universal lowerbound on the MSAE (respectively, ) of any direc-tion-of-arrival (DOA) estimator based on measurements fromany sensing hardware contained inside a sphere with radius is

whenever

(8)

(respectively, whenever ),where is the first positive zero of the Bessel func-tion of order 0, is the minimal wavelength of the incidentparticles, is without detected source par-ticles, and is the without detectedsource particles—which is uniformly attained by the MSAE ofa random estimator with uniform distribution over the US.

Proof: See Appendix C.Remark 7.1: The derivation of may be improved

for wideband energy/wavelength particles. (The partition of thewavelength bandwidth into several narrow bands should lead tothe replacement of in (8) by the inverse of the rms valueof .)

B. Universal Bound 2:

Another important universal bound for any DOA estimatorfor the source direction can be found using an interestingmethod similar to the derivation. This bound isobtained via the usual CRB expression; but instead of beingdependent on the measurement model of any specific sensor,it depends on the wave function for the energy-mo-mentum density distribution. Note that this is not the usual

for space–time density distribution, which can beseen by exploiting the fact that is proportional to the Fouriertransform (or characteristic function) of the usual . Then, min-imization of the CRB over possible functions (i.e., possiblesensors hardware) is equivalent to a maximization problemwith respect to . This gives a truly universal lower bound forthe of any possible hardware. This bound, whichmay be denoted as universal CRB or , is both looserand simpler than .

The need for the UCR stems from the fact that theis in fact a lower bound for the MSAE of the stochastic directionof the total momentum of the source particles with respect to ;i.e., if the direction of the total momentum may be regarded as“DOA estimator”—even though it cannot be observed directly,its MSAE with respect to can still be used as a lower boundfor any estimator of it. The problem with is that wetry to find a universal lower bound not for the MSAE of estima-tors for the total momentum direction, but instead for MSAE ofestimators of the source direction . Thus, another approach forsuch a universal bound is needed.

Theorem 7.2: A universal lower bound on the forany DOA estimator based on measurements from any sensinghardware contained inside a sphere with radius is

where

(9)

Proof: See Appendix C.Remark 7.2: Note that Remark 7.1 applies also to (9), and

VIII. CONCLUSION

In this part, we proposed a novel particle-detector array in-spired by the biological compound eye and statistically mod-eled its behavior. This array combines the advantages of bio-logical compound eyes (e.g., large field of view) and humaneyes (e.g., high spatial resolution). It can be used to localize op-tical, radioactive, or cosmic sources from the far field. To ana-lyze the array’s performance, we presented a mathematical min-imal model for the array’s measurements and gave an exampleto illustrate how the parameters in the model can be determinedin practice. We then defined the array’s physical variables andassociated them with the proposed mathematical model. In ad-dition, we discussed the performance measure of the array andderived its fundamental bounds. Part I forms the analytical basisfor the second part of this paper, in which we will use the pro-posed measurement model to analyze the performance of thearray, both analytically and numerically.

We believe that the proposed array potentially has a wideimpact on optics, security, and astronomy assisted applications.The fact that an artificial compound eye with similar structureand function to its biological counterpart was recently builtmakes this closer into reality.



A. Future Work

In future work, the current model can be extended by con-sidering the correlation effects among incident particles (e.g.,multipath scattering and polarization effects) by introducing acomplex transfer function, time-variant particle intensity (whichis essential for particle communication), particles with wide en-ergy spectrum (to extract more source parameters, e.g., temper-ature), adaptive optics, multifocal lenses and polarization filters.Other extensions may include eyelet and lens-shape optimiza-tion, adaptive estimation of the directions of multiple sources,and investigation of the channel capacity for particle communi-cation using the array.

APPENDIX AESTIMATION OF AND IN SECTION V

A. Estimation of

Assume that the array detects only visible photons, i.e., theelectromagnetic radiation within wavelength range ,where m and m.We recall Planck’s law of black-body radiation [23]:

, where is the power ofradiation per source surface area per wavelength (in Jm ), isPlanck’s constant Js , is the speed of light ina vacuum ms ), is the wavelength of a photon (inm), is the Boltzmann constant , andis the temperature on the source surface (in K). Also note thatthe energy of a single photon is ; thus the numberper second of photons emitted per source surface area m perwavelength (m) is computed as . The unit ofis Js m and that of is s m .

For simplicity, we also assume a candle flame to be roughlyspherical with radius 1 cm and the intensity of itsradiation to be isotropic. The surface temperature of theflame is roughly 1800 K [31]. Neglecting (temporarily)the atmospheric photon absorption and assuming isotropicmedium, it follows that the source photon intensity perunit wavelength at a distance from the source, is givenby . Hence,

.Using the notation , the substitution

gives , whereand . The last integral can be evaluated

as , using the function, ,

where the expansion for follows by termwise integrationof the integrand expansion (into powers of ). It can be shownthat for the truncated expansion, the error term is bounded by thefirst neglected term times .

Using the above values, we get m, ,and . The computation gives (using only the firstterm of the expansion) , hence(for 20 miles)

s m

This is the unattenuated value of .

Remark A.1: In order to evaluate the attenuated value ofdue to atmospheric absorption, we assume the attenuation factor(due e.g., to fog condition) to be 1.1 db/mile—resulting (for

20 miles) by a total 22 db (or 12.589) for the attenuationfactor, hence the attenuated (practical) value for is

s m (A.1)

B. Estimation of

We can write , where represents thecontribution from the air surrounding the sensor, representsthe contribution from our galaxy (milky way), and representsthe contribution from all the other galaxies in the universe.

Computation of : In this case, denote the source (sur-rounding air) temperature by . Then is given by thephoton intensity (due to surrounding air molecules) into thesensor surface divided by rad , since these photons arearriving to each point on the sensor surface from only onehemisphere of directions corresponding to this point. Hence,per unit wavelength is given by

. Therefore, . Usingthe notation , and the substitutions ,

, . We get .Assuming 280 deg K, we have m,

, , .Hence, s m rad .

Computation of : In this case, we have to estimate thecontribution of our galaxy’s stars (except the sun!) to . Sinceour solar system is located inside a spiral arm of our galaxy(milky way) whose local width in the neighborhood of our solarsystem is estimated by 200 pc (parsec) which is about 650 ly(light year), we approximate our galaxy contribution to bya spherical cluster of stars with radius 300 ly (about halfthe local width of the spiral arm), centered at our solar system.In this case we can treat the stars in this cluster as stationary(hence ignoring relativistic effects due to Hubble cosmologicalexpansion low).

The photon intensity per unit wavelength arriving to thesensor from a star at distance with radius and surface tem-perature is given by

, where . Integration over gives the photonintensity at the sensor due to a star at distance as

. The substitutions , ,gives .

The photon flux arriving from the star to the sensor is givenby where denotes the sensor radius. Thus, thephoton flux due to stars inside a spherical shell with radius

and width is ,where denotes the number of stars insidea spherical shell of radius and average density of starsper unit volume. Consequently, the photon flux arriving to thesensor from the spherical cluster (with radius ) of starsshould be .

The resulting photon intensity at the sensor due to thestar cluster is

. Since for each point on the sensor,



incoming photons are arriving from only one hemisphereof directions, it follows that the resulting is foundfrom by dividing it by rad ; hence .Using the above expressions for and , we get

. To estimate , weestimate the average number of stars per unit volume insidea sphere of radius 50 ly centered at our sun. It is known[32] that the number of such stars is about 1000; therefore,

50 ly m .Since our sun can be considered as an “average star,” we

estimate the star radius and its surface temperature bythese parameters of our sun; i.e., we choose m,

6000 deg K, and get m, ,, . Hence,

s m rad .Computation of : In this case, we have to estimate

the contribution of the other galaxies to . As we shall see,relativistic Doppler effect (due to relativistic cosmologicalexpansion) should be taken into account. Denote by thenumber of stars in the other galaxies (which is almost theirnumber in the universe!) and by the universe radius.For simplicity, we use the simplest cosmological expansionmodel (Hubble law) according to which the average recedingvelocity of a star at distance from an observer is given by

. Therefore, sufficiently far galaxies should haverelativistic velocities which should be taken into account. Wedenote by s the wavelength of a photon emitted from a starin the star frame, and by the wavelength of this photon asobserved in the detector frame. It follows from (relativistic)Doppler effect that s . Hence,by Hubble law s . Let

be the photon time period in the star frame, anddenote the photon time period in the detector

frame. Standard (even nonrelativistic) physical considerationshows that the number of incoming photons with bandwidth

to the detector during period in the starframe is identical to the corresponding number of incomingphotons with bandwidth to the detector during pe-riod in the detector frame. This equality can be writtenas , where

is the photon intensity per wavelength at distancein the star frame, while is the corresponding

photon intensity as observed in the detector frame. Therefore,. Since

, weget

. In the star system

, where . Since, we have

.Hence,

. This is the desired relativistic intensity expression in the de-tector frame.

Denote by the average number of stars per unit volume inthe universe. Clearly, where denotesthe number of stars in the universe and denotes the universe

radius. Since there are about galaxies in the universe andthe number of stars in an average galaxy is estimated by ,we estimate , while the current estimate for is

ly, which gives m .The photon intensity at the sensor due to a star at distance is

given by . The photon flux arrivingfrom the star to the sensor is therefore where de-notes the sensor radius. The photon flux due to a spherical shell(containing stars) with radius and width is ,where is the number of stars inside aspherical cluster of radius .

The photon intensity at the sensor due to all theother galaxies (except our galaxy) is therefore given by

. Using the aboveresults, it follows that

where

The substitution leads to.

Combining this expression and the above results, we get, where

. The substitutiongives .

where , . The integral foris therefore expressible in term of the function as

. Hence, thefinal expression for is

, where , .As we already found (for computation) ,

. The integral in the last expression for can be computednumerically approximating the function in the integrand byits truncated expansion. The approximate numerical value ofthe integral is 0.01448. Therefore,

s m rad , which givess m rad . This is the unattenuated

value of .Remark A.2: In order to evaluate the attenuated value of ,

we add to the attenuation of 22 dB due to fog condition an as-sumed additional attenuation factor of 3 due to clouds, i.e., atotal attenuation factor of 37.767. Hence, the attenuated (prac-tical) value for is

s m rad (A.2)



Fig. 6. (a) Top view of three associated cones of source directions of three contiguous eyelets; (b) planar approximation to the top view under large� assumption;and (c) side view of the radial cone and associated cones of the �th eyelet.

APPENDIX BPROOF OF THEOREM 6.1

Proof:a) The lower bound on : We derive the lower limit of the

angular expansion factor , assuming the number of eyeletsis sufficiently large (such that each eyelet’s associated cone ofsource directions is narrow). The lens center points on the sur-face of the spherical array are determined by the tables in [33].

We consider the associated direction cones of three con-tiguous identical lenses, namely, the th, th, and th.Fig. 6(a) shows the top view of these cones on the array surface.Under the large assumption, the edge of the cones (whichis a section of the spherical surface) can be approximated byplanar circles, as shown in Fig. 6(b), on which each point cor-responds to a source direction. Each solid circle represents theradial cone (of source direction) of the corresponding eyelet;the source can be seen from this eyelet without the lens if thedirection lies in this cone. Each dashed circle represents theassociated cone (of source direction) of the eyelet; the sourcecan be detected by this eyelet with the presence of the lens ifthe direction lies in this cone. The centers of the circles arelabeled as , , and , respectively. The shaded arearepresents the source direction set which is invisible to thearray, since it does not belong to any radial cones.

Obviously, to remove the “blind region” (shaded area), thecenter of the region (labeled by ) should be enclosed by atleast one associated cone under the lens effect. In other words,the “minimal” associated cone should be the dashed circles inFig. 6(b), which intersect at . Clearly, under thelarge assumption (i.e., ), or . On

the other hand, as shown in Fig. 6(c), generally (i.e., withoutthe large assumption) and

, where and are defined in the figure; i.e.,. It then follows that

(B.1)

Note that the value of in (B.1) is the minimal one that isnecessary to remove all the nonvisible regions on the unit sphere

surface. Hence, under the large assumption, ,which gives the bound in (6a).

b) The lower bound on : Let be the position ofa detected particle on a subarray DS, andbe the momentum vector of that particle. From the uncertaintyprinciple in quantum mechanics (see, e.g., [29] and [30]), wehave

(B.2)

where is the standard deviation of the component distri-bution of the detected particles positions on the subarray DS,and is the standard deviation of the statistical distributionof the component of their momenta . Under the azimuthalsymmetry assumption, the distributions of positions andmomenta are azimuth invariant; i.e., and, sim-ilarly, . We further assume that is approximatelyconstant over the DS (this is especially true for a “narrow” eyeletcone of associated directions); thus, the lower bound onshould be achieved at the center of the DS.8

It follows from (B.2) that , where equality isattained (from the derivation of the uncertainty principle usingthe Cauchy–Schwarz inequality) if and only if the wave functionis Gaussian (i.e., is Gaussian distributed), which indicates thatthe above lower bound on may be tight.

Now the problem becomes one of finding an upper boundon . Let denote the norm of the mo-mentum and by its maximal value. We assume that theparticle intensity is uniformly distributed over the inner lens sur-face, so that we may exactly compute the expectation

under this assumption as follows. Let the particle (mar-ginal) probability density at the lens inner surface, on a circlewith radius around the lens symmetry axis be

, where the random variable takes value in the interval. Note that , which validates the proba-

bility density distribution defined above. Hence,

8Even if � fails to be approximately constant over the DS, we still can tryto find a lower bound for it in the center of the DS. Then under the assumptionthat the lens is designed such that � is the smallest at the DS center, this lowerbound for � at the DS center still applies to any other point on the DS.



. Bychanging the integration variable to , we get

Since by azimuthal symmetry, we see thatthe lower bound on is given by

(B.3)

using the relationship . Note that when, is asymptotically equal to .

c) The lower bound on : According to the relationship, when is sufficiently small, we have the first

lower bound on :

(B.4)

where is given in (B.3). In the following we derive thesecond lower bound on corresponding to .

We assume the particles are arriving independently at thearray according to a common LSF, which is responsible for theircommon angular distribution, assumed to be invariant with re-spect to the azimuth. Thus, from each detected particle (whosedetection may be considered as a single experiment), one mayinfer the estimated angular distribution of the source direction(i.e., the probability distribution of the possible associated di-rections that lead to the detection at a specific point) using theLSF. However, from the definition of the LSF, this angular dis-tribution estimate of the source direction (estimated by the lens-shaping function knowledge) should have the same angular dis-tribution as the LSF itself! It follows that the MSAE for thedirection of the total momentum of the detected particles isasymptotic to as ESC tends to .

On the other hand, this MSAE has a universal bound wherethe lens is serving as a “particle sensor” whose cross sectionhas radius instead of . Hence, according to (8),

.It then follows from the above discussion that

(B.5)

Combining (B.4) and (B.5), we obtain the theoretical lowerbound on as follows:

(B.6)

This completes the proof.

APPENDIX CPROOFS OF THEOREMS 7.1 AND 7.2

A. Proof of Theorem 7.1

Proof: In order to prove the theorem, we introduce a defi-nition of a particle detection event as follows.

Definition C.1: A particle is said to be detected insidethe sensor detection domain (which is a certain space–timedomain), if the presence of a particle inside this domain isobserved (i.e., determined) and recorded by the hardware andsoftware of the sensor system.

Definition C.2: A particle-detection event is defined as thefirst point in space–time where a detected particlestarted (for the first time) its interaction with the sensor hard-ware (which leads eventually to its detection.)

Note that prior to its detection, a detected particle behavesas a “free particle” described by a quantum-mechanical wavefunction, which is a plane wave with expected (freeparticle) momentum: ,where is the gradient operator.Similarly, the free particle has an expected energy:

. To ensure that the sensor software is able toestimate the source direction, of the above “free particles”(i.e., prior to their detection event) must be estimated, since

is proportional to (the source direction)as they carry the same source direction information. Here, wedenote

(C.1)

Assume that in addition to detecting particles, the sensor isequipped with some software to estimate the momentum of theparticle exactly at its detection event, which leads to a “partialcollapse” of the wave function because of the observation act.This is a well known “observation principle” in quantum me-chanics in which the modified wave function may be regardedas a result of the former conditioned with respect to the de-tection and momentum observation.

Note that for the modified is the sameas that of a free particle (since the detection event is just priorto the interaction of the detected particle with the sensor hard-ware). Hence, due to the conservation of the momentum and en-ergy at the detection event (just before the sensor hardware be-gins) with respect to that of the free particles, we claim thatbased on the modified (i.e., the wave function describing theparticle at the random detection event conditioned on its obser-vation inside the sensor detection domain) is the same forthe former . Thus, for the modified , we have the followingthree constraints , where the right-hand-side(RHS) is the same as defined in (C.1) using the former .

Without loss of generality, we may redefine (by rotation) thesensor coordinate frame so that the source direction coincideswith the direction, and we get for the modified :

. Also, according to our assumptions, the particle detectionevent occurs only inside a sensor domain during a detectiontime interval . We denote by the corresponding “de-tection space-time domain,” and denote the quantum wave func-tion of the detected particles inside by , which



is a (complex-valued) function in with unit norm; i.e.,.

From a physical viewpoint, is continuous over and van-ishes outside (particularly on boundary).Moreover, is in with its derivatives with respect to

, , , in (such that the covariance matrix of its mo-mentum may be well defined).

The momentum components , , of each detected par-ticle are associated (as observables) with the following Her-mitian operators on : is associated with the operator

, and similarly for and with respect to and, respectively. Thus, the expectations of and are given

by, and its variance is

, which follows from the constraint . Similarexpressions apply for the , components.

For , denote by the MSAE for the direc-tion of the total momentum of detected source particles if ex-actly particles where detected within the time window. Clearly

is undefined for . For a particle with wave-length , the corresponding momentum is , thus ifis the minimal wavelength, then the total momentum of parti-cles is upper bounded by . Therefore

(C.2)

which follows again from the constraints . Notethat indicates the direction of source particles before enteringthe sensor domain (i.e., the expectation of their momentumcomponent along the direction from the source to the sensor),and ESC is the expected number of source particles entering thesensor domain during the detection time interval (i.e.,entering the “detection space-time domain” ), as definedpreviously. Clearly , where is the source par-ticle intensity around the sensor domain (it is assumed to beuniform there), and is the sensor cross section with respectto the source direction.

Remark C.1: If we denote by the projection of the three-dimensional domain on the , plane, then is a planardomain whose area is .

Since both the sensor hardware and the detection wave func-tion (especially inside ) are unknown in practice, we haveto find the minimum for the in (C.2), which depends(for given and ESC) only on the wave function , with re-spect to (subject to the restrictions on ).

Thus, we have the constraints , and

where the last equality follows from integration by parts. Hence,, where

in which the infimum is taken over all possible wave functionsin that satisfy the following restrictions: a) ;

b) is continuous over and vanishes outside ; c) hasfirst-order derivatives almost everywhere in ; d) the first-order derivatives of are in ; and e)

, which corresponds to .Denote by the MSAE of the direction of the

total momentum of the detected source particles duringthe time window provided that at least one particle is de-tected (i.e., ). Thus, is the expectation of

over all positive values. From the above weget . The lastconditional expectation of over the positive values of

can be estimated from below9 (after normalization of thePoisson distribution for only positive values of ) with theresult . In addition, itis easy to show that as defined in Theorem 7.1 isuniformly attained by the MSAE of a random DOA estimatoruniformly distributed over the unit sphere, which gives10

for any estimator whenever. Thus, by Assumption 7.1 and the above discussion

we have for any DOA estimator

To solve the infimum problem for , let be a space-time domain larger than (i.e., containing ). Clearly,any satisfying the above restrictions a)–e) in should alsosatisfy those in . Thus, the above infimum monotonicallydecreases with increasing ; i.e., ,which gives a lower bound for .

We then choose as the cylindrical domain ,where are restricted to the planar domain and

are unrestricted. In order to compute , weuse a classical expansion of any function ininto a Fourier–Bessel orthogonal expansion (see, e.g., [34,Sec. 3.7]) as ,

, are the Bessel functionswith order , and is the polar representation of the point

. The parameters are given bywhere is the th positive zero of . Note that the se-lection of ensures that whenever ;i.e., .

It follows that are orthogonal in and vanishon its boundary.11 Another well known fact is that the orthog-onal sequence in is complete,12 which justifiesthe observation that the above expansion of any functionin should converge in to itself. Most im-portant, functions result from the separation of variablesmethod for the 3-D Laplace equation in cylindrical coordinates,which is needed for the following derivation.

It can be easily seen (from, e.g., [34]) that after the coor-dinate is separated out, the remaining , dependence satisfies

9Using Jensen’s inequality �� .10Using the invariance of MSAE under rotations, and Fubini’s Theorem.11E.g., from the orthogonality relations (3.95) in [34].12E.g., w from the theory of Bessel functions.



the following , whose so-lutions are . Therefore, considering the above re-striction (b) that the solutions should vanish on the boundary

, we have , which leads to the quantizationcondition . Thus, for we have

(C.3)

Now, letting be any continuous function inwhich vanishes whenever , then for

any fixed , in , we may consider as ancontinuous function of , in the disk thatvanishes outside this disk. Thus, we have by the Fourier–Besselexpansion

(C.4)

where is the appropriate coefficient for this expan-sion.

Therefore, if is constrained to have unit norm in ,i.e., , we get

(C.5)

For “sufficiently regular” , applying the operatorto the expansion (C.4), we obtain from (C.3)

It then follows from integration by parts that

Now, since and (see e.g., (9.5.2)in [35] for interlacing properties for the zeros of Bessel func-tions ), using (C.5) we get the following:

(C.6)

where is the first positive zero of , and its numericalvalue is approximately 2.40482. The inequality (C.6) is true forsufficiently smooth and for any that satisfies the above fiveconstraints a)–e), with the equality attained if and only if the ,

dependence of is proportional to .

Thus, we conclude13 that . Since, we have

(C.7)

which implies the result. (The corresponding result foris similarly proved.)

B. Proof of Theorem 7.2

Proof: The idea of is to find the wave func-tion describing the momentum-energy probability den-sity of a source particle (instead of the traditional wave func-tion describing the space-time probability density); andthen to express the resulting CRB and as a functionof , which depends only on the hardware. Therefore, we justneed to minimize with respect to any possible

. Furthermore, since the function depends very simply onthe usual , we need only to minimize with respectto , which will be seen to be trivial.

The connection between and can be de-scribed as the following. First, note that the space-timeand the momentum-energy are dual spaces in the sensethat scaling down the space variables leads to (via the uncer-tainty principle) scaling up the momentum variables ; and viceversa. In other words, the vectors and are dual to each other.Similarly, and are also dual. A simple guess is that thewave function is proportional to the Fourier transformof , where the proportionality constant is determined sothat the normalization constraint is satisfied; i.e.,the probability to find the momentum and energy , of theparticle somewhere in the momentum-energy space is 1.

Simple evaluation of the normalization constant shows thatthe suggested expression for is given by

(C.8)

where is Planck’s constant, ST denotes the space-time do-main, denotes and is the standard measureover ST (i.e., over )

Note that i) is proportional to (and may be regardedas) the Fourier transform of ; and ii) since rep-resents the space-time pdf of a particle, it is integrable over ST(i.e., is in , or is in ).

Since is the Fourier transform of , it follows(by Plancherel’s theorem) that is in and by theinversion formula

(C.9)

where MES denotes the momentum-energy space, andis the standard measure over MES (i.e., over ).

13This result is a special case of the classical “Poincaré inequality.”



Thus, and are in 1 to 1 correspondence,which is in fact an isometry of onto itself—preservingits inner product and norm as an Hilbert space. In particular

if and only if (C.10)

i.e., the normalization is preserved as desired.Now, in order to justify our previous “guess” that the above

representation of is the correct one, we observe that byapplying the operator to (represented by (C.9)) we canget that the operator in space-time has its dual presentationin momentum-energy as , or

the operator in MES is represented by in ST (C.11)

Similarly, by applying the time derivative operator toas represented by (C.9), we get its representation asin MES, or

the operator in MES is represented by in ST

(C.12)

Repeating applications of these operators, we get that theabove representations hold also for all their moments, whichis in agreement with the basic quantum-mechanical principles.Since any PDF is uniquely determined by its moments, it fol-lows that the momentum-energy PDF is uniquely de-termined by the above guess of given by (C.8). Thisjustifies our guess for the PDF of the momentum-energy overMES, and is all we need to derive the corresponding CRB—thephase of is not needed.

Choosing the direction of the reference frame tobe in the source direction , we evaluate the “partialFIM” corresponding to and components of anyparticle momentum as what follows. Its componentsare (using the PDF )

,which can be derived by using the expression for . Similarexpression holds for , hence

(C.13)

(C.14)

Seeing that the corresponding CRB matrix can be found byinverting the FIM, we obtain a lower bound for theby using .Thus, lower bound is given in terms of (whichdepends of course on the specific hardware used), it is simplerto minimize the with respect to the usual insteadof the dual by observing that

the operator in MES is

represented by in ST (C.15)

the operator in MES is

represented by in ST (C.16)

Remark C.2: (C.15) and (C.16) can be proved similar to theproofs of (C.11) and (C.12).

We get from (C.13)–(C.16) [using the normalization (C.10)], . Thus, from

the lower bound expression

(C.17)

and , we are led to the minimizationproblem , such that becausethe detection event is inside a sphere of radius .

It is easy to verify that the minimizer satisfies. Inserting this into (C.17), we get

. Since this is true for each particle, and sincethe particles are independent, we finally get (using the expecta-tion of as in the proof of Theorem 7.1)

hardware (C.18)

which is the result.

APPENDIX DCONCEPTS, NOTATIONS, ASSUMPTIONS, AND TRANSFER

FUNCTION EXISTENCE FOR THE GENERAL MODEL

We assume that the polarization state of the particles is known(e.g., circular polarization of incoming electromagnetic radia-tion) in order to simplify the modeling; otherwise, the mod-eling should be generalized in terms of complex combinationsof quantum wave functions corresponding to independent po-larization states.

Definition D.1:a) US—denotes the two-dimensional unit sphere on

which each point may be regarded as a unit vectorrepresenting a direction, and is equipped with its standard(area) measure denoted as .

b) EDS—denotes the energy-direction space defined as the(Cartesian) product of (representing parti-cles energies) with US (representing particles directions).The EDS is equipped with the product measure of thestandard measures for and US, denoted as .Note that the EDS can also be identified with withoutits origin.

Definition D.2:a) —is defined as the expected number of

(virtual) particles, having their energies and directions ina measurable subset of the EDS and crossing (both in-ward and outward) the sphere with center and radiusduring the time window .

b) —is defined as the following limitwhenever it exists.

c) —is called the particle intensity for a set of en-ergies and directions at spatial point and time, and is defined as the rate with respect to (whenever it

exists) of .Note that the above definition of is independent of. This motivates the following assumption.Assumption D.1: exists for every measurable

contained in EDS and every space-time event . Hence,



can be regarded as a distribution (i.e., a positive mea-sure) over the EDS for each given space-time event .

Note that is influenced by the sensor. In order toavoid that effect we introduce the concept of exciting intensityby:

Definition D.3: —is called the exciting intensityand defined as the intensity without the presence ofthe sensor.

Note that is independent of the sensor presenceand not influenced (as ) by its scattering effects.

For simplicity, we assume that the exciting intensity is i)time-invariant so that in the EI notation can be ignored,14 andii) is uniform in the vicinity of the array due to the far-fieldsource assumption. Accordingly, we denote the time-invariantand uniform EI around the array as . Here, we assume thatiii) , interpreted as an exciting intensity distribution overthe EDS, is absolutely continuous with respect to the standard(product) measure over the EDS. Note that the assumption iii)is physically reasonable since by quantum mechanical princi-ples, the energies and moments of the particles have some jointprobability distribution whose density is determined by somewave function.

Definition D.4: —denotes the EI density with respectto the standard measure over the EDS, and is defined as theexciting particle intensity density at energy and directionover the EDS. In practice, we model the contribution of eachpoint-like source to by a “spherical function.”

Definition D.5: DS—denotes the detection surface, which isdefined as any smooth surface tightly enveloping the detectorsphysical surfaces endowed with its standard area measure.

Let denote (for any given subset of DS,subset of EDS, and EI) the expected particle counts withintime window due to particles hitting and resulting fromthe given EI distribution with energies and directions confinedto . Accordingly:

Definition D.6: —denotes the (time-variant)detection rate for given subset of DS, subset of EDS, andEI, and is defined as the rate with respect to (whenever it ex-ists) of at time .15 In what follows we as-sume it to be time-invariant, and denote this detection rate by

.We then make the following assumptions that are needed for

the model derivation.Assumption D.3: (Additivity Assumption of EI and DR With

Respect to the EDS): We assume that if and are dis-joint measurable sets in the EDS with being their union, thenwe have ; and for any measurablesubset on the detection surface,

.Note that this assumption is reasonable for deterministic

exciting particles, but for virtual particles this is not true ingeneral (due to interference effects.) Therefore, we need thefollowing assumption which is a sufficient condition for As-sumption D.3.

14Time-variant exciting intensities are useful in the modeling of particle com-munications.

15This definition of the detection rate is independent of � .

Assumption D.4: Interference effects among exciting parti-cles are negligible.

Remark D.1: As examples for Assumption D.4 we may con-sider a) the no scatterers case: in this case the exciting particlesare emitted independently of each other; and b) negligible cor-relations among sources and scatterers: in this case (appearinge.g., in applications with DOA including multipath effects) thecorrelations among the sources and scatterers (e.g., for EMwaves are described by constant phase differences among thevector phasors describing the EM waves from distinct sourcesand scatterers) are negligible.

Assumption D.5: For any fixed EI and measurable subsetof the DS, the detection rate —regarded as a pos-itive measure over measurable subsets of the EDS—is ab-solutely continuous with respect to the positive measureover the EDS. Note that this assumption is physically resonable(since it means that if then aswell).

We further define the following.Definition D.7: —denotes the detection

transfer function for EI and any given measurable subsetof the DS at point on the EDS, which is defined as thedensity of the measure with respect to .Considering the EI distribution as the sensor “input” and thedetection rate distribution as its “output,” we can see that

describes the sensor operation by giving itsinput–output relationship. Note that the existence of the density

is equivalent to assumption D.5.Now we make more reasonable assumptions (with justifi-

cations) in order to simplify the modeling and present para-metric structure suitable for analysis. The first step is to make

dependent only on the density (assuming its ex-istence) , with value only at and not the entiredistribution over the EDS.

Assumption D.6: (Continuity Assumption ofWith Respect to , ): We assume that for any given distri-bution EI and measurable subset of the DS, the function

as a function of is continuous over theEDS.

Remark D.2: Assumption D.6 is physically reasonable since, as a function of , is a result of physical

processes described by certain partial differential equations, andas such should be even smooth (and not merely continuous).

Assumption D.7: (Existence and Continuity Assumption of): We assume that each EI distribution over the EDS

has a continuous density with respect to the EDS stan-dard measure.

Remark D.3: The physical validity of Assumption D.7 lies inthe fact that since (by the uncertainty principle) the momentumof each particle is distributed, as a result both its energy anddirection are distributed as well. Note also that even in clas-sical physics, particle sources are distributed in space (with theonly “singular” object—a “black hole”—that cannot emit parti-cles, according to classical physics) and are not truly point-like.However, in this paper we may still use spherical distribu-tion for point-like particle sources, but it is only a mathemat-ical idealization of a truly distributed source. The continuity of

may be physically justified according to Remark D.2. There



is also a mathematical justification to Assumption D.7 in thesense that each EI distribution (as a positive absolutely con-tinuous Borel measure over EDS) can be approximated arbi-trarily closely by another distribution with a continuous density

. (Moreover, even by infinitely differentiablewith compact support in the EDS—i.e., with finite energy band-width—which follows from a well known approximation prop-erty of infinitely differentiable functions with compact supportin .)

Thus, under Assumptions D.6 and D.7 we have that(for given and EI) the detection rate(as a distribution over the EDS) has a density given by

. Moreover, byAssumption D.6 and D.7 this density should be continuouswith respect to and . Since depends (by itsdefinition) only on the local properties of the EI around thepoint , it may depend on EI only through (itsvalue at the specific point in the EDS). Hence, we maywrite it as to emphasize this dependenceon (which might be nonlinear in general).

Assumption D.8: We assume that for any fixed , , and, the density is a continuous functional of

over .The last step to achieve a “truly transfer function”

is to remove the above dependency by assuming:Assumption D.9: (Weak Intensity Assumption): We assume

that the total exciting intensity (TEI) is sufficiently small, suchthat (according to assumption D.8) we may putin —making independent of .Thus, we may write as —a truly transfer function.

Remark D.4: This assumption merely states that the sensoris working in its “linear zone.” For visible light, it is determinedmostly by the detectors, instead of the lens and possible opticalfilters whose operation is approximately linear for higher TEI.More precisely, . Notethat the above approximation holds (independently of the EI dis-tribution) whenever the total intensity is suf-ficiently small.

In the following we assume that Assumptions D.3 to D.9 holdso that the transfer function is well defined. More-over, we assume that is sufficiently smooth (e.g.,infinitely differentiable)—which is reasonable according to theremark to Assumption D.7.

Definition D.8: —denotes the associated energiesand directions for each set (of positive area) on the detec-tion surface, which is defined as the pair that maximizes

, assuming existence and uniqueness of the globalmaximizer. Note that the existence follows from the continuityof and the reasonable assumption that the transfer functiondecays to zero as the energy tends to infinity, while the unique-ness is a standard assumption.

Remark D.5: As stated above, the pair may be cal-ibrated (at least in principle) only in the case where is a unionof detectors.

Remark D.6: The transfer function is of funda-mental importance for the sensor modeling since it contains allthe information needed to describe and model the sensor oper-ation. The sensor may be described as an input–output system,

whose input is the EI distribution and output is the detection ratedistribution over measurable subsets of the detection

surface. Note that the only subsets for which can bereally observed are these which are union of some whole sur-face of detectors—however, we still have a definition for Borelmeasurable subsets .

In order to see that the transfer function describes com-pletely the behavior of the sensor as an input–output system,we show in the following how to use it to model the sensoroperation: let be any measurable subset of the EDS, we have

, whereis an element of the exciting particle intensity distributionover the EDS with respect to the coordinates. Clearly

, where is the ele-ment of the usual measure over the EDS.

Remark D.7: If is a detector with and as its asso-ciated energy and associated direction, then and are ob-servable and as a result can be calibrated. In conclusion, theassociated energy and associated direction are defined for anymeasurable on the detection surface with positive area; whilefor sets with zero area, is defined only when is asingle point (it is not defined e.g., for a one dimensional set likea curve).

Let denotes the detection surface for the th eyelet; andassume that to each point on there is a corresponding di-rection , where is a -diffeomorphism frominto US, such that the center of mass (with respect to the stan-dard measure) for each detector on is mapped byto the associated direction of . This assumption (about theexistence of such ) is reasonable since , in addition to beingsmooth, is required to satisfy only a finite set of interpolationrestrictions. Such will be called an interpolating diffeomor-phism. The image is called “the cone of associateddirections for the detector d,” and is called “thecone of associated directions for the th eyelet” with its interpo-lating parametrization induced by .

Remark D.8: For each given pair of the EDS, thetransfer function is a positive measure over theBorel measurable subsets of the DS, which is absolutelycontinuous with respect to the solid angle measure of theimage of under the interpolating diffeomorphism(as described by remark D.7). Therefore, under the assumptionthat the density of (as a measure over subsets )with respect to the solid angle measure of is continuousover , we get that is slowly varying overneighboring detectors. Thus, the transfer functionfor each detector in an eyelet is nearly proportional to thedetector’s solid angle . And by redefining the “detectorspecific transfer function” to be the ratio anddenoting it again by , we get that isalmost independent of the detector (irrespective of its solidangle ); then from remark D.6, we have

For the detection rate for each detector due to the densityof the exciting particle intensity distribution EI over the



EDS. Similar assumption exists for the noisy count rate for eachdetector, which should be nearly proportional to the detector’ssolid angle of its cone of associated directions.

Definition D.9: To each point on the DS, there correspondits “radial direction” defined as the unit vector in the direc-tion of the radius vector from the spherical array center to . Foreach measurable subset of the DS, denote by its “cone ofradial directions” which consists the radial directions for allpoints in . Clearly, this induces a “radial diffeomorphism”

from DS into US by for each point on theDS, whose restriction to is a diffeomorphism from onto

. By Remark D.7, there is another diffeomorphismfrom the DS into the US sending each point on the DS to its

associate direction . The Jacobian of the transformation fromthe radial directions to the associated directions

(for any point on the DS) is called “solid angleexpansion factor” with its square root called “angular expansionfactor” and denoted by . Note that though is defined locallyand may vary over the DS, for eyelets with sufficiently small an-gular aperture, we may assume to be (nearly) constant overthe eyelet’s associated cone of directions. The physical expla-nation for the angular expansion factor is mainly due to the lensfocusing effect.

APPENDIX ESPHERICAL GAUSSIAN FUNCTIONS, PG FUNCTIONS, AND

LPG FUNCTIONS

A spherical Gaussian function (of a unit vector ) over theUS is defined to be proportional to where is afixed unit vector, and is positive-semi-definite (PSD)quadratic form in the components of . More specifically,we denote (up to a normalization factor) by thefunction , where is a unit vector(the mean) and is a positive definite 3 3 matrix (the co-variance). Given any 2-dimensional local parametrization ofin a neighborhood of (i.e., parametrization of the 2-dimen-sional component of that is orthogonal to ),has a local 2 2 covariance matrix at with respect to anygiven such parametrization, defined as the inverse of the Hes-sian at (with respect to the given local parametrization) of

. Note that any sufficiently smooth functionover the US with a local maximum at , can be approx-

imated locally near (up to second order Taylor expansionterms) by a suitable spherical Gaussian function whose 2 2local covariance at agrees with the local covariance ofat (i.e., the inverse of the Hessian at of ).

Note that for we get a circular Gaussian func-tion on the US, which is invariant with respect to the choiceof reference frame. The normalization constant in this case is

. However, it is customary to define itas because its relative error is negligible for practicalvalues of .

A spherical polynomial over the US is a polynomial inthe components of the unit vector . It is well known (e.g., fromthe properties of the spherical harmonics) that each continuousfunction on the US is uniformly approximable arbitrarily closelyby spherical polynomials.

A PG function over the US is defined as a spherical poly-nomial multiplied by a spherical Gaussian function. Obviously,any continuous function on the US is uniformly approximablearbitrarily closely by PG functions.

LPG functions are linear combinations of PG functions andare very suitable for approximating sufficiently smooth func-tions with several maxima over the US. The method is to selectcertain “important” peaks of the given function and to approx-imate it in a neighborhood of the peak by a Gaussian functionwhose mean and local covariance at the peak is adapted to thatof the given function, then the approximating LPG functionis found as a linear combination of each Gaussian functionmultiplied by a suitable spherical polynomial of bounded orderso as to minimize this approximating error in sense.This gives a standard LS optimization for the LPG functioncoefficients.

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Zhi Liu (S’04) received the Ph.D. degree in electricaland system engineering from Washington Universityin St. Louis in 2007, the M.Sc. degree in informa-tion and control engineering from Tongji University,China, in 2001, and the B.Sc. degree in electrical en-gineering from the North China University of Tech-nology (NCUT), Beijing, China, in 1996.

Since 2007, he has been a Software Developer withOSISoft INC., San Leandro, CA. His current workincludes implementing signal processing techniquesin the development of data acquiring and processing

software. His research interests focus on statistical signal processing and its bi-ologically inspired application to sensor-array processing. He is also interestedin image processing, medical imaging (e.g., PET and SPECT), and biologicalvision.

Dr. Liu has been a member of the Phi Kappa Phi honor society by electionsince April 2004. He received the Siemens Prize for outstanding graduate studyin 2000.

Arye Nehorai (S’80–M’83–SM’90–F’94) receivedthe B.Sc. and M.Sc. degrees in electrical engineeringfrom the Technion—Israel Institute of Technology,Haifa, Israel, and the Ph.D. degree in electrical en-gineering from Stanford University, Stanford, CA.

From 1985 to 1995, he was a Faculty Memberwith the Department of Electrical Engineering atYale University. In 1995, he became a Full Pro-fessor the Department of Electrical Engineering andComputer Science at The University of Illinois atChicago (UIC). From 2000 to 2001, he was Chair of

the department’s Electrical and Computer Engineering (ECE) Division, whichthen became a new department. In 2001, he was named University Scholar ofthe University of Illinois. In 2006, he became Chairman of the Departmentof Electrical and Systems Engineering at Washington University in St. Louis.He is the inaugural holder of the Eugene and Martha Lohman Professorshipand the Director of the Center for Sensor Signal and Information Processing(CSSIP) at WUSTL since 2006.

Dr. Nehorai was Editor-in-Chief of the IEEE TRANSACTIONS ON SIGNAL

PROCESSING from 2000 to 2002. From 2003 to 2005, he was Vice President(Publications) of the IEEE Signal Processing Society (SPS), Chair of thePublications Board, member of the Board of Governors, and member of theExecutive Committee of this Society. From 2003 to 2006, he was the foundingeditor of the special columns on Leadership Reflections in the IEEE SignalProcessing Magazine. He was corecipient of the IEEE SPS 1989 Senior Awardfor Best Paper with P. Stoica, coauthor of the 2003 Young Author Best PaperAward and corecipient of the 2004 Magazine Paper Award with A. Dogandzic.He was elected Distinguished Lecturer of the IEEE SPS for the term 2004 to2005 and received the 2006 IEEE SPS Technical Achievement Award. He isthe Principal Investigator of the new multidisciplinary university research ini-tiative (MURI) project entitled Adaptive Waveform Diversity for Full SpectralDominance. He has been a Fellow of the Royal Statistical Society since 1996.

Eytan Paldi (S’87–M’94) received the B.Sc. andM.Sc. degrees in mathematics and the B.Sc. andD.Sc. degrees in electrical engineering from theTechnion—Israel Institute of Technology, Haifa,in 1977, 1984, 1978, and 1990, respectively. Heis currently working towards the Ph.D. degree inmathematics at the Technion.

From 1979 to 1986, he was a Mathematicianand Software Engineer with Elbit Computers, Inc.,working on R&D projects in avionics. From 1986 to1990, he was a Teaching Assistant at the Technion.

From 1990 to 1991, he was an HTI postdoctoral Fellow, and from 1991 to 1993he was a Postdoctoral Associate with the Department of Electrical Engineeringat Yale University, New Haven, CT. In 1994, he worked on a mathematicalresearch and development project at IBM, Haifa, Israel. Since 1995, he isan independent scientific consultant for industrial research and developmentprojects. His current research interests are in stochastic modeling, estimation,optimization, polynomial root clustering, rational non-negative definite kernels,and conformal surgery methods in complex analysis.

Dr. Paldi was awarded, during his studies, several Gutwirth Fellowships, theJury Prize, and the Kennedy-Leigh Prize.


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