Cramer-Rao bound and phase-diversity blind deconvolution performance versus diversity polynomials

Cramer–Rao bound and phase-diversity blind deconvolutionperformance versus diversity polynomials

Jean J. Dolne and Harold B. Schall

Information theoretic bounds on the estimated Zernike coefficients for various diversity phase functionsare presented. We show that, in certain cases, defocus diversity may yield a higher Cramer–Rao lowerbound (CRLB) than some other diversity phase functions. Using simulated images to evaluate theperformance of the phase-diversity algorithm, we find that, for an extended scene and defocus diversity,the phase-diversity algorithm achieves the CRLB for known objects. Furthermore, the phase-diversityalgorithm achieves the CRLB by a factor of �2 for unknown objects. © 2005 Optical Society of America

OCIS codes: 100.0100, 100.3010, 100.3020, 100.1830, 110.0110, 030.0030.

1. Introduction

Achieving fine image resolution at long ranges hasalways been of interest for intelligence and surveil-lance applications. The fine angular resolution re-quires, however, large optics.1–4 In some cases, thelarger optics consists of segments. To yield the max-imum image quality possible, the segments must bephased to within a fraction of a wave (typically� ��20 rms). Various methods exist to estimate thephase front of an optical system. Many of these meth-ods rely on a point source.5,6 For some applications, apoint source may not be readily available. One of theemerging techniques that uses an extended scene toestimate the phase front uses phase diversity7–9; inthis method more than one image of the same sceneis collected and an intentional phase diversity is in-troduced in one of the images. We note that the exactphase need not be known; only the functional form ofthe phase needs to be known.10 The phase diversitymost often used is defocus. This brings into questionwhether defocus is always the best diversity polyno-mial. In this paper we determine from an informationtheoretic limit the best diversity polynomial from thefirst 15 Zernike functions excluding piston and tilt. Inan earlier study, the authors of Ref. 11 computed thevariance on the estimated parameters that was due

to noise propagation. However, that study did notaddress the fundamental limit on the estimated pa-rameters. Furthermore, only defocus was consideredas diversity. In this paper we study the sensitivity ofthe estimated parameters for each of the diversitypolynomials for point and extended sources. We showthat the bound for a point source is lower (by morethan a factor of 30 for Poisson noise for the casesstudied) than that of an extended source, as theaberration sensitivity of the detected image is re-duced because of the convolution of the pristine objectand the point-spread function (PSF) operation. ACramer–Rao lower-bound12–14 study is performed forboth Gaussian and Poisson noise. Using simulatedimages, we compare the performance of the phase-diversity algorithm with the Cramer–Rao lowerbound (CRLB) on the aberrations for both known andunknown objects. We show that the phase-diversityalgorithm is efficient (achieves the CRLB) for most ofthe diversity polynomials considered. Even with anunknown object, the phase-diversity algorithm, incertain cases, performs close enough (within a factorof 2) to the CRLB bound to be considered efficient.

This paper is organized as follows: In Section 2 wedescribe the phase-diversity algorithm for known andunknown objects. The CRLB expressions are derivedin Section 3 for Gaussian and Poisson noise cases forknown objects. Results for various aberration setsand noise statistics are presented in Section 4 forpoint and extended scenes.

2. Algorithm

Diversity image id�x, y� at the image plane of an iso-planatic imaging system is given by15

The authors are with the Boeing Company, 6633 Canoga Ave-nue, MS WB63, P.O. Box 7922, Canoga Park, California 91309.J. J. Dolne’s e-mail address is [email protected].

Received 15 November 2004; revised manuscript received 28March 2005; accepted 27 April 2005.

0003-6935/05/296220-08$15.00/0© 2005 Optical Society of America

6220 APPLIED OPTICS � Vol. 44, No. 29 � 10 October 2005

id(x, y) � fdo � hd(x, y) � nd(x, y), (1)

where o�x, y� is the pristine object photocount map,hd�x, y� is the intensity PSF, R is the convolutionoperator, nd�x, y� is the independent and identicallydistributed noise process, and fd is the fraction of theaverage of total photocounts that goes to diversitysensor d with �d fd � 1, assuming no losses. Weconsider two diversity images in this paper. The firstdiversity image, i1�x, y�, and its PSF, h1�x, y�, are de-noted i�x, y� and h�x, y�, respectively. To conform tothe literature, we refer to the first diversity imageand PSF as the in-focus image and PSF, respectively.The other diversity image, i2�x, y�, and PSF, h2�x, y�,are denoted id�x, y� and hd�x, y� and are referred to asthe diversity image and PSF, respectively.

PSF h�x, y� and optical transfer function H�u, v�are related by a Fourier-transform relation as

h(x, y) � ��1[H(u, v)], (2)

where � represents the Fourier-transform operator.A similar relation holds for the diversity PSF and theoptical transfer function. Optical transfer functionterms H�u, v� and Hd�u, v� are expressed as autocor-relations of the generalized pupil function as fol-lows15:

H(u, v) � P(u, v)exp[i2��(u, v)]� P(u, v)exp[i2��(u, v)],

Hd(u, v) � P(u, v)exp{i2�[�(u, v) � �(u, v)]}� P(u, v)exp{i2�[�(u, v) � �(u, v)]},

(3)

where � is the correlation operator, P�u, v� is a bi-nary function with values of 1 inside the pupil and 0outside, and to reduce the optimization parameterspace we express optical path difference map��u, v� � �n cnZn�u, v� as a weighted sum of basispolynomials Zn�u, v�. The basis polynomials oftenused (and used in this paper) are Zernike polynomi-als, owing to their orthogonality over the interior of aunit circle and to the rotational symmetry of the op-tical system. More details of the properties of Zernikepolynomials may be found in Ref. 16. The intensityPSFs are also expressed as a function of amplitudePSFs a�x, y� and ad�x, y� by

h(x, y) � |a(x, y)|2, hd(x, y) � |ad(x, y)|2. (4)

The diversity amplitude PSF is the inverse Fouriertransform of the generalized pupil function and isexpressed as

ad(x, y) � ��1(P(u, v)exp{i2�[�(u, v) � �d(u, v)]}).(5)

To obtain the in-focus amplitude PSF we set �d�u, v�in Eq. (5) to zero.

The expressions for the Fourier transform of thein-focus and diversity images are I�u, v� � f1O�u, v� H�u, v� � N�u, v� and Id�u, v� � fdO�u, v�Hd�u, v�� Nd�u, v�, respectively, where we have

I(u, v) � �[i(x, y)], Id(u, v) � �[id(x, y)],

O(u, v) � �[o(x, y)],

H(u, v) � �[h(x, y)], Hd(u, v) � �[hd(x, y)],

N(u, v) � �[n(x, y)], Nd(u, v) � �[nd(x, y)].(6)

With uncorrelated Gaussian statistics assumed(Gaussian noise is a reasonable assumption whendetector thermal noise dominates or when the num-ber of detected signal photons per pixel is more than�20), the likelihood of obtaining the object and thePSF given D diversity images, each of which is of sizeX Y, is given by

l[o, h|i1(1, 1), . . . , iD(X, Y)] � �d�1, x�1, y�1

D, X, Y 1

�2�2

exp��[id(x, y) � fdo � hd(x, y)]2w(x, y)

22 , (7)

where 2 is the pixel variance and w�x, y� is a windowfunction that accounts for dead pixels or pixels at theedges of the images where ringing can be significant.

Log likelihood L, when a constant term is ne-glected, is given by

L � �1

22 �d, x, y

[id(x, y) � fdo � hd(x, y)]2w(x, y). (8)

The objective function that is equal to the negative ofthis log-likelihood function and is often used to findthe optimum parameters is expressed as7–10

E �1

22 �d, x, y

[id(x, y) � fdo � hd(x, y)]2w(x, y). (9)

As mentioned above, we consider two diversity im-ages in this paper: one in-focus and one diversityimage. We estimate the aberration coefficients forknown and unknown objects.

A. Known Object

For a known object, one finds the aberration param-eter estimates by minimizing the objective function[Eq. (9)], using, for instance, gradient search tech-niques.

10 October 2005 � Vol. 44, No. 29 � APPLIED OPTICS 6221

B. Unknown Object

Applying Parseval’s theorem to the objective function[Eq. (9)] yields the traditional objective function to beminimized. The object that minimizes the objectivefunction for w�x, y� � 1 is given by7–10,17

O(u, v) �f1H*(u, v)I(u, v) � fdHd*(u, v)Id(u, v)

f12|H(u, v)|2 � fd

2|Hd(u, v)|2, (10)

where * indicates complex conjugation and ˆ refers toan estimated quantity.

The object-independent objective function to be op-timized is given by7–10

E � �u, v

|fdI(u, v)Hd(u, v) � f1Id(u, v)H(u, v)|2

f12|H(u, v)|2 � fd

2|Hd(u, v)|2. (11)

As this error metric is object independent, it permitsfaster determination of the aberrations. However,this error metric is more suitable to objects that fitwell inside the detector’s field of view. In down-looking systems, the image extends beyond the de-tector’s field of view. This causes ringing at the edgesof the reconstructed object. For such cases it may bemore appropriate to use the objective function in thespatial domain as

E �1

22 �d, x, y

[id(x, y) � fdo � hd(x, y)]2w(x, y). (12)

The preceding objective function [Eq. (12)] is used toestimate the aberrations in this paper. The windowfunction w�x, y� is varied to minimize the differencebetween the estimated and the true aberration rms.This window function is set to 1 when the CRLB iscomputed in this paper, however.

3. Cramer–Rao Bounds

We derive in this section the CRLB on the aberrationparameters. In contrast to the usual definition of aCRLB as a lower bound on the mean-square error ofan estimator,13–14 we define the CRLB here to denotethe corresponding square root that bounds the rmserror of an estimator. Therefore the CRLB of an un-biased estimator of a parameter ci is expressed as��F�1�ii, where F is the Fisher information matrixwhose element ��, k� is given by

F�k � �E� �2L�c� � ck

, (13)

and E is the expectation operator. We mention thatthere is a separate CRLB for biased estimators,14,15

which, under certain conditions, can be lower thanthat for unbiased estimators. Therefore the CRLBcan be used to evaluate statistical and all reasonableestimators. One expects that at a high signal-to-noiseratio (SNR) all these estimators will approach the

CRLB. In this paper we consider the CRLB for unbi-ased estimators for cases for which thermal (Gauss-ian) noise or photon (Poisson) noise dominates.

A. Gaussian Noise Dominant

For Gaussian-noise-dominated processes, the deriv-ative of the log-likelihood function with respect to theaberration parameters is given by

�L�ck

�2

22 �x, y, d

[id � fdo � hd(x, y)]

fdo ��hd(x, y)

�ck�w(x, y). (14)

The derivative of the PSF with respect to the aber-ration coefficients in Eq. (14) when Eq. (4) is used isgiven by

�hd(x, y)�ck

� ad*(x, y)�ad(x, y)

�ck� c.c., (15)

where c.c. refers to the complex conjugate of the pre-ceding term and is equal to

�hd(x, y)�ck

� �2 imag{ad*(x, y)

��1[2�Zk(u, v)Cd(u, v)]}, (16)

with

Cd(u, v) � P(u, v)exp{i2�[�(u, v) � �d(u, v)]}. (17)

The imaginary part of a complex number z is desig-nated imag�z�.

The Hessian of the likelihood function with regardto aberration parameters is

�2L�c� � ck

�1

2 �x, y, d

�[id(x, y) � fdo

� hd(x, y)]fdo ��2hd(x, y)

�c� � ck�

� fd2 �

�k��, ko �

�hd(x, y)�c�k

�w(x, y), (18)

where

��k��, k

o ��hd(x, y)

�c�k�� o �

�hd(x, y)�c�

�o ��hd(x, y)

�ck�.

(19)


Substituting Eq. (18) into Eq. (13) yields

F�k �1

2 �x, y, d

�fd2�

x�, y�o(x�, y�)

�hd(x � x�, y � y�)�c�

� �

a, bo(a, b)

�hd(x � a, y � b)�ck

�w(x, y). (20)

Designating object o�x, y� by Kon�x, y�, where K isthe average of total number of photocounts and�x, y on�x, y� � 1, we express the Fisher informationmatrix elements as

F�k �K2

2 �x, y, d

�fd2�

x�, y�on(x�, y�)

�hd(x � x�, y � y�)�c�

� �

a, bon(a, b)

�hd(x � a, y � b)�ck

�w(x, y). (21)

Denoting the average number of photocounts perpixel K�XY and the combined-channel single-pixelSNR �SNRpix� K�XY, we rewrite Eq. (21) as

F�k � (SNRpix XY)2 �x, y, d

fd2

�x�, y�

on(x�, y�)�hd(x � x�, y � y�)

�c��

�a, b

on(a, b)�hd(x � a, y � b)

�ck�w(x, y),

(22)

which, after substitution of Eq. (16), yields

F�k � 4(SNRpix XY)2 �x, y, d

fd2(on � imag{ad*(x, y)

��1[2�Z�(u, v)Cd(u, v)]})

(on � imag{ad*(x, y)��1[2�Zk(u, v) Cd(u, v)]})w(x, y). (23)

The combined-channel single-pixel SNR SNRpixrefers to the SNR based on the total number of pho-tocounts over all diversity channels. We also notethat the definition of SNRpix does not include theeffect of fd, which makes it easier to have the SNRpixexpression outside the summations in Eqs. (22) and(23). In this way, one can compute the CRLB for aspecific SNRpix and easily scale the results for otherSNRpix, provided that the image size is the same. Thesingle-channel single-pixel SNR SNRpix,d, fdK�XY,differs from combined-channel single-pixel SNRSNRpix by fd. We can view Eq. (23) as the sum ofin-focus Fisher information and diversity Fisher in-formation.

B. Poisson Noise Dominant

When the noise process is Poisson, the Fisher infor-mation elements are12

F�k � K �x, y, d

� fd

on � hd(x, y)

�x�, y�

on(x�, y�)�hd(x � x�, y � y�)

�c��

�a, b


�ck�w(x, y). (24)

For Poisson noise, the combined-channel single-pixelSNR is given by �K�XY�1�2. We can, then, express Eq.(24) as

F�k � SNRpix2 XY �

x, y, d

fd

on � hd(x, y)

�x�, y�

on(x�, y�)�hd(x � x�, y � y�)

�c��

�a, b


�ck�w(x, y),

(25)

which, after substitution of Eq. (16), becomes

F�k � 4(SNRpix)2XY �

x, y, d

fd

on � hd(x, y)

(on � imag{ad*(x, y)��1[2�Z�(u, v)Cd(u, v)]})

(on � imag{ad*(x, y)��1[2�Zk(u, v)Cd(u, v)]}) w(x, y). (26)

We remark that the Fisher information is propor-tional to the total number of photocounts. We alsonote that, as before, this definition of SNRpix does notinclude the effect of fd. The single-channel single-pixel SNR SNRpix,d, �fdK�XY�1�2, differs from thecombined-channel single-pixel SNR SNRpix by �fd forPoisson noise.

4. Results

The various aberration sets used in the CRLB eval-uation are listed in Table 1. Aberration set C repre-sents a sample of aberrations for the optical systemthat is being analyzed here. This set was arrived atthrough internal communication about this opticalsystem. We remark that other optical systems willprobably have different aberration profiles. To fur-ther analyze the CRLB but to maintain it at compa-rable aberration strengths, we set all the nonzerovalues in sets A and B to a constant value, 0.1584�and 0.0792�, respectively. The rms of the wavefront(defined by ��i ci

2 for aberrations ci) for aberrationsets B and C is 0.25�; that for set A is 0.5�.

Figure 1 shows the pristine scene �128 128 pixels� used in the evaluation. The detectedimage was sampled at the Nyquist value. The nor-malized CRLB results presented in Figs. 2–6 are forK � 1, � 1�128, fd � 0.5, X � Y � 128, andSNRpix � 1�128. We can obtain the absolute CRLB bydividing the normalized CRLB by SNRpix �XY for


Gaussian and Poisson noise. We use the normaliza-tion factor SNRpix �XY instead of SNRpix XY forGaussian noise because � has been adjusted to haveequal values of SNRpix for Poisson and Gaussian noise.

We obtain the wavefront rms for any diversity poly-nomial used in Figs. 2–6 by finding the amount ofthat diversity aberration that minimizes the com-bined CRLB [defined as the root sum square (rss) ofthe individual CRLB of each of the aberrations] of the15 aberrations.

Figures 2 and 3 represent a combined CRLB of the15 aberrations for various diversity polynomials. Fig-ure 2 shows the combined CRLB for a point-sourcetarget for Poisson [Fig. 2(a)] and Gaussian [Fig. 2(b)]noise statistics for aberration sets A, B, and C. Thesefigures show that the diversity polynomial that pro-

duces the lowest bound is astigmatism for almost allthe aberration sets. These figures also show that thebound for Poisson noise is more than twice that forGaussian noise. Although aberration sets B and Chave equivalent total wavefront rms of 0.25 �, theresultant combined CRLBs are quite different. Fig-ure 3 shows the combined CRLB for Poisson [Fig.3(a)] and Gaussian [Fig. 3(b)] noise for an extendedsource. We remark that the combined CRLB for theextended source is worse than that for a point source.This is so because the image of the PSF is directlydependent on the aberrations, as opposed to the im-age of the extended source, which is a convolution of

Table 1. Aberration Sets Used in Phase-Diversity Evaluationa

Zernike Term

DiversityPolynomial

Number

Aberration Set(waves rms)

A B C

Defocus 4 0.1584 0.0792 �0.1590Astigmatism 5 0.1584 0.0792 �0.0412Astigmatism 6 0.1584 0.0792 �0.1720Coma 7 0.1584 0.0792 0.0123Coma 8 0.1584 0.0792 0.0193Elliptical coma 9 0.1584 0.0792 0.0293Elliptical coma 10 0.1584 0.0792 �0.0123Spherical 11 0.1584 0.0792 �0.0354Secondary astigmatism 12 0 0 0Secondary astigmatism 13 0.1584 0.0792 0.0469Quadratic 7th order 14 0.1584 0.0792 0.0354Quadratic 7th order 15 0 0 0Coma 5th order 16 0 0 0Coma 5th order 17 0 0 0Triangular 7th order 18 0 0 0

aRef. 6.

Fig. 1. 128 128 pixel pristine scene used in CRLB evaluation.The image is sampled at the Nyquist value.

Fig. 2. Combined CRLB on aberration parameters for a pointsource for aberration sets A, B, and C for (a) Poisson and (b)Gaussian noise. Average number of total photocounts, K � 1; � 1�128; X � Y � 128; fd � 0.5.

Fig. 3. Combined CRLB on aberration parameters for an ex-tended source for aberration sets A, B, and C for (a) Poisson and (b)Gaussian noise. Average number of total photocounts, K � 1; � 1�128; X � Y � 128; fd � 0.5.


the PSF and the extended scene. This convolutionoperation lessens the sensitivity of the image on theaberrations and, consequently, increases the com-bined CRLB. We remark also that the combinedCRLB for Gaussian noise is more than 1.5 times aslarge as that for Poisson noise, which is a reversal ofthe behavior illustrated in Fig. 2 for a point source.We may understand this result by looking at denom-inators 2 and on � hd�x, y� in Eqs. (21) and (24), re-spectively. As the scene size increases from point toextended, 2 remains constant, while the peak andthe minimum values of on � hd�x, y� decrease and in-crease, respectively. The diversity strength optimiza-tion process finds the on � hd�x, y� and

o ��hd(x, y)

�c��o �

�hd(x, y)�ck

�pair that results in the largest relative increase inFisher information (equivalently, a decrease inCRLB) for Poisson noise. This CRLB decrease can behigher than for Gaussian noise, for which

o ��hd(x, y)

�c��o �

�hd(x, y)�ck

�is optimized.

Figures 2 and 3 further suggest that, when thehigher-order Zernike polynomials are used as thephase diversity, the Zernike coefficients generally areestimated with higher errors. It is further noted thatastigmatism diversity polynomials 5 and 6 do notyield equivalent combined CRLB, in major part be-cause the secondary astigmatisms are different. Sec-ondary astigmatism polynomial 12 is 0, whereaspolynomial 13 changes for aberration sets A, B, andC.

Figures 4–6 show the individual CRLB for the first18 (excluding piston and tilt) Zernike parameters forvarious diversity polynomials. These curves showthat some aberrations are more sensitive than others.Therefore, one must be able to estimate certain ab-errations better than others. For aberration sets A(Fig. 4) and B (Fig. 5) the aberration functions thathave among the lowest bounds are 4 (defocus), 7(coma), 8 (coma), 11 (spherical), 12 (secondary astig-matism), 13 (secondary astigmatism), 16 (coma 5thorder), 17 (coma 5th order), and 18 (triangular 7thorder), irrespective of the diversity. The same trendcontinues for aberration set C (Fig. 6), except whenZernike polynomials 5 and 7–11 are used as diversitypolynomials, Zernike aberrations 4, 12, and 18 nolonger have among the lowest bounds.

Figure 7 displays comparisons of error rss fromaberrations estimated with the phase-diversity blinddeconvolution algorithm with the combined CRLB forvarious diversity polynomials, each 0.5� rms, for ab-erration set C, assuming Gaussian noise statisticswith SNRpix � 40. The initial values of the aberra-tions were all set to 0. For a known object the com-

bined CRLB follows closely the error rss for theaberrations. The rms error for the ith estimated ab-errations is calculated by

{E[(ci � ci)2]}1�2, (27)

where the mean-square error of the ith estimatedaberrations, E��ci � ci�2 , is approximated by the sam-ple variance as follows:

Fig. 4. CRLB on aberration parameters for an extended source asa function of Zernike term number for various diversity polynomi-als for (a) Poisson and (b) Gaussian noise for aberration set A.Average number of total photocounts, K � 1; � 1�128; X � Y� 128; fd � 0.5.

Fig. 5. CRLB on aberration parameters for an extended source asa function of Zernike term number for various diversity polynomi-als for (a) Poisson and (b) Gaussian noise for aberration set B.Average number of total photocounts, K � 1; � 1�128; X � Y� 128; fd � 0.5.


i2 � E[(ci � ci)

2] �1N �

k�1

N

(ci � ci,k)2. (28)

We note that ci,k is the estimated aberrated parame-ter i for the kth noise instance (there are N noiseinstances). The total error rss is given by ��i�1

Z i2�1�2,

where Z is the number of Zernike parameters thatare being estimated. For an unknown object for whichthe algorithm estimates both the object and the ab-errations, the phase-diversity algorithm’s perfor-mance comes close (within a factor of �2) to thecombined CRLB limit for certain diversity polynomi-als such as defocus. The difference between the error

rss and the combined CRLB for the other diversitypolynomials may become smaller when the combinedCRLB version is used for an unknown object. Thisissue needs to be studied further. We also observethat, for diversity polynomial 10, the rss of the wave-front aberration is smaller when the object is un-known. This may be a result of the limited number ofnoise instances (four noise instances were used inFig. 7 for diversity Zernike terms numbered 4–9 and11 because not much fluctuation in estimation errorswas observed. The number of instances of noise wasincreased to eight for term 10 to smooth out the fluc-tuations in estimation errors for the known objectcase) used in this simulation or to failure of the Mat-lab gradient search algorithm (fminunc) to convergeto the optimum aberration set. Further analysis willhelp to answer this question.

5. Conclusions

We have analyzed the Cramer–Rao lower bound onthe estimated aberration for various diversity poly-nomials. Although defocus seems to perform betterthe majority of times, we have shown that otherphase-diversity polynomials may be more appropri-ate in certain cases. Therefore one must examine theexpected aberrations of the optical system and deter-mine the appropriate diversity phase function thatminimizes the residual wavefront error. We havestudied the theoretical performance for Poisson andGaussian noise statistics. The bounds when a pointsource was used were found to be much lower thanthose when an extended target was used, a fact con-sistent with the findings presented in Ref. 12. Usingsimulated extended images, we computed the phase-diversity root-sum-square error of the aberration co-efficients and found it to satisfy the Cramer–Raolower bound for most of the diversity polynomialsanalyzed.

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Cramer-Rao bound and phase-diversity blind deconvolution performance versus diversity polynomials