Abstract

Polarimetric measurements are designed to obtain information pertaining to the system under study, however noise in the system limits the precision and hence information obtainable. Exploitation of a priori knowledge of the system allows for an improvement in the precision of experimental data. In this vein we present a framework for system design and optimisation based upon the Fisher information matrix, which allows easy incorporation of such a priori information. As such the proposed figure of merit is more complete than the commonly used condition number. Conditions of equivalence are considered, however a number of examples highlight the failings of the condition number under more general scenarios. Bounds on the achievable informational gains via multiple polarimeter arms are also given. Finally we present analytic results concerning error distribution in a Mueller matrix polar decomposition, allowing for a more accurate noise analysis in polarimetric experiments.

© 2008 Optical Society of America

1. Introduction

Polarimetry is the study and measurement of the polarisation state of light. Although measurement of the state of polarisation of light is often an important objective [1,2] such techniques are also frequently used to obtain information about an optical system, such as its birefringence [3]. One may then subdivide polarimetry into two broad categories; Stokes polarimetry and Mueller polarimetry. The former entails measuring the four Stokes parameters of light [4] whilst the latter is intended to measure the full Mueller matrix of a sample from which parameters of interest can be inferred. Much effort has been invested into determining optimal configurations of polarimeters in terms of their experimental setup [510], however invariably little consideration is given to a priori information that an experimenter may have about the system they are studying. Information theory however states that if exploited correctly such information can improve the accuracy of any measurements [11, 12]. For example if the position of an object is approximately known the field of view can be reduced, perhaps by using a confocal microscope giving rise to an increase in the bandwidth and hence resolution of the system [13]. To address this omission we consider how such a priori information can be represented and incorporated into system optimisation. In doing so we find that common optimisation procedures do not necessarily give the optimal polarimeter configuration. Furthermore our formulation naturally describes the distribution of errors amongst inferred polarisation parameters, such as diattenuation, retardance and depolarisation as may be obtained from a Lu-Chipman Mueller matrix decomposition [14]. These results could potentially be used for a more accurate noise analysis in polarimetry.

In Section 2 we first describe the system model of Stokes and Mueller polarimeters used throughout this text and a discussion of the noise model assumed. Section 3 then proceeds to discuss a priori information before we discuss its use in system optimisation in Section 4. A number of examples are given in Section 5 where we also consider and confirm the approach used when no a priori information is possessed. Finally in Sections 6 and 7 we discuss how our optimisation strategy can be extended and highlight the inherent description of noise propagation in polarimetry, before concluding in Section 8.

2. System description

2.1. Stokes polarimeter

A division of amplitude polarimeter (DOAP) as originally proposed by Azzam [15] can measure the Stokes vector S⃗=(S 0,S 1,S 2,S 3) of incident light by projecting it onto at least four independent polarisation basis states using a polarisation state analyser (PSA) and as such the vector of detected intensities D=(D0,D1,,DND1) can be written

D=TS

where T is a ND × 4 matrix with rows corresponding to the Stokes vectors of the ND polarisation basis states, known as the instrument matrix (see Fig. 1). It is by variation of the instrument matrix that we aim to optimise a Stokes polarimeter. Each row of the instrument matrix can be deduced from the polarisation elements present in each arm of the polarimeter setup used and is thus known. Given a set of measured intensities it is then possible to deduce the state of polarisation of the incident light by application of the inverse operation i.e.

S=T+D

where T + is the Moore-Penrose pseudoinverse of the instrument matrix.

Although a division of wavefront polarimeter (DOWP) [16] can also be used for polarimetric measurements we neglect this arrangement in this work since it requires beams that are uniformly polarised and that the beam intensity profile be known a priori; conditions which are generally not achieved in practice. DOWPs are hence rarely used.

 

Fig. 1. Schematic diagram of a polarimeter setup.

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2.2. Mueller polarimeter

A Mueller polarimeter builds on the principle of a DOAP by addition of a polarisation state generator (PSG) and a sample to the optical setup as shown in Fig. 1. The action of the sample on the incident polarised light can be described by a 4 × 4 Mueller matrix M [17] such that the light incident into the DOAP is described by the Stokes vector

S=MR

where R⃗ is the Stokes vector of the illuminating beam. Since the Mueller matrix has 16 elements, all of which must in general be determined, it is not sufficient to illuminate using a single polarisation state. At minimum four distinct polarisation states must be used to illuminate the sample. Under these circumstances R⃗ can be written as a 4×NR matrix, R, NR≥4 whose columns are the Stokes vectors of the input states. Consequently D⃗ becomes a ND×NR matrix whose columns correspond to the vector of detected intensities for each input polarisation state. We can hence write

D=TMR

from which the Mueller matrix can be found using the inverse operations i.e.

M=T+DR+

The incident polarisation matrix R provides additional degrees of freedom by which a Mueller polarimeter can be optimised.

2.3. Noise model

All practical systems are subject to noise of a stochastic nature. It is this unfortunate fact which limits the precision to which measurements can be made and hence the amount of information we can extract from said measurements. In optical systems noise can be attributed to two sources which Delaubert et. al [18] refer to as technical and quantum noise. The former arises from poor experimental setup and can in principle be reduced to an arbitrarily low level, however quantum noise, arising from the discrete nature of light and the stochastic nature of photon arrivals at the detector, can not. Despite being a fundamental limit to experimental precision the assumption of quantum noise is also valid for any shot noise limited setup, a limit that has been reached in many applications, for example single molecule detection, OCT and astronomy [1921]. Goodman’s discussion of the degeneracy parameter [22] also illustrates that at optical frequencies quantum noise is frequently dominant over other potential fluctuations in the source.

So as not to unnecessarily complicate calculations we adopt a semi-classical model of light whereby before incidence onto the detector light is modelled using traditional electromagnetic theory, whilst the final absorption process is necessarily quantum in nature. Under these assumptions it can be shown [22] that the number of photocounts registered by a photon counting detector is governed by a Poisson probability law

f(D|D¯)=D¯DD!exp(D)

where f (D|) denotes the probability density function (PDF) for registering D photocounts and the bar notation is used to highlight that the PDF is dependent on the ensemble average of possible measurements. Since the number of photocounts is proportional to the integrated intensity received on the detector according to D = ηW/hν¯ we refer to D simply as the detected intensity. Here η is the quantum efficiency of the detector, whilst hν¯ is the average energy of an incident photon.

DOAPs however have multiple detectors and it is necessary to consider the stochastic detection process on each. Assuming that the noise on each detector is independent the joint PDF is

f(D|D¯)=n=1NDNR(D¯n+Ind)DnDnexp(D¯nInd)

where an additional term I dn has been added to account for other potential additive sources of stray photons. The notation D can be read as either a matrix or vector of intensities, since the formulation in both cases is identical. A good discussion of such possible noise sources is given in [23], however we give the simple examples of detector dark count or a passive background. Although not necessary we make the simplifying assumption that these additional noise sources affect each detector equally such that Idn = Id.

3. Information in polarimetry

3.1. Deterministic systems

Optical measurements are intended to extract information about the system under study, however as discussed noise in the system limits the amount obtainable. This relationship can be placed on a more formal basis using the Cramér-Rao lower bound (CRLB) [24, 25] which states that KJ -1 w. Here and henceforth we assume we are estimating a set of parameters w⃗. K is then the covariance matrix of our estimate i.e. the obtainable precision, and J w is known as the Fisher information matrix (FIM) [26, 27] defined by

Jw=ED[lnf(D|w)wlnf(D|w)Tw]

ED[…] denotes the expectation or ensemble average with respect to D. In essence Fisher information is a statistical measure of the dependence of the observed data on the input parameters. A strong dependence implies that we can measure the parameters more precisely. Application of the chain rule to Eq. (8) quickly yields

Jw=GTJD¯G

where G=D¯w¯ is independent of D. Various authors use different conventions for matrix calculus however throughout this work we adopt the formalism described in [28]. From Eqs. (7) and (8) we arrive at the well known result that the FIM for estimation of the mean of a Poisson random process is given by

(JD¯)ij=δijDi+Id

where δij is the Kronecker delta. Eqs. (9) and (10) thus define the FIM associated with estimation of a parameter vector w⃗ from intensity measurements in the presence of Poisson noise. We can consider Stokes (or Mueller matrix) polarimeters by allowing w⃗ = S⃗(or M) to give the associated FIM, where by equating w⃗ to M we imply a vectorisation operation of M.

3.2. Stochastic systems

Equations given thus far are valid only for a particular value of w⃗ however the parameter values may differ between different experimental setups or measurements. The experimenter may however know from an existing model or earlier data that the object being studied belongs to a restricted class, that is to say they possess some a priori information about the parameters being measured. A fibre-optic communication channel provides a good example whereby it is known that during a measurement window either a pulse will be received or not with equal probability representing logical 1 and 0 respectively. Known restrictions on the possible values of w⃗ can be conveniently parameterised using a PDF f (w⃗) which describes the probability of each value of w⃗ occurring. As shown in the accompanying Appendix when w⃗ can vary the FIM takes the form

Jw=Ew[Jwnr]+Jwr

where J nrw is the deterministic FIM given by Eq. (9) and J rw depends only on our a priori knowledge via Eq. (46).

Although the CRLB and FIM are useful tools for defining a best obtainable precision it is worth questioning whether such a limit can practically be achieved. In answer we refer to [29] wherein it is shown that each photon can be considered as an independent measurement of the system. Furthermore the maximum-likelihood (ML) estimator [30] asymptotically achieves the CRLB as the number of samples increases. We calculate using the Central Limit Theorem that acceptable convergence requires ~8000 photons and hence under these conditions we can achieve the CRLB by using a ML estimator. If such a criteria is not meet then the CRLB is not the strongest bound and one must resort to alternative bounds, such as the Barankin bound as discussed in [3133]. Full exposition of these bounds is beyond the scope of this paper.

3.3. Channel capacity

Frieden [34] defined a metric Cw to quantify the ability of a system to transmit Fisher information about the system being studied, known as the Fisher channel capacity and given mathematically by the trace of the FIM. Parameter estimation can only be accomplished via some degree of data processing on the measured intensities. It is hence sufficient to consider the channel capacity in the context of estimating the mean intensities only i.e. tr(J ), since it is impossible for data processing steps to introduce additional information into the system. We consider first a Stokes polarimeter. From Eq. (10) we have

CD¯=n=1ND1Dn

which under the constraint 0<n=1NDD¯nS0 is a maximum when the intensity is equalised across the detectors, such that Dn = aS 0/ND, where a≤1 is a positive constant. We note that equalising the intensity on each detector also equalises the noise which is considered a desirable property in optimisation of polarimeters [5,35]. The channel capacity is thus bounded according to

CD¯ND2S0+NDId.

We thus see that although channel capacity initially increases quadratically with the number of detectors, this slows to a linear increase as the additive noise term Id becomes significant. We note here that if a Gaussian noise model were used, as is frequently done [18, 36], the channel capacity would increase linearly with ND. Since Fisher information is additive [34] the Fisher channel capacity for a Mueller polarimeter can be calculated by summing the capacities for each input polarisation state. Thus

CD¯NRND2R0+NDId

where R 0 is the intensity of the probing polarisation states We note that this bound is unlikely to be achieved since equality requires the sample to be perfectly transmitting to all incident polarisation states. The channel capacity for a Mueller polarimeter can be seen to scale only linearly with respect to the number of probing polarisation states.

Practically these results embody the intuitive result that by performing more measurements i.e. increase the sampling, we introduce greater redundancy into our experimental data hence allowing a better precision in our parameter estimates to be achieved. Henceforth we assume that ND = NR = 4 since this is the minimum number of measurements required to determine S⃗ or M uniquely. The associated FIMs are thus 4 × 4 and 16 × 16 respectively.

4. Optimisation of polarimeters

4.1. Ellipsoids of concentration

Consider a parameter column vector w⃗ (of length n) and an estimate made by an experimenter of this parameter vector denoted we. In general there will be a discrepancy between the estimated and true value of w⃗ due to noise in the system given by

Δw=wwe

We note that since w⃗e is derived from random experimental data Δw⃗ is a random variable. Subsequent experiments will give a different value for the vector Δw⃗. In the n-dimensional error space i.e. the space with coordinate axes given by Δw 1w 2, …,Δwn each realisation of Δw⃗ specifies a single point. The scatter of these points is determined by the variations in the experimenters estimate w⃗e as described by the associated covariance matrix K. Furthermore the extent of scatter of different errors can be visualised using concentration ellipsoids [30] defined by Δw⃗ K -1Δw⃗ T = c, the volume of which can be used as a metric of aggregate error. Via the CRLB the FIM defines the smallest possible ellipsoid of concentration, which has a volume of

vmin=vncJw1=vncJw

where Vn is the volume of the n-dimensional unit hypersphere. As such we use the determinant of the FIM as a figure of merit, whereby a larger determinant is preferable. This criterion is known as D-optimality [37, 38] and has the additional advantage that log|J w| is closely related to the Fisher channel capacity of the system, such that by maximising |J w| we are also maximising the channel capacity.

Setting w⃗ = S⃗ or M and using Eqs. (1), (4) and (9) it is possible to calculate the FIM for estimation of the Stokes parameters or the elements of the Mueller matrix respectively as

JS=TTJD¯T
JM=(RTT)JD¯(RTT)

where J is of the form of Eq. (11) and ⊗ denotes the Kronecker product. Hence

JS=T2JD¯
JM=R8T8JD¯

We now restrict to uniform distributions meaning J r = 0, so as to illustrate the general behaviour of these figures of merit, whereby

JS=T2Πi=14ES[1Di+Id]
JM=R8T8i=116EM[1Di+Id]

Closer inspection of Eqs. (18) reveals there are two factors which influence the amount of information received in an optical experiment. The first of these corresponds to the amount of information acquired during the physical measurement as described by the |J | term. This component also encompasses any a priori information that may be possessed.

The second, and perhaps the more familiar, influence is related to any subsequent data processing used to extract the Stokes vector or Mueller matrix from the measured intensities as per Eqs. (2) and (5). Any noise in the intensity measurements is amplified during data processing, the extent of which is often measured using the condition number of the associated matrices, namely T (and R). In this paper we define the condition number of the matrix X as κ X=‖XFX -1F where ‖…‖F denotes the Frobenius norm. Since the condition number of a matrix is inversely proportional to its determinant (see e.g. [39]) we see that noise amplification is described by the |T| (and |R|) factors of Eqs. (18).

Frequently the condition number of the instrument matrix (and input polarisation matrix) is used as a figure of merit for polarimeter optimisation [710]. Our discussion above however has highlighted the inadequacy of this strategy in general since it gives no regard to potential gains that can be made by improving the precision of the measurement itself or incorporation of a priori knowledge. We thus believe that our informational figure of merit is more holistic in terms of measuring the quality of a polarimeter.

4.2. Nuisance parameters

Hitherto it has been assumed that full knowledge of the Stokes vector or Mueller matrix was desired, however this may not always be the case as demonstrated in Section 5.3. Such a scenario thus warrants our attention, although we shall now give a full commentary since it is amply discussed in the literature e.g. [30].

Let us assume that the parameter vector w⃗ is formed via the concatenation of two vectors u⃗ and ν⃗ such that w⃗=(u⃗,ν⃗). u⃗ is assumed to contain the p parameters that we wish to know, whilst ν⃗contains q parameters that are of little interest, known as nuisance parameters. When nuisance parameters are present the relevant FIM J u is given by [40]

Ju=J11J12J22−1J21

where J 11 is p×p, J 12 is p×q, J 21 is q× p,, J 22 is q×q and

Jw=(J11J12J21J22)

is the FIM were all parameters to be estimated. Accordingly the matrix determinant on which to optimise the polarimeter setup can be shown via rules of matrix algebra to be

Ju=JwJ22

5. Examples

Having developed a more suitable framework within which both Stokes and Mueller polarimeters can be optimised we now give a number of examples to highlight some points of interest. We start by highlighting the circumstances under which our informational figure of merit is equivalent to the condition number, however further examples then illustrate that when a priori information is introduced this equivalence does not hold.

5.1. Maximal ignorance

Our first example considers the situation in which we are maximally ignorant of the likely incident polarisation states. Initially considering a Stokes polarimeter our a priori information (or our lack thereof) can be modelled by assuming each polarisation state is equally likely. The associated PDF is thus uniform over the Poincaré sphere. Assuming that all possible incident states have the same intensity S 0 and degree of polarisation P (defined as the fraction of the total intensity originating from totally polarised light) it is possible to show

Es[1Di+Id]=1S0Plog[(1+P)S0+2Id(1P)S0+2Id]
=2S0Parctanh[S0PS0+2Id]

Hence

JS=18T2(arctanh[S0PS0+2Id]S0P)4

It is thus apparent that to maximise the information obtained we must maximise |T| or equivalently make the condition number as small as possible. It can be shown geometrically that this corresponds to making the volume of the tetrahedron whose vertices on the Poincaré sphere are defined by T⃗i a maximum, i.e. making the tetrahedron regular [39]. Although the same conclusion has been previously reached via considerations of the structure of the instrument matrix and noise propagation [7, 35, 39, 41] our derivation based on information theory appears to be new. Since a maximum determinant corresponds to minimal noise amplification the SNR, given by

SNR=(S02Parctanh[S0PS0+2Id])12

is maximum. If IdS 0 this reduces to the familiar S 1/2 0 scaling associated with Poisson noise.

There are an infinite number of possible instrument matrices corresponding to the rotation of the tetrahedron within the Poincaré sphere about the origin, however given one optimal instrument matrix (as can easily be found numerically) e.g. [39]

T0pt=(110010.3330.8160.47110.33300.94310.3330.8160.471)

it is possible to find alternative and perhaps more practical instrument matrices by applying a suitable rotation matrix. For a Mueller matrix polarimeter the same result applies, however we must also maximise the determinant of R, that is to say make the incident polarisation states as orthogonal as possible. For the completely uniform distribution there is no relationship between T and R.

Finally we note with reference to Fig. 2 that the channel capacity is independent of the choice of instrument matrix and increases with the degree of polarisation as would be expected.

 

Fig. 2. Fig. 2. Dependence of channel capacity on degree of polarisation for S 0/Id = 104.

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5.2. Matched filter

Adopting the opposite extreme to maximal ignorance we now consider the polarimetric equivalent to the matched filter. Matched filters, which correlate a known signal (or template) with a measured signal so as to determine the presence or absence of the known signal against some background, are frequently encountered in signal processing since they maximise the SNR when the desired signal is present [42]. We here restrict our discussion to that of a Stokes polarimeter in which depolarised light constitutes the background signal.

Denoting the known polarisation state by its Stokes vector S⃗t=(St0,St1,St2,St3) our a priori knowledge can be represented by the PDF f (S⃗)=δ(S⃗-S⃗t) where δ(x⃗) is the multi-dimensional Dirac delta function. Since S⃗ is non-random J r is identically zero. Consequently (J )ij = δij/(i+Id) whereby

JS=T2i=141Di+Id

If the additive noise term Id is zero we see that we can obtain infinite information if Di = 0 on a single detector, corresponding to one arm of the polarimeter projecting the incident polarisation on to the basis state T⃗i∝(St 0,-St 1,-St 2,-St 3). Note the parallel with conventional matched filters whereby the filter corresponds to the template reversed in time. This result can be understood by noting that for a given state of polarisation there are only two PSA configurations capable of uniquely identifying that state, namely T⃗i∝(St 0St 1St 2St 3) corresponding to diametrically opposite points on the Poincaré sphere. For example only a horizontal or vertical polariser can unambiguously identify horizontally polarised light (giving a maximum or null intensity respectively). When taking a single measurement it is not in general possible to know which intensity level corresponds to the maximum, whilst a null intensity is clearly identifiable. Furthermore we have assumed an underlying Poisson process in which noise variations grow as the intensity grows and hence lower intensities give a better precision.

If present, a depolarised background necessitates a second, distinct polarimeter arm and also results in finite information. The situation is similar for non-zero Id. Additional polarimeter arms improve estimation precision as discussed earlier in Section 3.3. Although unnecessary, we maintain our assertion that ND = 4 to allow easy comparison. Once more we find there are an infinite number of possible instrument matrices that give rise to a maximum in the information, since it is possible to trade off precision in the intensity measurements (corresponding to higher light levels) with a reduction in the noise amplification associated with data processing i.e. smaller condition number. Three possible polarimeter configurations are shown in Fig. 3(a) for a template Stokes vector of (1,0,0,1). The first configuration (shown in green) gives the best condition number possible for a matched polarimeter (and consequently worse measurement precision), whilst the second (red) shows the opposite case, whereby the volume of the inscribed tetrahedron is significantly smaller i.e. larger condition number, yet the precision of measurement is increased since the total detected intensity is smaller. Practically this arrangement is unsuitable since it is highly sensitive to alignment errors in the PSA. The third configuration (blue) illustrates a more general arrangement.

 

Fig. 3. Poincaré sphere showing possible polarimeter configurations (a) for a Stokes polarimeter matched to the template Stokes vector (1,0,0,1) i.e. left circularly polarised light and (b) a linear polarimeter, assuming the ratio S 0/Id = 105. Each arrow denotes the basis Stokes vector of a polarimeter arm.

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5.3. Linear polarimeter

Our final example assumes the polarisation incident into a Stokes polarimeter is restricted to lie on the equator of the Poincaré sphere. This could for example correspond to studying the light from a family of polarisers. Such a model could be useful in polarisation multiplexed optical data storage [43, 44]. Using the PDF f(ε,θ)=δ(ε)/π where ε and θ are angles which define a position on the Poincaré sphere [4] the expectations can be evaluated analytically to give

E[1Di+Id]=1(S0+2Id)2+S02P2cos22αi

where αi is the equatorial angle on the Poincaré sphere for the i th basis Stokes vector of the instrument matrix. Considering we know a priori that the incident polarisation state is linearly polarised we have no need to estimate S 3 since this only describes the ellipticity of the light and can hence treat it as a nuisance parameter as described in Section 4.2. A linear Stokes polarimeter is thus optimised when

JS'=T2j=14[(S0+2Id)2+S02P2cos22αj]12j=14sin2(2αj)[(S0+2Id)2+S02P2cos22αj]12

is maximum, where S′ denotes the parameter vector (S 0,S 1,S 2). In agreement with [6] the maximum of this metric occurs when the measurement basis Stokes vectors T⃗i are equally spaced around the equator of the Poincaré sphere as shown in Fig. 3(b). When applied to Mueller polarimeters a similar analysis shows the optimal input polarisation states also be equally spaced about the equator of the Poincaré sphere, although their position need bear no resemblance to those defined by T⃗i.

6. Extension of optimisation results

The above results with regards to the optimisation of polarimeters holds not only for inference of the Stokes parameters or elements of a Mueller matrix but can be further extended for inference of parameters z⃗ from these quantities. Such a situation may arise when performing a Lu-Chipman polar decomposition [14] on a measured Mueller matrix as is heavily used in the literature [4547]. Eqs. (17) generalise to

Jz=sTzTTJD¯TSz
Jz=MTz(RTT)JD¯(RTT)Mz

Accordingly the volume of the ellipsoid of concentration as found from |J z| is modified by a factor of |∂w⃗/∂z⃗|2 (w⃗ = S⃗ or M), which is independent of both T and R. Its significance in terms of optimisation with respect to the experimental setup is thus null hence optimisation of these more complicated inference problems reduces to the optimisation procedure previously discussed.

Although in this work we discuss optimisation when the system is limited by quantum noise, the same technique can be used for alternative noise models. Adopting different noise models only requires a change in the deterministic FIM JDnr. For example the FIM when assuming a Gaussian noise model is JDnr=K1.

7. Noise propagation in Mueller matrix polar decomposition

7.1. Single element systems

Noise propagation in inference problems, that is to say how noise in experimental data manifests itself as errors in the parameters of interest, can also be considered by employing Eqs. (30). Although the mathematics is generally complicated we give here a result pertaining to polar decomposition of Mueller matrices [14]. Before considering the composite systems for which polar decomposition is relevant we must first describe noise propagation for single polarisation element systems, namely pure diattenuators, retarders and depolarisers.

A diattenuator is a non-depolarising polarisation element which preferentially transmits particular states of polarisation and has a Mueller matrix of the form [14]

MA=Tu(1ATAmA)

where Tu is the transmittance for unpolarised light, A⃗=(A 1,A 2,A 3) is the diattenuation vector whose magnitude A is known as the diattenuation and

mA=1A2I+(11A2)AATA2

I is the 3 × 3 identity matrix. Any decomposition algorithm will need to estimate all four unknown parameters z⃗=(A 1,A 2,A 3,Tu). Lengthy calculations give the derivatives required for evaluation of Eq. (30b) as

MAAk=Tu(0kTkmAAk),MATu=MATu

where k = 1,2 or 3, δkT=(δ1k,δ2k,δ3k) and

mAijAk=[AiδjkA2+AjδikA2][11A2]+Akδij1A2AiAjAkA4[2+A221A2]

where m Aij is the (i, j)th element of m A. Using the FIM, J A, as calculated from Eqs. (30)(34) and the CRLB we can calculate the best obtainable precision for estimation of the diattenuation parameters. The error on each parameter will in general be different, a point considered further in [48, 49].

Similarly we can consider a pure retarder which has a Mueller matrix of the general form

MR=(10T0mR)

where

mRij=δijcosR+RiRjR2(1cosR)+q=13εijqRqRsinR

Again (R 1,R 2,R 3) defines a retardance axis and has a norm of R known as the retardance and εijq is the Levi-Civita permutation symbol. Calculation of the FIM, J R, requires the derivatives

mRijRk=[RiδjkR2+RjδikR2](1cosR)RkδijRsinRRiRjRkR2[1cosRR+sinR]
+q=13εijqR2[RqRkcosR+(RδqkRkR)sinR]

The case of a depolariser is however much more difficult to tackle since in general an eigen-analysis of the system is required to find the pertinent depolarisation parameters. This can not be described analytically except in some special cases. For example if it were known a priori that the sample were a pure depolariser with Mueller matrix of the form

MΔ=(10000a0000b0000c),a,b,c1

where 1-|a|, 1-|b| and 1-|c| are the principal depolarisation factors we can easily calculate the derivatives

MΔija=δi2δj2,MΔijb=δi3δj3,=MΔijc=δi4δj4,

where M Δi j is the (i, j)th element of MΔ. The appropriate FIM JΔ is then given by substituting Eqs. (39) into Eq. (30b).

7.2. Composite systems

The single element results discussed above can be used for noise analysis when the experimenter has a priori knowledge about the structure of the Mueller matrix. If however this is not the case a Lu-Chipman decomposition is frequently performed so as to parameterise the sample. Fundamental to the Lu-Chipman decomposition is the fact that an arbitrary Mueller matrix can be written as the product of three distinct Mueller matrices corresponding to a depolariser, retarder and diattenuator i.e. M=MΔM R M A. Morio and Goudail [50] considered the importance of altering the order in which the product is evaluated and found that different decompositions gave either unphysical results or merely comprised of an appropriate rotation compared to the Lu-Chipman decomposition and thus is only a mathematical, not physical, difference. With these results in mind we adhere to the original formulation, since this ensures physicality and furthermore corresponds to common usage.

Calculation of the FIM for a Lu-Chipman decomposition can be achieved by application of the product rule to Eq. (30b) which yields

Jz=MATz(RMRTMΔTTT)JD¯(RTTMΔMR)MAz
+MRTz(MARMΔTTT)JD¯(RTMATTMΔ)MRz¯
+MΔTz(MRMARTT)JD¯(RTMATMRTT)MΔz¯

since the structure of the matrices dictates that the cross terms are identically zero. It is important to note that we stack the parameters of interest into a single parameter vector for example z⃖=(R 1,R 2, …,b,c). J z is then block diagonal

Jz=(JR0T0T0JA0T00JΔ)

where the order of the diagonal terms depends only on the ordering of the parameters in z⃗. Mathematically the FIMs J R, J A and JΔ are of the same form as the single element FIMs described in the previous section albeit for a slight modification in the effective input polarisation states and instrument matrix respectively as can be seen by comparing Eqs. (30b) and (40). Fortunately this makes physical sense considering the Mueller matrix polar decomposition models the system as a cascade of three independent polarisation elements. Once more we give the cautionary note that the derivatives required to calculate JΔ can not be found analytically in general.

8. Conclusions

In this paper we have addressed the question as to how a priori information we may possess about a system being measured can be used to improve the precision of our measurements. In doing so we used a metric based upon the FIM which takes a holistic approach by including the raw information obtained from a measurement and the noise amplification that occurs during data processing. We have shown the conditions under which our figure of merit is equivalent to the frequently used condition number of the instrument (and incident polarisation) matrix, namely maximal ignorance of the likely values of the parameters of interest. We then proceeded to show that when incorporating a priori knowledge to achieve a resolution improvement the condition number is unsatisfactory and does not give optimal results. This was illustrated by considering the polarimetric equivalent of a matched filter and linear polarimeters.

Calculation of the informational figure of merit requires calculation of a FIM, which is also beneficial since it provides a simple description of the noise propagation that may arise during data processing and how the resulting errors are distributed among the parameter values. Specifically we have given results pertaining to a Lu-Chipman polar decomposition since this is frequently used in polarimetric analysis.

Although formulated in terms of estimation of Stokes parameters and Mueller matrix elements in the presence of Poisson noise, optimisation with respect to Fisher information is easily extended to inference of alternative sample parameters and noise models and is thus applicable to a wide variety of optical experiments.

Finally we acknowledge the financial support of the EU via NANOPRIM Contract No. NMP3-CT-2007-033310.

A. Fisher information for random parameters

For a deterministic parameter the FIM is defined by Eq. (8) where f (D|w⃗) is the PDF of D conditioned on the value of w⃗. If however w⃗ can vary this definition becomes unsatisfactory since it does not account for our knowledge about the random nature of the parameter, which can be used to improve the precision of any measurement. Instead it is more appropriate to define the FIM in terms of the joint PDF of D and w⃗, namely f (D,w⃗)=f(D|w⃗) f(w⃗) Taking the logarithm gives

L(D,w)=Inf(Dw)+Inf(w)

so that the modified FIM is defined by

Jw=i=12j=12ED,w[LiwLjTw]
=J1+J2+J3+J4

where L 1(D,w⃗)=ln f(D|w⃗) and L 2(w⃗)=ln f(w⃗), and expectations are now with respect to both D and w⃗. Considering each of these terms in turn we have

J1=L1wL1Twf(D,w)dDdw
=[L1wL1Twf(Dw)dD]f(w)dw
=Ew[Jwnr]

where Ew[…] denotes the expectation with respect to w⃗ only and J nr w is the deterministic (or non-random) Fisher information matrix as defined by Eq. (8).

Adopting a similar treatment of J 2 we have

J2=L1wL2Twf(D,w)dDdw
=f(D,w)f(Dw)f(w)f(Dw)wf(w)wdDdw
=[wf(D|w)dD]f(w)Twdw
=0

where the last step follows from ∫f (D|⃗w)d D = 1. A similar result follows for J 3. Finally consider

J4=L2wL2Twf(D,w)dDdw
=L2wL2Twf(w)dw=Jwr

Combining these results we find

Jw=Ew[Jwnr]+Jwr

We thus see that the Fisher information matrix when trying to estimate a random parameter w⃗ is given by the average of the deterministic Fisher information with respect to w⃗ plus an additional term arising from our knowledge of the random behaviour of w⃗.

As a final point of interest we note here that if f (w⃗) is uniform then

Inf(w)w=0

such that J rw = 0 and J w = Ew [J nrw]. Furthermore if J nrw is not dependent on w⃗ then J w=J nrw

References and links

1. Planets Stars and Nebulae Studied With PhotopolarimetryT. Gehrels ed. (University of Arizona Press1974).

2. J. Tinbergen, “Interstellar polarization in the immediate solar neighbourhood,” Astron. Astrophys. 105, 53–64 (1982).

3. R. A. Chipman, Polarimetry, Handbook of OpticsVol 2, (McGraw-Hill, New York, 1995).

4. R.M.A. Azzam and N.M. Bashara, Ellipsometry and polarised light, (Elsevier, North Holland, 1987).

5. J. Zallat, S. Aïnouz, and M. Ph. Stoll “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A: Pure Appl. Opt. 8807–814 (2006). [CrossRef]  

6. J. S. Tyo, “Optimum linear combination strategy for an N-channel polarization sensitive imaging or vision system,” J. Opt. Soc. Am. A 15359–366 (1998). [CrossRef]  

7. J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. 41619–630 (2002). [CrossRef]   [PubMed]  

8. A. Ambirajan and D.C. Look, “Optimum angles for a polarimeter,” Opt. Eng. 341651–1658 (1995). [CrossRef]  

9. M. Smith, “Optimization of a dual-rotating-retarder Mueller matrix polarimeter,” Appl. Opt. 412488–2493 (2002). [CrossRef]   [PubMed]  

10. A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films , 455–456112–119 (2004). [CrossRef]  

11. D. Mendlovic and A. W. Lohmann “Spacebandwidth product adaptation and its application to superresolution: fundamentals,” J. Opt. Soc. Am. A 14 (1997).

12. T. J. Rothenberg, Efficient estimation with a priori information (New Haven, Yale University Press, 1973).

13. I. J. Cox and C. J. R. Sheppard, “Information capacity and resolution in an optical system,” J. Opt. Soc. Am. A 31152–1158 (1986). [CrossRef]  

14. S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996). [CrossRef]  

15. R. M. A. Azzam, “Division-of-amplitude Photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” J. Mod. Opt. 29685–689 (1982).

16. E. Collett, “Automatic determination of the polarization state of nanosecond laser pulses,” U.S. Patent No. 4158506 (1979).

17. H. Mueller, “The foundations of optics” J. Opt. Soc. Am. 38, 661–661 (1948).

18. V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, “Quantum limits in image processing,” J. Mod. Opt. 29685–689 (1982)

19. M. R. Foreman, S. S. Sherif, and P. Török, “Photon statistics in single molecule orientational imaging,” Opt. Express 1513597–13606 (2007). [CrossRef]   [PubMed]  

20. A. Dubois, K. Grieve, G. Moneron, R. Lecaque, L. Vabre, and C. Boccara, “Ultrahigh-Resolution Full-Field Optical Coherence Tomography,” Appl. Opt. 43, 2874–2883 (2004). [CrossRef]   [PubMed]  

21. B. J. Meers, “Recycling in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 382317–2326 (1988). [CrossRef]  

22. J. W. Goodman, Statistical Optics (Wiley, New York, 2000).

23. A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises,” J. Europ. Opt. Soc. Rap. Public. 308002 (2008). [CrossRef]  

24. H. Cramér, Mathematical Methods of Statistics, (Princeton Univ. Press. 1946), ISBN 0-691-08004-6.

25. C. Rao, “Information and the accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 3781–89 (1945)

26. R. Fisher, “On the mathematical foundations of theoretical statistics,” Phil. Trans. R. Soc. Lond. 222309–368 (1922) [CrossRef]  

27. R. Fisher, “Theory of statistical estimation,” Proc. Cam. Phil. Soc. 22700–725 (1925) [CrossRef]  

28. http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/calculus.html

29. H. H. Barrett, J.L. Denny, R.F. Wagner, and K.J. Myers “Objective assessment of image quality. II Fisher information, Fourier crosstalk, and figures of merit for task performance,” J. Opt. Soc. Am. A 12834–852 (1995) [CrossRef]  

30. L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, 1991)

31. E. W. Barankin, “Locally best unbiased estimates,” Ann. Math. Stat. 20477–501 (1949). [CrossRef]  

32. M. F. Kijewski, S. P. Müller, and S. C. Moore, “The Barankin bound: a model of detection with location uncertainty,” Proc. SPIE 1768153 (1992). [CrossRef]  

33. S. P. Müller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, “Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio,” Phys. Med. Biol. 503697–3715 (2005). [CrossRef]   [PubMed]  

34. B. R. FriedenPhysics from Fisher Information: A Unification, (Cambridge University Press1998). [CrossRef]  

35. J. S. Tyo, “Noise equalisation in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett. 251198–1200 (2000). [CrossRef]  

36. A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Precision of degree of polarization estimation in the presence of additive Gaussian detector noise,” Opt. Commun. 278264–269 (2007). [CrossRef]  

37. R. Mehra, “Optimal input signals for parameter estimation in dynamic systems—Survey and new results,” IEEE T. Automat. Contr. 19753–768 (1974). [CrossRef]  

38. E. Walter and L. Pronzatom “Qualitative and quantitative experiment design for phenomenological models - a survey,” Automatica 26195–213 (1990). [CrossRef]  

39. S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41965–972 (2002). [CrossRef]  

40. V. Bhapkar and C. Srinivasan, “On Fisher information inequalities in the presence of nuisance parameters,” Ann. Inst. Statist. Math 46593–604 (1994).

41. R. M. A. Azzam and F. F. Sudradjat, “Single-layer-coated beam splitters for the division-of-amplitude photopolarimeter,” Appl. Opt. 44190–196 (2005). [CrossRef]   [PubMed]  

42. A. D. Whalen, Detection of signals in noise, (Academic Press Inc., New York, 1971).

43. P. Török, M. Salt, E.E. Kriezis, P.R.T. Munro, H.P. Herzig, and C. Rockstuhl, “Optical disk and reader therefor,” Worldwide Patent No. WO 2006/010882 (2006).

44. P. R. T. Munro and P. Török, “Properties of confocal Mueller-matrix polarimeters,” (submitted to Opt. Lett.).

45. B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, “Mueller Polarimetric Imaging System with Liquid Crystals,” Appl. Opt. 43, 2824–2832 (2004) [CrossRef]   [PubMed]  

46. J. M. Bueno, “Depolarization effects in the human eye,” Vision Research 412687–2696 (2001). [CrossRef]   [PubMed]  

47. M. Floc’h, G. Le Brun, J. Cariou, and J. Lotrian, “Experimental characterization of immersed targets by polar decomposition of the Mueller matrices,” Eur. Phys. J. AP 3349–358 (1998). [CrossRef]  

48. S. M. Nee, “Error analysis for Mueller matrix measurement,” J. Opt. Soc. Am. A 201651–1657 (2003). [CrossRef]  

49. S. N. Savenkov and K. E. Yushtin, “Mueller matrix elements error distribution for polarimetric measurements,” Proc. SPIE 5158251–259 (2003). [CrossRef]  

50. J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. 292234–2236 (2004). [CrossRef]   [PubMed]  

References

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  1. T. Gehrels, ed., Planets Stars and Nebulae Studied With Photopolarimetry, (University of Arizona Press 1974).
  2. J. Tinbergen, "Interstellar polarization in the immediate solar neighbourhood," Astron. Astrophys. 105, 53-64 (1982).
  3. R. A. Chipman, Polarimetry, Handbook of Optics Vol 2, (McGraw-Hill, New York, 1995).
  4. R. M. A. Azzam and N. M. Bashara, Ellipsometry and polarised light, (Elsevier, North Holland, 1987).
  5. J. Zallat, S. Aï?nouz, and M. Ph. Stoll "Optimal configurations for imaging polarimeters: impact of image noise and systematic errors," J. Opt. A: Pure Appl. Opt. 8, 807-814 (2006).
    [CrossRef]
  6. J. S. Tyo, "Optimum linear combination strategy for an N-channel polarization sensitive imaging or vision system," J. Opt. Soc. Am. A 15,359-366 (1998).
    [CrossRef]
  7. J. S. Tyo, "Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error," Appl. Opt. 41, 619-630 (2002).
    [CrossRef] [PubMed]
  8. A. Ambirajan and D. C. Look, "Optimum angles for a polarimeter," Opt. Eng. 34,1651-1658 (1995).
    [CrossRef]
  9. M. Smith, "Optimization of a dual-rotating-retarder Mueller matrix polarimeter," Appl. Opt. 41, 2488-2493 (2002).
    [CrossRef] [PubMed]
  10. A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, "General methods for optimized design and calibration of Mueller polarimeters," Thin Solid Films 455-456,112-119 (2004).
    [CrossRef]
  11. D. Mendlovic and A. W. Lohmann "Spacebandwidth product adaptation and its application to superresolution: fundamentals," J. Opt. Soc. Am. A 14, 558-562 (1997).
  12. T. J. Rothenberg, Efficient estimation with a priori information (New Haven, Yale University Press, 1973).
  13. I. J. Cox and C. J. R. Sheppard, "Information capacity and resolution in an optical system," J. Opt. Soc. Am. A 3, 1152-1158 (1986).
    [CrossRef]
  14. S. Y. Lu and R. A. Chipman, "Interpretation of Mueller matrices based on polar decomposition," J. Opt. Soc. Am. A 13, 1106-1113 (1996).
    [CrossRef]
  15. R. M. A. Azzam, "Division-of-amplitude Photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light," J. Mod. Opt. 29, 685-689 (1982).
  16. E. Collett, "Automatic determination of the polarization state of nanosecond laser pulses," U.S. Patent No. 4158506 (1979).
  17. H. Mueller, "The foundations of optics" J. Opt. Soc. Am. 38, 661-661 (1948).
  18. V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, "Quantum limits in image processing," J. Mod. Opt. 29, 685-689 (1982)
  19. M. R. Foreman, S. S. Sherif, and P. Török, "Photon statistics in single molecule orientational imaging," Opt. Express 15, 13597-13606 (2007).
    [CrossRef] [PubMed]
  20. A. Dubois, K. Grieve, G. Moneron, R. Lecaque, L. Vabre, and C. Boccara, "Ultrahigh-Resolution Full-Field Optical Coherence Tomography," Appl. Opt. 43, 2874-2883 (2004).
    [CrossRef] [PubMed]
  21. B. J. Meers, "Recycling in laser-interferometric gravitational-wave detectors," Phys. Rev. D 38, 2317-2326 (1988).
    [CrossRef]
  22. J. W. Goodman, Statistical Optics (Wiley, New York, 2000).
  23. A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, "Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises," J. Eur. Opt. Soc. Rap. Pub. 3, 08002 (2008).
    [CrossRef]
  24. H. Cramér, Mathematical Methods of Statistics, (Princeton Univ. Press. 1946), ISBN 0-691-08004-6.
  25. C. Rao, "Information and the accuracy attainable in the estimation of statistical parameters," Bull. Calcutta Math. Soc. 37, 81-89 (1945)
  26. R. Fisher, "On the mathematical foundations of theoretical statistics," Phil. Trans. R. Soc. Lond. 222, 309-368 (1922)
    [CrossRef]
  27. R. Fisher, "Theory of statistical estimation," Proc. Cam. Phil. Soc. 22, 700-725 (1925)
    [CrossRef]
  28. http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/calculus.html
  29. H. H. Barrett, J. L. Denny, R. F. Wagner and K. J. Myers "Objective assessment of image quality. II Fisher information, Fourier crosstalk, and figures of merit for task performance," J. Opt. Soc. Am. A 12, 834-852 (1995)
    [CrossRef]
  30. L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, 1991)
  31. E. W. Barankin, "Locally best unbiased estimates," Ann. Math. Stat. 20, 477-501 (1949).
    [CrossRef]
  32. M. F. Kijewski, S. P. Muller, and S. C. Moore, "The Barankin bound: a model of detection with location uncertainty," Proc. SPIE 1768, 153 (1992).
    [CrossRef]
  33. S. P. Muller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
    [CrossRef] [PubMed]
  34. Physics from Fisher Information: A Unification, (Cambridge University Press 1998).
    [CrossRef]
  35. J. S. Tyo, "Noise equalisation in Stokes parameter images obtained by use of variable-retardance polarimeters," Opt. Lett. 25, 1198-1200 (2000).
    [CrossRef]
  36. A . Beniere, F . Goudail, M . Alouini, and D . Dolfi, "Precision of degree of polarization estimation in the presence of additive Gaussian detector noise," Opt. Commun. 278, 264-269 (2007).
    [CrossRef]
  37. R. Mehra, "Optimal input signals for parameter estimation in dynamic systems-Survey and new results," IEEE T. Automat. Contr. 19, 753-768 (1974).
    [CrossRef]
  38. E. Walter and L. Pronzatom "Qualitative and quantitative experiment design for phenomenological models - a survey," Automatica 26, 195-213 (1990).
    [CrossRef]
  39. S. N. Savenkov, "Optimization and structuring of the instrument matrix for polarimetric measurements," Opt. Eng. 41, 965-972 (2002).
    [CrossRef]
  40. V. Bhapkar and C. Srinivasan, "On Fisher information inequalities in the presence of nuisance parameters," Ann. Inst. Statist. Math 46, 593-604 (1994).
  41. R. M. A. Azzam and F. F. Sudradjat, "Single-layer-coated beam splitters for the division-of-amplitude photopolarimeter," Appl. Opt. 44, 190-196 (2005).
    [CrossRef] [PubMed]
  42. A. D. Whalen, Detection of signals in noise, (Academic Press Inc., New York, 1971).
  43. P. Torok, M. Salt, E. E. Kriezis, P. R. T. Munro, H. P. Herzig, and C. Rockstuhl, "Optical disk and reader therefor," Worldwide Patent No. WO 2006/010882 (2006).
  44. P. R. T. Munro and P. Torok, "Properties of confocal Mueller-matrix polarimeters," (submitted toOpt. Lett.).
  45. B. Laude-Boulesteix, A. De Martino, B. Drevillon, and L. Schwartz, "Mueller Polarimetric Imaging System with Liquid Crystals," Appl. Opt. 43, 2824-2832 (2004)
    [CrossRef] [PubMed]
  46. J. M. Bueno, "Depolarization effects in the human eye," Vision Research 41, 2687-2696 (2001).
    [CrossRef] [PubMed]
  47. M. Floc???h, G . Le Brun, J . Cariou, andJ . Lotrian, "Experimental characterization of immersed targets by polar decomposition of the Mueller matrices," Eur. Phys. J. AP 3, 349-358 (1998).
    [CrossRef]
  48. S. M. Nee, "Error analysis for Mueller matrix measurement," J. Opt. Soc. Am. A 20, 1651-1657 (2003).
    [CrossRef]
  49. S. N. Savenkov and K. E. Yushtin, "Mueller matrix elements error distribution for polarimetric measurements," Proc. SPIE 5158, 251-259 (2003).
    [CrossRef]
  50. J. Morio and F. Goudail, "Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices," Opt. Lett. 29, 2234-2236 (2004).
    [CrossRef] [PubMed]

2008

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, "Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises," J. Eur. Opt. Soc. Rap. Pub. 3, 08002 (2008).
[CrossRef]

2007

A . Beniere, F . Goudail, M . Alouini, and D . Dolfi, "Precision of degree of polarization estimation in the presence of additive Gaussian detector noise," Opt. Commun. 278, 264-269 (2007).
[CrossRef]

M. R. Foreman, S. S. Sherif, and P. Török, "Photon statistics in single molecule orientational imaging," Opt. Express 15, 13597-13606 (2007).
[CrossRef] [PubMed]

2006

J. Zallat, S. Aï?nouz, and M. Ph. Stoll "Optimal configurations for imaging polarimeters: impact of image noise and systematic errors," J. Opt. A: Pure Appl. Opt. 8, 807-814 (2006).
[CrossRef]

2005

S. P. Muller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
[CrossRef] [PubMed]

R. M. A. Azzam and F. F. Sudradjat, "Single-layer-coated beam splitters for the division-of-amplitude photopolarimeter," Appl. Opt. 44, 190-196 (2005).
[CrossRef] [PubMed]

2004

2003

S. M. Nee, "Error analysis for Mueller matrix measurement," J. Opt. Soc. Am. A 20, 1651-1657 (2003).
[CrossRef]

S. N. Savenkov and K. E. Yushtin, "Mueller matrix elements error distribution for polarimetric measurements," Proc. SPIE 5158, 251-259 (2003).
[CrossRef]

2002

2001

J. M. Bueno, "Depolarization effects in the human eye," Vision Research 41, 2687-2696 (2001).
[CrossRef] [PubMed]

2000

1998

M. Floc???h, G . Le Brun, J . Cariou, andJ . Lotrian, "Experimental characterization of immersed targets by polar decomposition of the Mueller matrices," Eur. Phys. J. AP 3, 349-358 (1998).
[CrossRef]

J. S. Tyo, "Optimum linear combination strategy for an N-channel polarization sensitive imaging or vision system," J. Opt. Soc. Am. A 15,359-366 (1998).
[CrossRef]

1997

D. Mendlovic and A. W. Lohmann "Spacebandwidth product adaptation and its application to superresolution: fundamentals," J. Opt. Soc. Am. A 14, 558-562 (1997).

1996

1995

1994

V. Bhapkar and C. Srinivasan, "On Fisher information inequalities in the presence of nuisance parameters," Ann. Inst. Statist. Math 46, 593-604 (1994).

1992

M. F. Kijewski, S. P. Muller, and S. C. Moore, "The Barankin bound: a model of detection with location uncertainty," Proc. SPIE 1768, 153 (1992).
[CrossRef]

1990

E. Walter and L. Pronzatom "Qualitative and quantitative experiment design for phenomenological models - a survey," Automatica 26, 195-213 (1990).
[CrossRef]

1988

B. J. Meers, "Recycling in laser-interferometric gravitational-wave detectors," Phys. Rev. D 38, 2317-2326 (1988).
[CrossRef]

1986

1982

R. M. A. Azzam, "Division-of-amplitude Photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light," J. Mod. Opt. 29, 685-689 (1982).

J. Tinbergen, "Interstellar polarization in the immediate solar neighbourhood," Astron. Astrophys. 105, 53-64 (1982).

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, "Quantum limits in image processing," J. Mod. Opt. 29, 685-689 (1982)

1974

R. Mehra, "Optimal input signals for parameter estimation in dynamic systems-Survey and new results," IEEE T. Automat. Contr. 19, 753-768 (1974).
[CrossRef]

1949

E. W. Barankin, "Locally best unbiased estimates," Ann. Math. Stat. 20, 477-501 (1949).
[CrossRef]

1948

H. Mueller, "The foundations of optics" J. Opt. Soc. Am. 38, 661-661 (1948).

1945

C. Rao, "Information and the accuracy attainable in the estimation of statistical parameters," Bull. Calcutta Math. Soc. 37, 81-89 (1945)

1925

R. Fisher, "Theory of statistical estimation," Proc. Cam. Phil. Soc. 22, 700-725 (1925)
[CrossRef]

1922

R. Fisher, "On the mathematical foundations of theoretical statistics," Phil. Trans. R. Soc. Lond. 222, 309-368 (1922)
[CrossRef]

Abbey, C. K.

S. P. Muller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
[CrossRef] [PubMed]

Aïinouz, S.

J. Zallat, S. Aï?nouz, and M. Ph. Stoll "Optimal configurations for imaging polarimeters: impact of image noise and systematic errors," J. Opt. A: Pure Appl. Opt. 8, 807-814 (2006).
[CrossRef]

Alouini, M

A . Beniere, F . Goudail, M . Alouini, and D . Dolfi, "Precision of degree of polarization estimation in the presence of additive Gaussian detector noise," Opt. Commun. 278, 264-269 (2007).
[CrossRef]

Alouini, M.

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, "Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises," J. Eur. Opt. Soc. Rap. Pub. 3, 08002 (2008).
[CrossRef]

Ambirajan, A.

A. Ambirajan and D. C. Look, "Optimum angles for a polarimeter," Opt. Eng. 34,1651-1658 (1995).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam and F. F. Sudradjat, "Single-layer-coated beam splitters for the division-of-amplitude photopolarimeter," Appl. Opt. 44, 190-196 (2005).
[CrossRef] [PubMed]

R. M. A. Azzam, "Division-of-amplitude Photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light," J. Mod. Opt. 29, 685-689 (1982).

Bachor, H. A.

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, "Quantum limits in image processing," J. Mod. Opt. 29, 685-689 (1982)

Barankin, E. W.

E. W. Barankin, "Locally best unbiased estimates," Ann. Math. Stat. 20, 477-501 (1949).
[CrossRef]

Barrett, H. H.

Beniere, A

A . Beniere, F . Goudail, M . Alouini, and D . Dolfi, "Precision of degree of polarization estimation in the presence of additive Gaussian detector noise," Opt. Commun. 278, 264-269 (2007).
[CrossRef]

Bénière, A.

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, "Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises," J. Eur. Opt. Soc. Rap. Pub. 3, 08002 (2008).
[CrossRef]

Bhapkar, V.

V. Bhapkar and C. Srinivasan, "On Fisher information inequalities in the presence of nuisance parameters," Ann. Inst. Statist. Math 46, 593-604 (1994).

Boccara, C.

Bueno, J. M.

J. M. Bueno, "Depolarization effects in the human eye," Vision Research 41, 2687-2696 (2001).
[CrossRef] [PubMed]

Cariou, J

M. Floc???h, G . Le Brun, J . Cariou, andJ . Lotrian, "Experimental characterization of immersed targets by polar decomposition of the Mueller matrices," Eur. Phys. J. AP 3, 349-358 (1998).
[CrossRef]

Chipman, R. A.

Cox, I. J.

De Martino, A.

B. Laude-Boulesteix, A. De Martino, B. Drevillon, and L. Schwartz, "Mueller Polarimetric Imaging System with Liquid Crystals," Appl. Opt. 43, 2824-2832 (2004)
[CrossRef] [PubMed]

A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, "General methods for optimized design and calibration of Mueller polarimeters," Thin Solid Films 455-456,112-119 (2004).
[CrossRef]

Delaubert, V.

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, "Quantum limits in image processing," J. Mod. Opt. 29, 685-689 (1982)

Denny, J. L.

Dolfi, D

A . Beniere, F . Goudail, M . Alouini, and D . Dolfi, "Precision of degree of polarization estimation in the presence of additive Gaussian detector noise," Opt. Commun. 278, 264-269 (2007).
[CrossRef]

Dolfi, D.

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, "Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises," J. Eur. Opt. Soc. Rap. Pub. 3, 08002 (2008).
[CrossRef]

Drevillon, B.

Drévillon, B.

A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, "General methods for optimized design and calibration of Mueller polarimeters," Thin Solid Films 455-456,112-119 (2004).
[CrossRef]

Dubois, A.

Fabre, C.

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, "Quantum limits in image processing," J. Mod. Opt. 29, 685-689 (1982)

Fisher, R.

R. Fisher, "Theory of statistical estimation," Proc. Cam. Phil. Soc. 22, 700-725 (1925)
[CrossRef]

R. Fisher, "On the mathematical foundations of theoretical statistics," Phil. Trans. R. Soc. Lond. 222, 309-368 (1922)
[CrossRef]

Foreman, M. R.

Garcia-Caurel, E.

A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, "General methods for optimized design and calibration of Mueller polarimeters," Thin Solid Films 455-456,112-119 (2004).
[CrossRef]

Goudail, F

A . Beniere, F . Goudail, M . Alouini, and D . Dolfi, "Precision of degree of polarization estimation in the presence of additive Gaussian detector noise," Opt. Commun. 278, 264-269 (2007).
[CrossRef]

Goudail, F.

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, "Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises," J. Eur. Opt. Soc. Rap. Pub. 3, 08002 (2008).
[CrossRef]

J. Morio and F. Goudail, "Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices," Opt. Lett. 29, 2234-2236 (2004).
[CrossRef] [PubMed]

Grieve, K.

Kijewski, M. F.

S. P. Muller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
[CrossRef] [PubMed]

M. F. Kijewski, S. P. Muller, and S. C. Moore, "The Barankin bound: a model of detection with location uncertainty," Proc. SPIE 1768, 153 (1992).
[CrossRef]

Laude, B.

A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, "General methods for optimized design and calibration of Mueller polarimeters," Thin Solid Films 455-456,112-119 (2004).
[CrossRef]

Laude-Boulesteix, B.

Le Brun, G

M. Floc???h, G . Le Brun, J . Cariou, andJ . Lotrian, "Experimental characterization of immersed targets by polar decomposition of the Mueller matrices," Eur. Phys. J. AP 3, 349-358 (1998).
[CrossRef]

Lecaque, R.

Lohmann, A. W.

D. Mendlovic and A. W. Lohmann "Spacebandwidth product adaptation and its application to superresolution: fundamentals," J. Opt. Soc. Am. A 14, 558-562 (1997).

Look, D. C.

A. Ambirajan and D. C. Look, "Optimum angles for a polarimeter," Opt. Eng. 34,1651-1658 (1995).
[CrossRef]

Lotrian, J

M. Floc???h, G . Le Brun, J . Cariou, andJ . Lotrian, "Experimental characterization of immersed targets by polar decomposition of the Mueller matrices," Eur. Phys. J. AP 3, 349-358 (1998).
[CrossRef]

Lu, S. Y.

Meers, B. J.

B. J. Meers, "Recycling in laser-interferometric gravitational-wave detectors," Phys. Rev. D 38, 2317-2326 (1988).
[CrossRef]

Mehra, R.

R. Mehra, "Optimal input signals for parameter estimation in dynamic systems-Survey and new results," IEEE T. Automat. Contr. 19, 753-768 (1974).
[CrossRef]

Mendlovic, D.

D. Mendlovic and A. W. Lohmann "Spacebandwidth product adaptation and its application to superresolution: fundamentals," J. Opt. Soc. Am. A 14, 558-562 (1997).

Moneron, G.

Moore, S. C.

S. P. Muller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
[CrossRef] [PubMed]

M. F. Kijewski, S. P. Muller, and S. C. Moore, "The Barankin bound: a model of detection with location uncertainty," Proc. SPIE 1768, 153 (1992).
[CrossRef]

Morio, J.

Mueller, H.

H. Mueller, "The foundations of optics" J. Opt. Soc. Am. 38, 661-661 (1948).

Muller, S. P.

S. P. Muller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
[CrossRef] [PubMed]

M. F. Kijewski, S. P. Muller, and S. C. Moore, "The Barankin bound: a model of detection with location uncertainty," Proc. SPIE 1768, 153 (1992).
[CrossRef]

Munro, P. R. T.

P. R. T. Munro and P. Torok, "Properties of confocal Mueller-matrix polarimeters," (submitted toOpt. Lett.).

Myers, K. J.

Nee, S. M.

Pronzatom, L.

E. Walter and L. Pronzatom "Qualitative and quantitative experiment design for phenomenological models - a survey," Automatica 26, 195-213 (1990).
[CrossRef]

Rao, C.

C. Rao, "Information and the accuracy attainable in the estimation of statistical parameters," Bull. Calcutta Math. Soc. 37, 81-89 (1945)

Réfrégier, P.

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, "Quantum limits in image processing," J. Mod. Opt. 29, 685-689 (1982)

Rybicki, F. J.

S. P. Muller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
[CrossRef] [PubMed]

Savenkov, S. N.

S. N. Savenkov and K. E. Yushtin, "Mueller matrix elements error distribution for polarimetric measurements," Proc. SPIE 5158, 251-259 (2003).
[CrossRef]

S. N. Savenkov, "Optimization and structuring of the instrument matrix for polarimetric measurements," Opt. Eng. 41, 965-972 (2002).
[CrossRef]

Schwartz, L.

Sheppard, C. J. R.

Sherif, S. S.

Smith, M.

Srinivasan, C.

V. Bhapkar and C. Srinivasan, "On Fisher information inequalities in the presence of nuisance parameters," Ann. Inst. Statist. Math 46, 593-604 (1994).

Stoll, M. Ph.

J. Zallat, S. Aï?nouz, and M. Ph. Stoll "Optimal configurations for imaging polarimeters: impact of image noise and systematic errors," J. Opt. A: Pure Appl. Opt. 8, 807-814 (2006).
[CrossRef]

Sudradjat, F. F.

Tinbergen, J.

J. Tinbergen, "Interstellar polarization in the immediate solar neighbourhood," Astron. Astrophys. 105, 53-64 (1982).

Torok, P.

P. R. T. Munro and P. Torok, "Properties of confocal Mueller-matrix polarimeters," (submitted toOpt. Lett.).

Török, P.

Treps, N.

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, "Quantum limits in image processing," J. Mod. Opt. 29, 685-689 (1982)

Tyo, J. S.

Vabre, L.

Wagner, R. F.

Walter, E.

E. Walter and L. Pronzatom "Qualitative and quantitative experiment design for phenomenological models - a survey," Automatica 26, 195-213 (1990).
[CrossRef]

Yushtin, K. E.

S. N. Savenkov and K. E. Yushtin, "Mueller matrix elements error distribution for polarimetric measurements," Proc. SPIE 5158, 251-259 (2003).
[CrossRef]

Zallat, J.

J. Zallat, S. Aï?nouz, and M. Ph. Stoll "Optimal configurations for imaging polarimeters: impact of image noise and systematic errors," J. Opt. A: Pure Appl. Opt. 8, 807-814 (2006).
[CrossRef]

Ann. Inst. Statist. Math

V. Bhapkar and C. Srinivasan, "On Fisher information inequalities in the presence of nuisance parameters," Ann. Inst. Statist. Math 46, 593-604 (1994).

Ann. Math. Stat.

E. W. Barankin, "Locally best unbiased estimates," Ann. Math. Stat. 20, 477-501 (1949).
[CrossRef]

Appl. Opt.

Astron. Astrophys.

J. Tinbergen, "Interstellar polarization in the immediate solar neighbourhood," Astron. Astrophys. 105, 53-64 (1982).

Automatica

E. Walter and L. Pronzatom "Qualitative and quantitative experiment design for phenomenological models - a survey," Automatica 26, 195-213 (1990).
[CrossRef]

Bull. Calcutta Math. Soc.

C. Rao, "Information and the accuracy attainable in the estimation of statistical parameters," Bull. Calcutta Math. Soc. 37, 81-89 (1945)

Eur. Phys. J. AP

M. Floc???h, G . Le Brun, J . Cariou, andJ . Lotrian, "Experimental characterization of immersed targets by polar decomposition of the Mueller matrices," Eur. Phys. J. AP 3, 349-358 (1998).
[CrossRef]

IEEE T. Automat. Contr.

R. Mehra, "Optimal input signals for parameter estimation in dynamic systems-Survey and new results," IEEE T. Automat. Contr. 19, 753-768 (1974).
[CrossRef]

J. Eur. Opt. Soc. Rap. Pub.

A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, "Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises," J. Eur. Opt. Soc. Rap. Pub. 3, 08002 (2008).
[CrossRef]

J. Mod. Opt.

R. M. A. Azzam, "Division-of-amplitude Photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light," J. Mod. Opt. 29, 685-689 (1982).

V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, "Quantum limits in image processing," J. Mod. Opt. 29, 685-689 (1982)

J. Opt. A: Pure Appl. Opt.

J. Zallat, S. Aï?nouz, and M. Ph. Stoll "Optimal configurations for imaging polarimeters: impact of image noise and systematic errors," J. Opt. A: Pure Appl. Opt. 8, 807-814 (2006).
[CrossRef]

J. Opt. Soc. Am.

H. Mueller, "The foundations of optics" J. Opt. Soc. Am. 38, 661-661 (1948).

J. Opt. Soc. Am. A

Opt. Commun.

A . Beniere, F . Goudail, M . Alouini, and D . Dolfi, "Precision of degree of polarization estimation in the presence of additive Gaussian detector noise," Opt. Commun. 278, 264-269 (2007).
[CrossRef]

Opt. Eng.

S. N. Savenkov, "Optimization and structuring of the instrument matrix for polarimetric measurements," Opt. Eng. 41, 965-972 (2002).
[CrossRef]

A. Ambirajan and D. C. Look, "Optimum angles for a polarimeter," Opt. Eng. 34,1651-1658 (1995).
[CrossRef]

Opt. Express

Opt. Lett.

Phil. Trans. R. Soc. Lond.

R. Fisher, "On the mathematical foundations of theoretical statistics," Phil. Trans. R. Soc. Lond. 222, 309-368 (1922)
[CrossRef]

Phys. Med. Biol.

S. P. Muller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005).
[CrossRef] [PubMed]

Phys. Rev. D

B. J. Meers, "Recycling in laser-interferometric gravitational-wave detectors," Phys. Rev. D 38, 2317-2326 (1988).
[CrossRef]

Proc. Cam. Phil. Soc.

R. Fisher, "Theory of statistical estimation," Proc. Cam. Phil. Soc. 22, 700-725 (1925)
[CrossRef]

Proc. SPIE

M. F. Kijewski, S. P. Muller, and S. C. Moore, "The Barankin bound: a model of detection with location uncertainty," Proc. SPIE 1768, 153 (1992).
[CrossRef]

S. N. Savenkov and K. E. Yushtin, "Mueller matrix elements error distribution for polarimetric measurements," Proc. SPIE 5158, 251-259 (2003).
[CrossRef]

Thin Solid Films

A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, "General methods for optimized design and calibration of Mueller polarimeters," Thin Solid Films 455-456,112-119 (2004).
[CrossRef]

Vision Research

J. M. Bueno, "Depolarization effects in the human eye," Vision Research 41, 2687-2696 (2001).
[CrossRef] [PubMed]

Other

A. D. Whalen, Detection of signals in noise, (Academic Press Inc., New York, 1971).

P. Torok, M. Salt, E. E. Kriezis, P. R. T. Munro, H. P. Herzig, and C. Rockstuhl, "Optical disk and reader therefor," Worldwide Patent No. WO 2006/010882 (2006).

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T. Gehrels, ed., Planets Stars and Nebulae Studied With Photopolarimetry, (University of Arizona Press 1974).

L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, 1991)

R. A. Chipman, Polarimetry, Handbook of Optics Vol 2, (McGraw-Hill, New York, 1995).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and polarised light, (Elsevier, North Holland, 1987).

J. W. Goodman, Statistical Optics (Wiley, New York, 2000).

T. J. Rothenberg, Efficient estimation with a priori information (New Haven, Yale University Press, 1973).

http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/calculus.html

E. Collett, "Automatic determination of the polarization state of nanosecond laser pulses," U.S. Patent No. 4158506 (1979).

H. Cramér, Mathematical Methods of Statistics, (Princeton Univ. Press. 1946), ISBN 0-691-08004-6.

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Figures (3)

Fig. 1.
Fig. 1.

Schematic diagram of a polarimeter setup.

Fig. 2.
Fig. 2.

Fig. 2. Dependence of channel capacity on degree of polarisation for S 0/Id = 104.

Fig. 3.
Fig. 3.

Poincaré sphere showing possible polarimeter configurations (a) for a Stokes polarimeter matched to the template Stokes vector (1,0,0,1) i.e. left circularly polarised light and (b) a linear polarimeter, assuming the ratio S 0/Id = 105. Each arrow denotes the basis Stokes vector of a polarimeter arm.

Equations (63)

Equations on this page are rendered with MathJax. Learn more.

D = T S
S = T + D
S = M R
D = T M R
M = T + D R +
f ( D | D ¯ ) = D ¯ D D ! exp ( D )
f ( D | D ¯ ) = n = 1 N D N R ( D ¯ n + I n d ) D n D n exp ( D ¯ n I n d )
J w = E D [ ln f ( D | w ) w ln f ( D | w ) T w ]
J w = G T J D ¯ G
( J D ¯ ) ij = δ ij D i + I d
J w = E w [ J w nr ] + J w r
C D ¯ = n = 1 N D 1 D n
C D ¯ N D 2 S 0 + N D I d .
C D ¯ N R N D 2 R 0 + N D I d
Δ w = w w e
v min = v n c J w 1 = v n c J w
J S = T T J D ¯ T
J M = ( R T T ) J D ¯ ( R T T )
J S = T 2 J D ¯
J M = R 8 T 8 J D ¯
J S = T 2 Π i = 1 4 E S [ 1 D i + I d ]
J M = R 8 T 8 i = 1 16 E M [ 1 D i + I d ]
J u = J 11 J 12 J 22 −1 J 21
J w = ( J 11 J 12 J 21 J 22 )
J u = J w J 22
E s [ 1 D i + I d ] = 1 S 0 P log [ ( 1 + P ) S 0 + 2 I d ( 1 P ) S 0 + 2 I d ]
= 2 S 0 P arctanh [ S 0 P S 0 + 2 I d ]
J S = 1 8 T 2 ( arc tanh [ S 0 P S 0 + 2 I d ] S 0 P ) 4
SNR = ( S 0 2 P arc tanh [ S 0 P S 0 + 2 I d ] ) 1 2
T 0 pt = ( 1 1 0 0 1 0.333 0.816 0.471 1 0.333 0 0.943 1 0.333 0.816 0.471 )
J S = T 2 i = 1 4 1 D i + I d
E [ 1 D i + I d ] = 1 ( S 0 + 2 I d ) 2 + S 0 2 P 2 cos 2 2 α i
J S ' = T 2 j = 1 4 [ ( S 0 + 2 I d ) 2 + S 0 2 P 2 cos 2 2 α j ] 1 2 j = 1 4 sin 2 ( 2 α j ) [ ( S 0 + 2 I d ) 2 + S 0 2 P 2 cos 2 2 α j ] 1 2
J z = s T z T T J D ¯ T S z
J z = M T z ( R T T ) J D ¯ ( R T T ) M z
M A = T u ( 1 A T A m A )
m A = 1 A 2 I + ( 1 1 A 2 ) A A T A 2
M A A k = T u ( 0 k T k m A A k ) , M A T u = M A T u
m A i j A k = [ A i δ j k A 2 + A j δ i k A 2 ] [ 1 1 A 2 ] + A k δ ij 1 A 2 A i A j A k A 4 [ 2 + A 2 2 1 A 2 ]
M R = ( 1 0 T 0 m R )
m R i j = δ ij cos R + R i R j R 2 ( 1 cos R ) + q = 1 3 ε i j q R q R sin R
m Rij R k = [ R i δ j k R 2 + R j δ i k R 2 ] ( 1 cos R ) R k δ ij R sin R R i R j R k R 2 [ 1 cos R R + sin R ]
+ q = 1 3 ε ijq R 2 [ R q R k cos R + ( R δ qk R k R ) sin R ]
M Δ = ( 1 0 0 0 0 a 0 0 0 0 b 0 0 0 0 c ) , a , b , c 1
M Δ ij a = δ i 2 δ j 2 , M Δ ij b = δ i 3 δ j 3 , = M Δ ij c = δ i 4 δ j 4 ,
J z = M A T z ( R M R T M Δ T T T ) J D ¯ ( R T T M Δ M R ) M A z
+ M R T z ( M A R M Δ T T T ) J D ¯ ( R T M A T T M Δ ) M R z ¯
+ M Δ T z ( M R M A R T T ) J D ¯ ( R T M A T M R T T ) M Δ z ¯
J z = ( J R 0 T 0 T 0 J A 0 T 0 0 J Δ )
L ( D , w ) = In f ( D w ) + In f ( w )
J w = i = 1 2 j = 1 2 E D , w [ L i w L j T w ]
= J 1 + J 2 + J 3 + J 4
J 1 = L 1 w L 1 T w f ( D , w ) d D d w
= [ L 1 w L 1 T w f ( D w ) d D ] f ( w ) d w
= E w [ J w nr ]
J 2 = L 1 w L 2 T w f ( D , w ) d D d w
= f ( D , w ) f ( D w ) f ( w ) f ( D w ) w f ( w ) w d D d w
= [ w f ( D | w ) d D ] f ( w ) T w d w
= 0
J 4 = L 2 w L 2 T w f ( D , w ) d D d w
= L 2 w L 2 T w f ( w ) d w = J w r
J w = E w [ J w n r ] + J w r
In f ( w ) w = 0

Metrics