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Blind deconvolution for thin-layeredconfocal imaging

Praveen Pankajakshan,1,* Bo Zhang,2 Laure Blanc-Féraud,1 Zvi Kam,3

Jean-Christophe Olivo-Marin,2 and Josiane Zerubia1

1Ariana Project-team, INRIA/CNRS, 2004 Route des Lucioles,BP 93, 06902 Sophia-Antipolis Cedex, France

2Quantitative Image Analysis Unit, Institut Pasteur, 25-28 rue du Docteur Roux, 75015 Paris, France3Department of Molecular Cell Biology, Weizmann Institute of Science, Rehovot 76100, Israel

*Corresponding author: [email protected]

Received 9 January 2009; revised 14 May 2009; accepted 15 May 2009;posted 3 June 2009 (Doc. ID 105374); published 27 July 2009

We propose an alternate minimization algorithm for estimating the point-spread function (PSF) of a con-focal laser scanning microscope and the specimen fluorescence distribution. A three-dimensional separ-able Gaussianmodel is used to restrict the PSF solution space and a constraint on the specimen is used soas to favor the stabilization and convergence of the algorithm. The results obtained from the simulationshow that the PSF can be estimated to a high degree of accuracy, and those on real data show betterdeconvolution as compared to a full theoretical PSF model. © 2009 Optical Society of America

OCIS codes: 100.1455, 180.1790, 100.3190.

1. Introduction

Most of the fluorescence microscopes that image auniformly illuminated three-dimensional (3D) objectby the optical sectioning technique are affected bysome out-of-focus fluorescence contributions. Second-ary fluorescence from the sections away from the re-gion of interest often interferes with the contrast andresolution of those features that are in focus. Let ustake the case of a single-photon (1-p) fluorescence mi-croscope, such as the wide-field microscope (WFM) orthe confocal laser scanning microscope (CLSM) [1].For the sake of simplicity, if we assume that the de-tectors are the same, then a WFM could be seen as aCLSM but with a fully open pinhole. The WFM cancollect more light even from the deeper sections of aspecimen but the data are sometimes rendered use-less as there is a significant amount of out-of-focusblur. Themaximum intensity in each plane decreasesas z−2, with z being the axial distance from the

source. A completely closed pinhole (diameter <1Airy units (AU); 1AU ¼ 1:22λex=NA), where NA is nu-merical aperture, on the other hand, confines thelight detected only to the in-focus plane but at theexpense of imaging low-contrast, highly noisy (signaldependent noise) images. The intensity from a pointsource in this case decreases as z−4 and the loss of in-focus intensity inhibits imaging of weakly fluores-cent specimens. Even with a usable pinhole diameterof 1AU, 30% of the light collected is from the out-of-focus regions. In addition, the microscope is inher-ently diffraction limited [1,2] and the image of a pointsource (the point-spread function (PSF)) displays alateral diffractive ring pattern (expanding with defo-cus) introduced by the finite-lens aperture.

Let OðΩÞ ¼ fo ¼ ðoxyzÞ : Ω⊂ℕ3→ ℝg denote all

possible observable objects on the discrete spatial do-main Ω ¼ fðx; y; zÞ : 0 ≤ x ≤ Nx − 1; 0 ≤ y ≤ Ny − 1; 0 ≤

z ≤ Nz − 1g and h : Ω↦ℝ the microscope PSF. If weassume that the imaging system is linear and shiftinvariant, then the interaction between h and o isa “3D convolution”: ðh � oÞðxÞ ¼ P

x0∈Ωhðx − x0Þoðx0Þ.From the perspective of computational methods, this

0003-6935/09/224437-12$15.00/0© 2009 Optical Society of America

1 August 2009 / Vol. 48, No. 22 / APPLIED OPTICS 4437

could be inverted with the knowledge of the scanningsystem properties and also by information about theobject being scanned. It is for this reason that theknowledge of the PSF h is of fundamental impor-tance. The nature of the PSF for fluorescence micro-scopes has been studied extensively [3–5]. We willintroduce the reader, in Subsection 2.B, to one suchtheoretical model based on the scalar diffractiontheory and to its parametric approximation inSubsection 3.B.

A. Problem Formulation

Restoration by deconvolution could be achieved byusing either a nonblind or a blind approach. Forthe nonblind case, the most common approach isan experimental procedure [6,7] that obtains thePSF by imaging a small fluorescent bead (so as to ap-proximate a point object) positioned in the coverslide. Although such a PSF should have been an idealchoice for a deconvolution algorithm, it suffers fromlow contrast (can be recorded only at finite defocusranges) and is contaminated by noise. Ways to sup-press the noise would be to either acquire severalbead data sets and then average them [8,9] or recon-struct them using Zernike polynomial moments [10].This approach is, however, handicapped by align-ment problems and also the whole process could takea long time. The alternative would be to use an ana-lytical model of the PSF [11,12] that takes into ac-count the acquisition system’s physical informationas parameters. This information, however, mightnot be available or might change during the courseof the experiment (for example, due to heating of livesamples).We hence arrive at the blind deconvolution ap-

proach of estimating the specimen and the unknownPSF parameters using a single observation of thespecimen volume. The problem of blind deconvolu-tion is thus reduced to answering the following ques-tion: “How does one estimate the original object andthe PSF, given only a single observation?”If we forget the effect of noise and consider the ob-

servation model ðh � oÞ in the Fourier space as F ðiÞ ¼F ðhÞ · F ðoÞ, several solutions for o and h answer thisproblem. For example, if ðh; oÞ is a solution, then thetrivial case is that h is a Dirac function and o ¼ i orvice versa. If h is not irreducible, there exist h1 andh2 such that h ¼ h1 � h2, and the couples ðh1 � h2; oÞand ðh1;h2 � oÞ are also solutions. Another ambiguityis in the scaling factor. If ðh; oÞ is a solution, thenðτh; 1=τoÞ∀τ > 0 are solutions, too. This last ambigu-ity can be waived, for example, by imposing a forcednormalization on h. Thus, broadly speaking, a way ofreducing the space of possible solutions and to regu-larize the problem is to introduce constraints on hand o. If the problem of deconvolution is ill-posed,that of blind deconvolution is underdetermined asthe number of unknowns to be estimated is increasedwithout any increase in the input observation data.Many methods use an iterative approach to esti-

mate the PSF and the object with no prior informa-

tion on the object [13,14]. Markham and Conchello[15] worked on a parametric form for the PSF anddeveloped an estimationmethod utilizing this model.The difficulty in using this model for our applicationis that the number of free parameters to estimate islarge and the algorithm is computationally expen-sive. Hom et al. [16] proposed a myopic deconvolutionalgorithm that alternates between iteration to decon-volve the object and estimate the PSF. In order tomyopically reconstruct the PSF, they introduce a con-straint on the optical transfer function (OTF) (theOTF and the PSF are Fourier Transform pairs).

This paper is organized in the following manner.We first discuss the nature of the noise, and thenits mathematical modeling and handling in Subsec-tion 2.A. The PSF modeling is introduced in Subsec-tion 2.B. Section 3 is dedicated to the proposed jointrestoration and estimation of the imaged object andthe microscope PSF using a Bayesian framework. Di-rect restoration from the observation data is very dif-ficult and, hence, it is necessary to define anunderlying model for both the object and the PSF.An alternate minimization (AM) algorithm is thenproposed to solve this particular problem. This AMalgorithm is then tested on images of degraded phan-tom objects and real data; the results obtained arepresented in Section 4. We then conclude in Section 5with a discussion and proposed future work. Thescope of this paper is restricted to restoring imagesfrom a CLSM given the spatial invariance natureof the diffraction-limited PSF.

2. Sources of Distortion and Their Modeling

A. Poissonian Assumption

In digital microscopy, the source of noise is either thesignal itself (so-called “photon shot noise”), or the di-gital imaging system. By tracking the photon-to-electron conversion at the detector, we can observethat the signal and the dependent noise follows anunderlying distribution which is Poissonian [17].Conversely, the imaging noise isolated in the absenceof any fluorescence source follows a Gaussian distri-bution [18,19]. The interested reader may refer to[1,20] for more details on this subject.

In this paper, we have assumed that there is noreadout or dark noise as the photomultiplier tubeis operating in the photon-counting mode. Whenthe imaging system has been a priori calibrated,there is almost negligible offset in the detector andthe illumination is uniform. Thus, if fiðxÞ : x ∈ Ωg(assumed to be bounded and positive) denotes the ob-served intensity of the volume, for the Poissonian as-sumption, the observation model can be expressed as

γiðxÞ ¼ Pðγð½h � o�ðxÞ þ bðxÞÞÞ; x ∈ Ω; ð1Þwhere Pð·Þ denotes a voxelwise noise function mod-eled as a Poissonian process. b : Ω↦ℝ is a uniformlydistributed intensity that models the low-frequencybackground signal caused by scattered photons and

4438 APPLIED OPTICS / Vol. 48, No. 22 / 1 August 2009

autofluorescence from the sample. 1=γ is known asthe photon conversion factor, and γiðxÞ is the ob-served photon at the detector.

B. Theoretical Diffraction-Limited Point-Spread FunctionModel

Among the enormous literature available on PSFmodeling, we highlight the work of Stokseth [11],who obtained the OTF for an aberration-free opticalsystem especially for large defocus. This model wasused to study the PSFs under different microscopesettings and also in validating the algorithm.If we consider a converging spherical wave in the

object space from the objective lens, the near-focusamplitude distribution hA can be written in termsof the amplitude OTF, OTFA, as hAðxÞ ¼

Rk OTFAðkÞ

expðjk · xÞdk, where j2 ¼ −1, and x and k are the 3Dcoordinates in the image and the Fourier space, re-spectively. By making the axial Fourier space coordi-nate kz as a function of lateral coordinates, kz ¼ðk2 − ðk2x þ k2yÞÞ1=2, the 3D Fourier transform is re-duced to hAðx; y; zÞ :¼

Rkx

RkyPðkx; ky; zÞ expðjðkxx

þkyyÞÞdkydkx, where k ¼ 2πμ=λ is the wavenumberof an illumination wave with a wavelength λ in va-cuum and in a medium of refractive index μ, andPð·; ·Þ describes the overall complex field distributionin the pupil of an nonaberrated objective lens [2,11].For an aberration-free microscope, the pupil func-

tion can be written as

Pðkx;ky;zÞ ¼�AðϕÞexpðjkψÞ; if

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2xþk2

y

q< ksinϕmax

0; otherwise;

ð2Þwhere ψ is the optical distance between the wave-front emerging from the exit pupil and the referencesphere measured along the extreme ray, ϕ ¼sin−1ðk2x þ k2yÞ1=2=k, and ϕmax is the maximum semi-aperture angle of the objective. The intensity pro-jected from an isotropically illuminating pointsource, such as a fluorophore, on a (flat) pupil planeis bound to be energy conserving. Therefore, the am-plitude AðϕÞ in the pupil plane for detection shouldvary as ðcosϕÞ−1=2 and the energy as ðcosϕÞ−1 [21].Conversely, for the illumination case, AðϕÞ variesas ðcosϕÞ1=2. Also, for small defocus, ψ in Eq. (2) couldbe approximated as ψ ¼ zð1� cosϕÞ [11]. To derivethe intensity distribution of a point source in the im-age space of a CLSM, we make use of the Helmholtzreciprocity theorem. Since, in induced fluorescence,the excitation (λex) and the emission wavelengths(λem) are different, the confocal PSF can be writtenas [22]

hðxÞ ¼ CjhAðx; λexÞj2

·ZZ

x21þy21≤D24

jhAðx� x1; y� y1; z; λemÞj2dx1dy1; ð3Þ

where C is a scaling factor andD is the backprojecteddiameter of the circular pinhole. This theoretical

model of the PSF does not take into account aberra-tions and assumes that diffraction effect predomi-nates the aberrations. However, this scalar modelcould be extended for other aberrations by modifyingthe pupil function expression in Eq. (2) to alsoinclude the additional phase term due to aberrations[20].

3. Bayesian Framework for the Alternate MinimizationBlind Deconvolution Algorithm

In this section we will use the Bayesian framework todescribe the method for the blind deconvolution.

A. Deconvolution

Since the advent of the nearest-neighbor deconvolu-tion algorithm [3], there have been numerous techni-ques proposed [23–26] for image restoration appliedto microscopy. These assume, however, that the noiseis Gaussian and are valid only for images with highSNR. Statistical methods [27,28], on the other hand,are extremely effective when the noise in theacquired 3D image is fairly strong. We propose hereone such nonlinear iterative algorithm that, al-though slightly computationally expensive (in com-parison to linear methods), can better restore the losthigher frequencies.

If we accept the Poissonian model approximationof Eq. (1), then the image i can be interpreted asthe realization of independent Poison processes ateach voxel. Hence the likelihood can be written as

Prðijo;hÞ ¼Yx∈Ω

½h � o�ðxÞiðxÞe−½h�o�ðxÞiðxÞ! ; ð4Þ

where the mean of the Poisson process is given by½h � o�ðxÞ. In all the derivations used henceforth,the background term has been excluded but the algo-rithm can be modified by changing the above mean to½h � oþ b�ðxÞ. The background fluorescence can be de-termined from the smoothed histogram of a single“specimen-independent” slice, and it is subsequentlyadded to the mean at every iteration of the maximumlikelihood (ML) algorithm Eq. (4) for o [20]. As itera-tive ML methods do not ensure any smoothness con-straints, if unchecked, they evolve to a solution thatdisplays many artifacts from noise amplification (forexamples see [29]). There are many remedies, suchas terminating the iteration (manually or by usinga statistical criterion) before the deterioration beginsor prefiltering the observation data. Onemight arguethat, by applying a low-pass filter as a preprocessingstep before deconvolution (as in [30]), the results areimproved in comparison to the deconvolved imageswith no prefiltering. The deconvolution algorithm ap-plied after denoising is less influenced by the priorterm of the object [31]. However, such prefiltering op-erations might influence the blind deconvolution al-gorithm as it is not clear how the resulting filtereddata is eventually mapped to the original object.The number of iterations for eventual convergenceof the deconvolution algorithm also increases and

1 August 2009 / Vol. 48, No. 22 / APPLIED OPTICS 4439

the final result need not be optimum. Such interven-tions are thus a post hoc method of regularizing theill-posed problem as it is a way of bringing someknowledge about the solution o. The maximum a pos-teriori (MAP) algorithm proposed in this paper usesthe prior model on the specimen and the PSF butwithin the Bayesian framework. We are hence ableto simultaneously denoise and deconvolve the obser-vation data without making any modifications what-soever.By using the Bayes theorem and assuming that o

and h are independent, the posterior joint probabil-ity is

Prðo;hjiÞ ¼ Prðijo;hÞPrðoÞPrðhÞPrðiÞ ; ð5Þ

where PrðoÞ is the global prior probability on the ob-ject and PrðhÞ is the global prior on the PSF. The nat-ure of the prior terms and their expressions arediscussed in Subsections 3.A.1 and 3.B. The esti-mates for o and h can be obtained by simultaneouslymaximizing the joint probability as

ðo; hÞ ¼ argmaxðo;hÞ

fPrðo;hjiÞg

¼ argminðo;hÞ

f− log½Prðo;hjiÞ�g: ð6Þ

As PrðiÞ does not depend on o or h, it shall hereafterbe excluded from all the estimation procedures thatinvolve either o or h. The minimization of the cologa-rithm of Prðo;hjiÞ in Eq. (6) can be rewritten as theminimization of the following energy functional:

J ðo;hjiÞ≡ J obsðijo;hÞ þ ðλoJ reg;oðoÞ þ λhJ reg;hðhÞÞ:ð7Þ

J obs : Ω↦ℝ is a measure of fidelity to the data and itcorresponds to the term Prðijo;hÞ, which is givenfrom the noise distribution. It has the role of pullingthe solution toward the observed data, while J reg;o :Ω↦ℝ and J reg;h : Ω↦ℝ are the prior terms on theobject and the PSF, which ensure smoothness ofthe solutions. λo and λh are positive parameters thatmeasure the trade-off between goodness of fit and theregularity of the solutions. For the Bayesian inter-pretation of regularization problems, we refer thereader to the paper by Demoment [32].Practically, simultaneous estimation of o and h

from Eq. (6) is a difficult task. A way to overcome thisdifficulty is to alternatively maximize the posteriorfirst with respect to o while assuming that the PSFh is known and fixed, and then update the PSF usingthe previous object estimate. This joint optimizationalgorithm is summarized as

oðnþ1Þ ¼ argmaxo

fPrðijo; hðnÞÞPrðoÞg;

hðnþ1Þ ¼ argmaxh

fPrðijoðnþ1Þ;hÞPrðhÞg: ð8Þ

The implementation strategy of this blind deconvolu-tion schema has been shown in Algorithm 1 inSubsection 4.A and the discussion follows in the sub-sequent sections.

1. A Priori Object Models

The ensemble model of an object class refers to anyprobability distribution PrðoÞ on the object spaceO ofthe following form:

PrðoÞ ¼ Z−1λo e

�λoEðoÞ; ð9Þ

where EðoÞ is a generalized energy and 1=λo (withλo > 0) is the Gibbs parameter for the prior term.We associate with each site ðx; y; zÞ ∈ Ω of the objecta unique neighborhood ηxyz⊆Ω∖ðx; y; zÞ, and we de-note the collection of all neighbors η ¼ fηxyzjðx; y; zÞ ∈Ωg as the neighborhood system. If we assume thatthe random field ðO ¼ oÞ on a domain Ω is Markovianwith respect to the neighborhood system η, thenPrðoxyzjoΩ∖x;y;zÞ ¼ PrðoxyzjoηxyzÞ. o is a Markov randomfield (MRF) on ðΩ; ηÞ, if o denotes a Gibbs ensembleon Ω and the energy is a superposition of potentialsassociated to the cliques (a set of connected pixels).Hence, EðoÞ ¼ P

C∈C VCðoÞ.We use in this paper the following first-order,

homogeneous, isotropic MRF, over a six-memberneighborhood ηx ∈ η (see Fig. 1) of the site x ∈ Ω:

Pr½O ¼ oðxÞ� ¼ Z−1λo e

−λoPx

j∇oðxÞj; ð10Þ

where j∇oðxÞj is the potential function and λo is theregularization parameter described above. Theestimation of this parameter is dealt with inSubsection 4.A.

From a mathematical perspective, j∇oðxÞj is notdifferentiable in zero. An approach to circumventthis problem is to regularize it, and instead to consid-er the (isotropic) discrete definition as

j∇oðx; y; zÞjϵ ¼ ððoðxþ 1; y; zÞ − oðx; y; zÞÞ2þ ðoðx; yþ 1; zÞ − oðx; y; zÞÞ2

þ ðoðx; y; zþ 1Þ − oðx; y; zÞÞ2 þ ϵ2Þ12;ð11Þ

where ϵ is an arbitrarily small value (<10−3). For thepartition function Zλo ¼

Po∈OðΩÞ expð−λo

Px j∇oðxÞjϵÞ

to be finite, we restrict the possible values of oðxÞ sothat the numerical gradient of∇oðxÞ is also bounded.When this model is used as a prior for the object,we have the following smoothed regularizationfunctional:

4440 APPLIED OPTICS / Vol. 48, No. 22 / 1 August 2009

J reg;oðoðxÞÞ ¼ λoXx

j∇oðxÞjϵ: ð12Þ

For numerical calculations, we will use the abovesmoothed approximation, and j∇oðxÞjϵ will hence-forth be simply written as j∇oðxÞj. From Eqs. (10)and (11), it can be inferred that sites with very highintensity gradients are more penalized and thosewith low gradients are less penalized. This is becauseit is more likely that high gradients correspond to thecase where there is less similarity between the site ofinterest and their closest neighbors.Tikhonov–Miller [33,34] introduced a regulariza-

tion based on the ℓ2 norm of the gradient of the im-age. However, we have used the total variation (TV)regularization (see Rudin et al. [35]) as it is able tobetter preserve discontinuities [28]. A direct 3D ex-tension of this TV algorithm for CLSM is describedin [29,36].

2. Estimation of the Object

For the time being let us assume that either the PSFor its parameters θ ∈ Θ are known (either by initial-ization or from previous estimates) and hence h is de-terminate. From Eqs. (4), (5), and (10) we get

Prðo; hjiÞ ¼ Z−1λo e

�λoPx

j∇oðxÞj·Yx∈Ω

½h �o�ðxÞiðxÞe−½h�o�ðxÞiðxÞ! :

ð13Þ

As in Eq. (6), by applying the − log operator to the aposteriori above, the cost function J ðo; hjiÞ to bemini-mized with respect to o becomes

J ðo; hjiÞ

≡

�Xx∈Ω

½h � o�ðxÞ −Xx∈Ω

iðxÞ log½h � o�ðxÞ

þXx∈Ω

logðiðxÞ!Þ�þ λo

Xx∈Ω

j∇oðxÞj þ log½Zλo �: ð14Þ

Richardson–Lucy algorithm with total varia-tion regularization

The Euler–Lagrange equation for minimizingJ ðo; hjiÞ in Eq. (14) with respect to o is

1 − hð�xÞ ��

iðxÞðh �oÞðxÞ

�� λodiv

�∇oðxÞj∇oðxÞj

�¼ 0; ð15Þ

where hð−xÞ is the Hermitian adjoint operation onhðxÞ and div stands for the divergence (see [28] fordetails). Inspired by the Richardson–Lucy (RL) algo-rithm [37,38], Eq. (15) can be solved for the object oby the following fixed-point iterative algorithm:

oðnþ1ÞðxÞ ¼�

iðxÞðoðnÞ � hÞðxÞ

� hð−xÞ�

·oðnÞðxÞ

1 − λodiv�

∇oðnÞðxÞj∇oðnÞðxÞj

� ; ð16Þ

where ð·Þ denotes the Hadamardmultiplication (com-ponent wise) and n is the iteration number for thedeconvolution algorithm. Equation (16) is similar tothe expectation-maximization (EM) algorithm [39]with an underlying statistical model of the process,and can be used for obtaining the MAP estimate ofthe object. The term divð∇oðnÞðxÞ=j∇oðnÞðxÞjÞ can benumerically implemented with the use of central dif-ferences and the minmod scheme [29].

Positivity and flux constraint for the objectestimate

The deconvolution algorithm that was describedabove suffers from an inherent weakness. For largevalues of λo, even when the starting guess oðnÞ (withn ¼ 0) is positive, the successive estimates need notnecessarily have positive intensities. We know thatthe true intensity of the object oðxÞ is always non-negative. Most algorithms truncate these negativeintensities to zero or a small positive value. This,however, is a crude manner of handling the esti-mated intensities as it can lead to loss of some essen-tial information and sometimes also introduces biasinto the calculations.

So how else can the problems associated with ne-gative intensity estimates be handled? Fortunately,the problem is entirely due to poor statistical meth-odology. The modification that we suggest is toinclude this knowledge of nonnegative true intensi-ties into the prior term of Eq. (10). The distributionthat would express precisely this condition is

Pr½oðxÞ� :¼�Z−1λo e

�λoPx

j∇oðxÞj; if oðxÞ ≥ 0

0; otherwise:ð17Þ

For the sake of numerical differentiability, we ap-proximate Eq. (17) using a sigmoid function as

Fig. 1. (Color online) MRF over a six-member neighborhood ηx.By permission of Ariana-INRIA/CNRS.

1 August 2009 / Vol. 48, No. 22 / APPLIED OPTICS 4441

Pr½oðxÞ� :¼ Z�1new;λoe

−λoPx

j∇oðxÞj

·�

1ð1þ expðβoðϵ − oðxÞÞÞÞ

�; ð18Þ

where ϵ is a small value close to zero and βo is a valuethat specifies the steepness of the sigmoid curve. Ty-pically, the values of βo and ϵ are chosen to be verylarge and small, respectively, as precision allows.Their values do not individually affect the algorithmand hence need not be known accurately. The costfunction of Eq. (14), the Euler–Lagrange Eq. (15),and the multiplicative algorithm in Eq. (16) are thusmodified as follows:

J ðo; hjiÞ≡�X

x∈Ω½h � o�ðxÞ −

Xx∈Ω

iðxÞ log½h � o�ðxÞ�

þ λoXx∈Ω

j∇oðxÞj þ log½Znew;λo �

− log�

1ð1þ expðβoðϵ − oðxÞÞÞÞ

�; ð19Þ

1 − hð�xÞ ��

iðxÞðh � oÞðxÞ

�� λodiv

�∇oðxÞj∇oðxÞj

�

� βoexpðβoðϵ − oðxÞÞÞ

1þ expðβoðϵ − oðxÞÞÞ¼ 0; ð20Þ

oðnþ1ÞðxÞ ¼�

iðxÞðoðnÞ � hÞðxÞ

� hð−xÞ�

·oðnÞðxÞ

1 − λodiv�

∇oðnÞðxÞj∇oðnÞðxÞj

�� βo expðβoðϵ−oðxÞÞÞ

1þexpðβoðϵ−oðxÞÞÞ

:

ð21Þ

Intuitively, the cost function in Eq. (19) ensures thatthe energy for negative intensity pixels (oðxÞ < ϵ) isvery high and, hence, is not reachable (or is not a pos-sible solution) during the iteration procedure.If the PSF is normalized such that jjhðxÞjj1 ¼ 1, in

the absence of a background signal, it is simple toshow that, for each iteration of the RL algorithm(see Eq. (16) with λo ¼ 0), the following property istrue: s ¼ P

x∈Ω iðxÞ ¼ Px∈Ω oðxÞ: This property is

known as the flux or global photometry conservationand it guarantees that the total number of counts ofthe reconstructed object is the same as the total num-ber of observation counts. However, this property islost with regularization and can be incorporated bymodifying the cost function in Eq. (14) to an additiveform or by enforcing it in the following manner afterevery iteration (except if the background signal is

nonzero): oðnþ1Þnew ðxÞ ¼ ðsð0Þ × oðnþ1Þ

old ðxÞÞ=sðnþ1Þ, wheresðnþ1Þ ¼P

x∈Ω oðnþ1Þold ðxÞ, sð0Þ ¼P

x∈Ω oð0ÞðxÞ ¼P

x∈Ω iðxÞ:B. Parametrization of the Point-Spread Function

When λo ¼ 0 in Eq. (14), theoretically speaking, theestimation method on the object and PSF shouldbe the same because h and o play a symmetric role.When no constraint is imposed on the PSF, the solu-tion is not always unique. Some reason that a regu-larization model on the PSF (J reg;hðhÞ) could also beargued along the same lines as the constraints intro-duced earlier for o [14,40]. First, a TV [35] kind ofregularization cannot model the continuity and reg-ularity in the PSF. A ℓ1 kind of norm is suitable onlyfor PSFs that have edges, such as motion blur [41].Second, in such cases the recovered PSF will be verymuch dependent on the object/specimen [42]. Separa-tion of the PSF and the object in this case becomesdifficult as they have the same or similar solutionspaces. Finally, the regularization parameter λh forsuch a model is highly dependent on the amount ofdefocus, and varies drastically from one image sam-ple to another. It is for these reasons that we are pro-posing to intrinsically regularize the PSF through aparametric model.

Because of the invariance property of ML estimate,we can say hMLðxÞ ¼ hðx; θMLÞ is the ML estimate ofthe PSF. θ ∈ Θ⊂ℝþ is the set of parameters that de-fines the PSF. In a more general manner, any PSFcan be written as the decomposition on a set ofbasis functions Φ as hðxÞ ≈ PNb

l¼1 wlΦlðxÞ ¼ hw;ΦðxÞi;∀x ∈ Ω, where wl denotes the corresponding weightsand Nb denotes the number of the basis functions.The imperfections in an image-formation system nor-mally act as passive operations on the data, i.e., theyneither absorb nor generate energy. Thus, when anobject goes out of focus, it is blurred but the volume’stotal intensity remains constant. Consequently, allenergy arising from a specific point in the fluorescentspecimen should be preserved and ‖hðxÞ‖1 ¼P

x∈Ω jhðxÞj ¼ 1. From Eq. (3), it is clear that the in-tensity distribution of a point source will always bepositive and so hðxÞ ≥ 0;∀x ∈ Ω. To satisfy the above-defined conditions, and an additional criterion ofcircular symmetry (i.e., hð�x;�yÞ ¼ hðx; yÞ;∀ðx; yÞ∈ ℝ2), the Gaussian kernel is chosen as the basis(see [43] for the two-dimensional case). This drasti-cally reduces the number of free parameters toestimate and yet retains a reasonable fit to the actualPSF. It was demonstrated by Zhang et al. [5] that, fora CLSM, a 3D separable Gaussian model gives arelative squared error (RSE) of <9% for a pinholediameter D < 3AU and when the PSF peaksare matched (i.e., ‖hðxÞ‖∞ ¼ 1), where we sayRSE :¼ ‖PSF − h‖2

2=‖PSF‖22.

Thus the diffraction-limited PSF (with restrictionson the pinhole diameter D) can be approximated as

hðxÞ ¼ ð2πÞ�32jΣj�1

2 exp��12ðx − μÞTΣ−1ðx − μÞ

�; ð22Þ

4442 APPLIED OPTICS / Vol. 48, No. 22 / 1 August 2009

where μ ¼ ðμx; μy; μzÞT is the mean vector, Σ ¼½σij�1≤i;j≤3 is the covariance matrix, and μð·Þ, σð·;·Þ ∈ Θ.As a first approximation, for thin-layered specimenimaging with no aberrations, the PSF is spatially in-variant and μ ¼ f0g. A mirror symmetry about thecentral xy plane results in a diagonal covariance ma-trix and, hence, its determinant is jΣj ¼ σ4rσ2z , whereσ11ð¼ σ22Þ ¼ σr and σ33 ¼ σz are the lateral and axialspreads, respectively. It can be shown that the para-meters that we are interested in estimating θ ¼fσr; σzg, are dependent on the following settings:wavelength λex, refractive index μ, and the NA [5].

1. Point-Spread Function Parameter Estimationon the Complete Data

Themethod outlined in Subsection 3.A.2 requires theknowledge of the PSF hðxÞ or hðx; θÞ. From Eqs. (4),(6), and (8), and with the invariance property of MLestimation described earlier, minimizing the energyfunction with respect to the PSF (J ðo;hjiÞ) or theparameters (J ðo; θjiÞ) are equivalent. Thus

J ðo; θjiÞ ¼ −Xx∈Ω

ðiðxÞ log½hðθÞ � o�ðxÞÞ

þXx∈Ω

½hðθÞ � o�ðxÞ: ð23Þ

If the true object o is assumed to be known a priori aso, then estimation of the true parameters of the PSFis straightforward as the cost function in Eq. (23) isconvex in the neighborhood of optimal θ ∈ Θ (seeFig. 2). The parameters of the PSF can hence be ob-tained by a gradient-descent (GD) algorithm [44].Analytically minimizing Eq. (23) with respect tothe parameters leads us to the following:

θlðnþ1Þ ¼ θl

ðnÞ− αðnÞ∇θlJ ðo; θlðnÞjiÞ; θ ¼ fσr; σzg; ð24Þ

where αðnÞ and ∇θlJ ðo; θlðnÞjiÞ are the step size andthe search direction at iteration n. The gradient ofthe cost function with respect to the parameterscan be calculated as

∇θlJ ðo; θljiÞ ¼Xx∈Ω

�∂

∂θlhðθÞ � o

�ðxÞ −

Xx∈Ω

iðxÞ½hðθÞ � o�ðxÞ

·�∂

∂θlhðθÞ � o

�ðxÞ; ∀θl > 0 ∈ Θ:ð25Þ

If we assume that the PSF is axially and radiallycentered, i.e., μ ¼ 0, ½∂hðθÞ=∂θl�θl¼σr ¼ ð−2=σr þ r2=σ3r ÞhðθÞ, and ½∂hðθÞ=∂θl�θl¼σz ¼ ð−1=σz þ z2=σ3z ÞhðθÞ. Theseparable nature of the Gaussian distributionreduces the complexity of the algorithm, as the con-volution with the 3D Gaussian PSF can be imple-mented as three successive 1D multiplications inthe Fourier domain. Only a single FFT of the objectestimate o needs to be performed as an analyticalclosed form expression for the Fourier transform ofthe Gaussian and its derivative exists and can hence

be numerically calculated. We stop the computationif the difference measure between two successiveiterations is smaller than ϵ (in practice 10−3 or10−4), and use the last estimate as the best one.

4. Results

In this section, we validate the proposed AM algo-rithm on some synthetic and real data.

A. Algorithm Analysis

The global procedure alternatively minimizesthe cost function in Eq. (14) first with respect to o

Fig. 2. (Color online) Variation of the energy function J ðo; θjiÞwith respect to (a) lateral (σr) and (b) with axial PSF parameter(σz). For this experiment, the true object o is known and the obser-vation is generated using a known 3D Gaussian model. The axialPSF parameter σz is varied by a factor �ϵ to monitor its effect onthe estimated parameter σr and vice versa. σð·;trueÞ is the true para-meter value. By permission of Ariana-INRIA/CNRS.

1 August 2009 / Vol. 48, No. 22 / APPLIED OPTICS 4443

Eq. (16) while keeping the PSF h fixed and then up-date the PSF in Eq. (24) using the previous object es-timate o. Since the iterative algorithm requires aninitial guess for the true object, we use the meanof the observed image (i.e., every site is assumedto have a uniform intensity and is, hence, equallylikely) for the initialization. For the PSF, as thereare no constraints on its spread or support, initiali-zation of the parameters to small values cannot guar-antee its convergence to the desired size (due to theDirac trivial solution). To avoid this problem, wechoose the initial parameters to be utmost 2κ−1 Re-sels and 6κ−1 Resels (1Resel ¼ 0:61λex=NA, κ ¼ 2:35)for the lateral case and the axial case, respectively,and descend down to the optimal value. BothJ obsðijo;hÞ and J reg;oðoÞ in Eq. (19) are convex,though not in the strict sense. Although the conver-gence of the algorithm to the optimal solution is the-oretically difficult to prove, numerical experimentsindicate that the global procedure does convergewhen the initialization is carried out as de-scribed above.A delicate situation is in the choice of the regular-

ization parameter λo; too small values yield overly os-cillatory estimates owing to noise or discontinuities,while too large values yield overly smooth estimates.The selection or estimation of the regularizationparameter is thus a critical issue on which there havebeen several proposed approaches [45]. However, weare looking for a simple technique that could be com-bined with the AM algorithm and also fits well withthe Bayesian framework. The difficulty in perform-ing marginalization with respect to λo is that the par-tition function is not easily computed. An approach tocircumvent this problem is by approximating thepartition function Znew;λo as λ−NxNyNz

o [46]. By assum-ing a uniform hyperprior on λo and maximiz-ing Eq. (19) with respect to λo leads to the optimalλo at iteration ðnþ 1Þ as λðnþ1Þ

o ¼ ðNxNyNzÞ=Px∈Ω j∇oðnÞðxÞj:

B. Numerical Experiments

For the numerical experiments in Fig. 3, we haveused a 3D simulated test object of dimensions

128 × 128 × 64, with XY and Z pixel sizes of 20nmand 50nm, respectively. The observed data was thengenerated by using an analytical model of the micro-scope PSF Eq. (3) (with a pinhole diameter of 1AU),and the noise was modeled as Poisson statistics[see Fig. 3(b); peak signal-to-noise ratio (PSNR),16:77dB]. The results of the AM algorithm are illu-strated in Figs. 3(c) and 3(d). The stopping thresholdϵ between two successive iterations was fixed as 10−4.Figure 4 shows the reduction in the cost functionwith iterations of the GD algorithm and the approachof the estimated lateral spread parameter σr to thestable value given the estimate of the object. Thequality of the restoration can be assessed by compar-ing with the original synthetic object using theI-divergence or generalized Kullback distance [29].For the AM algorithm, when the stopping criterionϵ was reached, the final I-divergence betweeno and o was 1.4334 (as opposed to 5.55 between oand i). Figure 5(a) compares the estimated 3D PSFwith the analytically modeled [11] PSF and the best3D Gaussian fit (in the least-squares sense) for theanalytical model. The PSFs are shown along onedirection of an off-central lateral plane, and a sectionof the plot can be viewed as an inset. The maximumof the residual error between the estimate and thetrue PSF is displayed on a logarithmic contraststretch in Fig. 5(b). Although the Gaussian modeldoes not capture the ringing sidelobes, as is evidentfrom the residue, the RSE was found to be <0:07%.

C. Experiments on Real Data

1. Imaging Setup and Sample Description

The Zeiss LSM 510 confocal microscope is mountedon a motorized inverted stand (Zeiss Axiovert 200M) and is equipped with an ArKr excitation laserof wavelength 488nm. The bandpass (BP) filtertransmits emitted light within the band 505–550nm.

The specimen that was chosen for the first experi-ment is an embryo of the Drosophila melanogaster[see Fig. 6(a)]. It was mounted and tagged withGreen Fluorescent Protein (GFP). This preparationis used for studying the sealing of the epithelial

Algorithm 1: Schema for the Proposed Blind Deconvolution Algorithm

1 beginInput: Observed volume i ∈ ℕ3.Data: Initial parameters θð0Þ (Subsection 4.A), convergence criterion ϵ.Output: Deconvolved volume o ∈ ℕ3, PSF parameters θ ∈ Θ⊂ℝ2þ.

2 Initialization: n←0, oðnÞðxÞ←MeanðiðxÞÞ, hðnÞðxÞ←hðx; θðnÞÞ [Eq. (22)].3 Estimate the background term b from the image histogram (Subsection 3.A).4 while jθðnÞ − θðn−1Þj=θðnÞ ≥ ϵ do5 Hyperparameter λo estimation: λðnÞo ←1=Meanðj∇onðxÞjÞ.6 Using the minmod scheme [29], calculate divð∇onðxÞ=j∇onðxÞjÞ.7 Deconvolution: Calculate oðnþ1Þ from Eq. (21).8 Projection Operation: Scale oðnþ1Þ for preserving the flux (Subsection 3.A.2).9 Parameter estimation: Calculate θðnþ1Þ from Eqs. (24) and (25).10 Assign: hðnþ1ÞðxÞ←hðx; θðnþ1ÞÞ and n←ðnþ 1Þ.11 end12end

4444 APPLIED OPTICS / Vol. 48, No. 22 / 1 August 2009

sheets (dorsal closure) midway during the embryo-genesis. The objective lens is a Plan-Neofluar with40× magnification having a NA of 1.3 and immersedin oil (Immersol 518 F, Zeiss, refractive index

μ ¼ 1:518). The pinhole size was 67 μm. The images(Institute of Signaling, Development Biology andCancer, Nice UMR6543/CNRS/UNS) were acquiredwith a XY pixel size of 50nm and a Z step size of170nm, and the size of the volume imaged is25:59 μm× 25:59 μm × 2:55 μm.

The second set of images [National Institute forAgricultural Research (INRA), Sophia-Antipolis,France] are the root apex of the plant Arabidopsisthaliana immersed in water (see Fig. 7). The dis-sected roots of the Arabidopsis thaliana plant weredirectly put on a microscope slide in approximately100 μl of water and this was then gently covered witha coverslip. This simple set up works very well whenthe image acquisition recording times are not toolong (about 30 min). The microscope specificationsare the same as that used for acquiring the first dataset, but the objective is a C-Apochromat water im-mersion lens with 63× magnification, 1.2 NA. Thelateral pixel dimensions are 113nm and the Z stepis 438nm. The pinhole was fixed at 110 μm. This pre-paration was used to studyNematode infection at thecenter of the root in the vascular tissue.

2. Deconvolution Results

A rendered subvolume of the observed and restoreddata for the Drosophila melanogaster is shown inFig. 6. The deconvolution algorithm was stoppedwhen the difference between subsequent estimateswas lower than ϵ ¼ 0:002. The AM algorithm con-verged after 40 iterations of the joint RL–TV andGD algorithm. The PSF parameters were initializedto 300nm and 600nm for the lateral and the axialcase, respectively, and the GD algorithm estimatedthem to be 257.9 and 477:9nm [47]. These are larger(by about 16% and 14.5% for the lateral and the axialcases, respectively) than their corresponding theo-retically calculated values [5]. These results arealso fully in line with an experimental studyperformed earlier [48] with subresolution beads,

Fig. 3. (Color online) 3D (a) phantom object (with false coloring),(b) observed image blurred by the PSFmodel in Eq. (3) and Poissonnoise (PSNR, 16:77dB; I-divergence, 5.55), (c) restoration afterRL–TV deconvolution with the estimated PSF (I-divergence,1.43), (d) estimated PSF. The intensities of the object, observation,and the restoration are on a linear scale while the PSF is on a loga-rithmic scale. By permission of Ariana-INRIA/CNRS.

Fig. 4. (Color online) Convergence of the cost function and lateralparameter by the GD method (when the original object is known).The Y axis is left-scaled for the cost function J ðθ; ojiÞ and right-scaled for the PSF parameter, respectively. By permission of Ari-ana-INRIA/CNRS.

Fig. 5. (Color online) (a) Full model (dashed curve), estimated (so-lid curve) and the best Gaussian fit (dashed-dotted curve) PSFs aredisplayed for one direction (off-central plane); the inset shows asection of the plot. (b) X-Z projection of the residual (RSE<0:07%) between the estimated and the full PSF model is dis-played on a log scale. By permission of Ariana-INRIA/CNRS.

1 August 2009 / Vol. 48, No. 22 / APPLIED OPTICS 4445

which indicated a large deviation between theoreti-cal aberration-free PSF models and empirically de-termined PSFs.Figure 8(a) shows a rendered subvolume (as indi-

cated in Fig. 7) of the observed root apex and the cor-responding restored result is shown in Fig. 8(b). It isevident from these results that the microtubules [asidentified by their specific binding proteins, microtu-bules binding domain (MBD)] are more easily dis-cerned in the restoration than in the original data.It was verified from the experiments on syntheticdata [47] that the proposed algorithm can not onlyestimate the actual PSF, but can also provide a muchbetter deconvolution result [49] in comparison to the-oretical microscope PSFs (generated using the micro-scope settings). Validation is very important as, in

some situations, artifacts might arise in the restoredimage. These artifacts would be hard to distinguishfrom biological structures unless some knowledgeabout the true image is available. However, the re-sults on real data are difficult to validate unless ahigher resolution image of the same sample is avail-able. Hence, we tested our deconvolution algorithmon images of spherical fluorescent shells (see [36])whose thickness was measured after deconvolutionand found to be closer to the true value specifiedby Molecular Probes.

5. Conclusions and Future Work

In this paper we have proposed and validated an AMalgorithm for the joint estimation of the microscopePSF and the specimen source distribution for aCLSM. We choose the RL algorithm for the deconvo-lution process as it is best suited for the Poisson data,and TV as the regularization model. A separable3D Gaussian model best describes the diffraction-limited confocal PSF, and is chosen as the a priorimodel for the PSF. We are able to achieve blind de-convolution by constraining the solution of the objectand the PSF to different spaces. The PSFapproxima-tion that is given in this paper is currently relevantto imaging thin samples. However, it could alsobe extended to encompass any PSF that can bedecomposed in a similar manner. We have experi-mented on simulated and real data, and the methodgives very good deconvolution results and a PSF es-timation close to the true value [29,47]. However, itshould be noted that all of the out-of-focus light can-not be rejected and some noticeable haze and axialsmearing remains in the images. This could beimproved by adding a Gamma prior on the PSFparameters.

This research was partially funded by the P2RFranco–Israeli Collaborative Research Programand the French National Research Agency (ANR)DETECFINE Project. The authors gratefully ac-knowledge Dr. Caroline Chaux (Université Paris-Est, France) and Prof. Arie Feuer (Technion, Israel)for several interesting discussions. We would alsolike to thank INRIA for supporting the Ph.D. ofthe first author through a state-subsidized INRIAdoctoral research contract (CORDI-S) fellowshipand CNRS for supporting the Ph.D. of the secondauthor. Additionally, our sincere gratitude goes to

Fig. 6. (Color online) (a) Rendered subvolume of the original spe-cimen (by permission of Institute of Signaling, DevelopmentalBiology & Cancer UMR6543/CNRS/UNS), and (b) restored image.The intensity is scaled between [0, 130] for display. By permissionof Ariana-INRIA/CNRS.

Fig. 7. (Color online) Observed root apex of an Arabidopsis thali-ana with a volume 146:448 μm× 146:448 μm× 30:222 μm (by per-mission of INRA). The subvolume chosen for restoration isemphasized.

Fig. 8. (Color online) Rendered subvolume of the (a) observed im-age slices in Fig. 7 (by permission of INRA) and (b) volume render-ing of the restored image slices (by permission of Ariana-INRIA/CNRS). ϵ ¼ 0:0001.

4446 APPLIED OPTICS / Vol. 48, No. 22 / 1 August 2009

Dr. Stéphane Noselli, Dr. Fanny Serman from the In-stitute of Signaling, Development Biology and Can-cer UMR 6543/CNRS/UNS, and Dr. Gilbert Engler(French National Institute for Agricultural Research(INRA) Sophia-Antipolis, France) for painstakinglypreparing the images presented in Subsection 4.C,and for the useful comments on their validation.

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4448 APPLIED OPTICS / Vol. 48, No. 22 / 1 August 2009

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