(r) ) ∑ ∑ - Joerg Enderlein

6836 J. Phys. Chem. A 2004, 108, 6836-6841Image Analysis of Defocused Single-Molecule Images for Three-Dimensional MoleculeOrientation StudiesDigambara Patra, Ingo Gregor, and Jo1rg Enderlein*Institute for Biological Information Processing I, Forschungszentrum Jülich, D-52425 Jülich, GermanyReceiVed: April 26, 2004; In Final Form: June 15, 2004An efficient algorithm for pattern matching has been developed based on least-squares analysis of fitting adiscrete set of master patterns against measured images. This algorithm has been applied to determine threedimensionalmolecule orientations in defocused single-molecule images. The developed algorithm exploitsthe excellent agreement between electrodynamic calculations of single-molecule emission and experimentallymeasured images. The procedure is found to be reliable and simple and can be applied to any kind of patternrecognition where the patterns to be recognized are precisely known a priori. The procedure works well evenfor noisy and low-intensity signals as usually encountered in single-molecule experiments.IntroductionFluorescence spectroscopy of single molecules has becomea routine technique for studying the physical and chemicalbehavior of individual molecules and has become an increasinglyimportant tool for molecular biology (for a recent overview ofthe field see ref 1). A topic of special interest has always beenthe experimental determination of the spatial orientation of themolecule’s absorption/emission dipole. The information aboutthis orientation is important for several reasons. On one hand,photophysical parameters of single molecules, such as fluorescencelifetime and observable emission intensity, are oftendependent on their orientation. On the other hand, singlemoleculeorientation can itself be a probe for studying photophysics2,3 or the structure of the embedding environment 4,5 orfor probing the orientation of labeled biomolecules. 6-10 In thepast, several techniques have been proposed for determiningsingle-molecule orientations by fluorescence imaging. The coreidea of these approaches is to obtain information about theangular distribution of a single-molecule’s fluorescence emissionby “deteriorating” the image of the molecule either by introductionof aberration, 11-14 by defined image defocusing, 15-17 orby imaging of the collected fluorescence light with a Bertrandlens. 18 In all cases, the intensity distribution of the blurred imagecontains information about the molecule’s emission-dipoleorientation. The defocusing method was proposed in refs 15and 16 and experimentally demonstrated for an immersionmirror objective used for imaging within a cryostat at lowtemperature. In ref 17, this concept was applied to imagesurface-bound molecules using a conventional CCD-imagingepi-fluorescence microscope with laser wide-field illumination.This method is easy to implement and allows for fast screeningof single-molecule orientations, without the necessity of scanning,excitation light modulation, or multiple channel detection.Thus, it is useful for applications where three-dimensionalorientations of single molecules constitute an important measurementparameter. However, for analyzing large numbers ofmolecule orientations, it was necessary to develop efficient* To whom correspondence should be addressed. Fax: +49-2461614216. E-mail: j.enderlein@fz-juelich.de.pattern recognition algorithms that can localize single-moleculepatterns within an image and determine their three-dimensionalorientation even under noisy low-signal conditions. The descriptionof such an algorithm and the demonstration of itsperformance is the aim of the present paper.TheoryImage Analysis. Let x jk be the intensity value of the pixelwith coordinates (j,k) in the original image. The task is toidentify and localize pre-defined patterns within the image. Thus,the algorithm has to compare all possible subregions of theimage with a discrete set of R predefined patterns p (r) jk whichshall have the size (2L + 1) × (2L + 1) (with 1 e r e Rcounting the patterns and -L e j e L and -L e k e L countingthe pixels of the patterns). Additionally, a uniform backgroundpattern b may be superimposed and the pattern-matching maybe restricted to nonrectangular subareas defined by the supportmatrixes s (r) jk of the same size as the patterns but having values1 and 0 only. Furthermore, it is assumed that, within any givensubarea of the image, only one of the R patterns can be present.Pattern comparison is done in the following way. The algorithmfits, by a least-squares method, each of the R patterns (plus aflat background) to all possible subareas of the image. For everysubarea, the algorithm chooses the pattern yielding minimalleast-squares error to be the most likely pattern present in thegiven subarea. Thus, for any given subarea of the image x withpixels m - L e j e m + L and n - L e k e n + L, the(r) (r)algorithm tries to find 2R parameters c nm and d nm so that the(r)least-squares errors e mn(r) ) ∑e mnLL∑j)-Lk)-Ls jk (x m+j,n+k - c (r) mn p (r) jk - d (r) mn b jk ) 2 (1)are minimized. Subsequently, it is assumed that the patterns(r)p jk as well as the background pattern b jk are all squarenormalizedso that10.1021/jp048188m CCC: $27.50 © 2004 American Chemical SocietyPublished on Web 07/24/2004

Image Analysis of Defocused Single-Molecule Images J. Phys. Chem. A, Vol. 108, No. 33, 2004 6837L∑L∑j)-Lk)-Ls jk (p (r) jk ) 2 ) 1 and ∑By differentiating with respect to the coefficients c (r) mn and d (r) mn ,these conditions lead to the 2R equations(r)∂e mn(r)∂c mn(r)∂e mn(r)∂b mnL) ∑) ∑∑j)-Lk)-LLL∑with 1 e r e R. Introducing the abbreviationsthe two equations can be rewritten in matrix form as( 1 P (r)P (r) 1 )( c (r)mn(r) ) ( ) Q (r)mnX mn)(5)with the explicit solution( c (r)mn(r)b mn) ( ) 1 P (r)P (r) 1 ) X mn) ) 1Lj)-Lk)-Lj)-Lk)-LThe square normalization of the patterns, together with thecondition that all pixel values of the patterns are non-negative(and at least one pixel value positive), ensures that 0 < P (r)

6838 J. Phys. Chem. A, Vol. 108, No. 33, 2004 Patra et al.light radiation into the glass along direction kˆ ){sin η cos ψ,sin η sin ψ, cos η} and solid angle sin η dη dψ is given by(see, e.g., refs 17 and 21)E(η,ψ) ) e p [cos βE ⊥ p (η) +sin βE | p (η)cos ψ] + e s sin βE | s (η)sin ψ (9)withE p ⊥ (η) ) sin η[e -inz cos η + R p (η)e inz cos η ]E | p,s (η) ) cos η[e -inz cos η - R p,s (η)e inz cos η ] (10)where β is the angle between the molecule’s emission dipoleaxis and the optical axis (being perpendicular to the interface);e p and e s are orthogonal unit vectors perpendicular to theradiation direction kˆ and with e s oriented perpendicularly to theoptical axis; n denotes the refractive index of glass (that of airis assumed to be 1); and the R p,s (η) are Fresnel’s reflectioncoefficients 22 for plane p- and s-waves at the glass/air interfacereflected along direction kˆ into the glass. Without restrictionof generality it was assumed that the molecule’s emission dipolelies within the plane ψ ) 0. The minus sign in the second ofeq 10 refers to p-waves, the plus sign to s-waves. After imagingthe radiation through an aplanatic imaging optics onto the CCDcamera, the electric and magnetic fields on the surface of theCCD chip are given by 17,23-25b j}and{ e xe y}{ b x{ E jB j} ) ∫ 0η′ maxdη′ sin η′cos η′n cos η{ e jb y}) i sin β ×2{cos η′(J 0 - J 2 cos 2ψ)E p-cos η′J 2 sin 2ψE | |p + J 2 sin 2ψE s}in′ sin β) ×2exp[ikδz cos η](11)| |+ (J 0 + J 2 cos 2ψ)E s +i cos β cos η′J 1 E p⊥{ cos ψsin ψ }{-cos η′J 2 sin 2ψE | |s + J 2 sin 2ψE pcos η′(J 0 + J 2 cos 2ψ)E | |s + (J 0 - J 2 cos 2ψ)E p}i cos βn′J 1 E p⊥{+-sin ψcos ψ } (12)where the J 0,1,2 denotes Bessel functions of the first kind 26 withfunctional argument k′F′ sin η′; the k,k′ are the wave vectors inglass and air, respectively. The angles η′ and η are connectedthrough the magnification M via Abbe’s sine condition (aplanticoptics), M sin η′ ) n sin η. The root factor in eq 11 ensuresenergy conservation during imaging and δz is the value of imagedefocusing (how much the imaging objective is moved out offocus toward the sample). Finally, the position-dependent lightintensity on the CCD chip is given by the z-component of thePoynting vectorS ) (c/8π)e z ‚(E × B*) (13)ExperimentThe measurement system consisted of an inverted microscope(IX70, Olympus) equipped with 1.2 NA, 100× plan fluoriteFigure 2. Measured defocused image of Cy5 molecules embedded inpoly(vinyl alcohol) on glass near the air/polymer interface. Defocusingwas achieved by moving the objective 1 µm toward the sample.apochromat oil immersion objective and a high-sensitive PeltiercooledCCD camera (CoolSnap HQ, Roper Scientific). Widefieldillumination for fluorescence excitation was done with 37-mW light of an Ar/Kr-ion laser (Stabilite 2018, Spectra-Physics)at 647.1 nm, using an additional excitation filter (647NB4)within the excitation path. Excitation light was conducted tothe objective using a multimode glass fiber. After passing themultimode fiber, the excitation light was nearly circularlypolarized. A dichroic mirror (650DLRP, Omega Optical)reflected the excitation light into the back focal plane of theobjective (Köhler illumination mode) for achieving homogeneouswide-field illumination of the sample. Fluorescence wascollected by the same objective and was passed through thedichroic mirror and an additional emission filter (690DF40,Omega Optical) for background suppression (Raman/Rayleighscattering). The CCD camera was positioned exactly at theimage plane of the microscope’s side port; the size of one CCDpixel is 6.45 × 6.45 µm 2 . The knowledge of this value allowsdirect conversion of pixel position into length and thus directcomparison between measured and calculated patterns. Axialpositioning of the objective was performed with a piezoelectrictransducer (PiFoc P-721-20, Physik Instrumente) with subnanometerresolution. The studied sample consisted of standardmicroscope cover slides (Menzel) of 170-µm thickness, on topof which 100 µL ofa10 -9 M solution of the cyanine dye Cy5(Amersham) in bidistilled water containing 1% wt/v poly(vinylalcohol) (CLARIANT GmbH) was spin-coated and dried,yielding a sparse distribution of single Cy5 molecules withinthe polymer matrix. Glass and poly(vinyl alcohol) have amatching refractive index of ca. 1.52. Images of the samplewere taken with an exposure time of 20 s. Image focusing/

Image Analysis of Defocused Single-Molecule Images J. Phys. Chem. A, Vol. 108, No. 33, 2004 6839Figure 3. Calculated defocused images for a fixed defocusing value of 1.2 µm with varying polymer film thickness. Above every pattern, valued of the film thickness is indicated. All other parameters used in the calculations are the same as those in the experiment.Figure 4. Calculated defocused images for 36 different orientations of an electric dipole emitter. Above every pattern, inclination angle θ betweendipole and optical axis as well as angle φ of in-plane dipole orientation are indicated. All parameters used in the calculations are the same as thosein the experiment.defocusing was adjusted by moving the objective with thepiezoelectric transducer.ResultsFigure 2 shows the complete CCD image of a defocusedmeasurement, displaying the patterns of roughly a hundredmolecules. The defocusing value for that image was 1.00 (0.25 µm; i.e., the objective was first set to sharp focus and thenmoved toward the sample by an amount of 1 µm. Theuncertainty in the absolute value of defocusing is caused bythe uncertainty to find the exact position of sharp focus.Furthermore, the thickness of the polymer film between themolecules and the polymer/air interface is not exactly known.Fortunately, it occurs that the precise shape of the defocusedpatterns of single molecules with the dipole axis parallel to theair/polymer interface (in-plane dipole orientation) is verysensitive to both defocusing value and film thickness, whichallows, by comparison between measurement and theoreticalcalculation, determination of both values simultaneously. Forexample, Figure 3 displays defocused patterns for varying values

6840 J. Phys. Chem. A, Vol. 108, No. 33, 2004 Patra et al.Figure 5. Example of correct and misidentified molecule positions (stars). Beside the exact position (circles, center of patterns), sometimes thealgorithm identifies one-half of the bilaterally symmetric pattern as an image of a molecule with high inclination angle (stars, off-center positions).of polymer film thickness (at a fixed defocusing value of 1.2µm), showing the dramatic changes in pattern shape withvarying film thickness. It occurred that a defocusing value of1.2 µm and a film thickness value of 300 nm are matching theobserved patterns best (compare Figure 3 with Figure 2).Moreover, because all molecules with in-plane dipole orientationdisplay the same interference patterns in Figure 2, they have tobe all located at the same distance to the air/polymer interface,assumedly adsorbed at the polymer/glass interface.Fluorescence intensity of the molecules varies significantlyfrom molecule to molecule, making it rather difficult to applyconventional image analysis methods which are based onintensity discrimination. To automatically identify single moleculesand to determine their orientations, 36 master patternsof size 31 × 31 pixel for 36 different possible orientations were(m)calculated (see Figure 4), giving the p jk of eq 1. Thedefocused patterns are not very sensitive to the inclination ofthe dipole axis toward the optical axis. The bilaterally symmetrictwo-lobed images convert into circular ringed interferencepatterns only at large inclination angles. Thus, patterns at onlyfour different inclination angles, θ ) 0, 30, 60, and 90°, werecomputed. This shows also the limitation of the defocusedimaging method for determination of dipole orientation: Theprecision of in-plane dipole orientation is much higher than thatfor out-of-plane inclination. A uniform background pattern b jkwas assumed; the support matrix s jk was chosen to be a disk of15 pixel radius. As the criterion for a positive match betweena master pattern and a molecule image, the relation c˜mn > 0.8ẽ mn was used (see step five of the algorithm in the first partof the Theory section). The center coordinates of such identifiedregions were taken as the center coordinates of identifiedmolecules. However, it occurred that such a molecule localizationmethod often generates spurious molecule positions besidesthe correct position: the typical error is that the algorithmidentifies one-half of a bilaterally symmetric pattern of an inplanemolecule as the image of a molecule with high inclinationangle (see Figure 5). To avoid this, the matrix c˜mn was smoothed(convoluted with a uniform disk of 7 pixel radius) to yield c˜mn/ ,/and only after that the criterion c˜mn > 0.6 ẽ mn was used formolecule localization. The algorithm identified 108 molecules,and the found patterns are displayed in Figure 6. Notice theexcellent agreement between theory and observed patterns,demonstrating nicely that single fluorescing molecules behaveFigure 6. Composite image displaying all identified molecules andtheir attributed patterns.indeed as ideal classical dipole emitters of light. As can be seen,the algorithm works quite well for sufficiently isolated moleculepatterns, even at low signal strength. It is feasible to safelyidentify inclination angles with ca. 30° resolution, and in-planeorientations with at least 15° resolution. The algorithm hasproblems to localize molecules and to correctly identify theirorientation if there is too much overlap between neighboringmolecule patterns. This is easy to understand when taking intoaccount that the algorithm tries to find the best match betweena given image region and a discreet set of master patterns. Imageregions with overlapping patterns that are not similar to any of

Image Analysis of Defocused Single-Molecule Images J. Phys. Chem. A, Vol. 108, No. 33, 2004 6841the master patterns will likely produce senseless results. Butthis will probably be a problem for any image processingalgorithm.ConclusionWe have presented an efficient pattern matching algorithmbased on a least-squares analysis of fitting a discrete set ofmaster patterns against measured fluorescence images of singlemolecules. We applied the algorithm to defocused images ofsingle fluorescing molecules, whose shapes depend strongly onthe three-dimensional orientation of the molecule. The algorithmworked well, even for the low-intensity signals as experiencedwhen imaging single molecules with defocused optics. It shouldbe noticed that the algorithm is extremely sensitive to an exactcorrespondence between the master patterns and the measuredpatterns: using master patterns that only slightly deviate fromthe measured ones quickly deteriorates the recognition qualityof the algorithm. Thus, the presented algorithm is especiallyuseful for pattern recognition problems where the patterns tobe recognized are precisely known a priori. In the case ofdefocused imaging of single molecules, this is indeed the casedue to the excellent agreement between electrodynamic andoptical theory and measurement.Acknowledgment. We thank Martin Böhmer for his technicalsupport, and we are grateful to Felix Koberling for lendingus the 1.2 N.A. oil immersion objective. We are much obligedto Benjamin Kaupp for his generous support of our work.Financial support by the Deutsche Forschungsgemeinschaft(Grant EN 297/7), and the Humboldt Foundation is gratefullyacknowledged.AppendixConvolution of an image x with pixel values x jk and pixelrange 1 e j e J, 1e k e K, with a smaller pattern p with pixelvalues p jk and pixel range - L e j,k e L (2L + 1 < J,K), canbe efficiently computed using fast Fourier transforms. Theconvolution q is defined byq mn ) ∑j)-Lk)-LBy introducing the new field p / jk with elementsand the new field x / jk with cyclic elementsthis convolution can be rewritten asLL∑x m+j,n+k p jk(A1)p / jk ){ p j-L-1,k-L-1 -L e j - L - 1 e L ∧ - L e k - L - 1 e L0 |j - L - 1| > L ∨|k - L - 1| > Lx / jk ) x [(j-L-2)mod J] + 1,[(k-L-2)mod K]+1J Kq mn ) ∑j)1k)1∑/ /x m+j,n+k p jkNext, the images x* and p* are represented by their twodimensionalFourier transforms x˜* and p˜* asx / jk ) 1∑ J KJKr)1∑ x˜rs exp[ ( / 2πi rjs)1p / jk ) 1∑ J KJKr)1∑ p˜rs exp[ ( / 2πi rjs)1J + sk K)]J + sk K)]Inserting the last equations into eq A1 yieldsq mn ) 1(JK) 2∑ j)1) 1∑ J KJKr)1∑ x˜rss)1J K J K J Kk)1∑∑ r)1∑s)1u)1∑V)1∑ x˜rs/ p˜uV/ ×exp[ ( 2πi rm + (u + r )jJ/ /p˜-r,-sexp[ 2πi ( rm J + sn K)]sn + (V +s)k+K)]showing that the Fourier transform q˜mn of the convolution hasthe simple formq˜rs ) x˜rs/ /p˜-r,-sThus, the convolution can efficiently be calculated by performingtwo inverse Fourier transformations x* f x˜*, p* f p˜* andone forward Fourier transformation q˜ f q.References and Notes(1) Zander, C., Enderlein, J., Keller, R. 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