HAL Id: hal-02424804
https://hal.inria.fr/hal-02424804
Submitted on 28 Dec 2019
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A Monte Carlo framework for missing wedge restoration
and noise removal in cryo-electron tomography
Emmanuel Moebel, Charles Kervrann
To cite this version:
Emmanuel Moebel, Charles Kervrann. A Monte Carlo framework for missing wedge restoration and
noise removal in cryo-electron tomography. Journal of Structural Biology: X, Elsevier, 2019, pp.1-18.
�10.1016/j.yjsbx.2019.100013�. �hal-02424804�
A Monte Carlo framework for missing wedge restoration and noise
removal in cryo-electron tomography
Emmanuel
Moebel,
Charles
Kervrann
Inria - Centre de Rennes Bretagne Atlantique, Campus Universitaire de Beaulieu, 35042 Rennes (France)
Keywords
:
cryo electron tomography
missing wedge
restoration
inverse problems
Monte-Carlo simulation
A B S T R A C T
We propose a statistical method to address an important issue in cryo-electron tomography
im-age analysis: reduction of a high amount of noise and artifacts due to the presence of a missing
wedge (MW) in the spectral domain. The method takes as an input a 3D tomogram derived from
limited-angle tomography, and gives as an output a 3D denoised and artifact compensated
vol-ume. The artifact compensation is achieved by filling up the MW with meaningful information.
To address this inverse problem, we compute a Minimum Mean Square Error (MMSE) estimator
of the uncorrupted image. The underlying high-dimensional integral is computed by applying a
dedicated Markov Chain Monte-Carlo (MCMC) sampling procedure based on the
Metropolis-Hasting (MH) algorithm. The proposed MWR (Missing Wedge Restoration) algorithm can be
used to enhance visualization or as a pre-processing step for image analysis, including
segmen-tation and classification of macromolecules. Results are presented for both synthetic data and
real 3D cryo-electron images.
1. Introduction
Cryo-electron tomography (cryo-ET) is generally used to explore the structure of an entire cell and constitutes a rapidly
growing field in biology. The particularity of cryo-ET is that it is able to produce three-dimensional views of vitrified
samples at sub-nanometer resolution, which allows observing the structure of molecular complexes (e.g. ribosomes)
in their physiological environment. Nevertheless, observation of highly resolved cellular mechanisms is challenging:
i/ due to the low dose of electrons used to preserve specimen integrity during image acquisition, the amount of noise
is very high; ii/ due to technical limitations of the microscope, complete tilting of the sample (90
𝑜
) is impossible,
resulting into a blind spot. As a consequence, projections are not available for a determined angle range, hence the
term “limited angle tomography”.
The blind spot is observable in Fourier domain, where the missing projections appear as a missing wedge (MW).
This separates the Fourier spectrum into two regions: the sampled region (SR) and the unsampled regions (MW).
The sharp transition between these two regions is responsible for a Gibbs-like phenomenon: ray- and side-artifacts
emanate from high contrast objects (see Fig.
1
), which can hide important structural features in the image. Another
type of artifact arises from the incomplete angular sampling: objects appear elongated in the direction of the blind
spot (see Fig.
1
), in other words the data has an anisotropic resolution (e.g. linear features perpendicular to the tilt
axis disappear). This elongation erases boundaries and makes it difficult to differentiate neighboring features. The
quality of tomograms can be improved if sophisticated algorithms such as MBIR (
Yan et al.
,
2019
) are applied instead
of conventional methods (e.g. WPB (
Radermacher
,
1992
), SIRT (
Gilbert
,
1972
)).
Filling up the MW with relevant data enables to potentially reduce or completely suppress these artifacts.
Experi-mentally this can partially be achieved during data acquisition by using dual-axis tomography (
Guesdon et al.
,
2013
),
where the sample is tilted with respect to the second axis. Consequently the blind spot is smaller and the MW becomes
a missing pyramid, which results into a smaller missing spectrum. However dual-axis tomography is technically
chal-lenging and requires intensive post-processing in order to correct tilt and movement bias in the microscope. Another
reconstruction approach consists in exploiting the symmetry of the underlying structure (
Förster and Hegerl
,
2007
),
but this can only be applicable to a limited number of biological objects (e.g. virus with either helical or icosahedral
structure). Another common computational approach amounts to combining several hundred or thousands views of
the same object, but with distinct blind spots. This so-called sub-tomogram averaging technique (
Förster and Hegerl
,
2007
) is routinely used in cryo-ET, and is continuously improved for structure determination (
Xu et al.
,
2018
). To
im-prove sub-tomogram averaging and compensate the remaining MW artifacts, tomographic reconstruction algorithms
with dedicated regularization have also been proposed in (
Paavolainen et al.
,
2014
;
Leary et al.
,
2013
).
The objective of our work is to design a statistical approach for the problem of recovering missing Fourier
coef-ficients from a single volume in the situation where low and high frequency coefcoef-ficients are missing in a specific and
large region of the 3D spectrum. A simple way of handling MW artifacts is described in (
Kováčik et al.
,
2014
), where
a dedicated spectral filter is used to smooth out the transition between SR and MW; ray- and side-artifacts are reduced
with this filter, but the object elongation remains in the resulting image. Inspired from (
Maggioni et al.
,
2013
), we have
rather investigated MCMC methods to compute a MMSE estimator based on any non-local image denoiser to recover
the missing information. We show that our Monte-Carlo sampling algorithm performs as well as the iterative method
(
Maggioni et al.
,
2013
) but converges faster. Nevertheless, our concept is more general since any denoising method
can be applied, included denoising algorithms dedicated to cryo-ET images (
Frangakis and Hegerl
,
2001
;
Fernandez
and Li
,
2003
;
Darbon et al.
,
2008
;
Wei and Yin
,
2010
;
Kervrann et al.
,
2008
;
Moreno et al.
,
2018
). In this paper, we
focus on the cryo-ET restoration problem but the proposed algorithm could be potentially used to address a large range
of applications including medical and seismic imaging, and other inverse scattering problems.
Related work
We first focus on computational methods designed for spectrum restoration and Fourier coefficients
recovering. Most of methods have been designed for 2D images and very few of them for 3D imaging. In general,
the corruption process is supposed to be known and the artifacts observed in the input image, are due to a set of
missing Fourier coefficients, well localized in the spectrum. First, several methods have been investigated to retrieve
partially-missing phases of complex coefficients from modulus of coefficients in electron microscopy (
Fienup
,
1982
)
Spatial domain
Fourier domain
Data
Corrupted
xy-view
yz-view
xz-view
Ray
artifacts
Object
elongation
Side
artifacts
Missing
wedge
Ground Truth
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Ray artifacts
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Side artifacts
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Missing wedge
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Figure 1:
The effects of the missing wedge on the Shepp-Logan phantom. Left: uncorrupted 3D data displayed as ortho-slices
(one central slice along each dimension). Right: data corrupted by the missing wedge. We display the data in spatial domain (top
row) and Fourier domain (bottom row).
and time-frequency signal analysis (
Krémé et al.
,
2018
). Here, we focus on another special case which consists in
ex-trapolating the band-limited spectrum of an image up to higher frequencies. Nevertheless, these problems are generally
formulated as denoising problems with specific reconstruction constraints. For instance,
Moisan
(
2001
) and
Guichard
and Malgouyres
(
1998
) investigated the Total Variation (TV) minimization to extend the band-limited spectrum of
an image. In (
Lauze et al.
,
2017
), the authors combine TV minimization and positivity constraints to reduce noise
and artifacts, providing an inpainting-like mechanism for the sinogram missing data in limited-angle tomography (see
also (
Frikel and Quinto
,
2013
;
Sentosum et al.
,
2017
)). The common objective is to create new frequencies while
preserving discontinuities and details in the restored image. Instead of explicitly imposing some regularity (e.g. Total
variation, or robust regularization (
Geman and Reynolds
,
1992
;
Charbonnier et al.
,
1997
)) on the solution, another
successful restoration approach consists in exploiting the spatial redundancy of the input image. In (
Chambolle and
Jalalzai
,
2014
), a non-local method was suggested in the framework of variational methods for image reconstruction.
In this approach, a patchwise similarity measure based on atoms corresponding to pseudo Gabor filters is designed
to compare corrupted regions. Meanwhile,
Maggioni et al.
(
2013
) adapted the concept of BM3D for recovering the
missing spectrum applied to MRI imaging with very promising results on synthetic data. BM3D (
Dabov et al.
,
2007
)
is a popular denoising algorithm which combines clustering of noisy patches, DCT-based transform and shrinkage
operation to achieve the state-of-the-art results for several years. In our approach, we also focus on patch-based
meth-ods (
Kervrann and Boulanger
,
2008
;
Kindermann et al.
,
2005
;
Lou et al.
,
2010
;
Katkovnik et al.
,
2010
;
Pizarro et al.
,
2010
;
Milanfar
,
2013
;
Sutour et al.
,
2014
) to restore the input image corrupted by noise and non-linear transform.
Indeed, it has been experimentally confirmed that the most competitive denoising methods are non-local and exploit
self-similarities occurring at large distances in images, such as BM3D (
Dabov et al.
,
2007
), NL-Bayes (
Lebrun et al.
,
2013
), PLOW (
Chatterjee and Milanfar
,
2012
), S-PLE (
Wang and Morel
,
2013
), PEWA (
Kervrann
,
2014
) and many
other adaptative filters (
Kervrann and Boulanger
,
2006
,
2007
;
Van De Ville and Kocher
,
2009
;
Deledalle et al.
,
2009
;
Louchet and Moisan
,
2011
;
Duval et al.
,
2011
;
Deledalle et al.
,
2012
;
Kervrann et al.
,
2014
;
Jin et al.
,
2017
), inspired
from the seminal N(on)L(ocal)-means algorithm (
Buades et al.
,
2005
).
To complete the brief overview of non-local methods, we mention that a noisy image can also be restored from a
set of noisy or “clean” patches or a learned dictionary. The statistics of a training set of image patches serve then as
priors for denoising (
Elad and Aharon
,
2006
;
Mairal et al.
,
2009
;
Zoran and Weiss
,
2011
). Another approach based
multi-layer perceptron (MLP) exploiting a training set of noisy and noise-free patches was also able to achieve the
state-of-the-art performance (
Burger et al.
,
2012
). Very recently,
Buchholz et al.
(
2018
) proposed to train content-aware
restoration networks for denoising cryo-transmission electron microscopy data. While all these machine learning
methods are attractive and powerful, computation is not always feasible in 3D because very large collection of 3D
“clean” patches are required. In our study, we focus on unsupervised denoising methods since they are more flexible
for real applications. They are less computationally demanding and are still competitive when compared to recent
machine learning methods.
Our approach is mainly inspired from (
Maggioni et al.
,
2013
), but can use any competitive denoising methods for
restoring the Fourier coefficients. The method proposed by
Maggioni et al.
(
2013
) works by alternatively adding noise
into the missing region and applying the BM4D algorithm which is the extension of BM3D (
Dabov et al.
,
2007
) to
volumes. The authors interpret this iterative restoration method in the framework of compressed sensing with two
information theory concepts in mind: sparsity of the signal in the transformed domain, and incoherence between the
transform and the sampling matrix. Actually, BM4D does rely on a transform where the signal is sparse. Moreover,
it is not clearly established that this transform is incoherent with the sampling matrix, defined by the support of the
sampling region. Therefore, the proof of convergence is not clearly established, even though the authors presented
convincing experimental results on synthetic images corrupted with white Gaussian noise. It remains unclear how
the concept performs on experimental data and non Gaussian noise. To generalize this idea, we propose a statistical
approach well-grounded in the Bayesian and MCMC framework and applied to challenging real data in cryo-ET. Our
contributions are the following ones:
1. We present a MMSE estimator dedicated to the problem of MW restoration.
2. We propose an original Monte Carlo Markov Chain (MCMC) sampling procedure to efficiently compute the
MMSE estimator.
Paper organization
The remainder of the paper is organized as follows. In the next section, we present an overview of
our computational method. In Section
3
, we give the theoretical background and formulate the reconstruction problem
as an inverse problem. We shortly describe the usual Bayesian approach to derive a MMSE estimator. Furthermore, a
Monte-Carlo Markov Chains method based on the popular Metropolis-Hastings algorithm is proposed to compute the
underlying high-dimensional integral. In Section
4
, we adapt this general Bayesian framework for MW restoration.
An original Metropolis-Hastings procedure is presented to explore the large space of admissible solutions and to select
relevant samples. Section
5
presents the experimental results obtained on simulated and real data. We illustrate the
potential of our MWR (Missing Wedge Restoration) algorithm with experiments on real cryo-tomogram images and
we compare our iterative approach to several competitive algorithms.
2. Overview of the MW restoration algorithm
In this section, we present our concept for MW restoration. First, we formulate the problem and introduce notation.
Second, we present the two-step algorithm which is embedded in a Monte-Carlo simulation framework.
2.1. Problem formulation and notation
Let us define a 𝑛-dimensional image 𝒙 ∶ 𝑆 ⊂ ℤ
3
→ ℝ
assumed to be periodic and defined over a cubic domain
Ω = [0, 1]
3
and 𝑛 = |Ω|. The discrete Fourier transform of 𝒙 = {𝑥(𝑠), 𝑠 ∈ 𝑆} is then as follows:
𝒙 ∶ 𝑘 →
∑
𝑠
∈𝑆
exp (−2𝑖𝜋𝑘 ⋅ 𝑠) 𝑥(𝑠),
(1)
where 𝑠 is the coordinate of point in spatial domain 𝑆. In our problem, one considers a corrupted image denoted
𝒚
= {𝑦(𝑠), 𝑠 ∈ 𝑆}
defined as
𝑦
(𝑠) =
∑
𝑘
∈𝑊
exp (2𝑖𝜋𝑘 ⋅ 𝑠)
𝒙[𝑘]
(2)
where 𝑊 is the sampled spectral region (SR) where the Fourier coefficients 𝒙[𝑘] are positive and non-zero. The
region 𝑊 is equivalent to the support of a binary mask 𝒎 ∈ {0, 1}
𝑆
such as 𝒎[𝑘] = 1 if 𝑘 ∈ 𝑊 and 0 otherwise:
𝑊
=
supp(𝒎). The so-called missing wedge 𝑊 is assumed to be symmetric with respect to the origin as illustrated
in Fig.
1
(bottom right), and 𝑆 = 𝑊 ∪ 𝑊 . In what follows, we assume that the clean image 𝒙 is known over the
region 𝑊 . Our objective is then to estimate 𝒙 ∶ 𝑆 → ℝ in the whole domain 𝑆 such that
∀𝑘 ∈ 𝑊 ,
𝒙[𝑘] = 𝒚[𝑘]
(3)
and ∀𝑠 ∈ 𝑆, 𝑥(𝑠) > 0. In other words, the set of known Fourier coefficients will be preserved by the restoration
procedure. The challenge is to recover the unknown set of low, middle and high frequencies in a large region in the
spectrum. This amounts to applying an interpolation operator 𝜑
𝑊
to the spectrum of 𝒚 to get an estimator ̂𝒙 of 𝒙:
̂
𝒙
=
−1
◦𝜑
𝑊
◦
𝒚.
(4)
2.2. Principles of the iterative two step-algorithm
Our computational method takes as input an noisy image 𝒚 and iterates two steps as illustrated in Fig.
2
. At the
initialization (𝑡 = 0), we set 𝒙
(0)
= 𝒚:
• In the proposal step and at iteration 𝑡, white Gaussian noise 𝜺 is added to the current image 𝒙
(𝑡−1)
. The Fourier
coefficients are then non-zero in the MW region 𝑊 . We substitute the original Fourier coefficients in the SR
region 𝑊 to the noisy Fourier coefficients to preserve information. Formally, this amounts to applying an
projection operator
𝑊
(⋅)
to the artificially noisy image. The objective is to randomly initialize the Fourier
coefficients in the MW. A denoising algorithm is then used to efficiently remove the majority of the noise while
preserving image structure. Due to its random nature, a small amount of the added noise will be close to the
signal and is thus "saved" after denoising. The denoising algorithm (⋅), which promotes smoothness while
preserving contours, acts as a spatial regularizer. In summary, a candidate image 𝒛 = (
𝑊
(𝒙
(𝑡−1)
+ 𝜺))
is
obtained by applying an projection operator in the Fourier domain and a denoising operator in the spatial
domain.
• In the second evaluation step, the candidate image 𝒛 is compared and is potentially substituted to the current
image 𝒙
(𝑡−1)
depending on its ability to satisfies several constraints, including fidelity to the original input image
𝒛
. A cost function (or energy) is used to accept/reject this candidate image with a certain probability established
in the Metropolis-Hastings simulation framework as presented in the next section.
By repeating these two steps iteratively, we collect several hundreds of images {𝒙
(𝑡)
}
to compute an average
corre-sponding to the final restored image. The correcorre-sponding MWR (Missing Wedge Restoration) algorithm is actually able
to diffuse information from SR into MW. It cannot retrieve unobserved data, but it merely makes the best statistical
t
Proposal
step
Evaluation
step
Denoising
Noise
Recover known Fourier coefficients
+
from original volume
Spatial domain
Spectral domain
Accept/reject
according to
Metropolis scheme
Compute
energy
x
t
+ ✏
P
W
(x
t
+ ✏)
D (P
W
(x
t
+ ✏))
z
x
t+1
y
Unsampled region
(Missing Wedge)
Sampled region
(SR)
00
PW
(x
(t)
+ ")
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y
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(t)
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D (PW
(x
00
(t)
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z
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