2038 J. Opt. Soc. Am. A/Vol. 10, No. 9/September 1993
Signal and noise transfer through imagingsystems consisting of a cascade of amplifying and
scattering processes
Hendrik Mulder
Research and Development Laboratories, Delft Instruments Medical Imaging BV Delft, The Netherlands
Received April 29, 1992; revised manuscript received January 14, 1993; accepted March 31, 1993
The transfer of signal and noise through an imaging system consisting of a cascade of amplifying and scatter-ing processes has been analyzed on the basis of the Burgess variance theorem. Up to second statisticalmoments, the results are identical to those deduced by Rabbani et al. [J. Opt. Soc. Am. A 4, 895 (1987)] withmultivariate moment-generating functions. An easy-to-visualize concept of detective quantum efficiency(DQE) that depends on object shape and size has been deduced that can be converted into the commonly usedDQE which depends on spatial frequency. A simple expression for the noise power spectrum of a system thatconsists of an arbitrary number of amplifying and scattering processes has been derived; noise added along thesystem can be taken into account.
1. INTRODUCTION
In imaging systems, particles such as x-ray quanta, pho-tons, and electrons can be converted from one kind intoanother by an amplifying process that is generally of astochastical nature. Up to the second statistical moment,each amplifying process can be characterized by the gainand the variance of gain. These parameters can be sto-chastic variables themselves; e.g., the mean number ofphotons emitted by an x-ray screen depends on the energyof the x-ray quantum absorbed. The Burgess variancetheorem' expresses the variance at the output of a stochas-tically amplifying process in the parameters of the processand in the input variance and mean.
A process that scatters particles can be described by thepoint-spread function or its Fourier transform, the opticaltransfer function (OTF). It is obvious that the OTFcauses correlation between regions that are spatiallyapart. This means that spatial integration to obtain thevariance of larger object areas has to take into accountthe covariance existing between different areas. To ac-count for this covariance, we shall introduce a correlationfunction, and its relation to the noise power spectrum(NPS) will be shown. We shall deduce relations for thevariance of an area of arbitrary size and shape.
In the case of stationary processes, relationships be-tween the input and the output NPS are derived that areidentical to those given by Rabbani et al.2 We shall deriverelations for the output NPS of a system having an arbi-trary number of consecutive stochastically amplifying andscattering processes in which noise along the system maybe added.
We shall introduce a detective quantum efficiency(DQE) that depends on object shape and size and showthat when applied to a periodic object, it can be convertedinto the commonly used DQE that depends on spatial fre-quency. The difference between these two concepts ofDQE is small, as will be illustrated with two examples.
2. LARGE AREA VARIANCE
We introduce the mean per unit area, e(x, y), and the vari-ance per unit area, o-2(x, y), of the number of particles inthe x-y plane (see Fig. 1). If no correlation exists betweenspatially separated areas, the variance of an arbitraryarea can be obtained by a simple integration of o-2 overthis area. In the general case with correlation, we writefor the variance of an area a
in which c(x 1,x 2,Yi,Y2) is a function per unit area thattakes into account the correlation between the areasaround (x1, yl) and ( 2 , y2). To keep our notation simplewe restrict ourselves to the one-dimensional case; exten-sion to two dimensions is straightforward. More funda-mentally, we restrict ourselves to the stationary situationin which c depends only on the difference x1 - x2.
We then obtain
Var(a) = f o'(xl)O-(x2)c(xl - 2)dxldx2 . (2)
In this one-dimensional case, e, cr2 , and c are numbers perunit length.
We adopt the following convention for Fourier trans-forms: a function in the spatial domain is denoted bya lowercase letter, while its counterpart in the fre-quency domain is denoted by an uppercase letter.The + Fourier transform of an arbitrary function h isdenoted F{h} or H, while the - Fourier transform is de-noted F.{h} or H*.
We introduce into expression (2) an aperture functiona(x) = 1 within a and a(x) = 0 elsewhere and obtain
Vol. 10, No. 9/September 1993/J. Opt. Soc. Am. A 2039
3. STOCHASTIC AMPLIFICATIONThe Burgess variance theorem expresses the variance atthe output of an amplifying process in terms of the inputmean and variance; the process parameters gain g, whichis the ratio of the output and the input mean; and vari-ance S2 of the gain. The Burgess theorem is
An integral without boundaries is supposed to extendfrom - to + o. Relation (3) can be rewritten in thefrequency domain as an integral with respect to the spa-tial frequency, fl:
Var(a) = F{au}F{ao-}rdf, (4)
with F{c} = F.Suppose that area a, situated around point x = s, has an
infinitely small length ; then we obtain from relation (3)
Var(a) = o2(s)12c(0).
The following relation should be consistent with thedefinition
lim Var(a) 2(s)
which is true if lim c(O) = 1; this implies a delta-functionbehavior of c(x) around x = 0.
To illustrate the meaning of correlation function c(x),suppose that a consists of an area al situated aroundpoint x = s and another area a2 around x = 2. Bothareas have infinitely small length 1, which with rela-tion (3) leads to
This result is consistent with the well-known expressionfor the variance of the sum of two stochastic variables:
Var(a) = 0-2 (Sl)l + cr2(s2)l + 2 cov(sl,s 2 )1.
From Eqs. (5) and (6) it follows that
C(S - S2)1 = COV(S1 , S2)1
0cs 1Slo(sD) cr(s2 )YV
the well-known correlation factor from statistics.
See Appendix A for the derivation of the Burgess variancetheorem and related expressions. From Eq. (3) we derivewith Eq. (A5) for the variance of an area a at the output ofan amplifying process, of which the parameters are sto-chastic variables themselves:
(3)Var(a)out =f ta(x)E(gl)o-(xl)a(X2)E(2)0'1(X2)
X C(X1 - x2)dxldx2
+ fa(x)[E(s2) + Var(g)]el(x)dx.
In the case in which the parameters of the process arespace invariant, we may write
Var(a)out = E2(g) ff a(xl)ol (xl)a(x2)o1 (x2)
X cl(x - 2)dxldx2 + [E(s2 ) + Var(g)]
x fa2(x)el(x)dx. (9)
In the last term we have replaced a(x) with a2(x), which ispermitted because of the definition of a(x). Rewritten inthe frequency domain, Eq. (9) transforms into
Var(a)..t = E2(g) f F.{acr}F{aorl}jFdfl
+ [E(s2) + Var(g)] F {ael}Adfl. (10)
If we restrict ourselves to the case of space-invariant -and el, Eq. (10) simplifies to
Var(a)out = E2(g)cr12f IA| 2 Fldn
+ [E(s2 ) + Var(g)]elfjIA2dfl
= f[E2(g)o. 1 2Fl + E(s 2)el + Var(g)el]IA12 dfl.
(11)
But according to Eq. (4),
Var(a)o0 t = f 0r22F2 Al 2dQ,
which can be true for any area a only if
0222 = E2 (g)or2rF + E(s 2)el + Var(g)ei.
(6)
(12)
If we interpret o-2r as the NPS, this expression is similarto Eq. (48) of Rabbani and Van Metter.3
y
(7)
Hendrik Mulder
(8)
2040 J. Opt. Soc. Am. A/Vol. 10, No. 9/September 1993
If there is no correlation in the first place, hence 1' = 1,then there will be no correlation as the result of an ampli-fying process, and consequently 12 = 1.
4. STOCHASTIC SCATTERING
Along the x axis we have a mean e(x) and a variancecr1
2(x) per unit length of particles emitted by the x axis.These particles are imaged on the u axis. Particles arespread according to a point-spread function t(u) along theu axis. The OTF is the Fourier transform; hence F{t} =T(fl). All particles arrive at the u axis, and we assumethat no amplification takes place during the scatteringprocess. This means that ft(u)du = 1. From an ele-ment dx at point x, the transfer of particles to an area awith length on the u axis is ruled by a binomial processwith gain
g(x;a) = ft(u - x)du = a(u)t(u - x)du = F_{AT*}.
(13)
Conforming Eq. (8), and with S2 = g(1 - g) and expres-sion (A4) below, we conclude for the variance of the area aon the u axis that
Var(u; a) = ffg(xl; a)- 1(xl)g(x2 ; a)- 1(x2)
X Cl(X - 2 )dxldx2
+ f g(x; a)[1 - g(x; a)]el(x)dx; (14)
or, rewritten in the frequency domain,
Var(u;a) = Fgol}F{gol}jrdfl + fG*Eldfl
- f F {gel}F{g}df1. (15)
If we restrict ourselves again to the case of space-invariant a, and el, make use of relations F{g} = G =AT* and F_{g} = G* A*T, substitute El = e(fl), andrealize that
G*(0) = A*(O)T(O) A*(O) = fa(u)du = fa2(u)du
- JIAI2df,
we deduce that
Again we interpret 2 r as the NPS; expression (17) isidentical to Eq. (27) of Rabbani et al.2 It is evident thatin the case in which the point-spread function t(u) dependson a stochastic parameter, the modulation transfer func-tion TI in Eq. (17) will be replaced by its expected valuewith respect to that parameter [see Eq. (63) of Ref. 3].Note that processes having a time response, e.g., delayedemission of photons by a scintillation screen, can be dealtwith in the same way (see Appendix B).
5. MULTISTAGE SYSTEM
Instead of the variance, s2, we prefer to use the quantityscintillation efficiency, A, for all kinds of processes. Thisquantity was introduced by Swank4 to account for thebroadening of the scintillation spectra of x-ray screens.A and s are related by
S2= (1/A - 1)g2.
In the case of a binomial process, i.e., a process in whichone input particle causes one or no output particle,s2 = g(1 - g), with g < 1, and consequently A = g. Inthe case of a Poisson process, S2 = g and hence A =g/(1 + g), with lim A = 1 as g > -. In the case of a de-terministic process s = 0, and consequently A = 1.
In the more general case in which the process parame-ters are stochastic variables themselves, we define thescintillation efficiency as
(A)= E(g)()=E(g2+ 2)'
(18)
We now continue with the two-dimensional case and de-fine some new quantities:
Zi = 1i r,-ei, (19)
Pi = (1/(Ai) - 11gi), (20)
in which the index, i, refers to input and parameters of theith process. In case of a binomial process, P = 0. Forthe sake of simplicity we write A and g instead of (A) andE(g), even if these parameters are stochastic variablesthemselves. Substitution of Eqs. (19) and (20) intoEq. (12) and making use of the relation ei+l = gie, give foran amplifying process
Zi+l= gi 2 (Z, + Pie1). (21)
The quantities Zi and ei are per unit area. If during thisprocess a geometrical demagnification Pi takes placesimultaneously, these quantities will be modified by a fac-tor of Pi 2; consequently, in general we obtain
Var(u; a) = f (2r, - el)AI2IT12 dfl + eiIA12d. (16)
Again according to Eq. (4),
Var(u; a) = f 022 2 A2 dQ,
which is true for every a only if
Z,+ = p12 g, 2 (Z, + Piei). (22)
Substituting Eq. (19) into Eq. (17) and making use of therelation ei+l = Pi 2ei, we obtain for the scattering process
Zi+l = pi2 jT 1j
2Z1 . (23)
From Eqs. (22) and (23) we deduce the general expression
Z,+l = pi 2gi 2jTji
2(Z, + Pie1). (24)
(17) In applying this expression we have to keep in mind that
Hendrik Mulder
0'2 2F = (0-,2rl - el)IT 12 + el.
Vol. 10, No. 9/September 1993/J. Opt. Soc. Am. A 2041
for amplification Til = 1 and that for scattering gi 1and Pi = 0. The last statement implies that Ai = gi = 1,which is to be expected of a scattering process, since itis binomial.
Before we write the expression for a multistage imagingsystem, we have to consider that a stage is a process andnot necessarily a physical component. For example, anx-ray image intensifier consists of at least three amplify-ing and four scattering processes:
* Absorption of an x-ray quantum in the x-ray phos-phor causes emission of photons,
* Photons are scattered in the x-ray phosphor,* Photons release electrons from the photocathode,* Photoelectrons are not ideally focused on the anode,* Electrons are scattered in the anode screen,* Electrons generate photons in the anode, and* Photons are scattered in the anode.
We have to split up a system into processes that have ei-ther amplification or scattering. Sometimes, as withoptical systems, we have on the one hand an amplifyingprocess (loss of photons as a result of limited numericalaperture and transmission) and on the other hand a scat-tering process (lens aberrations). Splitting up a systeminto an amplifying process followed by a scattering processis from a statistical point of view not identical to splittingit up in the reverse order (see, e.g., Ref. 5 and Section 10below); therefore one has to be careful in unravelingmixed processes.
For a multistage imaging system consisting of k con-secutive processes, one can deduce the following expres-sion, which can be proved by finite induction:
k
Zk+l = p62G21T12Z, + ,p2Gel Qi, (25)i=1
in which f3, G, and T represent the demagnification,the gain, and the OTF, respectively, of the completesystem and
k g1Qi = PiqrTgII 2 (26)J1=i
is the result of the multiplication of Pi with the gain andthe modulation transfer function squared of process i andof all consecutive processes.
From Eq. (25) and definition (19) we conclude for theNPS at the output that
k
NPSoUt = O-k+lk,+l/p2 = G2ITI2 Z, + Gel 2Qi,i=O
(27)
with Qo = 1.If the input signal has a Poisson distribution, which is
generally the case, Zi = ul2rl - el= cr12 - el = el -el = 0, and
k
NPSout = Gel 2 Qi. (28)i~O
By dividing by p2 we express the output NPS in the coor-dinates of the input plane, which makes comparison withthe input quantities easier. With Eq. (27) or (28) substi-tuted into Eq. (4) (with cr2 and e space invariant), one cancalculate the variance of any area in the output plane.
6. ADDITIONAL NOISESuppose that additional noise, characterized by Zj andmean per unit area njj, is fed into the jth process; thenaccording to Eq. (27) we have at the output of the kthprocess
k
Oejk+l Jk+l = jk Gjk ITjkl2Zjj + 6jk2Gjknj Q,izj-1
(29)
in which jk, Gjk, and Tjk represent demagnification, gain,and OTF, respectively, of the jth process together with allconsecutive processes; hence ,B = /
3 1k, G = Glk, and T =Tlk. From Eq. (29) we derive the NPS at the output, ex-pressed in the coordinates of the input plane:
k
NPSj.., = fIjk/3Gjk Iijk| 2Zjj + pjk /f3Gjknj I Qi.
i=j-l(30)
Note that the Qi's of Eq. (30) are identical to those ofEq. (27), but Qi = 1. Sources of noise, such as photo-cathode or CCD dark-current and scattered x rays or light,have in general a Poisson distribution; therefore in thosecases the first term of Eq. (30) vanishes. Furthermore,we may expect that to some extent these sources are uni-formly distributed over the image plane and that conse-quently the requirement of space invariance is fulfilled.However, noise caused by field emission or by uncon-trolled discharge in an image intensifier is generally notuniformly distributed and not Poissonian. The completeNPS is found by adding Eqs. (27) or (28) and Eq. (30) foras manyj as necessary for describing the noise behavior ofthe system adequately. If, at the input of a system, noisesuch as scattered x rays is added, then one must introducethis noise as n22 behind a first deterministic process(g = A 1 and T = 1) to prevent the noise from beingadded to el, thereby improving the DQE under certainconditions (see Section 9 below).
We now define signal and noise in the spatial domain inorder to calculate from input and output signal-to-noiseratio (SNR) the DQE according to
DQE= - SNRut2-Q SNRin2 (31)
The input has a space-invariant mean and variance el(Poisson distribution) per unit area, on which is super-imposed an object with area a having a mean and vari-ance eo per unit area. We define as the input signal
Si = E(a)i = aE[(el + eo) - el] = aeo. (32)
Introducing spatial frequencies v and w, we deduce fromEq. (4) for the two-dimensional case the variance atthe input:
Var(a)in = Oin2 ff FinIA 2dvdw,
making use of Oin = [(o2 + 12)]" 21 y, c in the case in
Hendrik Mulder
2042 J. Opt. Soc. Am. A/Vol. 10, No. 9/September 1993
which 0-02 = eo << o,2 = el. We find with i, = , = 1,
Var(a)i. = elffJAI2dvdw = ael. (33)
From the definition of the SNR we obtain with Eqs. (32)and (33)
SNRn 2 = S1. 2 /Var(a). = aeo2 /e,. (34)
As a result of the input signal, we have at the outputplane, superimposed on a space-invariant mean per unitarea Gel, a signal e(x, y) per unit area, for which holds theexpression
Gaeo = ff edxdy.
G represents the gain of the complete multistage system.As output signal we define
Sout = f e(x, y)dxdy =ff a(x, y)e(x, y)dxdy
= ffA*Edvdw, (35)
in which the object-contour function a(x, y) = 1 within aand 0 elsewhere. With E = GEoT and E0 = eA, T repre-senting the complete OTF of the system, we conclude forS0 ut that
(fi Tdvdw)DQE(a) = G2ae,
JJ NPSoutdvdw(40)
and in the case in which T = 1 for all frequencies or a isinfinitely large, Eq. (39) reduces to
DQE(a) = G2eT(0, 0) 2 G2e,NPSout(0, 0) NPSout(O,)
(41)
8. DETECTIVE QUANTUM EFFICIENCY(SPATIAL-FREQUENCY DEPENDENT)Let us take a periodic object with mean and variance ofthe particles e(x) = e cos(2rflox), superimposed upon aspace-invariant background with mean and variance el,both quantities el and e being per unit length andeo << el. As input signal we define the mean that passesa filter with transmission h(x), which satisfies h(x)cos(27rQox) if IxI < /21 and h(x) = 0 if Ixi > 1/21.
Hence
J+1/21
Sin(flo; ) = el cos(27rTQox)dx-1/21
r' /21~+1/2+ eO J
--1/21cos 2(2 rTfox)dx.
If is sufficiently large or Qlol is an integer, we obtain
Sin(Qo; 1) = /2eol.So0 t = Geo ff AI 2Tdudw. (36)
In the case in which T = 1 for all frequencies we obtainSout =Gaeo and in the case in which T -> 0 for all fre-quencies Sot - 0, results that one should expect with aproper definition of signal.
Because e << el, we may expect that the contributionof the signal to the variance at the output is negligible,and consequently we have almost a space-invariant situ-ation that makes the NPS significant. We thus get fromEq. (4)
Var(a)0 ,t = JJNPS0utIA12dudw, (37)
(42)
It is evident that in a similar way we obtain for the outputsignal
(43)
in which G and T represent the gain and the OTF, respec-tively, of the complete system. We take relation (8) as astarting point to calculate the variance; however, we omitthe second term because the filter is deterministic, andconsequently we have
and from Eqs. (36) and (37) we obtainVar(flo; l) =f F h}F{ho.}ordfl.
G2eo2(ff Al 2Tdvdw)
SNRout2 =
JJ NPSo0 tIAl 2 dudw
According to Eq. (31) we arrive with Eqs. (34) and (38) at
DQE(a) =G2e, 1/aH Al 2TdvdW
1/a ff NPS,,,IA2 2dvdw
In the case in which a is infinitely small, Eq. (39) reducesto
(38) At the input as well as at the output we neglect the contri-bution to the variance of the periodic signal; consequently,the situation becomes space invariant and the quantityNPS becomes significant. Relation (44) transforms into
Vol. 10, No. 9/September 1993/J. Opt. Soc. Am. A 2043
Substitution into Eq. (45) leads to
Var(fh; 1) = 1/412f NPS sinc2{7r(f - fQo)l}df1
+ /412fNPS sinc2{7r(fQ + flo)l}dfl
+ /212f NPS sinc{7r(fl - f10)l}
X sinc{7r(fl + flo)l}dQ,
which, in the case in which increases, approaches
Var(fl0 ; 1) = l/41NPS(flo)l f sinc2 tir(f1 - Qo)l}dQ
+ l/41NPS(-fQo)lf sinc2{r(fl + fQo)l}dfl
With f sinC2(irx)dx = 1 we obtain for the symmetricalcase c(-x) = c(x)
Var(fl0 ; 1) = /2lNPS(Qo). (46)
At the input we have NPSi 0(fl 0 ) = cr12r = e ( = 1),with Eq. (46) we get Varin(f0; 1) =
1/21el, and with Eq. (42)and definition (34) we obtain
SNR n2 = 1/2(e 2/el)l.
At the output we have
SNRout2
= 1/2G2 T2
eo2 l/NPS(flo).
From Eqs. (47) and (48) we conclude that
DQE(flo) = G2 T2el/NPS(flo).
(47)
(48)
and Eq. (50) substituted into Eq. (39) leads to
DQE(D)
= g1 (51)41 1/-J)1(1r/fD)exp(-a()df d
1 + 2(gIA - ) flJ,'(vflD)exp(-2afl)dfl
and substituted into Eq. (49) to
DQE(Ql) = g 1 +gexp(-2afl) IDQEf~)= g 1 + (g1 /Ai - 1)exp(-2afl) (52)
in which D represents the diameter of the object; hencea = 147rD2 . In Fig. 2, DQE(D)/Al and DQE(Q)/Al areshown; a/D and aQl are the respective variables and gl/A1is a parameter.
If the parameter g1/Ai increases, the influence of theOTF of the second process on the DQE lessens. As asecond example we interchange the sequence of the twoprocesses; hence T = exp(-afl), and then we have, withQ1 = 0,
NPS = g2e, + g2elQ2 = g2e1 (1 + g2P2),
which leads with Eq. (39) to
DQE(D) = A24 [f 1/f1Jl2(rD)exp(-afl)dfQ1,
and with Eq. (49) we have
DQE(fQ) = A2 exp(-2afQ).
(53)
(54)
(55)
(49)
This result is equivalent to the generalized definitionderived by Shaw6 and gives the DQE as a function of spa-tial frequency. Furthermore, this result is consistentwith Eq. (41) for fl0 = 0.
9. SOME REMARKS
Without noise added along the imaging system, the DQEdepends on system parameters only. If noise is added,however, the DQE becomes dependent also on the meanper unit area of the input particles. If this input meanincreases, the influence of the additional noise will lessenand the DQE will increase and approach the limiting valueof the situation without additional noise.
In the absence of scattered x rays, an x-ray imaging sys-tem incorporating an antiscatter grid will have a lowerDQE than a system without such a grid. In practice, how-ever, with scattered x rays, a system incorporating anantiscatter grid may give superior images.
DQE(D)/A,
0
DGE(Q )/A
- = -+ -- 1
a/D
(a)
7 H--r~~i | 1 _1- 1 I~~~~~~~~~~~~~~~~~~
\\S~~9 1-- - - -gjA1
-- - -I -I-1-
10. EXAMPLES
As a first example we take a system consisting of twoprocesses, the first one amplifying and the second onescattering with OTF T2= exp(-al). In this case wehave, according to Eq. (28), with Q2 = 0,
NPS= gle, + g1e1Q1 = giel(1 + glT 2 I2 PD), (50)
(b)Fig. 2. DQE of a system consisting of two processes: an ampli-fying one, with gain g1 and scintillation efficiency A1, followed bya scattering one, with OTF T2 = exp(-al); parameter gl/A1 hasvalues 1, 5, and 9; (a) DQE/A1 dependent on object diameter D asa function of a/D; (b) DQE/A 1 dependent on spatial frequency fQas a function of al.
Hendrik Mulder
I .\
2044 J. Opt. Soc. Am. A/Vol. 10, No. 9/September 1993
DQE(D)/A
DQE(S 2)/A,
0
- -- - A---
a/D 1(a)
i I I I
X~~~~~I-, -_
(b)
Fig. 3. DQE of a system consisting of two processes: a scatter-ing one, with OTF T1 = exp(-aQl), followed by an amplifying one,with gaing2 and scintillation efficiency A2; (a) DQE/A 2 dependenton object diameter D as a function of a/D; (b) DQE/A2 dependenton spatial frequency 1 as a function of afl.
If we substitute g1/A 1 = 1 in Eqs. (51) and (52) we obtain,respectively, Eqs. (54) and (55); hence in the case in whichthe amplifying process is binomial, the order of the pro-cesses is irrelevant.
In Fig. 3, DQE(D)/A2 and DQE(fl)/A2 are shown againwith variables a/D and a; the curves do not depend onthe parameter g2/A2 and are identical to those of Fig. 2 forthe case g1/A = 1.
11. CONCLUSIONS
The algorithms deduced for the NPS of imaging systemsconsisting of an arbitrary number of stochastically ampli-fying and scattering processes are simple; noise addedalong the system can be taken into account easily. Theconcept of DQE that depends on object shape and size isessentially similar to the concept of DQE that depends onspatial frequency; the latter DQE can be deduced from theformer straightforwardly. The probability of detection ofan object of specific shape and size in a noisy backgrounddepends on the SNR of the object; the DQE that is basedon object shape and size is a better starting point for calcu-lation of the SNR than is the frequency-dependent DQE.
When noise is added along a system, the DQE does notdepend on system parameters only; therefore input andnoise conditions should be specified.
APPENDIX AIf W(n; m) represents the probability that the stochasticvariable y equals n if x equals m, then the mean and thevariance of y, under the condition x = m, are, respectively,
E(y; m) = I nW(n; m),n
Var(y; m) = 2 n2 W(n; m) - E2(y; m)n
= E(y2 ; m) - E2 (y; m).
If P(m) represents the probability that x equals m, thenthe mean and the variance of y, under the condition that mcan take any value, are, respectively,
E(y) = EnP(m)W(n; m) = >.P(m)E(y; m)n m
= EE(y; x)}, (Al)
Var(y) = f n2 YP(m)W(n; m) - E2(y)n m
= Var{E(y; x)} + E{Var(y; x)}. (A2)
Suppose that x is the particles at the input and y those atthe output of a process of which the gain g is defined byg = E(y; 1) and the variance of the gain is defined by s2 =Var(y, 1). Then if k particles are fed into the process anddealt with independently, the mean and the variance of y,respectively, are given by
E(y; k) = kE(y; 1) = kg,
Var(y; k) = k Var(y; 1) = k 2 .
If k can take any value, the mean and the variance of ybecome with Eqs. (Al) and (A2), respectively,
The latter expression is the Burgess variance theorem.Suppose that W(t; m, n) represents the probability that
the stochastic variable y equals t if xi equals m and x2equals n and that P(m, n) represents the probability thatx1 equals m and x2 equals n; also suppose that xi and x2 arethe inputs of two different processes with parameters
2 2g1, si and g2, s22 , respectively, and that we add the outputs
to obtain y. Then with a reasoning similar to the one weused for the derivation of Eqs. (Al) and (A2), we can derive
With N inputs and N processes, we obtain for the outputsadded
N N N
Var(y) = > E gig, cov(xi, xj) + E si2E(xi),i-i j-1 i-1
(A4)
with cov(xi, xi) = Var(xi).Suppose that the inputs xi are the outputs of a branched
binomial process as a result of an input x and that thisprocess is characterized by gain ,ti and variance of gain/%(l - p.i), with
N
Ei = 1;i-1
hence
Hendrik Mulder
m
1
Vol. 10, No. 9/September 1993/J. Opt. Soc. Am. A 2045
E(xi) = gEWx,
Var(xi) = A,2 Var(x) + /,i(l -i)E(xi),
cov(xi, x;) = imj Var(x) - kjjE(x).
Then it is straightforward to deduce that
Var(y) = EA 2 (g)Var(x) + E (s2)E(x) + Var,(g)E(x),(A5)
with
N
E,(g) = gigi,i=1
N
Var. (g) = ,gi-E2()
N
EA(S2) = > S,2.i=l
The subscript ,u denotes that mean and variance aretaken with respect to the random gain of the branchedbinomial process. We call expression (A5) the extendedBurgess variance theorem.
APPENDIX B
The response of a process on an input particle can be de-scribed by a time-response function, r(t). If g representsthe mean of the total output, then the mean of the outputin a time interval, dt, at time t equals gr(t)dt. We assumethat
fr(t)dt = 1;
hence R(O) = 1, with R(f) the Fourier transform of r(t).If, first, no correlation exists between the space and thetime properties of the input signal; second, the timeresponse is independent of position; and third, the pa-rameters that describe the amplifying or the scatteringprocesses are independent of time; then the correlationfunction can be written as the product of a space-dependent and a time-dependent function; hence
C = c(x-X2)Ct(tl- t2 )-
If mean e, variance 2, and correlation function c, per unitarea and unit time, are space and time invariant, the vari-ance of the number of particles integrated in time within area a is
Var(ar) = fJJf Fabo-}F{abao}F8 tdvdwdf
cl.2fffIAI 2BI 2Fstdvdwdf,
in which b(t) = 1 within T and b(t) = 0 elsewhere.For a time-response process we find that, similar to a
scattering process,
cr22F2Ft2= (cr 2Fsirti - ei)1R12 + ei,
and if we interpret again t2 'j as the noise power spec-trum, now per unit area and unit time, we see that allexpressions for multistage systems remain the same; weonly have to replace the OTF T by the frequency-responseR(f).
ACKNOWLEDGMENT
The author is grateful to Henk van Elburg for helpful dis-cussions and criticism.
REFERENCES
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3. M. Rabbani and R. Van Metter, 'Analysis of signal and noisepropagation for several imaging mechanisms," J. Opt. Soc.Am. A 6, 1156-1164 (1989).
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