image enhancement by deconvolution - semantic scholar · 1.25 na, widefield 0.90 na, widefield 1.25...

INTRODUCTION

In this chapter we will try to provide a more intuitive (and lessmathematical) insight into image formation and practical imagerestoration by deconvolution methods. The mathematics of imageformation and deconvolution microscopy have been described ingreater detail elsewhere (see Chapters 11, 22, 23, and 24), so wewill limit our discussion to fundamental issues and gloss over mostof the mathematics of image restoration. We will also focus on prac-tical ways of assessing microscope performance and getting thebest possible data before applying more sophisticated image pro-cessing methods than are usually seen in the literature. Before welose the interest of the confocalist, we would point out there is nosuch thing as a real widefield microscope because real microscopeshave limited field of view and other apertures within the opticaltrain. These apertures introduce a limited degree of confocalityeven into widefield microscopes that, as we will see, improves thebehavior of the microscope in a way that makes quantitative imagerestoration both possible and very worthwhile. The methods (andsoftware) that improve images from conventional widefield (WF)microscopes will also significantly improve confocal images andour practical experience shows that the output of both confocal andmulti-photon microscope systems benefit from application ofimage restoration methods. For those who wish to know the con-clusion of this chapter without further reading, it is simply: decon-volve all data (if you can). To stress this point, we suggest that evena properly adjusted confocal microscope will not give the best pos-sible confocal image and the application of appropriate deconvo-lution methods will increase contrast, reduce noise, and evenimprove resolution. That such worthwhile effects are made possi-ble by computation resides in the fact that during deconvolution theinformation content of the image can be increased by the additionof information about the imaging process itself as well as that aboutthe statistics of photon capture. Thus, in our opinion, the applica-tion of deconvolution methods can always increase image quality,regardless of the source of the image.

BACKGROUND

The resolution of a measurement instrument is always limited sothat the recorded image of the sample is completely dependent onthe properties of the measurement device. Microscopy is simply ameasurement process used to quantify information about smallobjects, and the limits to resolution in microscopy are manifest asblurring in the acquired images. Blurring arises from the intensity

of signal at any point in the image being some weighted sum ofintensities from all points in the object space. With detailed under-standing of the weighting function, we can develop ways of cor-recting the image for the limiting behavior of the microscope. Inaddition, almost all microscopic samples are inherently three-dimensional while the detectors [such as the eye, charge-coupleddevice (CCD) camera, or photographic film] are two-dimensional.Because it is well known that two-dimensional representations ofthree-dimensional objects lead to artifacts and ambiguity, oneshould expect fundamental imaging problems in microscopy.

A number of phenomena contribute to the blurring process inlight microscopy. First, the wave nature of light dictates that as itpasses through apertures (e.g., the optical components making upthe microscope), diffraction occurs, which results in spreading andmerging of light rays in the image (Abbe, 1873; see Chapter 22).Second, samples can scatter light thereby removing the corre-spondence between the source of a light ray and its final locationin the image plane. This problem makes high-quality imaging inturbid, thick samples (such as brain slices) highly problematic. Weshould also note that this scattering also prevents certain light raysfrom passing through the optical train and thereby prevents someinformation from the object reaching the image. Similarly, a denseobject can obscure an (in-focus) object behind it and prevent infor-mation from it reaching the image. Finally, refractive index mis-matches in the light path (e.g., between the immersion medium andsample) and imperfections in the optical components (and theiralignment) can add to aberrations that will also cause light rays tobe misdirected.

The conventional WF microscope is designed to correctlyimage a sample at the focal plane of the objective; however anylight source within the conical volume of light collected by theobjective lens can contribute light to the image plane. While thisis not blurring, it will introduce a loss of contrast for in-planeobjects and contribute to the perception of a blurred image. Objectsoutside the plane of sharp focus are also seen in the image planeas blurred images that also reduce contrast for in-focus objects. (Itis the reduction of this effect by confocal microscopes that makesthem so useful.) Finally, the limited aperture of the microscopesystem (i.e., its limited light and information gathering ability) andthe limited wavelength of light itself lead to blurring of objects.With so many limiting factors, it is not surprising that real micro-scope imaging always represents a trade-off between various prob-lems. For example, increasing the numerical aperture of the systemmay decrease blurring and increase signal from in-focus objectsbut can also lead to a loss of image contrast due to the acceptanceof more scattered light from thick samples. In addition, high

25

Image Enhancement by Deconvolution

Mark B. Cannell, Angus McMorland, and Christian Soeller

488 Handbook of Biological Confocal Microscopy, Third Edition, edited by James B. Pawley, Springer Science+Business Media, LLC, New York, 2006.

Mark B. Cannell, Angus McMorland, and Christian Soeller • Department of Physiology, University of Auckland, 85 Park Road, Grafton, Auckland,New Zealand

Image Enhancement by Deconvolution • Chapter 25 489

numerical aperture objectives have limited working distances thatmay limit their ability for deep sectioning.

Image FormationThe transformation of information from a real object to blurredimage can be expressed mathematically using an operation calledconvolution. Deconvolution is the reverse operation, whosepurpose in microscopy is to remove the contribution of out-of-focus objects from the image plane as well as (partially) reverseaberrations arising from imperfections in the optical train. It shouldbe intuitive that, by using information about the imaging process,deconvolution techniques should be able to improve the quality ofthe image above that which could be achieved by any other methodwhich does not provide extra information (beyond that containedin the image plane itself). The transformation carried out by themicroscope on data from the object can be defined by a mathe-matical function called the point spread function (PSF). The PSFis the multi-dimensional image of a point in space and it can bepractically measured by imaging very small objects (which mustbe less than the wavelength of light in size) or computed from thephysical and optical properties of the imaging system (see Fig.25.1). Given imperfections in the imaging system (or put anotherway, limited knowledge of the real optical system) the latterapproach is inherently limited, but assumptions about the proper-ties of the PSF can help some blind deconvolution methods.

Put simply:

i(x, y, z, t) = o(x, y, z, t) ƒ psf(x, y, z, t) (1)

where x, y, z, t are the dimensions of space and time and i is therecorded image, o the actual underlying object. The ƒ operator isconvolution (see below and Fig. 25.2). If we take the Fourier trans-form (F{}) of this equation, the ƒ is replaced by multiplication:

F{i(x, y, z, t)} = F{o(x, y, z, t)} ¥ F{psf(x, y, z, t)} (2)

This simple equation shows that image restoration should bepossible and might be achievable by dividing the Fourier trans-form of the image by the Fourier transform of the PSF and thentaking the inverse Fourier transform. Before exploring this ideafurther, it should be noted that these equations remind us of theimportance of correct data sampling (image and PSF) and thebehavior of the PSF over time and space. For example, althoughwe generally assume that the microscopes do not change proper-ties as we focus through the sample and from day to day, is thisreally so? When objectives are damaged by lack of care (such asfrom a collision with slide/stage or accumulation of dirt), thegradual loss of imaging performance may go undetected. This isparticularly a problem for confocal microscopes whose pinholecan effectively hide a poor or deteriorating PSF at the expense ofsignal strength. Also note that coverslips (which are an integralpart of the optical system) vary in thickness from batch to batch and therefore introduce variable amounts of spherical aberration.

From here on, we assume that everything is time invariant.(This is not a trivial assumption as, in reality, samples are nevertime invariant, they move — especially if alive, they bleach, themicroscope focus drifts with temperature, etc.) So, from Eq. 2, ifone can measure or compute the PSF with sufficient accuracy, whycan images not be restored to the point where any desired resolu-tion is achieved? The major problem resides in the noise that isalways present in a physical measurement. The convolution orblurring of the object by the PSF is spatial frequency–dependentand attenuates high-frequency components (provided by small fea-tures and edges) more than low frequency (large, smooth) objects.Hence, deconvolution must boost high spatial frequency compo-nents more than low spatial frequency components. Because noise

-0.0 0.5 1.0

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1.5 XZ1

0.1

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FIGURE 25.1. Theoretical PSFs calculated usingwidefield equations. On the left, sections at 0.5 mmz-intervals from the plane of sharp focus are shownwhile xz sections are shown on the right. Panel (A)shows PSFs for two typical lenses. Note the effectof reducing NA on the axial extent of the PSF. Panel(B) shows equivalent PSFs for the above lenses in aconfocal configuration. Note that the confocal PSFlooks very similar to the widefield PSF with the out-lying wings removed. Again, the improved axialresponse of the higher NA lens is quite clear in theseimages. The gray scale is shown on the right and islogarithmic.

490 Chapter 25 • M.B. Cannell et al.

is present at all frequencies in the image right up to the spatial sam-pling frequency which is 1/(pixel size), noise components areboosted at all frequencies during deconvolution. At more than halfthe sampling frequency of the image, there is no real object dataretained (from the Nyquist sampling theorem) but the ever-presentnoise receives an even larger boost during deconvolution.

Put another way, because noise does not come directly fromthe object, but is introduced during the imaging process, andbecause noise is rich in high-frequency components, deconvolu-tion (which seeks to boost high-frequency content) can amplifynoise to the point where it masks, and renders useless, any infor-mation in the resulting deconvolved image. The addition of noisealso generates ambiguity in the image restoration (deconvolution)process so that more than one optimal restoration solution existswith no a priori method of determining which solution best rep-resents the real object. Current deconvolution algorithms workaround some of these limitations, often by making a number ofreasonable assumptions about the object (such as smoothness andnonnegativity) and include extra information about the noiseprocess itself.

Having introduced image restoration by deconvolution, wewill now describe the convolution and deconvolution processes inmore detail and give a guide to some of the practical issues regard-ing image deconvolution. We will also discuss how some populardeconvolution algorithms vary, and assess the utility of imagerestoration. We will show that deconvolution can substantiallyimprove WF microscopy, to the point where it can be used as aviable and practical alternative to confocal laser scanningmicroscopy (CLSM) and can also be used to good effect toimprove the quality of confocal and multi-photon microscopeimages.

FORWARDS: CONVOLUTION AND THE IMAGING SYSTEM

Before it is possible to develop an algorithm for reversing thedefects induced during the imaging process, it is necessary todevelop a basis for image formation. This section will give a short-ened version of that formalization and more detailed descriptionscan be found in Agard and colleagues (1989) and Young (1989).

The notation used generally follows the conventions of Press andcolleagues (1992).

The starting point for this formulation requires two assump-tions regarding the imaging process: linearity and shift invariance.The principle of linearity is met if the sum of the images of twoseparate objects is the same as the image of the combined objects.The assumption of linearity is generally safe in fluorescencemicroscopy provided that detectors are linear and we avoid self-quenching and self-absorption by the fluorophores (Chapters 16and 17, this volume). Shift invariance implies that the image of anobject will be the same regardless of where in the field of viewthat object lies. (While no real imaging system meets the require-ment of shift invariance, it is a reasonable assumption for a high-quality research microscope, subject to our being aware of thepotential complication introduced by changing aberrations acrossthe field and with focus.) Now, because any object can be repre-sented by a superposition of many delta functions (point lightsources) whose individual images are simply shifted copies of thePSF, it follows from the principles of linearity and shift invariancethat the whole image is made from the superposition (sum) ofappropriately scaled and shifted instances of the PSF. For clarity,we can express this in one dimension as an equation:

(3)

which is, in fact, mathematically the same as the convolution ofthe object with the PSF. In symbolic form:

i(x) = o(x) ƒ psf(x) (4)

which gives a basis for the first general equation given above. Asnoted above, convolution in the spatial domain is equivalent tomultiplication in the frequency domain so we can write (for three-dimensional objects):

F{i(x, y, z)} = F{o(x, y, z)} ¥ OTF(V, y, z) (5)

where OTF() is the optical transfer function and is the Fouriertransform of the PSF (see Figs. 25.3 and 25.4), x, y, and z are thespatial frequency coordinates of the OTF in Fourier space, derivedfrom the PSF sampled in x, y, and z, respectively. The OTF is intro-duced here because convolutions are most quickly computed in thefrequency domain by using the fast Fourier transform (FFT).Because the PSF is assumed to be shift invariant, only a singleFourier-transformed PSF is used for computations and the valuesof the PSF in the spatial domain per se have no practical value fordeconvolution. Thus, it is more convenient to describe the PSF ofa microscope in terms of its OTF rather than its PSF. The OTFdirectly describes how blurring affects the various frequency com-ponents making up an object and immediately gives insight intothe resolving power of the microscope. Although we usuallymeasure the PSF in the image plane (rather than at the rear aper-ture of the objective where the OTF exists) and most researchersare more familiar with an ideal PSF and can identify microscopedefects by inspecting it (see below), the OTF is actually moreuseful. Put another way, because the performance of the entiremicroscope is the result of the combination of the OTFs of eachoptical lens and aperture, it is easier to immediately determine theeffect of an aperture or lens in a microscope from its OTF. As anexample, because a circular aperture reduces the amplitude of theOTF with increasing spatial frequency, it is immediately clear thatthe ultimate determinant of resolution must be the most limitingaperture in the system [usually the numerical aperture (NA) of theobjective lens].

i x o x x psf x dx( ) = - ¢( ) ¢( ) ¢-•

+•

Ú

FIGURE 25.2. Schematic diagram demonstrating the convolution (ƒ) opera-tion with a 6 ¥ 6 pixel object and a 3 ¥ 3 pixel blurring kernel. The profilesabove show the maximum projection of the two-dimensional grids as wouldbe seen looking across the planes from above. Note how the contrast of thepeaks in the image is reduced and smeared across the image.


Spatial domain

Widefield

Confocal

Frequency domain

IFFT

FFT

PSF2 PSF

IFFT

FFT

FIGURE 25.3. The relationship between the PSF and OTF in confocal and widefield modes of a microscope. For simplicity, we assume that the pinhole func-tion leads to a simple squaring of the PSF, a reasonable approximation for an optimal pinhole with similar excitation and emission wavelengths. The convolu-tion of the widefield OTF with itself in the frequency (Fourier) domain is the equivalent of squaring in the spatial domain. Note that the inversion of theconvolution (inverted convolution symbol) is difficult in the frequency domain and is more easily accomplished in the spatial domain (by taking the square root).This illustrates that some operations are more simply carried out by operating in the appropriate domain; convolution is easier in the frequency domain whilemultiplication is simpler in the spatial domain. Such interchangeability between spatial and frequency domains is central to deconvolution methods.

A B

FIGURE 25.4. Volume rendered illustrations of the widefield PSF (A) and OTF (B). The PSF extends to infinity as a cone with ripples in the intensity in alldirections. The OTF appears smaller being closer to a toroidal shape except that the center values are non-zero. This leads to a cone of low values in the centerof the toroid. The toroid width in both directions is limited by diffraction. The gray scale shading is proportional to the logarithm of the intensity.


PROPERTIES OF THE POINT SPREAD FUNCTION

Because the PSF of a microscope determines how it creates imagesof an object, the shape of the PSF determines the way in whichimages of objects blur into each other in the final image. At thefocal plane of a WF microscope, the PSF [see Fig. 25.1(A)] existsas a central Airy disk, surrounded by dimmer concentric rings thatare the result of diffraction occurring as light passes through thecircular apertures of the microscope. Away from the plane-of-focus, the PSF spreads outwards, forming two apex-opposed coneswith diffraction ripples within them [Fig. 25.4(A)]. This patternreflects the fact that WF microscopes form images from cones oflight gathered by an objective lens, with the angle of the cone and thus the shape of the PSF being determined primarily by the NA of the objective lens (Chapters 11, 22, 23, and 24, thisvolume). The axial extent of the concentric cones of the widefieldPSF at z-levels well removed from the focal plane show how sig-nificant the out-of-focus contribution is to a WF image. This is notthe case for the CLSM PSF whose PSF approaches background levels with distance [Fig. 25.1(B)], a point which will be revisitedlater.

It is important to note that, unlike theoretically derived wide-field PSFs, measured widefield PSFs are rarely symmetric aroundthe focal plane, nor radially symmetrical about the optical axis.Asymmetry along the optical axis (z-axis) is commonly due tospherical aberration, which may result from refractive index mis-matches between the objective, immersion medium, and sample ortube length/coverslip thickness errors. Radial asymmetries in thePSF are commonly the result of misalignment of optical compo-nents about the z-axis, either as tilt or decentration (see Cagnet etal., 1962; Keller, 1995) (see Fig. 25.5).

The extent of the PSF determines the resolution of the micro-scope. PSF size is often expressed in terms of the width at which itis half the maximum intensity (full-width half-maximum, orFWHM). A typical FWHM for the widefield PSF of a high powerimmersion objective lens might be <0.4mm in the focal plane. Theaxial extent of a true widefield PSF should be infinite with constantenergy at all planes, but real microscopes lose signal energy withdistance because additional apertures (beyond the entrance pupil) inthe system lead to the more-distant wings of the widefield PSF beingclipped (or vignetted). Nevertheless, the extended axial response ofthe WF microscope results in a massive loss of in-plane contrast forextended specimens which are either fluorescent or scattering.

QUANTIFYING THE POINT SPREAD FUNCTION

Most deconvolution methods (i.e., not blind ones) require knowl-edge of the PSF relevant to the particular imaging conditionsencountered. PSFs can be estimated either mathematically or fromdirect measurement. In blind deconvolution methods an estimateof the PSF is also produced by the deconvolution algorithm. (Notethat if the object and image are known, the OTF is given by theFourier transform of the image divided by the Fourier transformof the object; see Eq. 5.) While it is relatively easy to quantify thePSF of a microscope under controlled conditions (see Fig. 25.5),matching the conditions encountered during an experimentalimaging situation is often impractical, and commonly someapproximation to the likely real PSF is used.

Most methods for calculating theoretical PSFs are based onequations in the definitive work of Born and Wolf (1980). A gooddescription of the calculations for a confocal PSF is given by vander Voort and Brakenhoff (1990), in which the PSF is calculated

A

B

–5 µm –4 µm –3 µm –2 µm –1 µm

0 µm +1 µm +2 µm +3 µm +4 µm

–10 µm –8 µm –6 µm –4 µm –2 µm

0 µm +2 µm +4 µm +6 µm +8 µm

FIGURE 25.5. Measuring the widefield PSF. (A)shows the appearance of a 0.2 mm bead with varyingdegrees of defocus. Note that although the mostfocused image is reasonably circular, with defocusextra structure appears in the Airy disk. Theseuncontrolled aberrations need to be measured forcorrect deconvolution. Note also that the aberra-tions are not symmetrical about the point of sharpfocus so that enforcement of symmetry in blinddeconvolution would be problematic. (B) shows athrough-focus series with spherical aberration.Asymmetry above and below best focus is evident.


from the NA of the objective, the illuminating and emitted wave-lengths, and the refractive index of the immersion medium in either(simpler) paraxial forms or with WF integrals (see Fig. 25.6).Numerically derived expressions for the PSF give an indication ofthe best possible resolution for a given objective but these limitsare not achievable. The real world limitations in alignment of theoptical path and lens aberrations inevitably lead to a real PSF thatis larger and has more asymmetric structure than predicted bymathematical approaches. Particular attention should be paid towhether a paraxial approximation is appropriate because there areconsiderable differences between the results of paraxial and WFcalculations for the high-NA objectives which are often used inexperiments (Fig. 25.6). The problem is confounded by softwarepackages that do not reveal what method is used to calculate thePSF. In our experience, real PSFs are typically >20% bigger thancalculated versions (based on the stated NA) and their shapes arerarely perfectly symmetrical about the optical axis and focal plane,even when we have tried to control refractive index mismatches.It should be intuitively obvious (from the preceding discussion)that the accuracy of the deconvolved image can be no better thanthat of the PSF that was used for the deconvolution. Errors in thePSF can produce bizarre artifacts in the image (such as hollowcones or slanting tails around bright objects) as well as the algo-rithm not properly converging to minimum noise (see Fig. 25.7).

The PSF of a microscope can be measured by taking imagesof commercially available fluorescent beads of <0.2mm diameter.These, being smaller than the diffraction limit of visible light, actlike point light sources, and their imaged size and shape thereforerepresent the microscope PSF. It is a common misconception thateven smaller beads give better estimates of the PSF. This is not thecase as the imaging process is a convolution and the convolutionis dominated by the larger of the two objects being convolved. Inaddition, the beads become much less bright as smaller beads areused (note that the signal intensity will fall with the cube of thediameter) so precise measurement of the entire PSF becomes verydifficult, especially as the PSF is recorded away from the focalplane. For those who feel uncomfortable using 0.2 mm spheres, itis possible to use deconvolution of the PSF image using a 0.2 mmsphere (the actual object) to yield the precise PSF (from Eq. 5),but in practice it is rarely necessary to go to such lengths. In ourexperience, the difference in the final results of deconvolutionusing bead images and PSFs derived by deconvolution of beadimages with spherical sources is negligible.

When imaging beads for the purposes of PSF measurement,the image data should be obtained at as high a resolution as pos-sible. In this case, Nyquist’s criterion of using twice the desired

resolution is insufficient, and pixel resolutions 0.05 mm (or evenbetter) are desirable. This has some advantages: (1) The fine dataspacing facilitates data resampling to enable the PSF resolution tomatch that of the recorded image. It is often the case, especiallywhen confocal variable zoom controls are used, that the final imageresolution may be different from that used to acquire the PSF. (2)Generous over-sampling also allows a greater dynamic range to berecorded as the received photons are recorded in more voxels.Even if the detector is only 8 bit, over-sampling by a factor of 2in all three dimensions raises the effective bit resolution to 11 bits.This increase in effective dynamic range is very useful as it enablesboth the bright center of the PSF as well as more dim distant fea-tures to be recorded. For deconvolution, low noise PSFs arerequired and this can be achieved by averaging a number of indi-vidual bead images, possibly from different beads (because theprinciple of shift invariance requires that the bead image — thePSF — is not affected by the position of the bead). This approachrequires an algorithm for aligning the centers of the image of eachbead so that the averaging operation can combine data points ineach bead image. The coordinates of the center of the 3D beadimage can be derived by calculating the center of signal mass ofan image that has been subjected to a thresholding procedure tocompletely remove any background signal. The most distal partsof the image should also be removed by the chosen threshold toprevent the estimate of the bead centroid depending on the posi-tion of the bead within the sampled data volume.

Perhaps the most problematic aspect of measuring PSFs usingfluorescent beads is matching the experimental imaging conditionsto which the PSF is to be applied. First, to account for the effectof any chromatic aberrations (as well as the direct effect of wave-length on resolution), PSFs should be measured from beads thatemit light of a similar wavelength to the experimental fluorophorebeing imaged. (Beads are available with a range of emission wave-lengths from 415 to 680nm.) Errors in refractive index, eitherbetween the objective and immersion medium or between immer-sion and bathing/mounting media, need to be consistent for correctreplication of the aberrated PSF. Because experimental samples areoften less well controlled than calibration bead samples, we oftenadd a few beads to critical experimental samples to provide a checkof the actual PSF in the biological sample. These beads are readilyrecognized in images as they usually stick to the outside of cells(in live-cell imaging) and do not bleach as rapidly as the experi-mental sample. Although it is possible to remove the bead fromthe experimental image digitally, we have never had to do this inpractice as it is usually possible to find a suitable field of view freeof beads.

HNA Paraxial

FIGURE 25.6. Comparison of widefield and paraxial com-putations of the PSF. The left panel shows the computed PSFfor a 1.25 NA objective using the integrated field equations(van der Voort and Brakenhoff, 1990). The right panel showsthe equivalent results for the paraxial equations which combines the electric vectors using the assumption thatsin(q) = q. The paraxial assumption leads to greater ringingin the response but a narrower Airy disk.


THE MISSING CONE PROBLEM

As noted above, while the PSF provides information about thespatial throughput of a microscope, its Fourier transform (the OTF)gives information about the frequency response of a microscopedirectly. While the NA of the microscope limits the highest spatialfrequency that can be recorded, extended sources (i.e., lowerspatial frequency objects) lose signal strength with defocusbecause source light rays must originate in the finite field of viewof the microscope. Thus, the OTF of the WF microscope looks likea toroid [Fig. 25.3 and 25.4(B)] with so-called missing cones aboveand below the focal plane. These missing cones of reduced signalreflect the cone of light acceptance of the objective (assuming it isthe limiting aperture in the microscope). The width of the torus inall planes is limited by the numerical aperture of the objectivegiving rise to a bandlimit equal to 2NA/l (as required fromNyquist sampling of the object using the spatial frequency of lightin the immersion medium). The axial band limit is NA2/2hl whilethe angle of the missing cone is arcsin(NA/h), where h is the lim-iting refractive index of the mounting media between objective andspecimen.

In a CLSM, the pinhole aperture restricts the ability of lightfrom out-of-focus objects to reach the detector. This means thelight-focusing pattern of the objective is used to determine boththe illumination and emission light paths, and so the PSF of aCLSM is approximately the square of the widefield PSF (assum-ing that the ratio of excitation to emission wavelength is close to1). The PSF of a two-photon microscope is similarly related to the

power of the widefield PSF because the probability of near simul-taneous multiple photon absorptions is proportional to the illumi-nating intensity raised to the nth power, where n is the nearestinteger ratio of illumination to excitation wavelengths (Denk et al.,1990). As a result, the PSFs [Fig. 25.2(A–C)] are ovoids with thelong axis corresponding to the optical axis of the microscope. Thecorresponding OTF does not have missing cones and is closer toan ovoid (Fig. 25.3). This can be explained by the fact that the con-focal PSF is the product of the illumination and detection PSFs(Wilson, 1990) and in Fourier space, multiplication becomes con-volution. Convolution of one toroid by another tends to fill in thecenter void.

As pointed out earlier, deconvolution can be thought of as the division of the Fourier transform of the image by the OTF ofthe microscope. The low values of the OTF in the neighborhoodof the missing cone and near the bandlimit greatly amplify anynoise in the image and this severely limits the ability of deconvo-lution methods to restore images. This problem is partly amelio-rated by the apertures in the optical train that limit the ability ofthe microscope to capture light from distant objects (whose in-plane components would have low OTF values); their attenuationprevents the massive amount of signal (and noise) in the integratedfar field from completely swamping in-focus data. Put anotherway, blurring of data from nearby objects is the biggest problem,while signal from more distant objects is attenuated because real microscopes are not true WF instruments. This problem isreduced in the CLSM because the missing cone does not exist,however, the loss of photons due to the pinhole reduces the signal-

k=0

A

B k=10–4 k=10–2

0%noise

1%

10%

FIGURE 25.7. Image restoration by Wiener filter-ing. Panel (A) shows the original test image and thePSF used to blur it. The resulting blurred image haslost contrast for all small features. Panel (B) showsthe results of image restoration with variableamounts of noise added to the blurred image forvarious values of k (see Eq. 10). On the left side,without noise the inverse operation perfectly restoresthe data. However, even with 1% noise (N = 0.01)inverse filtering yields fewer data than even the orig-inal blurred image. When k = 10-4, the effect of noiseis reduced but at the expense of high frequency per-formance. The loss of high frequency componentsleads to ringing near sharp edges and can be seen inthe small colored circles. When k = 5 ·10-2, noise iseven better controlled but the loss of high frequencyinformation is more severe. This can be clearly seenby comparing the centers of the spoked targets. Notethat the image where k = 10-4 and N = 0.01 is com-parable in appearance to the image k = 5 ·10-2 and N= 0.1. Thus increasing k controls increasing noise atthe expense of reduced spatial resolution.

to-noise ratio in the image and, during deconvolution, all noise isamplified.

NOISE

Photon noise is always present. The quantal nature of light meansthat a given illumination intensity is associated with statistical vari-ation in the time at which photons are detected. The number ofphotons observed over any period behaves as a Poisson processwhose variance is equal to the mean. In fluorescence microscopy,the number of photons that can be collected is limited by photo-damage (bleaching) and the signal-to-noise ratios are quite low. Inaddition, although the PSF and object are generally considered to be three-dimensional, the bleaching processes, fluctuations inillumination intensity, and even movement of the object make real objects four-dimensional. To remove this fourth dimension we need to take a three-dimensional snapshot to freeze time-dependent changes. This implies limited exposure periods must beused, which of course further limits light collection.

Photons need to be converted to electrical signals for quantifi-cation and this process also introduces noise. Even in the absenceof photons, all detectors give spurious signals (dark current) whileundetected photons lead to a reduced signal-to-noise ratio. Over arange of wavelengths, average detector efficiencies are generallybelow 50% and this figure is further degraded by significant lossesin the optical components in front of the detector (such as barrierfilters and mirrors). It is not unusual to see overall detection effi-ciencies well below 5% in real microscope systems at some wave-lengths and in the face of such signal losses the photon (shot) noisein the system can be severe. In addition, the detector may add elec-trical noise to the signal before digitization.

A third source of noise resides in the digitization of the signal.In most systems, the detector provides an analog signal that is dig-itized into discrete levels, and 8- to 12-bit detectors are mostcommon. There is, of course, a one-half–bit uncertainty in the dig-itization of any such signal and this typically leads to between0.2% to 0.01% noise. Although this may seem small (compared tophoton noise), this noise is also amplified during data processing.(To appreciate the potential contribution of this component weshould recall that the original in-focus data may represent <5% ofthe total signal.) The digitization noise does not apply to all digitalphoton-counting systems. Despite the concerns, photon noise generally remains the dominant noise source (Mullikin et al., 1994;Pawley, 1995; van Kempen et al., 1997) in microscopic fluores-cence images, at least when using PMTs. When using CCD detec-tors, other sources of noise are added. For a very good CCD with a readout noise of ±10 photoelectrons, readout noise willdominate until 100 photons are detected (at which point these noisesources are equivalent). For cameras, another source of noise isadded due to pixel-to-pixel variation in gain (and/or sensitivity).This flat-field noise is age and temperature sensitive, it increasesin direct proportion to signal strength, and it might be equivalentto photon noise at a signal strength of ~5000 photons. At high light levels this noise source dominates but can often be partiallycontrolled by correcting the image for the gain/sensitivity varia-tion (by normalizing the image to a reference image of a whitefield).

Finally, the thousands of computations required during decon-volution may lead to computational errors due to truncation. For-tunately, modern computers have quite long word lengths so thisproblem is largely disappearing, but errors in programming suchas inappropriate typecasting can lead to truncation errors propa-

gating. Computational errors are often hard to diagnose as theyusually appear as general noise in the image.

DECONVOLUTION ALGORITHMS

In the previous discussion we have shown that in fluorescencemicroscopy (in all its modes including widefield, confocal, and multi-photon) the imaging process can be mathematicallydescribed by a convolution. Convolution consists of replacing eachpoint in the original object with its blurred image (the PSF) in allthree dimensions and summing together overlapping contributionsfrom adjacent points to generate the resulting three-dimensionalimage. As pointed out above, the inverse operation or deconvolu-tion can be performed (at least in principle) by dividing the Fouriertransform of the recorded image by the Fourier transform of thePSF (this argument followed from Eq. 6). Although mathemati-cally correct, this inverse filter algorithm does not work well inpractice because the inevitable noise in the recorded data can beso strongly amplified as to render the deconvolved data useless. Itis this problem that more sophisticated deconvolution algorithmshave been developed to overcome and more robust methods arewell established. In this section we will describe the ideas behindthese improved methods without too much mathematics. Note that when using the term image here, we mean the full three-dimensional data set, often called an image stack. If desired, thereader can consult one of the Numerical Recipes in X texts (whereX may be Fortran, C or C++; see also http://www.nr.com for onlineaccess) for more detailed discussion of some of the fundamentalmathematical concepts involved (i.e., maximum likelihood, minimization of functions, etc.).

Nearest-Neighbor DeconvolutionDeconvolution algorithms vary greatly with respect to computa-tional intensity, that is, how long it actually takes to compute thedeconvolved image. Among the simplest algorithms that are ofpractical use are the nearest-neighbor approaches. Such algorithmsare based on the simplifying assumption that the loss of contrastin any given plane is only due to signal arising from objects inplanes immediately above and below it. To restore the data withthis assumption, we can subtract appropriately blurred versions ofthe object planes above and below the current focal plane from therecorded image data. The blurring of the adjacent planes is, ofcourse, determined by the PSF. For example, the signal contribu-tion from object structures in the plane immediately above can beobtained by a two-dimensional convolution:

oj+1 ƒ psfj+1 (6)

where, assuming that image planes are recorded a spacing Dz apart,psfj+1 denotes the values of the PSF in the next image plane, Dzfrom the focus. To calculate this contribution we would need toknow oj+1 (in addition to the three-dimensional PSF), but we onlyhave image data at this point. We therefore make a second approx-imation (which is no worse than the first) that oj+1 = ij+1. Thus, afraction of this blurred nearest-neighbor data is then subtractedfrom the data recorded in the current image plane, or, as an equation:

oj = [ij - c(ij-1 ƒ psfj-1 + ij+1 ƒ psfj+1)] (7)

where the index j identifies the current plane, j - 1 and j + 1 theplanes immediately below and above, o and i are the deconvolvedobject and recorded image, respectively, and c is a constant <1.


This algorithm increases in-plane contrast and blurring in the focalplane can also be reduced by applying a 2D inverse filter of thein-focus PSF (psfj

-1). For further details see Agard and colleagues(1989):

oj = [ij - c(ij-1 ƒ psfj-1 + ij+1 ƒ psfj+1)] ƒ psf -1j (8)

Note that all the convolutions are two-dimensional, makingthis method suitable for real-time implementation with FFTs. Onthe other hand, one should not be surprised that the enhancementobtained with this method is only moderate and (most important)not quantitative. These limitations reside in the subtractive natureof the process which will add noise and alter signal amplitudes.Nevertheless, where qualitative imaging at high speed is required,useful increases in in-plane contrast can be achieved with thismethod. For better results one needs to resort to fully three-dimensional algorithms.

An even faster implementation can be made by assuming thatthe data in adjacent planes is the same as that in the image plane,that is, ij±1 = ij. In this no-neighbors method, the PSF is used tosubtract a blurred contribution of the image plane from itself.Strictly speaking, this is not a deconvolution method at all but isformally equivalent to an unsharp-mask high-pass filter. Clearly,the gross assumption underlying this approach limits its applica-bility to very sparse three-dimensional objects and any intensityquantitation would be out of the question.

Wiener FilteringTurning now to full three-dimensional approaches, as noted abovethe object can be deconvolved by inverse filtering, that is, directdivision of the Fourier transform of the data by the OTF:

(9)

In this equation, the low values of the OTF in the region of themissing cone will increase noise in the restored image. Put anotherway, the data which is lost in the missing cone cannot be replacedunless we supply some other information that acts as a constrainton the restoration. Nevertheless, while the amplification of noisemakes direct inversion nearly useless (at least for microscopicimages), it can be improved by a simple modification (Shaw andRawlins, 1991):

(10)

where K is an additive correction factor (see Figure 25.7). Byadding K, noise amplification will be limited where the OTF issmall. It can be seen that if K = 0, the equation reduces to the orig-inal inverse filter (Eq. 9). The impact of K can be reduced in thelow frequency domain (where noise is less of a problem) bymaking it depend on the spectral characteristics of noise and objectdata. In this case, K can be mathematically optimized and thisoptimal filter is called the Wiener filter. Unfortunately, Wienerfilters do not generally provide optimal image quality because: (1)suppression of any part of OTF during restoration affects the entireimage, (2) where the OTF is small, noise will be amplified, possi-bly to an unacceptable degree, and (3) small errors or uncertain-ties in the OTF will lead to large errors in the reconstruction.Generally, for microscopic imaging purposes, Wiener and similarlinear filters have been superseded by nonlinear iterative decon-volution algorithms that have become more applicable withincreasing computer power.

F o F iOTF

OTF K{ } = { } ¥

+2

F oF iOTF

{ } = { }

Nonlinear Constrained Iterative Deconvolution AlgorithmsWhy use nonlinear methods? The problem resides in the fact thatlinear methods cannot guarantee that the deconvolved object ispositive. Rather, in the presence of noisy image data, some pixelsin the deconvolved object will assume negative values, a resultwhich is obviously incorrect. For fluorescent images all photonfluxes must be positive so nonlinear algorithms that ensure no neg-ative data values in the reconstructed object are clearly preferable.Such positivity constraints can be easily enforced within iterativealgorithms. In iterative algorithms, the deconvolved object is notcomputed in a single pass (as is the case in nearest-neighbor orWiener filtering) but the object is calculated in a series of stepsthat are then repeated. Each step results in a new estimate of theobject that should be somewhat closer to the actual object. Duringthis process, positivity can be enforced by explicitly setting nega-tive values in the current estimate to zero. The small error that isintroduced by this clipping procedure should be corrected in thenext iteration and, with a well-behaved algorithm, the estimateconverges to a nonnegative solution that closely approximates thetrue object. After a certain number of steps, the process is termi-nated by the user to yield the deconvolved data.

All iterative algorithms start from an initial estimate and wewill discuss reasonable starting choices briefly later. The algorithmthen enters the iterative loop in which a new estimate is calculatedfrom the current estimate and the recorded image data. The posi-tivity constraint is then enforced by setting all negative pixels inthe new estimate to zero. New estimates are calculated repeatedlyusing this scheme until the user (or the software implementationthat he uses) decides to terminate the computation and accept thelast calculated estimate as the deconvolved object.

Expressed as a piece of pseudo code:

1. Pick a starting estimate o0

a. Calculate OTF (if blind)2. Compute an initial new estimate (ok+1) from previous estimate

ok, recorded image data i and OTF3. Set all negative pixels in ok+1 to zero to give the current best

estimate and apply any other constraintsa. If blind, estimate a new OTF from F{i}/F{ok+1} applying any

constraints such as NA, symmetry and nonnegativity.4. Go to step 2 unless a stop criterion is reached

Differences between iterative algorithms primarily arise fromthe way in which the new estimate is calculated (step 2).

To calculate a better estimate of the object, we must have someway to find out if our current object guess (ok) is good (or not).Ideally, the estimate blurred by the PSF would be the same as ourrecorded image i except for some small difference due to the noisein the image data. Following this idea, we can calculate an errorsignal as the difference between our blurred guess and the actualimage data. So to update the estimate we can take a fraction oferror signal and add it to the current estimate (a Newton iterativeapproach). This approach underlies the Jansson–van Cittert algo-rithm (Jansson et al., 1976). As an equation:

ok+1 = ok + g(i - (ok ƒ psf)) (11)

and with a good choice for g (which can vary across the image)this scheme should lead to successively better estimates. However,in early implementations the choice of g was rather ad hoc(Jannson, 1970; Agard et al., 1989) and noise leads to uncertaintyas to whether the algorithm actually achieves the optimal solution.


In more recent iterative algorithms, the updating scheme isderived from more stringent mathematical reasoning. One class ofalgorithms that work well in practice are called maximum likeli-hood algorithms. The idea behind this approach is based on thenotion of the likelihood of a certain object o to underlie ourrecorded data i. In this context, we can view the deconvolutionproblem as a multi-dimensional fitting problem. In this view weanticipate that for the correct choice of fitting parameters (actuallythe pixels values in o), there should be a finite probability that ourdata should have been observed. Thus, we have to try to find theobject o which maximizes the likelihood that our data would haveoccurred given a specific noise model (which is required to math-ematically calculate these probabilities). Put simply, the idea is tofit a function (the blurred object guess) to the data with appropri-ate weighting for noise (which should be a concept familiar to allexperiment scientists); see also the discussion in Press and col-leagues (1992, section 15.1).

To deal with noisy fluorescence data, we can therefore calcu-late the probability that our data would have occurred, assumingthe noise follows a Poisson (or some other noise model) distribu-tion. Having found an explicit mathematical expression for thisprobability we then proceed to find the object o that maximizes thisprobability. Instead of trying to find this maximum directly we canmake life a bit easier by looking for the maximum of the logarithmof the probability. Due to the properties of logarithms, this log-likelihood is maximal at the same location that the likelihood itself is maximal, however, the mathematical expressions generallybecome easier to handle after taking the logarithm. To actually find the maximum, mathematicians use the observation that at amaximum the derivative of the log-likelihood will be zero. Cuttingthrough the mathematical derivation, one finally arrives at anexpression for the updating step in the presence of Poisson noise:

(12)

where IF{} is the inverse Fourier transform and OTF¢ is thecomplex conjugate of the OTF. (The Fourier transform yields realand imaginary parts so the OTF is complex.) By definition, thecomplex conjugate of a complex number x + iy is x - iy. Multipli-cation by the complex conjugate in Fourier space is known as cor-relation. Note that in this algorithm we do not use the additive errorI - ok ƒ psf but rather the relative difference i/(ok ƒ psf ). This algo-rithm is also known as the Richardson–Lucy algorithm after theauthors who first derived this scheme for astronomical applications(Richardson, 1972; Lucy, 1974). For an example of the imageenhancement possible with this algorithm see Figures 25.8 and25.9. Assuming Gaussian noise in the data (for large photon fluxes,the Poisson noise model approaches Gaussian noise) we arrive atan alternative updating step that uses the additive error signal:

(13)

With both types of maximum likelihood algorithms, we shouldstop after a certain number of iterations for best results as, even-tually, the error and/or noise starts to increase again.

This brings us back to the start and stopping choices that must be made with iterative deconvolution algorithms. Reasonablechoices for the first estimate are (a) the recorded image itself, (b)a constant estimate with every value set to the mean of the recordedimage, and (c) a smoothed version of the recorded image. Usingthe Richardson–Lucy maximum-likelihood algorithm (Eqs. 12,13) choices (b) or (c) are recommended. A more detailed discus-sion can be found in van Kempen and van Vliet (2000). We also

o o IF F i o psf OTFk k k+ = + - ƒ{ } ¥ ¢{ }1

o o IF Fi

o psfOTFk k

k+ = ¥

ƒÏÌÓ

¸˝˛

¥ ¢ÏÌÓ

¸˝˛

1

need a criterion of when to stop the iterative procedure and acceptthe current estimate as the deconvolved object. A reasonableapproach involves monitoring the relative change between esti-mates as the iteration proceeds. As the relative difference

(14)

between subsequent estimates falls below a chosen threshold (thatgenerally depends on the amount of noise in the images) the iter-ations should be stopped. For example, with some of the datashown in this chapter, thresholds of ~10-4 were used.

There are other issues that should be considered in a seriousimplementation of constrained-iterative deconvolution methods,for example, background estimation (van Kempen and van Vliet,2000) and prefiltering of data and PSF (van Kempen et al., 1997).The other approach to iterative deconvolution is to use a regular-ization term, often the Tikhonov functional, as set out by Tikhonovand Arsenin (1977), which aims to limit the generation of largeintensity oscillations in the deconvolved image (which will growas the loss of high frequency information in the OTF becomes lim-iting). A number of similar methods based on this approach exist,and these vary primarily in the approach taken to find the solutionto the regularization term which minimizes the signal oscillationsor roughness in the deconvolved image. One such solution is the iterative constrained Tikhonov–Miller (ICTM) algorithm, whichuses a minimization strategy based on conjugate gradients to min-imize the Tikhonov functional. Tikhonov-regularized forms of theRichardson–Lucy algorithm also exist, representing a consolida-tion of these two classes of solution.

Comparisons of a number of deconvolution algorithms (vanKempen et al., 1997; van Kempen, 1999) show that there is rela-tively little difference between their various deconvolved results.The regularized algorithms tend to be less sensitive to noise thanRichardson–Lucy-based methods, and reach an appropriate stop-ping point sooner. However the RL algorithm provides slightlybetter deconvolution results, as quantified by the mean square errorbetween the image and the PSF-blurred estimate of the decon-volved object. A useful improvement in the Richardson–Lucymethod can be achieved by prefiltering the image and PSF with aGaussian smoothing filter which reduces high frequency noisecontributions without affecting the deconvolution of lower fre-quency image components.

One other variant of the general deconvolution algorithm isblind deconvolution (Holmes and O’Connor, 2000 and Chapter 24,this volume), where the deconvolution is performed in the absenceof an independently measured (or calculated) PSF. Instead, the PSFis iteratively estimated along with the experimental data during thedeconvolution process simultaneously. Proponents of blind decon-volution suggest this method is superior to using PSF-dependentcalculations with a PSF rendered incorrect by sample-inducedaberrations and scattering. On the other hand, one can argue thatby supplying less information to the reconstruction, the final resultmust be inferior, if the PSF was known to sufficient precision. Inaddition, if the blind method makes use of symmetry in the PSFand/or the NA of the objective, artifacts will be produced if lensaberrations are present and/or the assumed NA is modified by, forexample, improper illumination of the rear aperture of the lens.

Comparison of MethodsA common question is: What method works best? Unfortunately,there is no single answer because all imaging methods represent a

o oo

k k

k

+ -1



A

C D

B

2 µm

5 µm

FIGURE 25.8. Maximum likelihood deconvolution of the cardiac t-system. (A) shows a two-photon image of a single, rat cardiac cell in dextran-linked fluo-rescein. The fluorescein has filled the t-tubules which are <~300nm in diameter. (B) shows an enlarged view from a part of (A). Note the relatively low signal-to-noise present in this image although tubules can be clearly seen. (C) shows the restored image. Note the improvement in signal-to-noise ratio as well as theincreased contrast. There has also been a useful but moderate increase in spatial resolution. The fine structure of the t-system can now be appreciated. (D) showsthe effect of deconvolving the image with an incorrect PSF. At first sight, the results appear reasonable, but closer inspection reveals the presence of artifactualdata when compared to (C). In view of the general problem of controlling spherical aberration, without good controls such artifact could easily lead to erroneousconclusions.

FIGURE 25.9. Enhancement of two-photon images by deconvolution. The panel at the left shows a neuronal dendrite in a living brain slice. The limited photonyield leads to little fine structure being visible in the volume-rendered image. After maximum likelihood deconvolution, synaptic spines are clearly visible andeven appear to have necks (arrows). In this case, deconvolution has been much more useful to control noise rather than increase optical resolution.

trade-off between the three fundamental limits in informationtransfer processes: resolution versus speed versus noise. Thus,improvements in one area will usually compromise another and itis up to the experimenter to consider what is actually desired ofthe imaging method. This principle is generally limiting evenwithin a chosen restoration method, because once resolution hasbeen increased to a certain point, further gains can only be madeby compromising noise and/or speed. As a guide, we suggest thatdeconvolved confocal/multi-photon images provide the ultimateimage in terms of resolution (see Figs. 25.8 and 25.9). In somecases, deconvolved WF images with iterative methods can achieve comparable resolution to confocal/multi-photon methods. Nearest-neighbor methods provide a lower level of image improvementthat can help control loss of contrast in WF images but suffer fromtheir non-quantitative nature (i.e., one should not compare bright-ness between different parts of the image). Deconvolution of WFimages may provide a resolution approaching that of confocalmethods but would be problematic in scattering specimens or inthe face of spherical aberration.

In terms of speed, nearest-neighbor methods are fastest withexecution times for interative-constrained methods being compa-rable regardless of whether the source images are WF or confo-cal/multi-photon. This pecking order also seems to apply in termsof signal-to-noise ratio (including errors in absolute signal levels)with nearest-neighbor methods being worst and iterative con-strained methods being best. In terms of noise, WF methods areinitially superior until resolution starts to become limiting, atwhich point differences between methods largely disappear.

Finally, because even after deconvolution, image quality iscritically dependent on the quality of the input data, we suggest 20tips for better imaging:

1. Store unused objectives lens down, clean, and dry in the manufacturer’s case.

2. Before using an objective, inspect it with a microscope eye-piece for cleanliness and damage — note any scratches acrossthe metal front of the lens — possibly resulting from colli-sions with stage or specimens. If spring-loaded, check theobjective front is not locked down.

3. Check the coverslips for thickness (check the coversliprequirement on the microscope barrel, usually 0.17mm) andthat they are suitable for the chosen objective. If the objec-tive has a correction collar, use it!

4. Get a supply of subresolution fluorescent beads in a widerange of excitation and emission wavelengths to measure thePSF and test microscope performance. Mount them in stan-dard slides (avoid mountants with harsh solvents); you canalso purchase slides already made up.

5. Oversample the PSF and subsequently resample to image res-olution by applying a Gaussian filter (to reduce noise) fol-lowed by a cubic-spline interpolation. If appropriate, apply acircular-symmetry constraint on the PSF to further reducenoise.

6. Add a small quantity of beads to samples before sealing the coverslip to enable visual inspection of the PSF in widefield.

7. Always seal wet-mounted specimens to prevent them dryingout.

8. Minimize spherical aberration by appropriate selection ofobjective lenses/mounting media. If the objective has a cor-rection collar, adjust it to make the observed PSF as sym-metrical as possible. This can be achieved by watching thebead image as the microscope is slightly defocused. The Airy

disk should grow symmetrically in both directions withdefocus (see Fig. 20.3, this volume).

9. If a high-power oil lens must be used with a water-based spec-imen, try raising the refractive index of the specimen withsucrose, mannitol, or salt if osmotic pressure is not a problem,but be careful not to allow crystal formation. It should benoted that an 84% weight/volume sucrose solution has thesame refractive index as glass (~1.5). Alternatively, useimmersion oil having a refractive index >1.52.

10. In confocal applications, always open the pinhole slightly toincrease light flux while maintaining a reasonable degree ofconfocality to achieve the desired axial resolution.

11. Use the highest voxel resolution possible in xy and z, but tryto avoid bleaching problems.

12. Minimize bleaching problems with antifade mountants andlatest-generation fluorochromes.

13. If imaging multiple wavelengths sequentially, image thelongest wavelength fluorochromes first.

14. Ascertain the minimum and maximum extents of the speci-men at the lowest possible illumination intensity (with thepinhole quite open in confocal applications). The idea is notto produce a nice image at this point but simply detect wherethe fluorescent specimen resides.

15. Keep a copy of the lens PSF with the image data and recordvoxel dimensions, objective, mounting media (some imagingsoftware allows comments to be placed in the image header).

16. Image stacks will usually fit within a single CD-ROM (do notuse any data compression unless you are sure it is lossless)so make a CD-ROM copy of the data before leaving the laboratory.

17. Avoid mechanical artifacts during imaging by keepingacoustic noise down, the door closed, and avoiding touchingany part of the microscope system. Vibration isolationsystems may be essential in laboratories with suspendedfloors. Air-cooled laser system fans should be mechanicallydecoupled from the microscope with a duct to a separateexhaust fan.

18. Use dimmable incandescent lighting in the laboratory anddim the lights while collecting data. On inverted microscopes,cover the specimen with a black cup.

19. Use a lookup table that makes it easy to spot over- and under-flows in the data. Make sure the black level of the imagingsystem is set to >0; this is even more important than havinga few pixels saturated.

20. Always have sample slides containing beads on hand andalways check the microscope image quality with them beforestarting imaging, every day.

21. Deconvolve everything!

REFERENCES

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Keller, H.E., 1995, Objective lenses for confocal microscopy, In: Handbook ofBiological Confocal Microscopy (J.B. Pawley, ed.), Plenum Press, NewYork, pp. 111–126 or Chapter 7, this volume.

Lucy, L.B., 1974, An iterative technique for the rectification of observed dis-tributions, Astronomy J. 79:745–765.

Mullikin, J.C., Van Vliet, L.J., Netten, H., Boddeke, F.R., van der Feltz, G.W.,and Young, I.T., 1994, Methods for CCD camera characterization. Proc.SPIE 2173:73–84.

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