Angular Uncertainty Refinement and Image Reconstruction
Improvement in Cryo-electron Tomography
Hmida Rojbani
1,2
,
´
Etienne Baudrier
1
, Benoˆıt Naegel
1
, Lo¨ıc Mazo
1
and Atef Hamouda
2
1
University of Strasbourg, 67412, Illkirch CEDEX, France
2
Faculty of Science of Tunis, University El-Manar, 2092, El Manar Tunis, Tunisia
K
eywords:
Electron Tomography, 3D Structures, Tilt Angles, Angular Uncertainty, Optimization, Conjugate Gradient.
Abstract:
In the field of cryo-electron tomography (cryo-ET), numerous approaches have been proposed to tackle the
difficulties of the three-dimensional reconstruction problem. And that, in order to cope with (1) the missing
and noisy data from the collected projections, (2) errors in projection images due to acquisition problems, (3)
the capacity of processing large data sets and parameterizing the contrast function of the electron microscopy.
In this paper, we present a novel approach for dealing with angular uncertainty in cryo-ET. To accomplish this
task we propose a cost function and with the use of the nonlinear version of the optimization algorithm called
Conjugate Gradient, we minimize it. We test the efciency of our algorithm with both simulated and real data.
1 INTRODUCTION
For a better understanding of the biological cells, sci-
entists use electron microscopy to investigate their in-
ner structures. The cryo-ET offers the possibility to
reconstruct the three dimensional (3D) volume of a
cell slice. The cryo-ET principle is to acquire two
dimensional (2D) projections with the Transmission
Electron Microscope (TEM) by tilting the sample
around an axis (cf. Figure 1). The tilting range is
[70
, 70
] with steps between 2
and 5
. Then the
projection angles are known and are used in the re-
construction of the 3D sample. Figure 2 presents the
four different steps of cryo-ET.
Figure 1: Transmission electron microscopy acquisition
mechanism.
Nevertheless, there are several limits to the reso-
lution of the reconstruction: the missing data corre-
sponding to the uncovered projection space, the low
electron dose used during the acquisition to minimize
the weight of radiation damage (Egerton et al., 2004),
the thickness of the ice used in the sample fixation
that affects also the final reconstructed volume reso-
lution (Stagg et al., 2006) and the error made on the
projection parameters. The missing data, the electron
dose and the ice thickness are an acquisition phase
problem that we cannot correct them directly in our
work, but we try to minimize their effect on the re-
construct results. The classical reconstruction pro-
cess involves as a first step an image alignment pro-
cess. The two well common alignment techniques are
(Frank, 2006): (1) marker tracking, where gold par-
ticles are implanted into the samples before the ac-
quisition of the images and then the calculation based
on these markers is used to align the images (Brandt
and Ziese, 2006), (Sorzano et al., 2009); (2) cross-
correlation methods, where alignment is carried out
by the cross-correlation calculation between each suc-
cessive couple of images (Zheng et al., 2010).
The second step of the cryo-ET is the 3D re-
construction. Numerous reconstruction methods are
used in cryo-ET, mainly belonging to two families.
First, the analytic family, with methods as filtered
back-projection or direct Fourier inversion, imple-
mented in Fourier’s space (Penczek, 2010). The sec-
ond family is the algebraic family, with iterative real-
96
Rojbani, H., Baudrier, É., Naegel, B., Mazo, L. and Hamouda, A.
Angular Uncertainty Refinement and Image Reconstruction Improvement in Cryo-electron Tomography.
DOI: 10.5220/0005680600940100
In Proceedings of the 11th Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2016) - Volume 3: VISAPP, pages 96-102
ISBN: 978-989-758-175-5
Copyright
c
2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
space methods such as ART (Gordon et al., 1970) or
SIRT (Gilbert, 1972).
Besides the reconstruction difficulties, the acqui-
sition process rises different type of problems such
as the specimen movement (translation, rotation) over
the carbon support especially when the tilting reaches
the highest values, the blur caused by the Contrast
Transfer Function (CTF), and the uncertainty on the
angular information due to mechanical imprecision of
the microscope (Colliex, 1998). All these problems
affect the quality of resolution of the reconstructed
object. Many approaches are proposed to overcome
all this mentioned problems except for the angular
uncertainty, due to the fact that the resolution of the
reconstruction is mediocre, so the angular uncertainty
does not have the effect over it. Now days, and due to
the improvement of the reconstruction techniques, the
resolution become important especially when we aim
to reconstruct and locate small particles such as ribo-
some and nucleosomes. Treating the angular uncer-
tainty will push as forward to a better reconstruction
resolution.
We begin in our work by concentrating on the
angular uncertainty problem, which we prove in our
tests that even a slight change in the set of angles use
in the reconstruction has its affect over the accuracy
of the resolution of the reconstructed 3D volume. Our
approach is based on optimization problem where we
use a cost function has the set of angles as parameters
to be minimized. To gain more accuracy we include
also the reconstructed object as parameters into the
function. Using the two sets of parameters simulta-
neously in our approach provides more accurate re-
sults than optimizing the angular uncertainty and the
reconstructed object separately.
Figure 2: Different Steps of cryo-ET.
2 RELATED WORK
In recent years, the use of simultaneous optimization
is increased in many fields. One of those fields is
the alignment phase in the reconstruction. For exam-
ple, in the field of the singular particles reconstruc-
tion, a large set of small projections used in the re-
construction of an object are from unknown orien-
tations. Thus, to improve the estimation of the un-
known projection orientations, Yang et al. (Yang
et al., 2005) proposed to use a Quasi-Newton opti-
mization based algorithm to minimize a cost function
between the projection angles and the reconstructed
3D object starting from a rough reconstruction.
In cryo-ET, the context is different: the data is a
small set of large projections with known tilt angles.
However, this tilt angles can be erroneous due to the
malfunction of the tilt mechanism of the object holder
in the TEM. In this field, Tran et al. (Tran et al.,
2013) proposed a hierarchical method to correct the
reconstruction and the alignment problems in alter-
nating way. They treat the transformation parameters
(translation, rotation, scale). Inspiring from the cross-
correlation method of alignment, they begin by find-
ing the first set of transformationparameters by an op-
timization between each successive pair of projection
images. After that, a first reconstruction is applied
followed by refinement of the set of transformation. If
the method has not yet converged,they raise a new re-
construction. Infact, the reconstruction is based on an
optimization between the actual projections and the
resulting projections of the reconstructed object. In
the same way, the phase of refinement of transforma-
tion parameters represents an optimization between
the processing parameters found from the current pro-
jections and those of the previous iteration.
We propose in our work to optimize the recon-
structed object and the transformation parameters si-
multaneously at the same level. In addition, we take
into account the error of the angular uncertainty of
projection. Indeed, we make the correction on the
reconstructed object itself, instead of those made on
projections. The idea is to try for each projection ori-
entation used to reproduce the same projection pro-
vided by the TEM. In this paper, we only present our
work for the refinement of the projection angles and
the reconstructed object.
The rest of the paper is structured as follows: in
Section 3, the projection algebraic model of the ac-
quisition is given; in Section 4, the proposed approach
is detailed and the associated cost function is defined;
the results are shown and discussed in Section 5. Fi-
nally, we summarize and give some perspectives.
Angular Uncertainty Refinement and Image Reconstruction Improvement in Cryo-electron Tomography
97
3 PROJECTION ALGEBRAIC
MODEL
In the cryo-ET, The reconstruction is three-
dimensional problem, but in the section we will
give a general n-dimensional presentation for the
problem. To see the reconstruction problem as an
algebraic model, we must change the reconstructed
volume f and the projections images p
i
to their
respective vectors forms.
Let f be the vector representationof n-dimensional
image f, f = (f
1
, f
2
, . . . , f
N
)
T
. N is the number of vox-
els of f, in other words N = Π
n
d=1
N
d
, where N
d
is the
size of f in the dimension d.
Let p
θ
i
= (p
θ
i
1
, p
θ
i
2
, . . . , p
θ
i
M
)
T
is the vector represen-
tation of the n1-dimensional projection image p
i
of
the image f according to the orientation θ
i
. M is the
number of pixels of p, in other words M = Π
n1
d=1
M
d
,
where M
d
is the size of p
i
in the dimension d.
In fact, p is an element of a bigger vector P,
which contains all the elements of the S projec-
tions images. S is the numbers of the angles used
in the tilting process. We can assemble these an-
gles in a set called Θ, and then we have P =
(p
θ
1
, p
θ
2
, . . . , p
θ
i
, . . . , p
θ
S
), θ
i
Θ.
We define the relation between the projection vec-
tor P and the image vector f as:
P = Wf, (1)
Which can be seen also as :
p
θ
i
= W
θ
i
f, (2)
Thereby, the projection problem is modeled as an
equation system whose matrix W
θ
i
is sparse. The ma-
trix W
θ
i
holds the coefficients of the projections ac-
cording to the angle θ
i
. In fact, each line of this matrix
describes one of the projection lines passing throws
the image. Many methods are proposed to calculate
this matrix such as voxel driven” and ”ray driven”
(Joseph, 1982). However, in our work, the calcula-
tion of W
θ
i
is based on the ”distance driven” method
proposed by (Man and Basu, 2004). Note that the non
null coefficients change when θ
i
changes but the ma-
trix W
θ
i
remains sparse. The equation (2) cannot be
straightly inversed. Then we propose thereafter a re-
constructed volume refinement method that simulta-
neously acts on the reconstructed volume and the tilt
angles.
4 PROPOSED APPROACH
When we investigate the reconstruction problem, we
find that the only real data that we have it is the pro-
jections images and a set of uncertain angles of pro-
jection. Hence, the idea to create a function that can
calculate the difference or we can call it also the dis-
tance between the real projection data and the projec-
tion data found after re-projecting the reconstructed
volume according to the current set of angles. Thus,
by refining these two, we can assure having a much
accurate reconstructed objet.
We begin by defining the cost function C , which
presents the Euclidian distance between the real set of
projection data Π and the re-projection data from the
reconstructed object P, so we have :
C (f, Θ) =
1
2
kΠ Pk
2
2
=
1
2
S
i=1
kπ
i
p
θ
i
k
2
2
(3)
This equation can be developed to :
C (f, Θ) =
1
2
S
i=1
M
j=1
(π
i
j
p
θ
i
j
)
2
(4)
=
1
2
S
i=1
M
j=1
(π
i
j
N
k=1
w
θ
i
, j
k
f
k
)
2
(5)
where w
θ
i
, j
k
is the projection coefficient of the k-th
pixel by the j-th projection line according to the θ
i
.
The cost function C depends on three parameters,
the actual projections, the current reconstructed vol-
ume and the current tilt angles.
We use the square in the equation (3) to ensure the
positivity of the function. The similarity between the
two types of projections reaches its maximum when
the function C reaches its minimum. In this case, the
solution set of our problem (
b
f,
b
Θ) is :
(
b
f,
b
Θ) = argmin C (f, Θ) (6)
The equation (6) is convex in f (due to the using of
a quadratic sum) and Θ is close to the solution by hy-
pothesis . We can therefore assume that in this case,
the optimization problem of Eq. (6) is convex. More-
over, it is plain that the cost function C is not linear
according to (f, Θ). Thus, we have to use a non-linear
minimization algorithm. Among the knowndetermin-
istic optimization algorithms mentioned in the litera-
ture (Nocedal and Wright, 2006), we chose the non-
linear version of the Conjugate Gradient (CG) algo-
rithm (Dai and Yuan, 1999). The CG method is the
most prominent iterative method for solving sparse
systems of equations. We preferred this algorithm for
its convergence rate, its simple implementation and
since it appeals only to the first order derivative.
The steps below constitute one of the CG itera-
tion of movement along a conjugate direction d
i
. The
algorithm starts by initializing d
0
= g
0
= C (χ
0
)
(Dai and Yuan, 1999), then :
VISAPP 2016 - International Conference on Computer Vision Theory and Applications
98
find α
i
the length of the descent step that mini-
mizes C (χ
i
+ α
i
d
i
),
χ
i+1
= χ
i
+ α
i
d
i
,
g
i+1
= C (χ
i+1
),
β
i
= max{
g
T
i+1
(g
i+1
g
i
)
g
T
i
g
i
, 0},
updating the conjugate direction d
i+1
= g
i+1
+
β
i
d
i
.
The first step to use the CG is to calculate the gra-
dient of our cost function C . In order to facilitate this
phase, we have changed the equation (4) by :
C (f, Θ) =
1
2
S
i=1
M
j=1
(C r(f, θ
i
, j))
2
(7)
with C r(f, θ
i
, j) = π
i
j
N
k=1
w
θ
i
, j
k
f
k
.
Now we calculate the C r then we will deduct the
C . Due to the discrete nature of the cost function,
we use the finite difference to estimate the gradient
values.
For h = 1, . . . , N, the f
h
-th partial derivative of C
is given by
C r
f
h
= w
θ
i
, j
h
which leads to,
(C r)
2
f
h
= 2w
θ
i
, j
h
π
i
j
N
k=1
w
θ
i
, j
k
f
k
!
.
Then,
C
f
h
=
S
i=1
M
j=1
w
θ
i
, j
h
π
i
j
N
k=1
w
θ
i
, j
k
f
k
!
.
In the same, we have for the h-th derivative ac-
cording to Θ, h = 1, . . . , M.
C r
∂θ
h
=
1
2
θ
N
k=1
(w
(θ
h
θ
), j
k
w
(θ
h
+
θ
), j
k
)f
k
!
where w
(θ
h
+
θ
), j
k
, w
(θ
h
θ
), j
k
are the projection coeffi-
cients according to θ
h
+
θ
, θ
h
θ
,
θ
here is the fi-
nite difference used in the gradient calculation. which
leads to,
(C r)
2
∂θ
h
=
1
θ
N
k=1
(w
(θ
h
θ
), j
k
w
(θ
h
+
θ
), j
k
)f
k
!
×
π
i
j
N
k=1
w
θ
i
, j
k
f
k
!
.
Then,
C
∂θ
h
=
1
2
θ
M
j=1
N
k=1
(w
(θ
h
θ
), j
k
w
(θ
h
+
θ
), j
k
)f
k
!
×
π
i
j
N
k=1
w
θ
i
, j
k
f
k
!
.
We noticed that according a small value to the pa-
rameter
θ
may produce a null derivative for some
angles because of the discrete nature of the rotation
which may boil down to the identity map for the small
angle changes. Therefore, after many experiments we
found that the most suitable value is 1
.
After computing the gradient, we can proceed to
the optimization phase. Firstly, we reconstruct the ini-
tial object; in our case, we chose to use the algebraic
reconstruction method SIRT for this phase. However,
any algorithm either analytic or algebraic is can be
used. Secondly, we take the initial reconstructed ob-
ject and the projection data with the tilt angles used in
the acquisition as input for the iterative CG algorithm.
The algorithm of our approach is showed in Alg.1.
Algorithm 1: CG refinement algorithm.
initialization f
0
= f
init
, Θ
0
= Θ
init
, cond=0
while iter < iter
max
AND cond=0 do
compute gradient C (f
iter
, Θ
iter
)
compute f
iter+1
, Θ
iter+1
with the CG
C = C (f
iter
, Θ
iter
) C (f
iter+1
, Θ
iter+1
)
if C < ε then
cond=1
end if
f
iter
f
iter+1
, Θ
iter
Θ
iter+1
, iter iter+ 1
end while
return f
iter+1
, Θ
iter+1
5 RESULTS
Several experiments were conducted to assess the ef-
ficiency of the proposed method on 2D and 3D data.
The tilt angles used for 2D tests are between 70
and 70
with a 2
tilt step. For each angle a random
error in [1.5
1.5
] is added. For the 3D tests, We
used a different set of angles, one similar to the set
used in cryo-ET. The boundaries are the same, but the
tilt step is different. The step used between 50
and
50
is 5
and it is 2
for the others. One can notes that
the number of tilt angles for the 3D case is less than
the 2D, which makes the refinement harder. Along
the projection noise, we used three sets of tilt angles:
the real one, one with absolute value of the random
error 1
and one with random error 2
.
Angular Uncertainty Refinement and Image Reconstruction Improvement in Cryo-electron Tomography
99
To measure the quality of reconstruction, we pro-
pose two evaluation criteria. The first is the cross-
correlation coefficient (equation (8)). It is a criterion
to measure the degree of similarity between the orig-
inal image and the reconstructed image. This coef-
ficient is equal to 1 when the reconstructed image
coincides exactly with the original image and zero
otherwise. The second criterion is Normalized Root
Mean Square Error (NRMSE) defined by the equa-
tion (9). Unlike the correlation coefficient, the zero of
the NRMSE means a better result. We use NRMSE to
measure the quality of angles correction produced by
our approach. The equation (10) shows the Measure
of Angle Correction (MAC). More the MAC value is
close to zero, the more the angles found by our ap-
proach are corrects.
Corr =
N
k=1
(f
k
M(f))(
b
f
k
M(
b
f))
q
N
k=1
(f
k
M(f))
2
N
k=1
(
b
f
k
M(
b
f))
2
(8)
NRMSE =
q
S
i=1
(θ
i
b
θ
i
)
2
S
max(Θ) min(Θ)
(9)
MAC = 1
NRMSE
init
NRMSE
final
NRMSE
init
(10)
With NRMSE
init
and NRMSE
final
are respectively
the NRMSE between the true angles, the angles be-
fore the optimization and the angles after the opti-
mization. Thus, if the angles after the optimization
are totally corrected NRMSE
final
will be zero, which
means the MAC will be zero also. However, if the er-
rors are not corrected or even are amplified, the MAC
will be greater than 1.
The 2D experiments involve 122 2D synthetic
gray-level images randomly generated at different
resolutions N×N with N = 32, 64, 128, 256, 512. A
sample of the synthetic images used are shown in the
figure 3. The same number of iterations are applied
on all the experiments images.
Figure 3: 2D synthetic images.
All the projections were corrupted with different
levels of Gaussian noise σ and a random error has
been assigned to each of the tilt angles (some samples
are shown in Fig. 5).
The mean results of the synthetic data are gathered
in Tab. 1 and the tilt angles refinements of one of the
tests are presented in Fig. 4.
Figure 4: Error enhancement between the original tilt an-
gles and the corrected ones for one of the test images.
For all the tested resolutions, the error that re-
mains after optimization is under 30% of wrong pix-
els and under 20% of wrong projection angles. In our
future work, we will investigate deeper the relation
between the image size and the number of iterations
to reach the minimum. We notice also that more the
gray-levels we use, more the discrete research space
is connected and more the method is robust to noise.
Table 1: Evaluation of 2D synthetic data.
Noise variance σ 0 2 4 6 8
Corr 0.97 0.93 0.88 0.87 0.83
MAC 0.09 0.12 0.15 0.18 0.19
Sinogram SIRT Our method
σ = 0σ = 4σ = 8
Figure 5: The sinogram of a single image is corrupted with
Gaussian noise of different variances σ. The reconstruction
by SIRT (Penczek, 2010) and our method is shown for each
noise level.
The 3D experiments were done on synthetic
and real data. A set of 3D synthetic gray-level
volumes were created with different resolutions
(N = 16, 32, 64, 128). Fig. 7 shows a 3D volume
with its results. The real data are projections of the
Orf-parapoxvirus: Fig. 6: presents some of those pro-
jections.
Figure 6: Orf-parapoxvirus projections respectively accord-
ing to the angles -60
, -40
, -10
, 20
, and 50
.
The mean results according to deferent angular er-
ror (AE) and noise variance σ sing the same evalua-
VISAPP 2016 - International Conference on Computer Vision Theory and Applications
100
Figure 7: Reconstruction of a synthetic 3D; left: 3D volume
(N = 32), center: with our method and right: with SIRT.
tion measures used for the 2D evaluation are given in
the Tab. 2 and Tab. 3.
In this two table, one can notice the promising re-
sults, the correlation is very high over all the testing
seniors. The same for the MAC results that are low
for the most of the testing seniors. We notice also,
that when we use the true angles, they are slightly cor-
rupted by our algorithm.
Table 2: Corr results for 3D synthetic data.
σ
AE
0
1
2
0.5 0.97 0.89 0.88
2 0.90 0.84 0.85
5 0.88 0.77 0.75
Table 3: MAC results for 3D synthetic data.
σ
AE
0
1
2
0.5 0.01 0.09 0.08
2 0.05 0.13 0.14
5 0.07 0.18 0.21
For the Orf-parapoxvirus reconstruction, we cal-
culate the improvement rate of resolution, which was
21.86%, which we think it is a good result consider-
ing that we used volume and angular refinement only.
Also, we sought an opinion of an expert in the biol-
ogy field and he validate the amelioration of the re-
construction object generated by our approach. Some
of the 2D slice of the reconstructed object by our ap-
proach and SIRT are presented in Tab. 4.
Table 4: Some of the 2D slices of the reconstructed object
by SIRT and our approach.
SIRT
Our approach
6 CONCLUSION
In this paper, we have described a method to refine
the reconstruction of an object from a set of its to-
mographic projections and simultaneously correct the
errors over the used tilt angles. The starting point of
the proposed method is an approximation of the ob-
ject (provided by some reconstruction algorithm) to-
gether with a set of uncertain tilt angles used to ac-
quire the projections. Then, considering both the an-
gles and the values of the reconstructed volume as
parameters, we minimize the Euclidean distance be-
tween our ground truth (the original tomographic pro-
jections) and the re-projections of the estimated object
according to the current corrected set of angles. For
the optimization process, a non-linear conjugate gra-
dient algorithm is used. Our experiments shows that
the proposed method improves the reconstruction of
the object compared to using SIRT directly. Further
improvements of the reconstruction can still be ob-
tained in the frame of our method. In our future work,
we plan also to incorporate the other transform pa-
rameters of the alignment process, also, the correction
of the contrast transfer function in the optimization
process.
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