Dynamic speckle illumination microscopy with translated versus …users.df.uba.ar/dkunik/hilo/mertz_2006.pdf · 2011-09-13 · Dynamic speckle illumination microscopy with translated

Dynamic speckle illuminationmicroscopy with translated versus

randomized speckle patterns

Cathie Ventalon and Jerome MertzBoston University, Department of Biomedical Engineering, 44 Cummington St.,

Boston, MA, 02215

[email protected], [email protected]

Abstract: Dynamic speckle illumination (DSI) microscopy is a widefieldfluorescence imaging technique that provides depth discrimination. Thetechnique relies on the illumination of a sample with a sequence of specklepatterns. We consider an image processing algorithm based on a differentialintensity variance between consecutive images, and demonstrate that DSIsectioning strength depends on the dynamics of the speckle pattern. Trans-lated speckle patterns confer greater sectioning strengththan randomizedspeckle patterns because they retain out-of-focus correlations that leadto better background rejection. We present a theory valid for arbitrarypoint-spread-functions, which we corroborate with experimental results.

© 2006 Optical Society of America

OCIS codes: (180.1790) Confocal microscopy; (180.2520) Fluorescence microscopy;(180.6900) Three-dimensional microscopy.

References and links1. T. Wilson and C. Sheppard, Theory and practice of scanningoptical microscopy, Academic Press, London (1984).2. G. Q. Xiao, T. R. Corle, and G. S. Kino, “Real-time confocal scanning optical microscope,” Appl. Phys. Lett.53,

716-718 (2000).3. E. M. McCabe, D. T. Fewer, A. C. Ottewill, S. J. Hewlett, andJ. Hegarty, “Direct-view microscopy: Optical

sectioning strength for finite-sized, multiple-pinhole arrays,” J. Microscopy184, 95-105 (1996)4. T. Wilson, R. Juskaitis, M. A. A. Neil and M. Kozubek, “Confocal microscopy by aperture correlation,” Opt.

Lett. 21, 1879-1981 (1996).5. R. Juskaitis, T. Wilson, M. A. A. Neil and M. Kozubek, “Efficient real-time confocal microscopy with white light

sources,” Nature383, 804-806 (1996).6. P. J. Verveer, G. S. Hanley, P. W. Verbeek, L.J. Van Vliet, and T. M. Jovin, “Theory of confocal fluorescence

imaging in the programmable array microscope (PAM),” J. Microscopy 189, 192-198 (1998).7. M. A. A. Neil and T. Wilson, “Method of obtaining optical sectioning by using structured light in a conventional

microscope,” Opt. Lett.22, 1905-1907 (1997).8. M. A. A. Neil, A. Squire, R. Juskaitis, P. I. H. Bastiaens and T. Wilson, “Wide-field optically sectioning fluores-

cence microscopy with laser illumination,” J. Microscopy197, 1 (2000).9. C. Ventalon and J. Mertz, “Quasi-confocal fluorescence sectioning with dynamic speckle illumination,” Opt.

Lett. 30, 3350-3352 (2005).10. J. W. Goodman, Statistical Optics, Wiley, New-York (1985).11. P. A. Stokseth, “Properties of a defocused optical system,” J. Opt. Soc. Am.59, 1314-1321 (1969).12. T. Yoshimura, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am. A3, 1032-1054 (1986)13. H. T. Yura, S. G. Hanson , R. S. Hansen, and B. Rose, “Three-dimensional speckle dynamics in paraxial optical

systems,” J. Opt. Soc. Am. A16, 1402-1414 (1999)

#71882 - $15.00 USD Received 12 June 2006; revised 28 July 2006; accepted 30 July 2006

(C) 2006 OSA 7 August 2006 / Vol. 14, No. 16 / OPTICS EXPRESS 7198

1. Introduction

Confocal fluorescence microscopy [1] is a well established technique in the bioimaging com-munity, mainly because it provides optical sectioning. Implementations with laser illuminationusually involve scanning of a single laser focal spot, whereas implementations without laserillumination typically make use of multi-spot array scanning [2, 3]. More light-efficient alterna-tives have been proposed such as aperture correlation microscopy [4, 5], or programmable arraymicroscopy [6] that involve illuminating the sample with patterns of light defined by illumi-nation masks, and collecting the fluorescence through thesesame masks. Alternative strategiesthat allow direct collection of the fluorescence without a return passage through the illuminationmask include the simple and well-established technique of incoherent structured illuminationmicroscopy [7, 8].

We recently proposed a technique similar to structured illumination microscopy wherein theincoherent grid illumination pattern is replaced with a dynamic speckle illumination (DSI) pat-tern from a laser, and we demonstrated that this new technique confers optical sectioning inbiological tissue [9]. In our initial implementation, the sample was illuminated with a sequenceof statistically independent (random) speckle patterns. The root-mean-square (rms) of the corre-sponding sequence of fluorescent images comprised the optically sectioned image. A drawbackof this technique is that its sectioning strength is weaker than that of a confocal microscope.In this paper, we show that the sectioning strength can be significantly improved by using thesame image processing algorithm but with a better control ofhow the DSI pattern is rendereddynamic. While in our initial implementation, the speckle pattern was completely randomizedbetween image acquisitions, here we apply small lateral translations to the speckle pattern, onthe order of a speckle grain size, and demonstrate both theoretically and experimentally that theresulting sectioning strength is near-confocal.

2. Principle of dynamic speckle illumination microscopy

Figure 1 depicts speckle illumination inside a sample and the detection point-spread-function(PSFdet) for an arbitrary pixel in the imaging CCD camera. The photocounts recorded by thispixel are plotted as a function of time, or, more precisely, image framek. These photocounts canbe thought of as arising from two contributions: signal fluorescence arising from the in-focusplane (denoted byS) and background fluorescence arising from out-of-focus planes (denoted byB). Our goal with DSI microscopy is to estimate the fluorophoreconcentration at the in-focusplane based only on these photocounts. Our strategy for achieving this is to render the speckleillumination dynamic and to exploit ana-priori knowledge on how the temporal statistics ofSandB differ.

In particular, there is a fundamental difference between how the photocounts from zonesSandB are acquired. Because the in-focus PSFdet is narrow and the out-of-focus PSFdet is wider,we may roughly think of PSFdet as locally sampling the in-focus fluorescence while spatiallyaveraging the out-of-focus fluorescence. Correspondingly, when the speckle pattern is rendereddynamic, different realizations of the speckle pattern arelocally sampled in zoneS whereasthey are spatially averaged in zoneB.

The temporal average of the detected photocounts isId = S + B. Assuming that the differentrealizations of the speckle pattern are statistically independent forS (though not necessarily forB) thenS provides a direct estimate of the fluorophore concentrationat the in-focus plane. Ourgoal, therefore, is to distinguishS from B. The temporal average of the photocounts does not, byitself, provide enough information to do this. However, in the case of DSI the additional knowl-edge garnered from the temporal variance of the photocountsdoes provide enough information.In particular, it is known that detected photocounts from locally sampled independent specklepatterns obey Bose-Einstein statistics [10]. Accordingly, the temporal variance ofS is given by



B BS

CCD camera

Id

k

Fig. 1. Schematic view of the laser speckle (blue ovals) inside the sample and the detectionPSF (green lines) for an arbitrary CCD pixel. We define two regions delimited by the dashedvertical lines: the in-focus region corresponding to the signalS we want to measure, and theout-of-focus region corresponding to the backgroundB we want to reject.Id is the recordedpixel intensity at each camera framek.

Var(S) = S2+ S, which can be interpreted as arising from both “classical”local intensity fluctu-ations of speckle (which obey negative-exponential statistics – first term) and “quantum ”shotnoise associated with photodetection (which obeys Poissonian statistics – second term). In con-trast, the local intensity fluctuations inB are spatially averaged and hence contribute little to itstemporal variance, giving Var(B) ≈ B. The total variance is then Var(Id) ≈ S2 + S + B, and anestimate ofS becomes straightforward:S ≈

√

Var(Id)− Id . This estimate is our intended goal.A caution should be made regarding this simple idea since theseparation between zonesS

andB in the sample is nowhere near as clear-cut as suggested in Fig. 1. In particular, we haveassumed that PSFdet in zoneB is so wide as to completely spatially average out the “classi-cal”intensity fluctuations of speckle inB and extinguish their contribution to Var(B). This isan approximation. In fact, a simple variance algorithm cannot completely extinguish residualintensity fluctuations, and better algorithms for their reduction constitute the main motivationfor this paper.

3. DSI sectioning with translated versus randomized speckle patterns

The algorithm we consider in this paper is based not on an intensity variance, as suggestedabove, but rather on a structure functionD defined as half the differential intensity variance:

D =1

2N

N

∑k=1

(Id,k+1− Id,k)2 (1)

whereId,k is the intensity recorded by the CCD camera on pixeld in framek (we consider only“classical”intensities for the moment), andN +1 is the total number of acquired images. Thisalgorithm was introduced in ref. [9] and has the advantage ofbeing insensitive to long-termpower variations in the illumination beam. We note that if consecutiveId,k’s are statisticallyindependent, thenD is formally equivalent to the intensity variance Var(Id). Such statistical in-dependence is achieved, for example, if the speckle patternis completely randomized betweencamera frames, as was done in ref. [9]. However, if correlations are induced between consec-utive Id,k’s, thenD can be much smaller than Var(Id). In particular, if correlations are inducedonly in the out-of-focus background and not in the in-focus signal, they can lead to significantlyenhanced background rejection. A simple technique for introducing such background correla-tions is with DSI based on speckle translation rather than speckle randomization. In this section



we present an intuitive argument for why background rejection is improved, and defer a morerigorous argument to section 4.

To quantify DSI sectioning strength, we consider as we did inref. [9], a sample that is athin uniform fluorescent plane and calculate the expected structure functionD as a functionof the sample’s axial positionzc (see Fig. 2). To gain an intuitive understanding, we adopt thefollowing simplifications:

1) We approximate the lateral profile of PSFdet(~ρ,z) to be a circular top-hat with a diameterthat expands linearly with distancez away from the focal plane. PSFdet(~ρ,z) is normalized suchthat PSFdet(0,0) = 1 and

∫

PSFdet(~ρ,z)d2~ρ = Ad , independently ofz. That is,Ad correspondsto the area of PSFdet at the focal plane.

2) We further approximate the speckle pattern to consist of amosaic of speckle “grains ”ofequal lateral areasAs (the speckle correlation area) and of uniform intensities within these areas.The intensity of each speckle grain is statistically independent of its neighbors’, as illustratedin Fig. 2(c). The speckle grain intensities are denotedIs,n and obey the negative-exponentialstatistics of a fully developed speckle pattern [10]. Hence, the spatial variance of these grainintensities sampled over many grain areas is given by the average speckle intensity squaredIs

2(again, we neglect shot-noise for the moment). We note that for widefield speckle illumi-

nation, as is the case in DSI microscopy, the speckle correlation area (or grain area) does notsignificantly depend onz [9].

Fluorescent

plane z

(a)

Fluorescent

plane

Focal

plane

zc

Random: Translation:

s

(b) (e)

s

Random: Translation:

(c)

(d)

(f)

In focus Out of focus

Fig. 2. Simple picture to evaluate the translation and random DSI signals produced by auniform fluorescent plane located at defocus positionzc. When the plane is in focus (a),PSFdet is narrow and the detected speckle pattern is completely renewed upon random-ization (b) or translation (c). When the plane is out of focus (d), PSFdet is wide and thedetected speckle pattern is completely renewed upon randomization (b) but only partiallyrenewed upon translation (c).∆ρs is the translation step size.



To calculate the intensity detected by a CCD pixel, we defineNs to be the number of specklegrain areas “seen ”by PSFdet. That is, when the sample plane is in focus thenNs = 1 (weassumeAd ≤ As); when it is out of focus thenNs is the ratio of the PSFdet area to the specklegrain area, which increases asz2

c . The intensity detected by a CCD pixel is then given by thespatial summation

Id =CAd

Ns

Ns

∑n=1

Is,n (2)

whereC is the fluorophore concentration andIs,n is the intensity of speckle grainn. We can nowreadily calculate the structure functionD for the cases when the speckle pattern is randomized(DR) versus when it is translated (DT ). Our final DSI image in both cases is given by

√D.

We consider the random case first. As mentioned above,DR can be directly interpreted as thevariance of the detected intensities, and therefore reads:

DR =(CAdIs)

2

Ns(3)

When the sample is at the focal plane, the DSI image is simply given by the product of theaveraged illumination intensityIs, the concentrationC of fluorophores, and the area of PSFdet

at the focal plane, as we would obtain with a confocal microscope. When the sample is out offocus, thenNs is proportional toz2

c and the final DSI image (√

DR) decays as 1/ |zc|. Spatialaveraging caused by PSFdet therefore reduces the out-of-focus speckle fluctuations.

For the translation case, we first define∆Id,k as the difference between two consecutive inten-sities (∆Id,k = Id,k − Id,k−1). For a uniform fluorescent plane (as considered here), a translationof the speckle pattern relative to a fixed PSFdet is equivalent to a translation of PSFdet relativeto a fixed speckle pattern, allowing us to write:

∆Id =CAd

Ns

[

∆Ns

∑n=1

Is,n −∆Ns

∑n′=1

Is,n′

]

(4)

where∆Ns is the (zc dependent) number of speckle grain areas encompassed in either of thedifferential areas between PSFdet and its shifted replica. These differential areas are colored ingreen and red in Figs. 2(c) and 2(f), green corresponding to the gain in new speckle grain areas(left term in Eq. (4)) and red corresponding to the loss in oldspeckle grain areas (right term inEq. (4)). The corresponding structure functionDT is given by:

DT = (CAdIs)2 ∆Ns

N2s

(5)

If the lateral shift∆ρs of the speckle pattern is larger than the width of PSFdet (as is the case, forexample, in Fig 2(f)), then∆Ns = Ns andDT = DR. Two consecutive raw imagesId,k are thencompletely uncorrelated and the final DSI image is the same regardless of whether the specklepattern is translated or randomized.

In contrast, if the lateral shift∆ρs is smaller than the width of PSFdet, then∆Ns < Ns andhenceDT < DR. In this case, two consecutive raw images are highly correlated because of theoverlap in the areas covered by PSFdet and its shifted replica, meaning consecutive raw imagescomprise many of the same speckle grains. Assuming that∆ρs is small (in practice it is of orderthe speckle grain size) then∆Ns scales linearly withzc for large|zc| (i.e. out of focus) and thefinal DSI image

√DT decays as 1/ |zc|3/2. We therefore expect better DSI sectioning strength

with translating speckle patterns than with randomized speckle patterns.We recall that the signal from a fluorescent plane scales as 1/z2

c for a confocal microscope [1].The DSI sectioning strength for randomized speckle patterns (1/ |zc|) was previously referred



to as “quasi-confocal”[9]. For lack of a better term, we refer to the DSI sectioning strength fortranslated speckle patterns (1/ |zc|3/2) as “near-confocal”.

4. DSI sectioning for arbitrary PSF

In this section, we present a rigorous calculation of the structure functionDT that is validfor arbitrary illumination and detection PSF’s (denoted PSFdet and PSFill respectively). Thiscalculation assumes a translating speckle pattern, however we note that in the limit where thetranslation step size is infinite, thenDT reduces toDR. Hence, this calculation applies to bothDSI techniques.

In general, the fluorescence intensity at a position~ρd on the CCD detector plane is given by

Id(~ρd) =∫ ∫

PSFdet(~ρd −~ρ,−z)C(~ρ,z)Is(~ρ,z)d2~ρdz (6)

where Is(~ρ,z) is the speckle intensity in sample andC(~ρ,z) the fluorophore concentration.We normalize our PSF’s such that PSFill (0,0) = PSFdet(0,0) = 1,

∫

PSFill (~ρ,z)d2~ρ = As and∫

PSFdet(~ρ,z)d2~ρ = Ad .Again we consider a uniform fluorescent plane located at axial positionzc. The fluorophore

concentration is thenC(~ρ,z) = C δ (z− zc), and the intensity detected at the CCD camera canbe simplified to

Id(~ρd) = C∫

PSFdet(~ρd −~ρ,−zc)Is(~ρ,zc)d2~ρ. (7)

Denoting as∆Id the difference between two consecutive images, we find,

∆Id(~ρd) = C∫

PSFdet(~ρd −~ρ,−zc)Is(~ρ,zc)d2~ρ −C

∫

PSFdet(~ρd −~ρ,−zc)Is(~ρ − ~∆ρs,zc)d2~ρ(8)

where ~∆ρs is the speckle pattern translation vector. Equivalently,

∆Id(~ρd) = C∫

[

PSFdet(~ρd −~ρ,−zc)−PSFdet(~ρd −~ρ + ~∆ρs,−zc)]

Is(~ρ,zc)d2~ρ. (9)

For ease of notation we recast the structure functionDT (Eq. (1)) asDT (~ρd) = 12∆Id(~ρd)2,

where the overline indicates an average over many successive speckle patterns. Expanding, weobtain

DT (~ρd) =C2

2

∫ ∫

[

PSFdet(~ρd −~ρ,−zc)−PSFdet(~ρd −~ρ + ~∆ρs,−zc)]

[

PSFdet(~ρd −~ρ ′,−zc)−PSFdet(~ρd −~ρ ′ + ~∆ρs,−zc)]

Is(~ρ,zc)Is(~ρ ′,zc)d2~ρd2~ρ ′

At this point, we invokea-priori knowledge on the speckle statistics. Assuming that the specklepattern is fully developed and that the speckle size (correlation length) is roughly constant overa large depth of field [9], we have [10]:

Is(~ρ,z)Is(~ρ ′,z) = Is2[1+PSFill (∆ρ,0)] (10)

where∆ρ = |~ρ −~ρ ′|. DefiningRdet as the lateral autocorrelation of PSFdet:

Rdet(∆~ρ,zc) =∫

PSFdet(~ρd −~ρ,−zc)PSFdet(~ρd −~ρ +∆~ρ,−zc)d2~ρ (11)



and noting that∫

Rdet( ~∆ρ + ~∆ρs,zc)d2 ~∆ρ = A2d , we obtain finally:

DT (~ρd)= Is2C2

∫

[

Rdet( ~∆ρ,zc)− 12Rdet( ~∆ρ + ~∆ρs,zc)− 1

2Rdet( ~∆ρ − ~∆ρs,zc)]

PSFill (∆ρ,0)d2 ~∆ρ(12)

This expression is valid for arbitrary PSFill and PSFdet and we can progress no further with-out specifying these. In the case where both PSF’s exhibit Gaussian-Lorentzian profiles, thenDT (~ρd) can be solved analytically (see below). However, in practice, PSFdet is usually definedby a circular pupil. To consider this latter more realistic case, we assume thatRdet( ~∆ρ,zc) issmooth on the scale of a speckle grain size, defined by PSFill (∆ρ,0) (we are mostly concernedhere with out-of-focus background rejection, and for largeenough|zc| this assumption is rea-sonably valid). We then make the approximation PSFill ( ~∆ρ,0)≈ Asδ (∆ρ), and Eq. (12) readilysimplies to

DT (~ρd) = Is2C2As

[

Rdet(0,zc)− 12Rdet( ~∆ρs,zc)− 1

2Rdet(− ~∆ρs,zc)]

(13)

Equation (13) is equally valid for translated or randomizedspeckle patterns. In the case ofrandomized patterns(∆ρs → ∞), then only the first term in Eq. (13) is non-zero, and we recoverour previous result (Eq. (3)). Note: to recover this result,we invoke the general definition forthe lateral area of PSFdet at depthzc, given by

Area[PSFdet] =

[∫

PSFdet(~ρ,zc)d2~ρ]2

∫

PSF2det(~ρ,zc)d2~ρ

=A2

d

Rdet(0,zc), (14)

which is valid for arbitrary PSFdet.Further insight into the behavior ofDT (~ρd) is gained by reformulating Eq. (13) in terms of

the 2D Optical Transfer Function (OTF):

OTFdet(~k⊥,z) =1

Ad

∫

PSFdet(~ρ,z)e−i~ρ ~·k⊥d2~ρ (15)

where~k⊥ is a lateral spatial frequency. We note that

Rdet(∆~ρ,zc) =1

(2π)2A2d

∫

∣

∣

∣OTFdet(~k⊥,zc)

∣

∣

∣

2e−i∆~ρ ~·k⊥d2~k⊥ (16)

from which we readily obtain:

DT (~ρd) =Is

2C2As

(2π)2A2d

∫

∣

∣

∣OTFdet(~k⊥,zc)

∣

∣

∣

2[

1−cos( ~∆ρs · ~k⊥)]

d2~k⊥ (17)

Once again, this equation is equally valid for translated orrandomized speckle patterns. Inthe case of randomized patterns(∆ρs → ∞), then the term cos( ~∆ρs · ~k⊥) can be replaced by itsaverage value, namely 0. The random case expression forDT thus simplifies to

DR(~ρd) =Is

2C2As

(2π)2A2d

∫

∣

∣

∣OTFdet(~k⊥,zc)

∣

∣

∣

2d2~k⊥. (18)

The advantage of expressingDT andDR in terms of OTFdet is that a well-established analyt-ical approximation for OTFdet is available for circular pupils, given by [11]

OTFdet(k⊥,z) = g

(

k⊥∆k⊥

)

2J1(zk k⊥(∆k⊥− 1

2k⊥))zk k⊥(∆k⊥− 1

2k⊥)(19)



whereJ1 is Bessel function of the first kind of order 1, 2∆k⊥ = 2ksinα is the cutoff frequencyfor incoherent imaging (sinα corresponds to the detection numerical aperture andk = 2π/λwhereλ is the optical wavelength), and

g

(

k⊥∆k⊥

)

= 1−0.69

(

k⊥∆k⊥

)

+0.076

(

k⊥∆k⊥

)2

+0.043

(

k⊥∆k⊥

)3

. (20)

Figure 3(a) shows plots of∣

∣

∣OTFdet(~k⊥,zc)

∣

∣

∣

2for different values ofzc. This function is always

equal to 1 at zero spatial frequency, but then rapidly decaysat higher frequencies. The greaterthe defocus, the faster the decay.

Numerical integration of Eq. (18) indicates thatDR decays as 1/z2c (see Fig. 3(b)). The ran-

dom DSI out-of-focus background rejection (√

DR) therefore scales as 1/ |zc|, as expected fromour simple model in section 3.

In contrast, numerical integration of Eq. (17) indicates that DT decays as 1/z2c for |zc| .

12∆ρsk/∆k⊥ and as 1/ |zc|3 for |zc| & 1

2∆ρsk/∆k⊥ (see elbow in Fig. 3(b)). This transition de-focus can be interpreted as the defocus where the width of PSFdet is roughly equal to∆ρs.When |zc| is less than this transition defocus, the speckle translation is effectively seen as arandomization. However when|zc| is greater than this transition defocus, the speckle trans-lation preserves correlations in consecutive speckle patterns. To keep this transition defocusas tightly confined to the focal plane as possible,∆ρs must be chosen to be small (thoughno smaller than the speckle grain size!). In practice,∆ρs is chosen to be of order the specklegrain size. The translation DSI out-of-focus background rejection (

√DT ) therefore scales as

1/ |zc|3/2, also as expected from our simple model in section 3. The reason for this improved

background rejection is clear from Fig. 3(a), and stems fromthe extra factor[

1−cos( ~∆ρs.~k⊥)]

in the integrand of Eq. (17). This term effectively quenchesthe low-frequencies contributions

to∣

∣

∣OTFdet(~k⊥,zc)

∣

∣

∣

2, which decay only weakly (if at all) with|zc|.

Having developed a model that is valid for arbitrary PSF, we now examine how the DSIsectioning strength is influenced by the PSF shape. In the case of speckle randomization, DSIsectioning strength seems to depend little on the exact profile of PSFdet since both circular (seeabove) and Gaussian (see ref. [9]) pupil functions lead to quasi-confocal sectioning.

The situation is different, however, in the case of speckle translation. For example, ifPSFill and PSFdet are both defined by a same Gaussian–Lorentzian profile: PSF(~ρ,z) =

11+ζ 2 e−2ρ2/w2

0(1+ζ 2) (whereζ = λ zπw2

0andw0 is the PSF waist), and we calculateDT directly

from Eq. (12), we find

DT (~ρd) =2A2

3+2ζ 2c

[

1− e− 2∆ρs2

w20(3+2ζ2

c )

]

(21)

whereA = πw20/2. (this expression reduces to that found in ref. [9] when∆ρs → ∞). In the case

of small speckle translations thenDT (~ρd) ≈ ( 2A∆ρsw0(3+2ζ 2

c ))2 and we find that the DSI signal

√DT

exhibits truly confocal sectioning (as opposed to near-confocal sectioning for a circular pupilfunction). This suggests that we could improve out-of-focus background rejection by usingGaussian apodization in the detection pupil.

5. Experimental results

Our experimental setup, shown in Fig. 4, is based on a standard widefield microscope with anargon laser as a light source. A spatial light modulator (SLM- Holoeye LC-R-768) is used toapply a random binary phase mask to the laser beam, by randomly assigning to each pixel a



0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

10 1001E-5

1E-4

1E-3

0.01

0.1

1

|OT

F(k

, z

c)|

2

k / k

/ k

12 / k

80 / k

DT [n

orm

.]

zc

k2/ k

1-cos(kx s)

Fig. 3. DSI sectioning strength for a circular aperture OTF (Eq. (19)).Panel (a):|OTF(k⊥,zc)|2 for zc∆k⊥

2/k = 0 (solid blue line), 10 (dashed red line) and 40 (dash-dottedgreen line). The dotted black line corresponds to the modulation factor[1− cos(kx∆ρs)]for a speckle translation in thex direction of step size∆ρs = π/∆k⊥. Panel (b): Numer-ical evaluation ofDT (Eq. (17)) for a uniform plane sample as a function of defocuszc, for ∆ρs = π/∆k⊥ (bottom blue solid line),∆ρs = 12π/∆k⊥ (middle black solid line)and∆ρs = 80π/∆k⊥ (top green solid line), on a logarithmic scale. For∆ρs = π/∆k⊥ and∆ρs = 80π/∆k⊥, the results are fitted by straight lines of slopes−2 and−3 respectively.For ∆ρs = 12π/∆k⊥ there is a break between these slopes. Note: traces in panel (b) arenormalized so that the random DSI signal is unity atzc = 0.

phase-shift of 0 orπ. The SLM is imaged onto the back focal plane of the microscopeobjective,thereby producing widefield speckle illumination. The fluorescence emitted at the focal planeof the objective is then imaged onto a CCD camera (Retiga 2000R, Q-Imaging).

Various strategies can be used to translate the speckle pattern, the simplest of which is toapply a linear phase-gradient across the SLM (a phase slope in the objective back focal planeleads to an amplitude translation in the front focal plane).Unfortunately, the dynamic range ofour SLM was limited to aboutπ, and hence we had to resort to an alternative strategy: It is wellknown that a lateral translation of the random phase mask applied at the SLM plane will resultin a translation of the speckle pattern everywhere except atthe corresponding SLM Fourierplane [12, 13]. We therefore decided to displace this Fourier plane away from the objectivefocal plane by illuminating the SLM with a diverging beam instead of a collimated beam. TheSLM Fourier plane is then located at the geometric image of the effective illumination pointsource (see Fig. 4), and with this configuration a simple translation of the SLM phase maskinduces a translation of the speckle pattern in the region ofthe objective focal plane.

A more detailed schematic of this illumination configuration is shown in the inset of Fig. 4where, for simplicity, we removed the 1×1 telescope and depicted the SLM directly at the ob-jective back focal plane. The dotted lines illustrate the conjugation between the effective pointsource and the SLM Fourier plane. Moreover they provide a simple geometric picture relating aphase-mask translation∆ρM at the SLM plane to the corresponding speckle pattern translationsinside (or outside) the sample. At the SLM Fourier plane, this corresponding translation is equalto zero, and the speckle pattern is said to be “boiling”[12].Away from the SLM Fourier plane,the speckle is almost purely translated. The amount of translation ∆ρs is linearly proportionalto the distanced between the observation plane and the SLM Fourier plane:∆ρs = ∆ρMd/ fo



Fluorescence

Sample

Objective

Dichroic

beamsplitter

CCD camera

f3=200mm

f2=200mmf1=200mmArgon

laser

Focal plane

Back focal

planeSLM

Fourier plane

Effective

point

source fo

dFP

Focal plane

fo fo dFP

fo

M s

Fig. 4. Experimental setup. A spatial light modulator (SLM) imparts a random phase maskon an argon laser beam. The SLM is imaged onto the back focal plane of the microscopeobjective, so as to create a widefield speckle illumination (the SLM is used in reflection butdrawn here in transmission for simplicity). The fluorescent light emitted from the sampleis imaged onto a CCD camera. The SLM is illuminated with a diverging beam so as todisplace the SLM Fourier plane away form the objective focal plane. A translation of theSLM phase mask then results in a translation of the speckle pattern inside the sample (seetext for details). The inset is a simplified schematic of the illumination geometry.

(where fo is the objective focal length)[12]. The SLM Fourier plane can thus be regarded asa pivot plane about which the 3D speckle pattern becomes sheared. In our experimental setupfo = 4.5mm,dFP ≈ 100µm, and we choose∆ρM such that the speckle translation∆ρs at theobjective focal plane is on the order of a speckle grain diameter (< 1µm). We note that as longas|∆ρs/dFP| ≪ sinα (the case here), then the speckle translation always remains smaller thanthe width of the out-of-focus PSFdet, as required to ensure out-of-focus correlations betweenconsecutive speckle patterns.

We note that the theory presented in sections 3 and 4 is equally valid for translations com-prising directed walks (fixed translation steps and directions) or random walks (fixed translationsteps but random directions). The former was found to cause residual streaking in the final DSIimage, and we generally opted for the latter in practice (we verified experimentally that bothwalks lead to the same depth sectioning strength).

For demonstration purposes, we directly compare randomization versus translation DSIimaging by applying a speckle update protocol that alternates between randomization and trans-lation. In this way,Id,k+1 andId,k are statistically independent whenk is odd (by randomization)and statistically dependent whenk is even (by translation), and the correspondingDR andDT

are evaluated from:

DT =1

2N

N/2

∑k=1

(Id,2k+1− Id,2k)2 and DR =

12N

N/2

∑k=1

(Id,2k − Id,2k−1)2 (22)



-20 -15 -10 -5 0 5 10 150.0

0.2

0.4

0.6

0.8

1.0

1 10

0.01

0.1

1

D1

/2 [norm

.]

zc [ m]

D1

/2 [norm

.]

zc [ m]

Fig. 5. Experimental measurement of the DSI sectioning strength√

D for a uniform fluo-rescent plane sample (thickness less than 1.5µm), for a translation step size∆ρs of approx-imatively one speckle grain size (blue trace) and 30 speckle grain sizes (green trace). Thered trace corresponds to speckle randomization. The objective used inthis experiment hasa numerical aperture of 0.65 (Olympus 40× dry). Traces are shown in linear scale (a) andlogarithmic scale (b), and are normalized such that the random DSI signal is unity atzc = 0.For a large∆ρs, the sectioning trace is identical to that obtained with random DSI. In panel(b), the experimental traces are fitted with straight lines of slope−3/2 for small∆ρs, and−1 for a large∆ρs and random DSI.

Such an alternating speckle update protocol would not normally be used in practice, howeverit has the advantage here of allowing us to compare the sectioning strength ofDR andDT withrigorously identical data sets.

We emphasize that we have only considered “classical”speckle fluctuations throughout sec-tions 3 and 4, and have neglected the contributions of shot-noise toDR andDT . In practice, toremove these shot-noise contributions we must subtract them from DR andDT before applyingthe square-root operation to retrieve our final DSI images, as prescribed in section 2 (and beingcareful to take into account CCD-camera gain and offset).

To experimentally quantify the DSI sectioning strength, wemeasure the signal from a thinfluorescent plane as a function of its axial positionzc. Figure 5 displays the measured sig-nal

√DT for two different translation step sizes∆ρs, and for randomization. The green trace

corresponds to a step size∆ρs that is larger than the maximun out-of-focus PSFdet width. Asexpected, such a large step size is equivalent to randomization, and we observe the same quasi-confocal sectioning strength for translation and randomization (red trace), where

√DT decays

as 1/ |zc|. In contrast, when∆ρs is of order a speckle grain size (blue trace), we confirm thatDSI sectioning is improved to near-confocal and that

√DT decays as 1/ |zc|3/2, as expected

from theory.Finally, we present images of a fluorescent pollen grain (Fig. 6). From the same set of raw

images, we compute three different images: the translationDSI image (a), the random DSI im-age (b) and the widefield image (c) which is obtained simply byaveraging all the raw images.The translation∆ρs is set to approximatively 1 speckle grain size. Again, we observe that trans-lation DSI confers better sectioning than random DSI: the center of the pollen grain is darker,the details appear finer, and the overall image quality is significantly improved. Figure 6(d)illustrates a 3D image of the pollen grain reconstructed from a z-stack of 75 2D images.



(b)(a)

(c)

10 µm10 µm

10 µm

(d)

10 µm

Fig. 6. Images of a fluorescent pollen grain obtained using an objectiveof numerical aper-ture 1.3 (Olympus 40× oil). Panel (a): signal

√DT obtained with translating speckle (one

image from full z-stack - avi movie 2.27MB). Panel (b): signal√

DR obtained with ran-domized speckle. Panel (c): widefield image (i.e. average of the raw images). Panel (d): 3Dreconstruction from z-stack (using Image J - avi movie 1.75MB). 128raw images (acquisi-tion time 150 ms per image) were used for each sectioned DSI image. Images (a), (b) and(c) were calculated from the same set of raw images (see text for detail).

6. Conclusion

In conclusion, we have studied the sectioning capacity of DSI microscopy based on a (half)differential intensity variance algorithm, and have shownthat this sectioning capacity can beimproved by controlling the dynamics of the speckle. We havedemonstrated both theoreticallyand experimentally that out-of-focus background rejection scales as 1/ |zc|3/2 when we slightlytranslate the speckle pattern and as 1/ |zc| when we randomize it, wherezc is the defocus dis-tance. The improved sectioning strength with translation DSI is near-confocal. In addition, wedisplayed images of a fluorescent pollen grain and demonstrated a significant improvement inimage quality with translation DSI as compared to random DSI. This work is a step towardfurther improvements in DSI microscopy algorithms designed to increase sectioning strengthwhile reducing the number of raw images required for DSI image processing.

Acknowledgment

We thank Rainer Heintzmann for helpful discussions on shot noise and its subtraction. Thiswork was funded by the Whitaker Foundation.



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Dynamic speckle illumination microscopy with translated versus …users.df.uba.ar/dkunik/hilo/mertz_2006.pdf · 2011-09-13 · Dynamic speckle illumination microscopy with translated