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The contrast-imaging function for tilted specimens
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Ultramicroscopy 107 (2007) 202–212 www.elsevier.com/locate/ultramic
The contrast-imaging function for tilted specimens Ansgar Philippsena,, Hans-Andreas Engelb, Andreas Engela a
Maurice E. Mu¨ller Institute for Structural Biology, University of Basel, Klingelbergstr. 70, 4056 Basel, Switzerland b Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Received 12 December 2005; received in revised form 20 July 2006; accepted 20 July 2006
Abstract A theoretical description of the contrast-imaging function is derived for tilted specimens that exhibit weak-phase object characteristics. We show that the tilted contrast-imaging function (TCIF) is a linear transformation, which can be approximated by the convolution operation for small tilt angles or for small specimens. This approximation is not valid for electron tomography, where specimen tilts are above 601 and specimen dimensions amount to some 10 mm. The approximation also breaks down for electron crystallography, where atomic resolution is to be achieved. Therefore, we do not make this approximation and propose a generalized algorithm for inverting the TCIF. The implications of our description are discussed in the context of electron tomography, single particle analysis, and electron crystallography, and the improved resolution is quantitatively demonstrated. r 2006 Elsevier B.V. All rights reserved. PACS: 42.30.Lr; 42.30.Va; 87.64.Ee Keywords: CTF; Space-variant point spread function; 3D reconstruction
1. Introduction Imaging with a far field optical system, such as an electron microscope, is best described by the linear systems theory: One assumes a space invariant point-spreadfunction (PSF) that is convoluted with the input object. Thus, the Fourier transform (FT) of the measured image is the product of the FT of the input object with the FT of the PSF, the so-called contrast transfer function (CTF). The deconvolution of the measurement into the original input involves division of the FT of the image by the CTF followed by inverse FT (FT1), and is referred to as ‘‘CTFcorrection.’’ Since in the imaging process information is only lost for spatial frequencies where the CTF has zeros, deconvolution is widely used for image restoration. To optimize information recovery in the presence of noise, Wiener filtering can be applied [1]. For the structural analysis of biological specimens, electron microscopy is one of the most useful methods. Corresponding author. Tel.: +41 612671476; fax: +41 612672109.
E-mail address: [email protected] (A. Philippsen). 0304-3991/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2006.07.010
As a result of technical progress, modern electron microscopes, equipped with coherent electron sources, provide atomic resolution routinely, even when operated far from focus. However, for organic matter atomic resolution is difficult to reach, because the electron beam quickly damages this matter. Therefore, atomic structure determination of biomolecules by electron microscopy has only been demonstrated so far for two-dimensional (2D) crystals, where it is possible to acquire information to a resolution of about 3 A˚ by taking advantage of crystallographic methods [2]. However, it remains a challenging task to grow highly ordered 2D crystals and to collect and extract accurate information at such high resolution. Suitable modern instruments keep the specimen at a temperature of a few Kelvin and illuminate it with a highly coherent beam, providing optimal prerequisites for data collection at atomic scale. Furthermore, to extract the structural information at high resolution, a rigorous correction of the optical artifacts is indispensable. Since different 2D projections are required to extract 3D information, one needs to tilt the specimen plane with respect to the optical axis. This leads to a particular optical problem, because the defocus changes in
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Nomenclature
l
CTF FT PSF TCIF a b
N z0 W0 c v q Q
Cs D
contrast transfer function Fourier transform point spread function tilted contrast-imaging function tilt angle of specimen out of plane orientation of tilt axis within image plane, b ¼ 0 means tilt axis is along y-axis spherical aberration constant of instrument sampling distance in digitized image
the direction perpendicular to the tilt axis. Thus, the PSF is no longer space invariant, and one can no longer apply the established concept of CTF deconvolution to correct the optical artifacts. This problem was recognized several years ago, and Henderson and coworkers developed a commonly used, heuristic algorithm for processing of 2D crystal images [3]. More recently, Winkler and Taylor [4] introduced a numerical solution based on solving a Fredholm integral equation of the first kind to perform a global correction. Both of these approaches will be discussed further below in the context of the derivation presented herein.
ing along the optical axis. Here, s is the 2D position vector in the image plane. Let us consider a single specimen point (r; z). The contribution of this single point to the total image amplitude at s is cr ðsÞ ¼ vðrÞhz ðs=M rÞ, where v(r) is the amplitude of the wave exiting from the specimen at (r; z), and M denotes the magnification. This description is valid for a thin object for which we can take v(r) to be the integral of all object contributions along the projection direction at position r, i.e. the true projection of the object. For the sake of simplicity we set M ¼ 1. The total image amplitude is the linear superposition of all individual specimen point contributions, i.e. it is obtained by integrating over the specimen area,
3 4
plzp2 Þ
.
Z
Z
cðsÞ ¼
dr vðrÞhz ðs rÞ.
dr cr ðsÞ ¼ S
The geometrical layout and the measurement process of the specimen is illustrated in Fig. 1. We consider a specimen point at (r; z), with r ¼ (x, y) and defocus z. The contribution to the image is blurred, as described by the defocus-dependent PSF hz(r). The contribution from each point is spread over a considerable area, as illustrated in Fig. 2; the larger the absolute value of the defocus, the farther reaching the spread, easily reaching hundreds of A˚. Generally, there is no closed form for the PSF hz(r). However, its FT, Hz(p) (the CTF), is based on the Scherzer formula [5] and is derived in standard textbooks (e.g. [6] or [7]). Considering only defocus and spherical aberration, it becomes H z ðpÞ ¼ AðpÞeiððp=2ÞC s l p
electron wavelength, calculated from acceleration voltage number of pixels in digitized image base defocus (defocus at center of specimen) Scherzer formula at the base defocus image amplitude specimen function measured image intensity transformation describing tilted contrast imaging 1010 m
A˚
2. Derivation of the tilted contrast-imaging function (TCIF) 2.1. Point spread and pupil function
203
(2)
S
r1
r2
Specimen
z(r1) z(r2)
Focal Plane Lens
(1)
Cs is the spherical aberration coefficient, l the electron wavelength, z the defocus, and A(p) is a masking function with a cut-off pmax, i.e. A(p) ¼ 1 for pppmax, A(p) ¼ 0 for p4pmax, and we take l51=pmax . Here, the 2D frequency vector is denoted by p ¼ ðpx ; py Þ, and its magnitude is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ¼ p2x þ p2y . 2.2. Image amplitude c Next, we evaluate the image amplitude cðsÞ 2 C of a tilted thin specimen illuminated by a plane wave propagat-
hz (r2)
hz (r1) Image
Mr2
Mr1
Fig. 1. Schematic PSF imaging. Schematic imaging of a tilted 1D specimen, with a lens-induced point spread function hz(r) and a varying defocus distance z from the focal plane. The coordinate system of r is parallel to the lens plane. The graphical representation is somewhat misleading, as the PSF and the resulting image amplitude are complex functions.
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Fig. 2. Defocus dependent point spread. For a single specimen point, the spread on the measured images can become considerable for large defocus values. For each defocus, the intensity profile in real-space along the line y ¼ 0 is plotted. For z ¼ 0 and z ¼ 800 nm, the full 2D image is given on the right-hand side.
In other words, due to the linearity of the imaging process, we can consider each specimen point (r; z) individually, i.e., each specimen point can be understood as a sample with zero extent and thus has a well-defined defocus z and PSF hz. Assuming a planar specimen, we write the defocus as zðrÞ ¼ z0 þ r d tan a, where z0 denotes the defocus offset, a the specimen tilt, d ¼ ðcos b; sin bÞ and where b is the orientation of the tilt axis in the specimen plane. Substituting z(r) into Eq. (1) yields Z 2 hz ðs rÞ ¼ dp AðpÞ e2piðsrÞp eiW 0 ðpÞ eiplrdp tan a (3) with W 0 ðpÞ ¼ ðp=2ÞC s l3 p4 plz0 p2 . From Eqs. (2) and (3), and by writing v(r) in terms of its FT V(k), we find the following expression for the image amplitude: Z Z Z cðsÞ ¼ dp AðpÞ e2pisp eiW 0 ðpÞ dr dk e2pikr
S 1 2pirðp2dp2 l tan aÞ V ðkÞe .
ð4Þ
Now we integrate over r and obtain a d function dðk fðpÞÞ with fðpÞ ¼ p 12dp2 l tan a. Integration over k yields Z cðsÞ ¼ dp AðpÞ e2pisp eiW 0 ðpÞ V ðfðpÞÞ. (5) Eq. (5) is valid for any planar, tilted specimen. We now apply the ‘‘weak object’’ approximation. It is valid for thin organic specimens, where elastic scattering by light atoms produces only small phase shifts, and the loss term is small as well. Thus, we use vðrÞ ¼ 1 ðrÞ þ ifðrÞ with ; f51, i.e. V ðpÞ ¼ dðpÞ EðpÞ þ iFðpÞ, where E(p) and F(p) are the FTs of e(r) and f(r), respectively. Note that f(p) vanishes at p0 ¼ 0 and at p1 ¼ 2 cotðaÞ d=l; but because p14pmax we can use AðpÞdðfðpÞÞ ¼ AðpÞdðpÞ. In the following, we assume
that all contributions to the image will be negligible for p4pmax and hence omit A(p). We obtain from Eq. (5): Z cðsÞ ¼ 1 þ dp e2pips eiW 0 ðpÞ EðfðpÞÞ þ iFðfðpÞÞ . (6) Further, we note that W0(p) is an even function. Also, e(r) and f(r) are real, thus EðpÞ ¼ EðpÞ and FðpÞ ¼ FðpÞ. 2.3. Specimen measurement The experimental measurement process records the intensity of the image amplitude, qðsÞ ¼ jcðsÞj2 ¼ cðsÞ cðsÞ . Ignoring the loss term e and higher order terms in f we obtain from Eq. (6) the image intensity Z qðsÞ const: þ dp e2pips i eiW 0 ðpÞ FðfðpÞÞ eiW 0 ðpÞ FðfðpÞÞ Z ¼ const: þ dp e2pips QðpÞ. ð7Þ We neglect the constant term and note that Q(p) is the FT of the measured image intensity q(s) with QðpÞ ¼ i eiW 0 ðpÞ F p 12dp2 l tan a ð8Þ eiW 0 ðpÞ F p þ 12dp2 l tan a . Eq. (8) is the main result of this derivation. We call this the TCIF, which is a linear transformation on F(k), i.e. QðpÞ ¼ TCIF½FðkÞ, as shown schematically in Fig. 3A.1 1
We note that for the special case of an untilted specimen, we have f(p) ¼ p and recover the well-known expression Qa¼0 ðpÞ ¼ 2F ðpÞ sinðW 0 ðpÞÞ.
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Fig. 3. Schematic representation of various TCIF transformations. (A) The 1D case of the linear transformation QðpÞ ¼ TCIF½FðkÞ, described by Eq. (8). The value at Q(p) is a linear combination of the two values F(k1) and F(k2) (each multiplied with a phase factor, see text), which are located in equal distance to the left and right of p, proportional to p2. F and Q represent specimen and measured image, respectively. (B) For the Fourier transform of a tilted 1D crystals image, F(k), the peaks will appear split in the measurement Q(p), according to Eq. (8). The two split peaks in Q are found at positions p1 and p2, as explained in the text. The distances are D1;2 ¼ 12p21;2 l tan a, with D24D1, i.e., the original position q is not located in the center of the split peaks. The gray bars indicate locations k0 1 and k0 2 from where F would also contribute to the measurement Q(p1) and Q(p2); however, for an image of a 2D crystal, it is generally the case that Fðk0 11 Þ ¼ Fðk0 22 Þ ¼ 0, since k0 1 and k0 2 are not on the reciprocal lattice of the crystal. Thus, the value of each Q(p) results from a F(k) with a well-defined k, in contrast to the generic case where contributions from k and k0 are present. (C) Effect of nested transformation Qð2Þ ¼ TCIF½TCIF½F. For a perfect crystal, the contributions from the left and right position (k1 and k2) generally have a contribution of zero (see text), and hence only the original value at k ¼ p is recovered.
2.4. Reduction to 1D Without loss of generality, we set b ¼ 0 (taking the y-axis parallel to the tilt axis). Writing FðkÞ ¼ Fðkx ; ky Þ, we note in Eq. (8) that d ¼ (1,0), thus each py corresponds to a fixed value of ky. Therefore, in Fourier space, the TCIF transformation is reduced to one dimension. Each line (with fixed py ¼ ky) of Q(p) as a function of px is given by Qpy ðpx Þ ¼ ieiW 0 ðpÞ F px 12p2 l tan a; py ieiW 0 ðpÞ F px þ 12p2 l tan a; py . ð9Þ Each such line can now be treated separately.
measurement, after the TCIF transformation described by Eq. (8). However, for evaluating Q ¼ TCIF½F for a discretely sampled weak-phase object fðxn ; yn Þ, with n ¼ 0 . . . N 1, the FT needs to be calculated at continuous frequencies px, rather than only for integer multiples of ðNDx Þ1 . Hence, to evaluate Eq. (9), we calculate FT[f] along each dimension as Fðpx Þ ¼ PN1 2pipx nD to obtain n¼0 fn e Fðpx 12p2 l tan a; py Þ ¼
N x 1 X n¼0
N y 1 X
fðxn ; ym Þe2pi
1 px 2p2 l tan a nDx 2pi py mDy
e
.
ð10Þ
m¼0
2.5. Interpolation of discretized data Upon actual recording of the measurement a discretized image qðxn ; yn Þ is obtained, whose FT gives Q at discretized values of p. We note that due to the coordinate transformation f(p), the frequency components p 12dp2 l tan a in Eq. (8) will usually not coincide with a discretized frequency, but rather take some intermediate values. This does not modify our above derivation of a physical measurement process, since this discretization happens at the very end of the
The FFT algorithm [8] may be used for calculating the discrete FT in y, yielding Qpy ðpx Þ ¼
NX x 1
ðsin x sin xþ Þ þ iðcos x cos xþ Þ
n¼0
~ p ðxn Þ F y
ð11Þ
with x ¼ 2p px 12p2 l tan a nDx W 0 ðpÞ
~ p ðxn Þ ¼ FTy fðxn ; ym Þ 2 C. F y
ð12Þ
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(13)
changes of the PSF, i.e. the resolution is limited by d, as illustrated in Fig. 4A. Let us now consider a given resolution d, i.e. we need pmax ¼ 1=d. Then, our description holds for samples of thickness
(14)
Dp
Equivalently, Eq. (11) may be written as Qpy ðpx Þ ¼ 2
N x 1 X
~ p ðxn Þ e2pipx nDx H~ py ðpx ; nÞ F y
n¼0
with
H~ py ðpx ; nÞ ¼ sin W 0 ðpÞ pp2 lnDx tan a 2 R.
In accordance with the nomenclature used above in Eq. (1), we use H~ in Eq. (14) to denote conceptually a generalized CTF that describes the effect of the tilt, whereby positions and frequencies are mixed and the description of a convolution no longer applies. The term nDx tan a corresponds to the defocus Dz at the spatial position given by index n. We will discuss H~ below. 2.6. Generalized inversion In order to extract the specimen information from the measurement, we must calculate F ¼ TCIF1 ½Q, i.e. Eq. (13) must be solved for F. For each py, it is a system of Nx linear equations, which can be solved to obtain Fpy , as demonstrated below for some artificial data. To calculate ~ the instrument parameters Cs, l, z0, and a must be H, known at a sufficient level of accuracy (see below) that can be estimated using this theoretical framework. 3. Discussion We have derived an analytical expression for the image intensity distribution q(s) of a tilted weak-phase specimen, illuminated by a plane wave and imaged by a lens with spherical aberration Cs. It is shown that a linear 1D transformation describes this process in Fourier space, and we call this transform the TCIF. This now allows the image formation under tilted conditions to be simulated, errors of the conventional CTF correction procedure to be estimated, and an accurate reconstruction of the original specimen projection v(r) from the measured intensity q(s).
d 2 cos a 0:61l
(16)
which is plotted in Fig. 4B. A very similar relation was found by DeRosier [9], who considered the deviation from the projection approximation based on the curvature of the Ewald Sphere. He derived p anffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi estimate for the maximum resolution given by pmax ¼ 2C=ðDlÞ, where D is the thickness of the sample and C a constant to be determined. He estimated the latter by a series of simulations of different specimen models, where a phase error of 661 was used as the maximum tolerable error, yielding C ’ 0:7 as a ‘‘quick rule of thumb’’. Rewriting of Eq. (16)preveals immediately that ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi our estimation gives pmax ’ 1:64=ðD0 lÞ, where D0 ¼ D= cos a is the tilt corrected thickness; thus we have a corresponding C of 0.82, and the two estimates are in agreement. An alternative way to estimate the resolution limit can be seen in Fig. 4C, D, where individual point spread functions are summed over the range of the sample thickness, and the Fourier transform of this sum reveals a first diminution of the overall signal at approximately the same resolution limit as estimated in Eq. (16) above. 3.2. Classical CTF deconvolution To what extent does the TCIF differ from the CTF, i.e. what errors are introduced by ignoring the tilt and correcting an image assuming constant defocus? From Eq. (14) (or indeed already from Eq. (1)) an estimate of the error is found by noting that for proper reconstruction the phase deviation must be less than p/2. Thus, the CTF approximation is valid for
3.1. Sample thickness limitations
2p2 lNDx tan ao1.
To derive the analytical expression we considered a thin sample of thickness D. We have implicitly assumed that the PSF does not change on the scale of the attempted resolution when integrating the sample contribution along the projection axis at a given coordinate r. The radius rPSF is approximately rPSF yz, where y 0:61lpmax dictates the resolution (after CTF correction the optical system is diffraction limited to a first approximation). Moving along the projection axis, the sample has thickness D= cos a, and the radius of the PSF changes by
The error of the CTF approximation is illustrated in Fig. 5, where the difference between H and H~ is plotted against the frequency p for several values of x ¼ NDx . Further, if a too small sampling area (in real space) is used, the TCIF and CTF can no longer be discriminated. This is seen in frequency space when the two contributions p 12p2 l tan a (Eq. (8) and Fig. 3A) collapse on the same pixel for
yD . (15) d¼ cos a As the features of the PSF scale uniformly (to a first approximation), Eq. (15) corresponds to the spatial
p2 l tan ap
1 . ND
(17)
(18)
A conclusion to draw from these estimates is that for a small image extent, the CTF approximation will only fail at high resolution, whereas for a large image extent, proper TCIF correction is required at fairly low resolution already.
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Fig. 4. Resolution estimation based on sample thickness for l ¼ 2 pm. (A) Graphical construction of the estimated resolution limit for the projection approximation. In this case, a sample thickness of D ¼ 200 A˚ at a tilt of 601 is assumed, with the length of the projection path 400 A˚. One-half of the PSF is plotted for each of the three indicated defocus values, and the difference in the extent of the PSF, judged by the distance between two corresponding peaks, is indicated by d. (B) Log–log plot of the estimated resolution limit against the maximal sample thickness, as given by Eq. (16). The abscissa contains the shift d, as given in (A) and explained in the text. The ordinate displays the maximal thickness of the sample where the projection approximation may be considered fulfilled at the given resolution and the given tilt angle. For example, at a tilt of 601, a resolution of 6 A˚ is achievable with a sample thinner than 120 nm (indicated by the dotted lines). (C) Summed, normalized PSF—in steps of 1 nm for the range of 1020–980 nm—reveals how the accumulation of slightly varying PSFs per slice modifies the overall PSF. (D) Fourier Transform of (C) exhibits a first diminution of the overall shape at around 2 A˚1, corresponding to the estimation in (B), namely that the projection estimation of a 20 nm thick sample tilted at 601, and hence with a projection path of 40 nm, can be considered valid up to a resolution of 2 A˚.
3.3. Artificial 1D specimen and measurement To illustrate further aspects of the TCIF transformation described here, we transform a 1D artificial specimen using Eq. (8), with py ¼ 0. The specimen function f is based on a sum of Gaussians, and we use analytical (non-interpolated) values F(p), see Appendix. To obtain a simulated measurement, the resulting analytical transform is evaluated at evenly sampled x, with spatial sampling D ¼ 1 A˚ and N ¼ 512 sampling points, as illustrated in Fig. 6. Here, the tilt angle is a ¼ 601, therefore the specimen occupies about one quarter of the discretized measurement area. According to the above error estimation (Fig. 5), for 300 kV electrons (l ¼ 2 pm), the resolution limitation of the traditional deconvolution method is about 10 A˚, which is in good agreement with our simulation in Fig. 6, where
deconvolution has been used as a possible correction approach: this approach breaks down at frequencies representing a resolution better than 6 A˚. In contrast, the reconstruction of the original specimen function utilizing the inversion approach described above yields the original input function within the numerical error limitations. Winkler and Taylor [4] apply similar ideas to their derivation, in particular the reduction to one dimension and regarding the effect in the frequency domain. They obtain a generic expression corresponding to a Fredholm integral equation of the first kind, which contains a spatially dependent kernel H(y, v) (their Eq. (3)). They point out that such inversion schemes are inherently ill posed, and they provide a recipe to control the singular value decomposition with a regularization parameter to
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~ x ¼ NDx Þ, the TCIF, is plotted for four different object Fig. 5. Error estimation for CTF vs. TCIF. The difference between H(p), the CTF, and Hðp; dimensions against the spatial frequency p. The larger the difference (ordinate), the more error will be introduced by assuming a constant defocus for the correction of the optical distortions. For samples on the order of a few thousand A˚, where the distance from the base defocus is in the order of 1000 A˚, significant differences start to occur below 10 A˚ resolution. For larger specimen, the proper inclusion of the TCIF effect must occur already at lower resolutions. The error estimation given in Eq. (17), namely the p/2 criterion, is indicated as a vertical line for each sample size.
avoid small terms from dominating. In our simulation in Fig. 6, no such regularization was necessary. However, Winkler and Taylor do not derive an exact expression for a particular defocus model, but rather utilize a numerical solution based on a matrix equation, where each matrix element is explicitly calculated from H(y, v). Also, they do not provide the explicit formula they use for their subsequent calculations. 3.4. Robustness against noise and uncertainties in optical parameters In order to have a first assessment of the robustness of our inversion scheme under more realistic conditions of experimental data, we performed inversions of the TCIF following the setup from Fig. 6, either after adding noise to the simulated measurement or using slightly modified optical parameters. When adding 2% white noise (based on the mean sample value), our procedure is robust against such noise, i.e. it does not seem to amplify high-frequency noise contributions; in contrast to the standard deconvolution method where filtering would be required. When evaluating the sensitivity to variations in optical parameters, we consider our inversion scheme as successful if it leads to a result closer to the original sample than the deconvolution method with the exact optical parameters. As expected, the dependence on the parameter errors is non-linear; however, we can estimate the thresholds as follows. Errors of 0.5% for l, 7% for Cs, 1% for z0, and 20% for a still yield successful inversions according to our criteria (plots not shown). Thus, our procedure is also
robust against errors in the optical parameters, and can be applied to experimental data. We defer a discussion of concrete experimental data to a future work. 4. Implications for biological electron microscopy The three major methods that utilize electron microscopy of biological specimens are electron tomography, single-particle analysis, and electron crystallography. We now discuss the implications of the optical distortions in imaging tilted specimens for these three methods. 4.1. Electron tomography Electron tomography attempts to produce 3D maps of entire cells or thin sections thereof at a resolution of ultimately 20 A˚ [10]. In this case, the specimen size may reach 10 mm, and when such a sample is tilted to 601, the defocus ranges over 4.3 to 4.3 mm; at both of these limits, the first zero-crossings2 of the CTF are found at approximately (30 A˚)1. Therefore, appropriate correction of the optical defects is required to achieve the anticipated resolution. Indeed, the above estimation (Eq. (17)) gives corresponding resolution limits: for this example, conventional CTF correction applied to the whole image will limit the resolution to about 50 A˚. When tilt angles are p151, the maximum resolution will be 20 A˚, but as result of the missing cone artifact [11] tilting to higher angles is a prerequisite for obtaining a meaningful 3D reconstruction. The related gain in resolution upon appropriate correction 2
sin (W0) ¼ 0, with l ¼ 2 pm and CS ¼ 2 mm.
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Fig. 6. Simulated measurement and inversion of an artificial 1D specimen. The artificial 1D original specimen is based on a sum of Gaussians, from which the projection (tilted specimen) and TCIF transformation (measurement simulation) is calculated analytically (see text and Eq. (8)). The curves in the frequency domain represent real and imaginary part, respectively. The values for the simulated image were discretized into 1 A˚ steps and subjected to ‘‘classical’’ deconvolution and full TCIF inversion, as described in the text. We used singular value decomposition for solving Eq. (13); however, we note that no singularities were encountered in this inversion, i.e., there were no vanishing eigenvalues. While the deconvolution approach introduces considerable errors, particularly for high-resolution frequencies (o10 A˚), the inversion based on the TCIF transformation is able to completely restore the specimen. Furthermore, we note that our inversion scheme is robust against small amounts of measurement noise; as an example, we show the inversion after adding 2% white noise to the measurement simulation (based on the sample average), which results in a reconstructed sample that is still better than the one from the classical deconvolution without noise.
of the tilt effect has been demonstrated by Winkler and Taylor [4], or using an alternative stripe-based approach by Fernandez et al. [12]. However, fundamental limitations of the achievable resolution actually result from the beam sensitivity of vitrified preparations. Grimm et al. [13] have estimated the highest achievable resolution as function of the sample thickness for 300 kV. For specimens whose thickness corresponds to multiples of the mean path between electron scattering events, the attainable resolution is far worse than the limitations of the optical system discussed here, e.g. the attainable resolution is about 90 A˚ for an 800 nm thick sample. Therefore, to
achieve 20 A˚ resolution as required for identification of macromolecular complexes within a cell, samples need to be thinner than 250 nm [13]. Progress in the preparation of frozen sections [14] opens an avenue to produce such samples and to overcome the thickness limitations. In this case the goal of 20 A˚ could be reached provided that optical defects inherent to images of tilted samples are properly corrected. 4.2. Single particle For 3D reconstruction of single particles, tilting the specimen is not required if the particles are randomly
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oriented within the vitrified solution layer, and hence non-tilted images are sufficient to obtain all orientations needed for reconstruction. However, in the case of samples that assume a preferred orientation, the 3D reconstruction is achieved by a tomographic approach using the random conical tilt method [15]. This is the case for membrane bound complexes that are kept in solution by detergents, which requires a carbon support film rather than a fenestrated film [16,17]. Random conical tilt is also of advantage if assignment of the projection direction is difficult. Typical sample areas have a diameter of 1 mm, at 601 the defocus range is 70.43 mm, and corresponding first zero-crossings of the CTF are at around (9 A˚)1. In pursuit of the goal to achieve atomic scale resolution, appropriate correction of the tilt-related optical defects is needed here as well. The commonly used correction scheme starts with the extraction of rectangular areas (subimages), centered on each particle of interest, and subsequent isolated processing of each subimage. Irrespective of the tilt, the ideal extent of each subimage is given by the following conditions: (i) include all spread information due to the PSF (see Fig. 2); (ii) exclude information from any other particle within its boundaries. This effectively results in an optimal separation of particles that is about twice the spatial extent of the PSF. Assuming that such subimages are indeed obtained, the correction for non-tilted specimens may proceed by conventional CTF correction. For tilted specimens two alternative approaches may be taken. One is a full TCIF correction across the subimage (as outlined above). The other is to simply ignore the tilt across the subimage and go ahead with the convential CTF correction. Since the defocus variation across a subimage is small, the latter approach is feasible and will yield acceptable results (see Eq. (16)), cf. Fig. 6. However, if the distribution of particles is so dense that image contributions from differents particles overlap as a result of the extended PSF, the best way to correct for optical artifacts is the full TCIF correction of the complete image for tilted specimens.
4.3. Electron crystallography Regarding electron crystallography, we discuss the implications only for a 1D crystal; the extension to 2D is straightforward (Eq. (9)). As shown in Fig. 3B, a diffraction peak F(k) of a periodic specimen f(x) will contribute to the TCIF-transformed specimen Q(p) at two different values py ¼ p1 ; p2 , given by the conditions p1 þ 1 2 1 2 2 2p2 l tan a ¼ k, which can be 2p1 l tan a ¼ k and pp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2kl tan a 1 ðl tan aÞ1 and solved to give p1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 ¼ 1 1 2kl tan a ðl tan aÞ1 , with p1 ; p2 ; kX0 and a; l40. The distance between the split peaks (plotted
Fig. 7. Peak split distance for crystallographic data. Based on Eq. (8), the distance between peaks in the FT of tilted 2D crystal images may be calculated. Here we show the predicted distance for image of 10 000 pixels, with a sampling of 1 A˚. The abscissa of the main graph is displayed with the frequency encoded as resolution from 100 to 2 A˚, the abscissa of the inner graph from 20 to 4 A˚, reflecting the resolution range where the peak separation becomes visible. The ordinate is in absolute pixels. The four curves represent tilt angles of 151, 301, 451 and 601.
in Fig. 7) is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 p1 ¼ 2 1 2kl tan a 1 þ 2kl tan a ðl tan aÞ1 ; in the 2D case, the vector between the split peaks is perpendicular to the tilt axis. It should be noted that the center of the split peaks 12 p1 þ p2 does not exactly correspond to the frequency k of the original sample peak, but is offset by approximately k3 l2 tan2 a. In practical terms, this offset is much smaller than the reciprocal pixel spacing and may thus be neglected. Henderson and coworkers [18] were the first to describe this effect and offer a correction mechanism that is applicable to crystalline samples. They obtain the same expression for H~ (Eq. (14)) by taking the Scherzer formula (Eq. (1)) with position dependent defocus z(x) and using it in the CTF. Accordingly, the consequences of imaging a tilted sample is a ‘‘phase grating’’ effect; the apparent specimen function is fapp ðxÞ ¼ fðxÞ sinðc1 k4 þ c2 k2 þ c3 kxÞ for a specific reflection at spatial frequency k, whose contrast sinusoidally varies as a function of the spatial position x, and the period of the sinusoidal varying as a function of the particular frequency k. The heuristic correction approach is somewhat unconventional, as the already transformed image is again (numerically) trans-
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~ as justified by the phase grating formed with the same H, effect. Based on Eq. (8) derived above, this approach can be understood as a nested TCIF transformation of the form Qð2Þ ¼ TCIF½Q ¼ TCIF½TCIF½F, which can be approximated by only keeping terms linear in (pl), yielding
Qð2Þ ðpÞ 2F½p e2iW 0 ðpÞ F p dp2 l tan a
þ e2iW 0 ðpÞ F p þ dp2 l tan a . ð19Þ At each frequency, the original value is combined with values from two neighboring frequencies, as shown in Fig. 3C. For a crystalline specimen, the two neighboring contributions are zero, since p p2 l tan a is generally not equal to a reflection frequency. Consequently, under the assumption of a perfectly ordered crystal, where only a few, widely separated scattering angles form the final image, the nested TCIF transformation is a good approximation to the inverse TCIF, and the correction approach taken by [18] is valid. However, to correct images from real, imperfect datasets, it is necessary for high resolution to perform an ‘‘unbending’’ procedure [18] to construct a perfect crystal. The drawback of unbending first and correcting for the tilt afterwards, however, is that during unbending the continuity of the PSF is disrupted; this violates the assumption made for the PSF, and full information is not restored during tilt correction. For this reason, even for the case of 2D crystals, it is imperative that the correction of the TCIF is done prior to unbending (however, now Q(2) is no longer a good approximation, because in Eq. (19) the contributions from the two neighboring frequencies might not vanish). Therefore, the established scheme [3], where the unbending precedes the contrast function correction, is not suitable if high-resolution information is to be extracted. Finally, for the realistic case that each diffraction spot is really formed by a collection of closely related scattering angles, the above-described general inversion may be applied to a region surrounding diffraction spots only, without the need to perform a global image correction. Their extent depends on the crystal order, the tilt angle and the resolution to be achieved. Crystal order and tilt angle need to be determined from the experimental data, most likely at medium resolution. The resolution to be reached then dictates the size of the regions to be processed.
5. Outlook So far, we have limited our derivation to a specific set of approximations, in particular we have (a) ignored the effects of elastic scattering beyond pmax and of inelastic scattering, (b) considered only the defocus and spherical aberration for the contrast-imaging function, and (c) not fully explored the effects of noise and uncertainty in the optical parameters. We intend to relax these assumptions and to include such properties to our mathematical description in a future work.
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The suggested TCIF inversion approach harbors two particular limitations. First, we do not yet know to what accuracy the pertinent experimental parameters are required prior to reconstructing specimens from actual measurements of biological specimens. Second, when solving the linear equation system described above, for every sample line perpendicular to the tilt axis, the time requirement scales with N3; this becomes a significant computational effort for large samples (e.g. N ¼ 10 000 pixels for a crystal of 1 mm diameter). The numerical implementation of our inversion approach can be enhanced by more refined numerical methods, reducing time requirements and including parameter estimation and noise handling. The simple estimates of the errors introduced by the conventional CTF correction protocol presented here will also serve as guide to decide whether or not full TCIF correction is necessary or if conventional CTF is sufficient. 6. Conclusion Our derivation of the contrast-imaging function for tilted, thin, weak-object specimens serves as a theoretical framework upon which concrete correction algorithms may be build upon, and the discussed error estimations will help to decide when such a correction is feasible and necessary. Acknowledgements We thank Dirk van Dyck, Philip Koeck, and Andreas Schenk for discussions and for critical reading of the manuscript. This work was supported by the EU 6th framework (3DEM-NoE), the NCCR Structural Biology, and by NSF Grants nos. PHY-01-17795 and DMR-02-33773. Appendix. Sum of Gaussians TCIF transform We define a Gaussian with area A,pmean ffiffiffiffiffiffi m, and2 standard 2 deviation s, i.e. Gðx; A; s; mÞ ¼ ðA=s 2pÞ eðxmÞ =ð2s Þ , and its corresponding Fourier transform is G FT ðp; A; s; mÞ ¼ 2 2 2 Ae2pimp e2p s p . Thus, for a sum of N Gaussians having different parameters Ai, si, and mi, we obtain G QG ðpx Þ ¼ FG þ Fþ sin W 0 ðpx Þ G ð20Þ þ i FG Fþ cos W 0 ðpx Þ P N 1 2 with FG ¼ i¼1 G FT ðp 2px l tan a; Ai ; si ; mi Þ. Deconvolution Given a noise-free, artificial measurement, using known instrument parameters, we deconvolute using 8 QðpÞ < for j2 sin W 0 ðpÞjX; 2 sin W 0 ðpÞ FDeconv ðpÞ ¼ (21) : QðpÞ otherwise; where e is a lower-cutoff to avoid division by zero.
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Inversion Solving the linear equation system given in Eq. (13) was implemented as an algorithm in IPLT [19], using the linear algebra routines from the GNU scientific library [20]. References [1] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes in FORTRAN, Cambridge University Press, 1992. [2] Y. Fujiyoshi, Adv. Biophys. 35 (1998) 25. [3] R. Henderson, J.M. Baldwin, T.A. Ceska, F. Zemlin, E. Beckmann, K.H. Downing, J. Mol. Biol. 213 (1990) 899. [4] H. Winkler, K.A. Taylor, J. Struct. Biol. 143 (2003) 24. [5] O. Scherzer, J. Appl. Phys 20 (1949) 20. [6] L. Reimer, Transmission Electron Microscopy, Springer, Berlin, 1997. [7] J.C.H. Spence, High-Resolution Electron Microscopy, Oxford University Press, Oxford, 2003. [8] J.W. Cooley, O.W. Tukey, Math. Comput. 19 (1965) 297. [9] D.J. DeRosier, Ultramicroscopy 81 (2000) 83.
[10] O. Medalia, I. Weber, A.S. Frangakis, D. Nicastro, G. Gerisch, W. Baumeister, Science 298 (2002) 1209. [11] A.J. Koster, R. Grimm, D. Typke, R. Hegerl, A. Stoschek, J. Walz, W. Baumeister, J. Struct. Biol. 120 (1997) 276. [12] J.J. Fernandez, S. Li, R.A. Crowther, Ultramicroscopy 106 (2006) 587. [13] R. Grimm, H. Singh, R. Rachel, D. Typke, W. Zillig, W. Baumeister, Biophys. J. 74 (1998) 1031. [14] A. Al-Amoudi, J.J. Chang, A. Leforestier, A. McDowall, L.M. Salamin, L.P. Norlen, K. Richter, N.S. Blanc, D. Studer, J. Dubochet, EMBO J. 23 (2004) 3583. [15] M. Radermacher, T. Wagenknecht, A. Verschoor, J. Frank, J. Microsc. 146 (1987) 113. [16] M. Chami, I. Guilvout, M. Gregorini, H.W. Remigy, S.A. Muller, M. Valerio, A. Engel, A.P. Pugsley, N. Bayan, J. Biol. Chem. 11 (2005) 37732. [17] N. Eifler, M. Vetsch, M. Gregorini, P. Ringler, M. Chami, A. Philippsen, A. Fritz, S.A. Muller, R. Glockshuber, A. Engel, U. Grauschopf, EMBO J. 25 (2006) 2652. [18] R. Henderson, J.M. Baldwin, K.H. Downing, J. Lepault, F. Zemlin, Ultramicroscopy 19 (1986) 147. [19] A. Philippsen, A.D. Schenk, H. Stahlberg, A. Engel, J. Struct. Biol. 144 (2002) 4. [20] Galassi et al., GNU Scientific Library Reference Manual, http:// www.gnu.org/software/gsl.
Ultramicroscopy 107 (2007) 202–212 www.elsevier.com/locate/ultramic
The contrast-imaging function for tilted specimens Ansgar Philippsena,, Hans-Andreas Engelb, Andreas Engela a
Maurice E. Mu¨ller Institute for Structural Biology, University of Basel, Klingelbergstr. 70, 4056 Basel, Switzerland b Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Received 12 December 2005; received in revised form 20 July 2006; accepted 20 July 2006
Abstract A theoretical description of the contrast-imaging function is derived for tilted specimens that exhibit weak-phase object characteristics. We show that the tilted contrast-imaging function (TCIF) is a linear transformation, which can be approximated by the convolution operation for small tilt angles or for small specimens. This approximation is not valid for electron tomography, where specimen tilts are above 601 and specimen dimensions amount to some 10 mm. The approximation also breaks down for electron crystallography, where atomic resolution is to be achieved. Therefore, we do not make this approximation and propose a generalized algorithm for inverting the TCIF. The implications of our description are discussed in the context of electron tomography, single particle analysis, and electron crystallography, and the improved resolution is quantitatively demonstrated. r 2006 Elsevier B.V. All rights reserved. PACS: 42.30.Lr; 42.30.Va; 87.64.Ee Keywords: CTF; Space-variant point spread function; 3D reconstruction
1. Introduction Imaging with a far field optical system, such as an electron microscope, is best described by the linear systems theory: One assumes a space invariant point-spreadfunction (PSF) that is convoluted with the input object. Thus, the Fourier transform (FT) of the measured image is the product of the FT of the input object with the FT of the PSF, the so-called contrast transfer function (CTF). The deconvolution of the measurement into the original input involves division of the FT of the image by the CTF followed by inverse FT (FT1), and is referred to as ‘‘CTFcorrection.’’ Since in the imaging process information is only lost for spatial frequencies where the CTF has zeros, deconvolution is widely used for image restoration. To optimize information recovery in the presence of noise, Wiener filtering can be applied [1]. For the structural analysis of biological specimens, electron microscopy is one of the most useful methods. Corresponding author. Tel.: +41 612671476; fax: +41 612672109.
E-mail address: [email protected] (A. Philippsen). 0304-3991/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2006.07.010
As a result of technical progress, modern electron microscopes, equipped with coherent electron sources, provide atomic resolution routinely, even when operated far from focus. However, for organic matter atomic resolution is difficult to reach, because the electron beam quickly damages this matter. Therefore, atomic structure determination of biomolecules by electron microscopy has only been demonstrated so far for two-dimensional (2D) crystals, where it is possible to acquire information to a resolution of about 3 A˚ by taking advantage of crystallographic methods [2]. However, it remains a challenging task to grow highly ordered 2D crystals and to collect and extract accurate information at such high resolution. Suitable modern instruments keep the specimen at a temperature of a few Kelvin and illuminate it with a highly coherent beam, providing optimal prerequisites for data collection at atomic scale. Furthermore, to extract the structural information at high resolution, a rigorous correction of the optical artifacts is indispensable. Since different 2D projections are required to extract 3D information, one needs to tilt the specimen plane with respect to the optical axis. This leads to a particular optical problem, because the defocus changes in
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Nomenclature
l
CTF FT PSF TCIF a b
N z0 W0 c v q Q
Cs D
contrast transfer function Fourier transform point spread function tilted contrast-imaging function tilt angle of specimen out of plane orientation of tilt axis within image plane, b ¼ 0 means tilt axis is along y-axis spherical aberration constant of instrument sampling distance in digitized image
the direction perpendicular to the tilt axis. Thus, the PSF is no longer space invariant, and one can no longer apply the established concept of CTF deconvolution to correct the optical artifacts. This problem was recognized several years ago, and Henderson and coworkers developed a commonly used, heuristic algorithm for processing of 2D crystal images [3]. More recently, Winkler and Taylor [4] introduced a numerical solution based on solving a Fredholm integral equation of the first kind to perform a global correction. Both of these approaches will be discussed further below in the context of the derivation presented herein.
ing along the optical axis. Here, s is the 2D position vector in the image plane. Let us consider a single specimen point (r; z). The contribution of this single point to the total image amplitude at s is cr ðsÞ ¼ vðrÞhz ðs=M rÞ, where v(r) is the amplitude of the wave exiting from the specimen at (r; z), and M denotes the magnification. This description is valid for a thin object for which we can take v(r) to be the integral of all object contributions along the projection direction at position r, i.e. the true projection of the object. For the sake of simplicity we set M ¼ 1. The total image amplitude is the linear superposition of all individual specimen point contributions, i.e. it is obtained by integrating over the specimen area,
3 4
plzp2 Þ
.
Z
Z
cðsÞ ¼
dr vðrÞhz ðs rÞ.
dr cr ðsÞ ¼ S
The geometrical layout and the measurement process of the specimen is illustrated in Fig. 1. We consider a specimen point at (r; z), with r ¼ (x, y) and defocus z. The contribution to the image is blurred, as described by the defocus-dependent PSF hz(r). The contribution from each point is spread over a considerable area, as illustrated in Fig. 2; the larger the absolute value of the defocus, the farther reaching the spread, easily reaching hundreds of A˚. Generally, there is no closed form for the PSF hz(r). However, its FT, Hz(p) (the CTF), is based on the Scherzer formula [5] and is derived in standard textbooks (e.g. [6] or [7]). Considering only defocus and spherical aberration, it becomes H z ðpÞ ¼ AðpÞeiððp=2ÞC s l p
electron wavelength, calculated from acceleration voltage number of pixels in digitized image base defocus (defocus at center of specimen) Scherzer formula at the base defocus image amplitude specimen function measured image intensity transformation describing tilted contrast imaging 1010 m
A˚
2. Derivation of the tilted contrast-imaging function (TCIF) 2.1. Point spread and pupil function
203
(2)
S
r1
r2
Specimen
z(r1) z(r2)
Focal Plane Lens
(1)
Cs is the spherical aberration coefficient, l the electron wavelength, z the defocus, and A(p) is a masking function with a cut-off pmax, i.e. A(p) ¼ 1 for pppmax, A(p) ¼ 0 for p4pmax, and we take l51=pmax . Here, the 2D frequency vector is denoted by p ¼ ðpx ; py Þ, and its magnitude is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ¼ p2x þ p2y . 2.2. Image amplitude c Next, we evaluate the image amplitude cðsÞ 2 C of a tilted thin specimen illuminated by a plane wave propagat-
hz (r2)
hz (r1) Image
Mr2
Mr1
Fig. 1. Schematic PSF imaging. Schematic imaging of a tilted 1D specimen, with a lens-induced point spread function hz(r) and a varying defocus distance z from the focal plane. The coordinate system of r is parallel to the lens plane. The graphical representation is somewhat misleading, as the PSF and the resulting image amplitude are complex functions.
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Fig. 2. Defocus dependent point spread. For a single specimen point, the spread on the measured images can become considerable for large defocus values. For each defocus, the intensity profile in real-space along the line y ¼ 0 is plotted. For z ¼ 0 and z ¼ 800 nm, the full 2D image is given on the right-hand side.
In other words, due to the linearity of the imaging process, we can consider each specimen point (r; z) individually, i.e., each specimen point can be understood as a sample with zero extent and thus has a well-defined defocus z and PSF hz. Assuming a planar specimen, we write the defocus as zðrÞ ¼ z0 þ r d tan a, where z0 denotes the defocus offset, a the specimen tilt, d ¼ ðcos b; sin bÞ and where b is the orientation of the tilt axis in the specimen plane. Substituting z(r) into Eq. (1) yields Z 2 hz ðs rÞ ¼ dp AðpÞ e2piðsrÞp eiW 0 ðpÞ eiplrdp tan a (3) with W 0 ðpÞ ¼ ðp=2ÞC s l3 p4 plz0 p2 . From Eqs. (2) and (3), and by writing v(r) in terms of its FT V(k), we find the following expression for the image amplitude: Z Z Z cðsÞ ¼ dp AðpÞ e2pisp eiW 0 ðpÞ dr dk e2pikr
S 1 2pirðp2dp2 l tan aÞ V ðkÞe .
ð4Þ
Now we integrate over r and obtain a d function dðk fðpÞÞ with fðpÞ ¼ p 12dp2 l tan a. Integration over k yields Z cðsÞ ¼ dp AðpÞ e2pisp eiW 0 ðpÞ V ðfðpÞÞ. (5) Eq. (5) is valid for any planar, tilted specimen. We now apply the ‘‘weak object’’ approximation. It is valid for thin organic specimens, where elastic scattering by light atoms produces only small phase shifts, and the loss term is small as well. Thus, we use vðrÞ ¼ 1 ðrÞ þ ifðrÞ with ; f51, i.e. V ðpÞ ¼ dðpÞ EðpÞ þ iFðpÞ, where E(p) and F(p) are the FTs of e(r) and f(r), respectively. Note that f(p) vanishes at p0 ¼ 0 and at p1 ¼ 2 cotðaÞ d=l; but because p14pmax we can use AðpÞdðfðpÞÞ ¼ AðpÞdðpÞ. In the following, we assume
that all contributions to the image will be negligible for p4pmax and hence omit A(p). We obtain from Eq. (5): Z cðsÞ ¼ 1 þ dp e2pips eiW 0 ðpÞ EðfðpÞÞ þ iFðfðpÞÞ . (6) Further, we note that W0(p) is an even function. Also, e(r) and f(r) are real, thus EðpÞ ¼ EðpÞ and FðpÞ ¼ FðpÞ. 2.3. Specimen measurement The experimental measurement process records the intensity of the image amplitude, qðsÞ ¼ jcðsÞj2 ¼ cðsÞ cðsÞ . Ignoring the loss term e and higher order terms in f we obtain from Eq. (6) the image intensity Z qðsÞ const: þ dp e2pips i eiW 0 ðpÞ FðfðpÞÞ eiW 0 ðpÞ FðfðpÞÞ Z ¼ const: þ dp e2pips QðpÞ. ð7Þ We neglect the constant term and note that Q(p) is the FT of the measured image intensity q(s) with QðpÞ ¼ i eiW 0 ðpÞ F p 12dp2 l tan a ð8Þ eiW 0 ðpÞ F p þ 12dp2 l tan a . Eq. (8) is the main result of this derivation. We call this the TCIF, which is a linear transformation on F(k), i.e. QðpÞ ¼ TCIF½FðkÞ, as shown schematically in Fig. 3A.1 1
We note that for the special case of an untilted specimen, we have f(p) ¼ p and recover the well-known expression Qa¼0 ðpÞ ¼ 2F ðpÞ sinðW 0 ðpÞÞ.
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Fig. 3. Schematic representation of various TCIF transformations. (A) The 1D case of the linear transformation QðpÞ ¼ TCIF½FðkÞ, described by Eq. (8). The value at Q(p) is a linear combination of the two values F(k1) and F(k2) (each multiplied with a phase factor, see text), which are located in equal distance to the left and right of p, proportional to p2. F and Q represent specimen and measured image, respectively. (B) For the Fourier transform of a tilted 1D crystals image, F(k), the peaks will appear split in the measurement Q(p), according to Eq. (8). The two split peaks in Q are found at positions p1 and p2, as explained in the text. The distances are D1;2 ¼ 12p21;2 l tan a, with D24D1, i.e., the original position q is not located in the center of the split peaks. The gray bars indicate locations k0 1 and k0 2 from where F would also contribute to the measurement Q(p1) and Q(p2); however, for an image of a 2D crystal, it is generally the case that Fðk0 11 Þ ¼ Fðk0 22 Þ ¼ 0, since k0 1 and k0 2 are not on the reciprocal lattice of the crystal. Thus, the value of each Q(p) results from a F(k) with a well-defined k, in contrast to the generic case where contributions from k and k0 are present. (C) Effect of nested transformation Qð2Þ ¼ TCIF½TCIF½F. For a perfect crystal, the contributions from the left and right position (k1 and k2) generally have a contribution of zero (see text), and hence only the original value at k ¼ p is recovered.
2.4. Reduction to 1D Without loss of generality, we set b ¼ 0 (taking the y-axis parallel to the tilt axis). Writing FðkÞ ¼ Fðkx ; ky Þ, we note in Eq. (8) that d ¼ (1,0), thus each py corresponds to a fixed value of ky. Therefore, in Fourier space, the TCIF transformation is reduced to one dimension. Each line (with fixed py ¼ ky) of Q(p) as a function of px is given by Qpy ðpx Þ ¼ ieiW 0 ðpÞ F px 12p2 l tan a; py ieiW 0 ðpÞ F px þ 12p2 l tan a; py . ð9Þ Each such line can now be treated separately.
measurement, after the TCIF transformation described by Eq. (8). However, for evaluating Q ¼ TCIF½F for a discretely sampled weak-phase object fðxn ; yn Þ, with n ¼ 0 . . . N 1, the FT needs to be calculated at continuous frequencies px, rather than only for integer multiples of ðNDx Þ1 . Hence, to evaluate Eq. (9), we calculate FT[f] along each dimension as Fðpx Þ ¼ PN1 2pipx nD to obtain n¼0 fn e Fðpx 12p2 l tan a; py Þ ¼
N x 1 X n¼0
N y 1 X
fðxn ; ym Þe2pi
1 px 2p2 l tan a nDx 2pi py mDy
e
.
ð10Þ
m¼0
2.5. Interpolation of discretized data Upon actual recording of the measurement a discretized image qðxn ; yn Þ is obtained, whose FT gives Q at discretized values of p. We note that due to the coordinate transformation f(p), the frequency components p 12dp2 l tan a in Eq. (8) will usually not coincide with a discretized frequency, but rather take some intermediate values. This does not modify our above derivation of a physical measurement process, since this discretization happens at the very end of the
The FFT algorithm [8] may be used for calculating the discrete FT in y, yielding Qpy ðpx Þ ¼
NX x 1
ðsin x sin xþ Þ þ iðcos x cos xþ Þ
n¼0
~ p ðxn Þ F y
ð11Þ
with x ¼ 2p px 12p2 l tan a nDx W 0 ðpÞ
~ p ðxn Þ ¼ FTy fðxn ; ym Þ 2 C. F y
ð12Þ
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206
(13)
changes of the PSF, i.e. the resolution is limited by d, as illustrated in Fig. 4A. Let us now consider a given resolution d, i.e. we need pmax ¼ 1=d. Then, our description holds for samples of thickness
(14)
Dp
Equivalently, Eq. (11) may be written as Qpy ðpx Þ ¼ 2
N x 1 X
~ p ðxn Þ e2pipx nDx H~ py ðpx ; nÞ F y
n¼0
with
H~ py ðpx ; nÞ ¼ sin W 0 ðpÞ pp2 lnDx tan a 2 R.
In accordance with the nomenclature used above in Eq. (1), we use H~ in Eq. (14) to denote conceptually a generalized CTF that describes the effect of the tilt, whereby positions and frequencies are mixed and the description of a convolution no longer applies. The term nDx tan a corresponds to the defocus Dz at the spatial position given by index n. We will discuss H~ below. 2.6. Generalized inversion In order to extract the specimen information from the measurement, we must calculate F ¼ TCIF1 ½Q, i.e. Eq. (13) must be solved for F. For each py, it is a system of Nx linear equations, which can be solved to obtain Fpy , as demonstrated below for some artificial data. To calculate ~ the instrument parameters Cs, l, z0, and a must be H, known at a sufficient level of accuracy (see below) that can be estimated using this theoretical framework. 3. Discussion We have derived an analytical expression for the image intensity distribution q(s) of a tilted weak-phase specimen, illuminated by a plane wave and imaged by a lens with spherical aberration Cs. It is shown that a linear 1D transformation describes this process in Fourier space, and we call this transform the TCIF. This now allows the image formation under tilted conditions to be simulated, errors of the conventional CTF correction procedure to be estimated, and an accurate reconstruction of the original specimen projection v(r) from the measured intensity q(s).
d 2 cos a 0:61l
(16)
which is plotted in Fig. 4B. A very similar relation was found by DeRosier [9], who considered the deviation from the projection approximation based on the curvature of the Ewald Sphere. He derived p anffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi estimate for the maximum resolution given by pmax ¼ 2C=ðDlÞ, where D is the thickness of the sample and C a constant to be determined. He estimated the latter by a series of simulations of different specimen models, where a phase error of 661 was used as the maximum tolerable error, yielding C ’ 0:7 as a ‘‘quick rule of thumb’’. Rewriting of Eq. (16)preveals immediately that ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi our estimation gives pmax ’ 1:64=ðD0 lÞ, where D0 ¼ D= cos a is the tilt corrected thickness; thus we have a corresponding C of 0.82, and the two estimates are in agreement. An alternative way to estimate the resolution limit can be seen in Fig. 4C, D, where individual point spread functions are summed over the range of the sample thickness, and the Fourier transform of this sum reveals a first diminution of the overall signal at approximately the same resolution limit as estimated in Eq. (16) above. 3.2. Classical CTF deconvolution To what extent does the TCIF differ from the CTF, i.e. what errors are introduced by ignoring the tilt and correcting an image assuming constant defocus? From Eq. (14) (or indeed already from Eq. (1)) an estimate of the error is found by noting that for proper reconstruction the phase deviation must be less than p/2. Thus, the CTF approximation is valid for
3.1. Sample thickness limitations
2p2 lNDx tan ao1.
To derive the analytical expression we considered a thin sample of thickness D. We have implicitly assumed that the PSF does not change on the scale of the attempted resolution when integrating the sample contribution along the projection axis at a given coordinate r. The radius rPSF is approximately rPSF yz, where y 0:61lpmax dictates the resolution (after CTF correction the optical system is diffraction limited to a first approximation). Moving along the projection axis, the sample has thickness D= cos a, and the radius of the PSF changes by
The error of the CTF approximation is illustrated in Fig. 5, where the difference between H and H~ is plotted against the frequency p for several values of x ¼ NDx . Further, if a too small sampling area (in real space) is used, the TCIF and CTF can no longer be discriminated. This is seen in frequency space when the two contributions p 12p2 l tan a (Eq. (8) and Fig. 3A) collapse on the same pixel for
yD . (15) d¼ cos a As the features of the PSF scale uniformly (to a first approximation), Eq. (15) corresponds to the spatial
p2 l tan ap
1 . ND
(17)
(18)
A conclusion to draw from these estimates is that for a small image extent, the CTF approximation will only fail at high resolution, whereas for a large image extent, proper TCIF correction is required at fairly low resolution already.
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Fig. 4. Resolution estimation based on sample thickness for l ¼ 2 pm. (A) Graphical construction of the estimated resolution limit for the projection approximation. In this case, a sample thickness of D ¼ 200 A˚ at a tilt of 601 is assumed, with the length of the projection path 400 A˚. One-half of the PSF is plotted for each of the three indicated defocus values, and the difference in the extent of the PSF, judged by the distance between two corresponding peaks, is indicated by d. (B) Log–log plot of the estimated resolution limit against the maximal sample thickness, as given by Eq. (16). The abscissa contains the shift d, as given in (A) and explained in the text. The ordinate displays the maximal thickness of the sample where the projection approximation may be considered fulfilled at the given resolution and the given tilt angle. For example, at a tilt of 601, a resolution of 6 A˚ is achievable with a sample thinner than 120 nm (indicated by the dotted lines). (C) Summed, normalized PSF—in steps of 1 nm for the range of 1020–980 nm—reveals how the accumulation of slightly varying PSFs per slice modifies the overall PSF. (D) Fourier Transform of (C) exhibits a first diminution of the overall shape at around 2 A˚1, corresponding to the estimation in (B), namely that the projection estimation of a 20 nm thick sample tilted at 601, and hence with a projection path of 40 nm, can be considered valid up to a resolution of 2 A˚.
3.3. Artificial 1D specimen and measurement To illustrate further aspects of the TCIF transformation described here, we transform a 1D artificial specimen using Eq. (8), with py ¼ 0. The specimen function f is based on a sum of Gaussians, and we use analytical (non-interpolated) values F(p), see Appendix. To obtain a simulated measurement, the resulting analytical transform is evaluated at evenly sampled x, with spatial sampling D ¼ 1 A˚ and N ¼ 512 sampling points, as illustrated in Fig. 6. Here, the tilt angle is a ¼ 601, therefore the specimen occupies about one quarter of the discretized measurement area. According to the above error estimation (Fig. 5), for 300 kV electrons (l ¼ 2 pm), the resolution limitation of the traditional deconvolution method is about 10 A˚, which is in good agreement with our simulation in Fig. 6, where
deconvolution has been used as a possible correction approach: this approach breaks down at frequencies representing a resolution better than 6 A˚. In contrast, the reconstruction of the original specimen function utilizing the inversion approach described above yields the original input function within the numerical error limitations. Winkler and Taylor [4] apply similar ideas to their derivation, in particular the reduction to one dimension and regarding the effect in the frequency domain. They obtain a generic expression corresponding to a Fredholm integral equation of the first kind, which contains a spatially dependent kernel H(y, v) (their Eq. (3)). They point out that such inversion schemes are inherently ill posed, and they provide a recipe to control the singular value decomposition with a regularization parameter to
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~ x ¼ NDx Þ, the TCIF, is plotted for four different object Fig. 5. Error estimation for CTF vs. TCIF. The difference between H(p), the CTF, and Hðp; dimensions against the spatial frequency p. The larger the difference (ordinate), the more error will be introduced by assuming a constant defocus for the correction of the optical distortions. For samples on the order of a few thousand A˚, where the distance from the base defocus is in the order of 1000 A˚, significant differences start to occur below 10 A˚ resolution. For larger specimen, the proper inclusion of the TCIF effect must occur already at lower resolutions. The error estimation given in Eq. (17), namely the p/2 criterion, is indicated as a vertical line for each sample size.
avoid small terms from dominating. In our simulation in Fig. 6, no such regularization was necessary. However, Winkler and Taylor do not derive an exact expression for a particular defocus model, but rather utilize a numerical solution based on a matrix equation, where each matrix element is explicitly calculated from H(y, v). Also, they do not provide the explicit formula they use for their subsequent calculations. 3.4. Robustness against noise and uncertainties in optical parameters In order to have a first assessment of the robustness of our inversion scheme under more realistic conditions of experimental data, we performed inversions of the TCIF following the setup from Fig. 6, either after adding noise to the simulated measurement or using slightly modified optical parameters. When adding 2% white noise (based on the mean sample value), our procedure is robust against such noise, i.e. it does not seem to amplify high-frequency noise contributions; in contrast to the standard deconvolution method where filtering would be required. When evaluating the sensitivity to variations in optical parameters, we consider our inversion scheme as successful if it leads to a result closer to the original sample than the deconvolution method with the exact optical parameters. As expected, the dependence on the parameter errors is non-linear; however, we can estimate the thresholds as follows. Errors of 0.5% for l, 7% for Cs, 1% for z0, and 20% for a still yield successful inversions according to our criteria (plots not shown). Thus, our procedure is also
robust against errors in the optical parameters, and can be applied to experimental data. We defer a discussion of concrete experimental data to a future work. 4. Implications for biological electron microscopy The three major methods that utilize electron microscopy of biological specimens are electron tomography, single-particle analysis, and electron crystallography. We now discuss the implications of the optical distortions in imaging tilted specimens for these three methods. 4.1. Electron tomography Electron tomography attempts to produce 3D maps of entire cells or thin sections thereof at a resolution of ultimately 20 A˚ [10]. In this case, the specimen size may reach 10 mm, and when such a sample is tilted to 601, the defocus ranges over 4.3 to 4.3 mm; at both of these limits, the first zero-crossings2 of the CTF are found at approximately (30 A˚)1. Therefore, appropriate correction of the optical defects is required to achieve the anticipated resolution. Indeed, the above estimation (Eq. (17)) gives corresponding resolution limits: for this example, conventional CTF correction applied to the whole image will limit the resolution to about 50 A˚. When tilt angles are p151, the maximum resolution will be 20 A˚, but as result of the missing cone artifact [11] tilting to higher angles is a prerequisite for obtaining a meaningful 3D reconstruction. The related gain in resolution upon appropriate correction 2
sin (W0) ¼ 0, with l ¼ 2 pm and CS ¼ 2 mm.
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Fig. 6. Simulated measurement and inversion of an artificial 1D specimen. The artificial 1D original specimen is based on a sum of Gaussians, from which the projection (tilted specimen) and TCIF transformation (measurement simulation) is calculated analytically (see text and Eq. (8)). The curves in the frequency domain represent real and imaginary part, respectively. The values for the simulated image were discretized into 1 A˚ steps and subjected to ‘‘classical’’ deconvolution and full TCIF inversion, as described in the text. We used singular value decomposition for solving Eq. (13); however, we note that no singularities were encountered in this inversion, i.e., there were no vanishing eigenvalues. While the deconvolution approach introduces considerable errors, particularly for high-resolution frequencies (o10 A˚), the inversion based on the TCIF transformation is able to completely restore the specimen. Furthermore, we note that our inversion scheme is robust against small amounts of measurement noise; as an example, we show the inversion after adding 2% white noise to the measurement simulation (based on the sample average), which results in a reconstructed sample that is still better than the one from the classical deconvolution without noise.
of the tilt effect has been demonstrated by Winkler and Taylor [4], or using an alternative stripe-based approach by Fernandez et al. [12]. However, fundamental limitations of the achievable resolution actually result from the beam sensitivity of vitrified preparations. Grimm et al. [13] have estimated the highest achievable resolution as function of the sample thickness for 300 kV. For specimens whose thickness corresponds to multiples of the mean path between electron scattering events, the attainable resolution is far worse than the limitations of the optical system discussed here, e.g. the attainable resolution is about 90 A˚ for an 800 nm thick sample. Therefore, to
achieve 20 A˚ resolution as required for identification of macromolecular complexes within a cell, samples need to be thinner than 250 nm [13]. Progress in the preparation of frozen sections [14] opens an avenue to produce such samples and to overcome the thickness limitations. In this case the goal of 20 A˚ could be reached provided that optical defects inherent to images of tilted samples are properly corrected. 4.2. Single particle For 3D reconstruction of single particles, tilting the specimen is not required if the particles are randomly
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oriented within the vitrified solution layer, and hence non-tilted images are sufficient to obtain all orientations needed for reconstruction. However, in the case of samples that assume a preferred orientation, the 3D reconstruction is achieved by a tomographic approach using the random conical tilt method [15]. This is the case for membrane bound complexes that are kept in solution by detergents, which requires a carbon support film rather than a fenestrated film [16,17]. Random conical tilt is also of advantage if assignment of the projection direction is difficult. Typical sample areas have a diameter of 1 mm, at 601 the defocus range is 70.43 mm, and corresponding first zero-crossings of the CTF are at around (9 A˚)1. In pursuit of the goal to achieve atomic scale resolution, appropriate correction of the tilt-related optical defects is needed here as well. The commonly used correction scheme starts with the extraction of rectangular areas (subimages), centered on each particle of interest, and subsequent isolated processing of each subimage. Irrespective of the tilt, the ideal extent of each subimage is given by the following conditions: (i) include all spread information due to the PSF (see Fig. 2); (ii) exclude information from any other particle within its boundaries. This effectively results in an optimal separation of particles that is about twice the spatial extent of the PSF. Assuming that such subimages are indeed obtained, the correction for non-tilted specimens may proceed by conventional CTF correction. For tilted specimens two alternative approaches may be taken. One is a full TCIF correction across the subimage (as outlined above). The other is to simply ignore the tilt across the subimage and go ahead with the convential CTF correction. Since the defocus variation across a subimage is small, the latter approach is feasible and will yield acceptable results (see Eq. (16)), cf. Fig. 6. However, if the distribution of particles is so dense that image contributions from differents particles overlap as a result of the extended PSF, the best way to correct for optical artifacts is the full TCIF correction of the complete image for tilted specimens.
4.3. Electron crystallography Regarding electron crystallography, we discuss the implications only for a 1D crystal; the extension to 2D is straightforward (Eq. (9)). As shown in Fig. 3B, a diffraction peak F(k) of a periodic specimen f(x) will contribute to the TCIF-transformed specimen Q(p) at two different values py ¼ p1 ; p2 , given by the conditions p1 þ 1 2 1 2 2 2p2 l tan a ¼ k, which can be 2p1 l tan a ¼ k and pp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2kl tan a 1 ðl tan aÞ1 and solved to give p1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 ¼ 1 1 2kl tan a ðl tan aÞ1 , with p1 ; p2 ; kX0 and a; l40. The distance between the split peaks (plotted
Fig. 7. Peak split distance for crystallographic data. Based on Eq. (8), the distance between peaks in the FT of tilted 2D crystal images may be calculated. Here we show the predicted distance for image of 10 000 pixels, with a sampling of 1 A˚. The abscissa of the main graph is displayed with the frequency encoded as resolution from 100 to 2 A˚, the abscissa of the inner graph from 20 to 4 A˚, reflecting the resolution range where the peak separation becomes visible. The ordinate is in absolute pixels. The four curves represent tilt angles of 151, 301, 451 and 601.
in Fig. 7) is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 p1 ¼ 2 1 2kl tan a 1 þ 2kl tan a ðl tan aÞ1 ; in the 2D case, the vector between the split peaks is perpendicular to the tilt axis. It should be noted that the center of the split peaks 12 p1 þ p2 does not exactly correspond to the frequency k of the original sample peak, but is offset by approximately k3 l2 tan2 a. In practical terms, this offset is much smaller than the reciprocal pixel spacing and may thus be neglected. Henderson and coworkers [18] were the first to describe this effect and offer a correction mechanism that is applicable to crystalline samples. They obtain the same expression for H~ (Eq. (14)) by taking the Scherzer formula (Eq. (1)) with position dependent defocus z(x) and using it in the CTF. Accordingly, the consequences of imaging a tilted sample is a ‘‘phase grating’’ effect; the apparent specimen function is fapp ðxÞ ¼ fðxÞ sinðc1 k4 þ c2 k2 þ c3 kxÞ for a specific reflection at spatial frequency k, whose contrast sinusoidally varies as a function of the spatial position x, and the period of the sinusoidal varying as a function of the particular frequency k. The heuristic correction approach is somewhat unconventional, as the already transformed image is again (numerically) trans-
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~ as justified by the phase grating formed with the same H, effect. Based on Eq. (8) derived above, this approach can be understood as a nested TCIF transformation of the form Qð2Þ ¼ TCIF½Q ¼ TCIF½TCIF½F, which can be approximated by only keeping terms linear in (pl), yielding
Qð2Þ ðpÞ 2F½p e2iW 0 ðpÞ F p dp2 l tan a
þ e2iW 0 ðpÞ F p þ dp2 l tan a . ð19Þ At each frequency, the original value is combined with values from two neighboring frequencies, as shown in Fig. 3C. For a crystalline specimen, the two neighboring contributions are zero, since p p2 l tan a is generally not equal to a reflection frequency. Consequently, under the assumption of a perfectly ordered crystal, where only a few, widely separated scattering angles form the final image, the nested TCIF transformation is a good approximation to the inverse TCIF, and the correction approach taken by [18] is valid. However, to correct images from real, imperfect datasets, it is necessary for high resolution to perform an ‘‘unbending’’ procedure [18] to construct a perfect crystal. The drawback of unbending first and correcting for the tilt afterwards, however, is that during unbending the continuity of the PSF is disrupted; this violates the assumption made for the PSF, and full information is not restored during tilt correction. For this reason, even for the case of 2D crystals, it is imperative that the correction of the TCIF is done prior to unbending (however, now Q(2) is no longer a good approximation, because in Eq. (19) the contributions from the two neighboring frequencies might not vanish). Therefore, the established scheme [3], where the unbending precedes the contrast function correction, is not suitable if high-resolution information is to be extracted. Finally, for the realistic case that each diffraction spot is really formed by a collection of closely related scattering angles, the above-described general inversion may be applied to a region surrounding diffraction spots only, without the need to perform a global image correction. Their extent depends on the crystal order, the tilt angle and the resolution to be achieved. Crystal order and tilt angle need to be determined from the experimental data, most likely at medium resolution. The resolution to be reached then dictates the size of the regions to be processed.
5. Outlook So far, we have limited our derivation to a specific set of approximations, in particular we have (a) ignored the effects of elastic scattering beyond pmax and of inelastic scattering, (b) considered only the defocus and spherical aberration for the contrast-imaging function, and (c) not fully explored the effects of noise and uncertainty in the optical parameters. We intend to relax these assumptions and to include such properties to our mathematical description in a future work.
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The suggested TCIF inversion approach harbors two particular limitations. First, we do not yet know to what accuracy the pertinent experimental parameters are required prior to reconstructing specimens from actual measurements of biological specimens. Second, when solving the linear equation system described above, for every sample line perpendicular to the tilt axis, the time requirement scales with N3; this becomes a significant computational effort for large samples (e.g. N ¼ 10 000 pixels for a crystal of 1 mm diameter). The numerical implementation of our inversion approach can be enhanced by more refined numerical methods, reducing time requirements and including parameter estimation and noise handling. The simple estimates of the errors introduced by the conventional CTF correction protocol presented here will also serve as guide to decide whether or not full TCIF correction is necessary or if conventional CTF is sufficient. 6. Conclusion Our derivation of the contrast-imaging function for tilted, thin, weak-object specimens serves as a theoretical framework upon which concrete correction algorithms may be build upon, and the discussed error estimations will help to decide when such a correction is feasible and necessary. Acknowledgements We thank Dirk van Dyck, Philip Koeck, and Andreas Schenk for discussions and for critical reading of the manuscript. This work was supported by the EU 6th framework (3DEM-NoE), the NCCR Structural Biology, and by NSF Grants nos. PHY-01-17795 and DMR-02-33773. Appendix. Sum of Gaussians TCIF transform We define a Gaussian with area A,pmean ffiffiffiffiffiffi m, and2 standard 2 deviation s, i.e. Gðx; A; s; mÞ ¼ ðA=s 2pÞ eðxmÞ =ð2s Þ , and its corresponding Fourier transform is G FT ðp; A; s; mÞ ¼ 2 2 2 Ae2pimp e2p s p . Thus, for a sum of N Gaussians having different parameters Ai, si, and mi, we obtain G QG ðpx Þ ¼ FG þ Fþ sin W 0 ðpx Þ G ð20Þ þ i FG Fþ cos W 0 ðpx Þ P N 1 2 with FG ¼ i¼1 G FT ðp 2px l tan a; Ai ; si ; mi Þ. Deconvolution Given a noise-free, artificial measurement, using known instrument parameters, we deconvolute using 8 QðpÞ < for j2 sin W 0 ðpÞjX; 2 sin W 0 ðpÞ FDeconv ðpÞ ¼ (21) : QðpÞ otherwise; where e is a lower-cutoff to avoid division by zero.
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