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INSTITUTE OF PHYSICS PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 49 (2004) 509–522 PII: S0031-9155(04)69260-8
Transfer function restoration in 3D electron
microscopy via iterative data refinement
C O S Sorzano
1,2
, R Marabini
2,3
, G T Herman
4
, Y Censor
5
and J M Carazo
2,3
1
Escuela Polit
´
enica Superior, Universidad San Pablo-CEU, Campus Urb Montepríncipe,
s/n, 28668 Boadilla del Monte, Madrid, Spain
2
Centro Nacional de Biotecnolog
´
ıa, Campus Universidad Aut
´
onoma s/n, 28049 Cantoblanco,
Madrid, Spain
3
Escuela Polit
´
ecnica Superior, Universidad Aut
´
onoma de Madrid, Campus Universidad
Aut
´
onoma s/n, 28049 Cantoblanco, Madrid, Spain
4
The Graduate Center, The City University of New York, 365 Fifth Avenue, New York,
NY 10016-4309, USA
5
Department of Mathematics, University of Haifa, Mt Carmel, Haifa 31905, Israel
Received 12 September 2003
Published 27 January 2004
Online at stacks.iop.org/PMB/49/509 (
DOI: 10.1088/0031-9155/49/4/003)
Abstract
Three-dimensional electron microscopy (3D-EM) is a powerful tool for
visualizing complex biological systems. As with any other imaging device,
the electron microscope introduces a transfer function (called in this field
the contrast transfer function, CTF) into the image acquisition process that
modulates the various frequencies of the signal. Thus, the 3D reconstructions
performed with these CTF-affected projections are also affected by an implicit
3D transfer function. For high-resolution electron microscopy, the effect of
the CTF is quite dramatic and limits severely the achievable resolution. In
this work we make use of the iterative data refinement (IDR) technique to
ameliorate the effect of the CTF. It is demonstrated that the approach can be
successfully applied to noisy data.
1. Introduction
The analysis of macromolecular complexes and their dynamics is one of the most interesting
challenges in molecular biology. A promising future is awaiting the electron microscopist due
to the possibilities of visualizing molecular machines, reconstructing unique (as opposed to
averaged) objects and imaging dynamic processes. The road to achieving these possibilities is
via three-dimensional reconstruction from electron-microscopic images of the macromolecular
complexes.
There are many methods for reconstructing a three-dimensional object from its line
integrals (Herman 1980, Natterer and W
¨
ubbeling 2001). Typically, the line integrals are
estimated for a set of parallel lines from a projection image that is obtained by some instrument.
0031-9155/04/040509+14$30.00 © 2004 IOP Publishing Ltd Printed in the UK 509
510 C O S Sorzano et al
-1.5
-1
-0.5
0
0.5
1
1.5
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
CTF
Frequency (1/A)
Figure 1. Radial profile of the CTF used in the cryomicroscopy simulations.
Figure 2. Amplitude of a typical astigmatic CTF.
A difficulty that arises in electron microscopy is that the image that is produced by the
instrument corresponds to the convolution of the ideal projection image with a point spread
function (PSF). The PSF is usually described by its Fourier transform that is commonly called
the contrast transfer function (CTF); for example, see figures 1 and 2.
The CTF severely limits the achievable resolution in the three-dimensional reconstruction.
In particular, it filters both the high and the low frequencies, introduces zones of alternate
contrast and eliminates all information at certain frequencies. It is, therefore, desirable to
Transfer function restoration in 3D electron microscopy via iterative data refinement 511
replace the reconstruction obtained by a ‘real’ microscope by a reconstruction that would be
obtained from images that would be produced by an ideal, aberration-free microscope. In order
to achieve this goal several methods have been proposed: Frank and Penczek (1995) applied
Wiener filtering in the three-dimensional space to the reconstructed volume; Zhu et al (1997)
incorporated a three-dimensional PSF into the data model and used a regularized steepest-
descent technique; Stark et al (1997) applied inverse CTF filtering to the reconstructed volume;
Skoglund et al (1996) incorporated a two-dimensional CTF, particular to each projection to the
projection, model in a maximum-entropy reconstruction algorithm; Grigorieff (1998) provided
a Fourier reconstruction algorithm in which the CTF for each projection is considered in a
Wiener-like fashion; Ludtke et al (1999) proposed a CTF correction applied to the individual
projections with a weighting function in the Fourier space computed from a set of images
sharing a common CTF; Ludtke et al (2001) added a Wiener filter to the weighting function
defined in Ludtke et al (1999). An alternative is to explicitly introduce the effect of the CTF
in the reconstruction equations; this was done by Zubelli et al (2003), who then reformulated
the problem so that Chahine’s method became applicable to it. The existence of these multiple
approaches is indicative of the fact that there is no agreed standard technique for the correction
of CTF effects in three-dimensional electron microscopy (3D-EM) of single particles and the
search for superior methods is still active.
In this work we apply the technique of iterative data refinement (IDR)—introduced in
Censor et al (1985) and further studied in Herman (1989), Herman and Ro (1990), Losada and
Navarro (1998) and Ro et al (1989)—to reduce the effect of the CTF and, thus, to obtain high-
resolution structural information about the macromolecules under study. As opposed to many
of the approaches discussed in the previous paragraph, our proposed algorithm can handle the
case of differing CTFs in the projections and does not require estimation of the signal-to-noise
ratio (SNR). The potential benefit of the method is illustrated by an experiment that involves
realistic simulation of the electron microscopic imaging of a biological macromolecule.
2. Mathematical background
2.1. Contrast transfer function
Image formation by an electron microscope is due to several physical processes of electron
interaction with the specimen. These effects combine to produce a single CTF, see Frank
(1996, ch 2.II). A parametric model of this transfer function has been used in the simulations
presented in this work. This parametric model accounts for the various effects involved in the
CTF (Zhu et al (1997) and Frank (1996,ch2.II))andwasalsousedinZhouet al (1996).
Basically, the microscope transfer function is a real-valued function in Fourier space
formed by a damped harmonic function. The ‘sine’ part of this function comes from the phase
change that electrons undergo when interacting with the sample specimen. A detailed study
of the electronic interaction in the image formation plane shows that the transfer function of
an electron microscope can be usefully approximated by
CTF(ω) = E(ω)
sin(πf |ω|
2
) − Q
0
cos(πf
|
ω
|
2
)
(1)
where ω is the spatial frequency, f is the defocus and Q
0
is a factor accounting for the
loss of electrons during the image formation process. Usually Q
0
is a small number, which
implies that the DC component of the projection Fourier transform is nearly removed and, thus,
the absolute density values in the projections are not meaningful. For this reason, usually only
the relative values are taken into consideration when interpreting a 3D-EM reconstruction. The
damping envelope, E(ω), models microscope imperfections such as chromatic aberration,
512 C O S Sorzano et al
spherical aberration, current and voltage instabilities, angular aperture, etc; see, e.g.,
Zhou et al (1996)orFrank(1996,ch2.II).
The model explained so far defines the shape of the profile of the CTF (for a typical
example, see figure 1). Many studies assume that this profile is radially symmetric, although
this is not necessarily true. Astigmatism is a well-known effect which turns the circles
produced by the radial symmetrization of the CTF profile into ellipses (see figure 2). This
results in a different defocus along every radial line of the Fourier space.
2.2. Phase flipping
Notice (in figure 1) that the sign changes at the zero-crossings result in a contrast inversion
in the projection image and cause the complete elimination of the information at certain
frequencies. This is a very limiting factor in electron microscopy, since without CTF correction
all reconstructions are unreliable at frequencies beyond the one where the CTF first becomes
zero. Nevertheless, biologically useful results can sometimes be obtained even without CTF
correction (B
´
arcena et al 2001, Sorzano et al 2001). However, this can only be the case if the
important biological information is not in the high-frequency part of the reconstruction, since
(as illustrated in our experiment reported below) in a reconstruction without CTF correction,
the information regarding frequencies beyond the first zero-crossing of the CTF is incorrect.
A simple method to alleviate this problem consists of multiplying the projection Fourier
transform by the sign of the CTF (this approach is named phase flipping (Frank 1996, p 45))
to produce the corrected projection data to which the reconstruction algorithm is then applied.
Thus the correction of the CTF sign is simple, as it only needs to adjust the sign at those
frequencies where it is flipped. However, amplitude correction is more difficult, as it requires
either dividing by the transfer function (avoiding zeros by using, for instance, a Wiener filter)
or the incorporation of the CTF operator into the reconstruction algorithm, allowing each
projection to have its own CTF. The IDR approach of this paper addresses this more difficult
problem. It will be set up based on the assumption that the data had been already corrected by
phase flipping.
2.3. Computational representation of volumes and projections
For the computational procedures of this paper we need to establish conventions for
representing volumes and projections by finite sets of numbers. In this subsection we present
our conventions and explain the operators which are incorporated into our algorithms.
We approximate arbitrary volumes by finite series expansions of the general form
J
j=1
c
j
b(r − r
j
). (2)
In this formula r is the point at which the volume is being approximated, the r
j
are fixed points
in space, b is a function of three variables and the c
j
are the coefficients of the expansion. In
any application, b and the r
j
are fixed, it is the c
j
that distinguish one volume from another.
Following Lewitt (1990, 1992) and Matej and Lewitt (1995, 1996), we use a generalized
Kaiser–Bessel window function (also called a blob)forb and a finite subset of the body
centered cubic grid for the points r
j
. Such a representation was found useful in electron
microscopy applications; see B
´
arcena et al (2001), Marabini et al (1997, 1998) and Sorzano
et al (2001). The specific choice that we adopt for the blob and the grid is the one referred to
as the ‘standard blob’ in Matej and Lewitt (1996).
We approximate a projection by a two-dimensional array of numbers, each representing a
projection value at a point of a square grid. To bring this into the electron microscopy context,
Transfer function restoration in 3D electron microscopy via iterative data refinement 513
we think of the square grid as lying on a projection plane that is perpendicular to the direction
of the electrons. Assuming that we have m different projection planes, we use g
i
(1 i m)
to denote the array of numbers associated with the ith projection. We use g to denote the
complete set of the m projections, meaning that g is the concatenated vector of all the g
i
s.
Given a volume representation as in (2) it is easy to calculate the ideal projection (line
integrals along lines perpendicular to the projection plane and passing through the points of
the square grid). This is so because the integration can be brought inside the summation and
can be analytically evaluated for the known blob b and grid point r
j
.For1 i m,we
define an ideal projection operator P
i
that associates with the J -dimensional vector c (whose
jth component is c
j
) the vector representing the ideal projection of the volume onto the ith
projection plane. Note that, in practice, P
i
is a matrix of J columns and as many rows as the
number of grid points in the ith projection plane.
Each projection also has its own CTF operator that we denote by H
i
. In practice, given
the projection g
i
, H
i
g
i
is computed by taking the discrete Fourier transform of g
i
, multiplying
it point-wise by the phase-flipped CTF associated with the ith projection and then taking the
inverse discrete Fourier transform.
2.4. Iterative data refinement
The measuring device (the electron microscope) provides data that only approximate what we
intend to measure. The discrepancy between the actual data (under our assumptions, corrupted
by the phase-flipped CTF) and data that are idealized (uncorrupted by the CTF, henceforth
called ideal data) can be estimated from the actual data and knowledge of the measuring
process, leading to a better approximation of the ideal data. This new approximation can then
be used to estimate the new discrepancy, and the process can be repeated. Our knowledge
of the measurement process is insufficient to obtain the ideal data exactly, but the original
discrepancy is significantly reduced by just a few of such iterative steps. This process is
accomplished by the iterative data refinement (IDR) methodology of Censor et al (1985).
Here we briefly review the fundamentals of IDR and describe our specific implementation of
it for the CTF removal problem. We then supply a short discussion that puts the approach in
perspective and relates it to the current literature.
In the following we use R to denote a recovery operator (in our case a three-dimensional
reconstruction algorithm) that produces, for a complete set of projections g, a vector c that
represents a volume using (2). For now it is not important to specify our choice of R;we
will do so below. An important assumption about R (well justified by the known behaviour
of reconstruction algorithms (Herman 1980)) is that if it is applied to the ideal data
ˆ
g (that
is, the concatenated vector of all the P
i
cs), then
ˆ
c = R
ˆ
g is an acceptable approximation to c.
The problem is that, in practice, the actual data
˜
g are corrupted by the CTF and so R
˜
g is not
a satisfactory approximation to c. The IDR approach aims at estimating the ideal data
ˆ
g from
the actual data
˜
g, assuming knowledge of the CTFs.
IDR produces a sequence of vectors g
k
(k = 0, 1, 2,...); the aim is that they should be
improving estimates of the ideal data. We denote by g
k
i
the part of g
k
that is associated with
the ith projection. With this notation, our version of the IDR algorithm is formulated as the
following iterative process. (Figure 3 depicts a block diagram of the algorithm.)
Algorithm: iterative data refinement (IDR) for 3D electron microscopy
Initialization: Take g
0
=
˜
g, the actual data (i.e., the experimentally measured data corrected
by phase flipping).
514 C O S Sorzano et al
Figure 3. Block diagram for the IDR algorithm (based on the one originally published by Censor
et al (1985)).
Iterative step: Given the current iterate g
k
=
g
k
i
m
i=1
, calculate the next iterate g
k+1
=
g
k+1
i
m
i=1
by using, for all i = 1, 2,...,m,the formula
g
k+1
i
= µ
k
g
0
i
+ (P
i
− µ
k
H
i
P
i
)Rg
k
(3)
where {µ
k
}
∞
k=0
is a sequence of the so-called relaxation parameters.
This algorithm generates an iterative sequence {g
k
}
∞
k=0
which is guaranteed to converge
to the ideal data
ˆ
g under some stringent conditions, see Censor and Zenios (1997, proposition
10.5.8). However, in practice, it has been shown (see Censor et al (1985) and references
therein) that, even when convergence cannot be guaranteed, the early iterates produced by the
IDR algorithm are closer to the ideal data vector
ˆ
g than g
0
is. This property of IDR is the basis
of the present study.
The underlying idea of IDR is to refine the data iteratively in a way that bridges the gap
between an accurate model of data collection (but one for which we do not have a reconstruction
algorithm) and an approximate model that leads to a reconstruction algorithm (which would
Transfer function restoration in 3D electron microscopy via iterative data refinement 515
work if the model were correct). This is quite different from just being another reconstruction
method, since it iterates on the data rather than on the unknowns of the reconstruction problem.
The IDR method is a member of the family of ‘iterative defect-correction methods’ much used
in the field of differential equations, see, e.g., B
¨
ohmer et al (1984) and Stetter (1978). For
additional applications of the IDR approach, consult section 2 of Censor et al (1985), where
beam hardening correction in x-ray computerized tomography, attenuation correction in single
photon emission computed tomography (SPECT) and image reconstruction with incomplete
data are described along with references to the original studies. See also Herman and Ro
(1990) for a study of the connection between IDR and phase retrieval algorithms.
2.5. Reconstruction algorithm
We now return to the choice of the recovery operator R that was left unspecified in the previous
subsection. We emphasize that the basic approach of IDR is independent of this choice; any
good reconstruction algorithm could be used. In our study we used for the recovery operator R
the reconstruction algorithm called block-ART with blobs in which each block corresponds to
one projection. We note that this algorithm has been found efficacious for 3D reconstruction
from electron microscopic data (B
´
arcena et al 2001, Marabini et al 1997, 1998, Sorzano et al
2001).
Block-ART with blobs is an iterative algorithm. Given a data vector g, it produces a
sequence of iterates {c
k
}
∞
k=0
, each of which defines a volume using (2).
Algorithm: iterative block-ART for volume recovery
Initialization: Take c
(0)
= 0, the zero vector.
Iterative Step: Given the current iterate c
k
, calculate the next iterate c
k+1
by the formula
c
k+1
= c
k
+ P
T
i
k
(g
i
− P
i
c
k
) (4)
where i=k mod m + 1.
Here P
T
i
is the transpose of the ideal projection operator P
i
, and each
k
is a relaxation
matrix whose exact nature is explained by Eggermont et al (1981), who also provide
convergence results for the iterative block-ART algorithm (their theorem 1.3). In the
experiments, reported below, we do not run the algorithm to convergence, but use only
one full cycle of it; i.e., we define Rg to be c
m
. (This is justified by previous experience in this
application area; see, e.g., Marabini et al (1998).) Also, we selected each
k
to be a diagonal
matrix, the value of each entry on the diagonal is a constant λ divided by the square of the
norm of the corresponding row of P
i
; see (2.18) in Eggermont et al (1981). (This is also
justified by previous experience. It was also found that for the data collection that we used for
the experiments reported below, λ = 0.05 is a good choice.)
The recovery operator R is used not only in the IDR process, but also to produce the final
reconstruction from the (possibly corrected) projection data. For this purpose, it needs to be
extended by an additional step, since further analysis of the reconstructions requires that they
should be evaluated at the points of a cubic grid. Such an evaluation is done using (2), yielding
a three-dimensional array of numbers that we consider to be the reconstruction.
There is an interesting alternative to be considered here: why not incorporate the effect
of the CTF into the model and then use iterative block-art for volume recovery directly on
the actual data? (In practice, this means that in (4) P
i
has to be replaced by H
i
P
i
in both
places where it occurs, with corresponding changes in the calculations of the entries of the
diagonal matrix
k
. Computationally this is some additional burden, but not too much: since
516 C O S Sorzano et al
H
i
is symmetric, H
T
i
= H
i
and the computation can be carried out by the method described at
the end of the subsection on computational representation of volumes and projections.) In the
current paper the choice of the recovery operator is not essential: it can be replaced in the
IDR process by any good reconstruction algorithm. We leave to the future the investigation
of the efficacy of IDR relative to applying directly to the actual data either iterative block-art
for volume recovery or the alternative algebraic approach proposed by Zubelli et al (2003).
3. Evaluation methodology
To compare the performance of IDR with no CTF correction and with phase flipping, we
adapted the methodology proposed in Furuie et al (1994) and previously applied to electron
microscopy in Marabini et al (1997, 1998) and Sorzano et al (2001). In the experiments
described below, many sets of simulated electron microscopy projections of a particular
molecule are taken and reconstructions are produced from each of these sets using the
approaches to be compared. The success of the approaches is determined by using figures of
merit (FOMs). We now provide details of this outline.
3.1. Projection data generation
The volume used in our tests was created from an atomic structure deposited in the protein data
bank (PDB), see Berman et al (2000), namely the Halobacterium halobium bacteriorhodopsin
(PDB id: 1BRD, Henderson et al (1990)). For comparison purposes this volume was evaluated
for points of a 64 ×64 ×64 cubic grid, with distance 3.5
˚
A between neighbouring grid points.
We refer to the resulting three-dimensional array as the phantom.
Several sets of 2,000 projections were created with signal-to-noise ratio 1/3; this resembles
cryomicroscopy conditions. This noise in the measurements was combined with other sources
of inconsistency in the form of random translations (by moving the projection plane parallel to
itself by a distance randomly selected from a zero-mean Gaussian distribution with standard
deviation 7
˚
A) and random rotations (by adding a zero-mean 5
◦
standard deviation Gaussian
noise to each of the Euler angles that defined the orientation of the projection planes).
In addition to noise, the projections were convolved with a CTF. The parameters for a
circularly-symmetric simulated CTF, see (1), were given values typically found in experimental
conditions: Q
0
=−0.06 and f =−20 000
˚
A, and the factors that influence the damping
envelope E(ω) were selected as acceleration voltage =100 kV, spherical aberration =5.5 mm,
chromatic aberration =6 mm, energy loss =9.9 eV, convergence cone =0.2 mrad, longitudinal
displacement = 80
˚
A(seeVel
´
azquez-Muriel et al (2003) for a description of these parameters).
The radial profile of this CTF is shown in figure 1. The same CTF was used in all projections,
since this is the worst case that can occur because it makes it difficult to compensate for
missing information in one projection by data from other projections.
A surface rendering of the ideal volume and a selection of projections are shown in
figures 4 and 5, respectively.
3.2. Figures of merit
Figures of merit (FOMs) are numerical measures of the reconstruction quality that are based on
specific aspects. A simple but often used measure is the sum of the squares of the differences
between the individual values in the reconstructions and the corresponding values in the
phantom. In Sorzano et al (2001), this measure was divided by the number of points in the
cubic grid and the result was called the FOM scL2.
Transfer function restoration in 3D electron microscopy via iterative data refinement 517
Figure 4. Side and top view of the isosurface of the bacteriorhodopsin phantom.
Figure 5. A selection of projections simulating cryomicroscopy images from the bacteriorhodopsin
phantom.
More sophisticated measures are provided by the Fourier shell correlation (FSC), as
described in equation (3.65) of Frank (1996) with F
1
the phantom and F
2
the reconstruction.
The FSC indicates, for every shell of frequencies (determined by a frequency k and a shell
thickness k), how well the reconstruction correlates with the phantom for all frequencies
within that shell. Thus, the FSC provides a separate FOM for every shell. The FSC can also
be used to provide the additional single FOM of resolution by defining it as the frequency at
which the FSC falls below 0.5.
The presence of many reasonable FOMs (Sorzano et al (2001) lists 24 of them) makes
exhaustive comparisons difficult. We have developed a methodology that applies multivariate
518 C O S Sorzano et al
Figure 6. Difference in the scL2 FOM between the reconstruction after kth and (k − 1)th IDR
iterations for 30 tests.
statistics to obtain a single FOM that in some senses captures the essence of what is provided
by the full set of FOMs; it is described in detail in the doctoral dissertation of Sorzano (2002).
Here we give a brief description of the use of this methodology for selecting an optimal
range for a parameter µ, such as one of the relaxation parameters µ
k
in (3). To do this, a
number of training data sets are created and each one of them is processed using a number
of values of the parameter µ. Then the method for producing a single representative FOM
proceeds in five steps: first, each of the FOMs that we wish to consider in producing the
single FOM is normalized to have mean 0 and standard deviation 1; second, those FOMs
that show no dependency (as indicated by a 1-way analysis of variance, ANOVA for short)
with µ are removed; third, all FOMs showing a similar dependency with µ are clustered
by a hierarchical classification; fourth, the dimensionality of the clusters is reduced using a
principal component analysis (PCA) and a single representative is selected for each cluster and
fifth, the cluster representatives are combined into a single FOM. We refer to it in this paper
as the training FOM.Theoptimal range for µ is considered to be the maximal range within
which the performance (as measured by the training FOM applied to the results obtained from
the training data sets) is not statistically significantly different from the optimal performance.
3.3. Training
We first give the details of the training methodology for selecting the optimal range of µ
0
.
Preliminary tests indicated that we should not be looking outside the range [1.4, 2.4]. Within
this range we investigated values of µ
0
at 0.1 increments. For each of this 11 values, ten
complete actual (noisy) data sets were generated by the method described above and a training
FOM was produced based on these 110 reconstructions. We found that the corresponding
optimal range for µ
0
is [1.7, 1.9].
To select µ
1
, we essentially repeat this process, but now the evaluation is based on the
reconstructions produced from g
2
. (For each of the 110 data sets, µ
0
was randomly selected
from its optimal range.) The results were similar, namely the optimal range for µ
1
turned out
Transfer function restoration in 3D electron microscopy via iterative data refinement 519
Figure 7. From top to bottom, slices corresponding to central sections of: bacteriorhodopsin
phantom, reconstruction without CTF correction, reconstruction with phase flipping, reconstruction
with IDR after one and six iterations.
to be the same as for µ
0
. In fact, repeating this process for µ
k
,for2 k 15, the same
optimal range was found each time.
To determine the stopping criterion, 30 new complete actual data sets were generated
and the IDR algorithm was run for 15 iterations. For each of the 30 actual data sets
˜
g,
reconstructions were produced from g
k
,for0 k 15. The FOM scL2 was calculated for
each reconstruction and, for 1 k 15, the difference between scL2 values for that iteration
and the previous iteration (from the same data set) was calculated. The results, plotted in
figure 6, show that there appear to be no significant changes produced by the IDR iterations
beyond the sixth one. In this case, the scL2 value to which the algorithm converged was 0.998.
4. Results
Figure 7 illustrates our results: four central slices of the phantom and of reconstructions
(from a new data set generated by the previously described rules) are shown. In figure 8 we
plot the associated Fourier shell correlations (see section 3) for assessing the reconstruction
quality. The improvement by any of the corrections over the uncorrected reconstruction is
highly noticeable, even in the case of only phase flipping. Note that using only the FOM
‘resolution’ (as defined in section 3), there is no significant difference between phase flipping
and IDR (regardless of the number of iterations), since all the CTF corrected reconstructions
520 C O S Sorzano et al
0
0.2
0.4
0.6
0.8
1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Fourier Shell Correlation
Frequency (1/A)
No CTF correction
Phase correction
IDR iteration 1
IDR iteration 6
Figure 8. Fourier shell correlation curves for the reconstructions without CTF correction, with
phase flipping and IDR after one and six iterations.
have a resolution at around 0.06
˚
A
−1
(as opposed to the resolution of the reconstruction
from uncorrected data that is around 0.035
˚
A
−1
). However, examining the full FSC curves
instead of concentrating only on resolution reveals a clear improvement in the IDR-corrected
reconstructions over the phase-flipped reconstruction in the range in which the CTF is inverted
(between 0.037
˚
A
−1
and 0.052
˚
A
−1
). Furthermore, we can see the improvements produced by
additional IDR iterations. All this is visually confirmed in figure 7.
The discussion in the previous paragraph is anecdotal: it is based on reconstructions from
a single data set. To be able to assign statistical significance to our claim of superiority of the
IDR reconstruction, over the phase-flipped reconstruction, we generated 30 additional actual
data sets (using the same rules as before) and compared the phase-flipped reconstructions with
the reconstructions after the sixth iteration of the IDR algorithm. The results are summarized
in figure 9, which plots (for 19 shells) the average (over the 30 data sets) of the FSC value
for IDR after six iterations less the FSC value for phase flipping. Standard deviations of these
differences over the 30 experimental outcomes are also indicated. The standard deviation of
the average difference is 1/
√
30 times the standard deviation indicated in the figure. Hence it
is clear that, for each of the 19 shells, one can extremely confidently reject the null hypothesis
that the expected value of the FSC for the IDR reconstruction after six iteration is the same as
the expected value of the FSC for the phase-flipped reconstruction in favour of the alternative
hypothesis that the expected value of the FSC is higher for the IDR reconstruction. In fact, for
each of the 19 shells, the value of the average difference is more than ten times the standard
deviation of the average difference (thus providing us with a P value less than 10
−23
).
IDR’s ability of removing the effect of an instrumental transfer function was first
demonstrated by Ro et al (1989) for magnetic resonance imaging. We have shown here
that IDR is also efficacious in electron microscopy, which presents much nastier transfer
functions and extremely noisy images. Our results demonstrate that it is possible to recover
much of the information that is lost near the zeros of the CTF and that the quality of the
Transfer function restoration in 3D electron microscopy via iterative data refinement 521
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Difference in Fourier Shell Correlation
Frequency (1/A)
Average
Average-Std.Deviation
Average+Std.Deviation
Figure 9. Plot of the values for 19 shells of the average over 30 actual data sets (±1 standard
deviation) of the FSC for IDR after six iterations less the FSC for phase flipping.
reconstruction from electron microscopic data can be significantly improved by iterative data
refinement.
Acknowledgments
Partial support is acknowledged to the Comisi
´
on Interministerial de Ciencia y Tecnología
of Spain through projects BIO98-0761 and BIO2001-1237 and to National Institutes of
Health through grant HL70472. The work of Y Censor was done in part at the Center
for Computational Mathematics and Scientific Computation (CCMSC) at the University of
Haifa and supported by Research Grant 592/00 from the Israel Science Foundation founded
by the Israel Academy of Sciences and Humanities.
The authors would like to thank DrJJFern
´
andez for fruitful discussions and advice on
the manuscript.
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