HAL Id: hal-01737088
https://hal.archives-ouvertes.fr/hal-01737088
Submitted on 19 Mar 2018
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Sparse reconstruction methods in x-ray CT
Juan F P J Abascal, Monica Abella, Cyril Mory, Nicolas Ducros, Claudia de
Molina, Eugenio Marinetto, Françoise Peyrin, Manuel Desco
To cite this version:
Juan F P J Abascal, Monica Abella, Cyril Mory, Nicolas Ducros, Claudia de Molina, et al.. Sparse
reconstruction methods in x-ray CT. Developments in X-Ray Tomography XI, Aug 2017, San Diego,
United States. �10.1117/12.2272711�. �hal-01737088�
1
Sparse reconstruction methods in X-ray CT
J. F. P. J. Abascal
*a
, M. Abella
b,c
, C. Mory
a
, N. Ducros
a
, C. de Molina
b,c
, E. Marinetto
b,c
, F. Peyrin
a
,
M. Desco
b,c,d
a
Univ.Lyon, INSA-Lyon, Université Claude Bernard Lyon 1, UJM-Saint Etienne, CNRS, Inserm,
CREATIS UMR 5220, U1206, F-69100, Lyon, France;
b
Dep. Bioingeniería e Ingeniería Aeroespacial, Universidad Carlos III de Madrid, Madrid, Spain;
c
Instituto de Investigación Sanitaria Gregorio Marañón (IiSGM), Madrid, Spain;
d
Centro de Investigación en Red de Salud Mental (CIBERSAM), Madrid, Spain
ABSTRACT
Recent progress in X-ray CT is contributing to the advent of new clinical applications. A common challenge for these
applications is the need for new image reconstruction methods that meet tight constraints in radiation dose and
geometrical limitations in the acquisition. The recent developments in sparse reconstruction methods provide a
framework that permits obtaining good quality images from drastically reduced signal-to-noise-ratio and limited-view
data.
In this work, we present our contributions in this field. For dynamic studies (3D+Time), we explored the possibility of
extending the exploitation of sparsity to the temporal dimension: a temporal operator based on modelling motion
between consecutive temporal points in gated-CT and based on experimental time curves in contrast-enhanced CT. In
these cases, we also exploited sparsity by using a prior image estimated from the complete acquired dataset and assessed
the effect on image quality of using different sparsity operators. For limited-view CT, we evaluated total-variation
regularization in different simulated limited-data scenarios from a real small animal acquisition with a cone-beam micro-
CT scanner, considering different angular span and number of projections. For other emerging imaging modalities, such
as spectral CT, the image reconstruction problem is nonlinear, so we explored new efficient approaches to exploit
sparsity for multi-energy CT data.
In conclusion, we review our approaches to challenging CT data reconstruction problems and show results that support
the feasibility for new clinical applications.
Keywords: Sparsity, compressed sensing, L1-norm, split Bregman, contrast-enhanced CT, respiratory gated-CT, limited
view CT, spectral CT
1. INTRODUCTION
Recent progress in X-ray CT is contributing to the advent of new clinical applications. A common challenge for these
applications is the need for new image reconstruction methods that meet tight constraints in radiation dose and
geometrical limitations in the acquisition.
The revolutionary discovery of compressed sensing (CS) provides a framework that enables accurate image
reconstruction from few noisy projections using convex optimization, provided that the image is sufficiently sparse in a
transformed domain and that uniform random sampling and an incoherence condition hold.
1
Given the success of CS,
sparse regularization has become a preferred choice for designing iterative regularized schemes even if the CS
assumptions are not met. Recent theoretical work also aims to extend the CS framework to more practical measurement
conditions and for nonlinear measurement operators.
2,3
Specifically, in x-ray imaging, sparsity has been proposed to
improve image quality in cases of reduced signal-to-noise-ratio and limited-view data.
4,5
In this work, we present our contributions to exploiting sparsity for solving challenging problems in X-ray CT. Most of
our contributions focus on high dimensionality problems, for which data redundancy and image sparsity can be highly
*
Juan.abascal@creatis.insa-lyon.fr
2
exploited. For dynamic studies (3D+Time), we have explored the possibility of extending the exploitation of sparsity to
the temporal dimension. In this case, the Prior Image Constrained Compressed Sensing (PICCS) algorithm leads to
superior results than the widely used total variation method, as images are highly sparse when subtracted from a prior
image obtained by combining all data.
6–8
In respiratory gated-CT, we have explored several methods. We assessed the effect on image quality of using different
sparsity operators within PICCS algorithm. An improved method obtained a sparser temporal representation by
modelling motion between consecutive temporal points. We proposed two different motion-based reconstruction
approaches. PRIMOR method modelled motion using B-splines and solved the optimization problem using the Split
Bregman formulation.
9
The second approach, named MA-ROOSTER, was applied to free breathing respiratory cone
beam CT data. The optimization framework is empirical: it alternates between iterations of conjugate gradient to
minimize a 4D data-attachment term, and several denoising steps: enforcing non-negativity, canceling motion outside the
patient, spatial 3D TV, and temporal 1D TV along curved trajectories determined by a priori motion.
10
The need for a
reasonable a priori motion, e.g. estimated from a previously existing 4D reconstruction, is critical for MA-ROOSTER. In
radiation therapy, its main target, patients undergo a 4D CT before treatment, therefore the prior information on motion
is available in clinical routine.
In contrast-enhanced CT, we proposed an alternative formulation to regularization based on replacing the original
problem by a new one that is well conditioned. This approach was specifically designed for the case of slow rotating
scanners for preclinical imaging. The idea was inspired on previous work where the solution was a combination of few
temporal basis functions that modelled the contrast uptake.
11
We reformulated the reconstruction problem as the recovery
of piecewise cubic polynomials in the temporal dimension, which significantly reduced the number of degrees of
freedom.
12
In limited-view CT, where only a few projections are taken in a reduced angular span, analytical reconstruction methods
provide low quality images with lots of artifacts. We evaluated the viability of total variation method to solve this
problem using numerical simulations based on geometrical description of a real scanner.
13,14
For other emerging imaging modalities, such as spectral CT, the image reconstruction problem is commonly divided in
two steps, the material decomposition problem, which is nonlinear, and the tomographic step. Mostly all studies that
proposed regularized- or sparsity-based approaches focused on the tomographic step. The material decomposition step is
nonlinear and so it is more challenging to solve. In this problem, we have recently proposed an efficient regularized
Gauss Newton approach to exploit specific regularization for each material.
15
2. RESPIRATORY GATED CT
Respiratory gating helps to overcome the problem of breathing motion in cardiothoracic imaging, which is relevant in
several clinical situations.
16,17
The gating process assigns acquired projections to the different respiratory phases, which
leads to insufficient data for the reconstruction of each phase when using standard reconstruction algorithms. Hence,
using conventional reconstruction methods requires acquiring more data, which results in a substantial increase in the
radiation dose. In this section, we propose several sparse reconstruction methods that exploit sparsity in the temporal
dimension.
2.1 Investigation of different sparsity transforms for PICCS algorithm
Prior image constrained compressed sensing (PICCS) has been proposed for reconstructing gated CT data from highly
undersampled and noisy data.
7,8
PICCS has led to significant improvements in image quality with respect to the total
variation method, in a wide variety of applications. Its success is because reconstructed images are highly sparse when
subtracted from a prior image, which is obtained by combining all data. PICCS solves the convex constrained
optimization problem
2
2
2
1
1
st. )( )1(min
pRuuuu
p
u
(1)
where u represents the reconstructed gates, u
p
represent the prior image, p represents the raw data for all respiratory
phases, R represents the forward operator, σ
2
accounts for noise in the data, and α weights the image prior penalty
function. The first and second terms in (1) impose spatial sparsity and temporal sparsity with respect to the prior image,
3
respectively. The common choice for the transformations Ψ and Ф is the spatial discrete gradient that leads to total
variation functional. However, other functional may lead to superior results.
In this work, we evaluated the suitability of different sparsity transforms for Ф (unitary, gradient, and symlet wavelet
transform) within the PICCS formulation for reducing dose in CT respiratory gating for small-animal imaging.
18
This
lead to three different flavors of PICCS: L1-PICCS, TV-PICS and WT-PICCS. Methods were assessed in different
scenarios, corresponding to different X-ray flux levels and number of projections. Problem (1) was solved using the Split
Bregman formulation, which efficiently solves L1-regularized problems.
19,20
All flavors of PICCS performed very
similarly in terms of noise, spatiotemporal resolution, and streak reduction. Nevertheless, the wavelet transform led to
superior image texture than the other transforms (figure 1). The gradient and unitary transforms led to staircase-like
artefacts and pixel-like artefacts, respectively. PICCS led to significantly improvement in image quality with respect to
filtered back-projection and allowed to reduce the respiratory motion artefact.
Figure 1. Zoomed-in images of reconstructed images with the different methods for one respiratory phase. Gated data consisted
of 120 projections per respiratory phase and X-ray flux corresponding to a number of photons I
0
= 4.5∙10
4
.
2.2 PRIMOR - A prior- and motion-based reconstruction method
In this work, we proposed a prior- and motion-based compressed sensing (PRIMOR) method for respiratory gating in
small-animal CT.
21
PRIMOR is a combination of both prior- and motion-based reconstruction methods. Thus, it benefits
from the advantages of these two approaches. On the one hand, it exploits the available prior image, and on the other
hand, it takes into account motion between consecutive respiratory gates, which leads to sparser representation in the
temporal dimension.
PRIMOR is implemented as a two-step approach. In the first step, motion is estimated using a nonrigid registration
method based on hierarchical B-splines.
22
In the second step, only image variation with respect to the prior is
reconstructed using the previously estimated motion. PRIMOR method solved the problem
2
2
21
1
1
)( st. )( )(min
pvuRvuvvu
ppp
v
(2)
where u
p
is the prior image, v is the image variation with respect to the prior image, u= u
p
+v, and T is an operator that
encodes motion between respiratory phases. PRIMOR was solved using the split Bregman formulation and was
compared to an equivalent prior-based method without motion estimation on different simulated scenarios (MATLAB
code for PRIMOR method can be found https://github.com/HGGM-LIM/prior-motion-reconstruction-CT). Both
PRIMOR and prior-based methods greatly improved FDK reconstruction in all scenarios. The prior-based method was
prone to streak artefacts and noise when using a low number of projections or low dose. PRIMOR corrected for these
effects, leading to better contrast recovery, less error and improved motion artefact correction (figure 2).
4
Figure 2. Zoomed-in of reconstructed images with FDK, prior-based method (PBR) and PRIMOR method, for different scenarios
corresponding to different number of projections and number of photons (I
0
= 4.5∙10
4
). Yellow arrows indicate where
artifacts are more noticeable: an increase in streak artifacts (1), blurred edges for bone tissue (2), blurred edges for soft tissue
(3), loss of contrast resolution within bone tissue (4).
2.3 MA-ROOSTER – Motion-Aware RecOnstructiOn using Spatial and TEmporal Regularization
MA-ROOSTER assumes that a motion mask, i.e. a rough segmentation of the region where movement is expected to
occur, is available.
10,23
As motion can occur outside the lungs, since the rib cage and the abdomen move during
breathing, the simplest option is to use the whole patient mask. It also assumes that a prior estimation of the patient’s
breathing motion is available. This prior motion estimation need not be accurate: if the temporal regularization is not too
strong, MA-ROOSTER will partially correct for motion misestimation.
The algorithm alternates between several optimization goals. It consists in solving the following five subproblems at
each iteration of the main loop:
- Minimizing the data-attachment term,
∑
α
‖
R
α
S
α
f
−
p
α
‖
2
2
, by 4D conjugate gradient (CG)
- Enforcing positivity, by setting all negative voxels of f to zero
- Removing motion where it is not expected to occur, by averaging along time outside the motion mask
- Enforcing the spatial gradient’s sparsity in each frame using 3D total variation (TV) denoising
- Enforcing the temporal gradient’s sparsity for each spatial position, by one-dimensional (1D) TV denoising along time
In order to account for the estimated breathing motion, all frames are warped to a single reference before temporal TV
denoising. The warped frames only differ where the motion estimation was inaccurate, or where the attenuation varies
over time as a consequence of tissue density change (typically, the lungs are denser at end-exhale than at end-inhale).
After temporal denoising, each frame is warped back to its original state.
Each supbroblem’s output is used as the input for the next subproblem. This constitutes one iteration of the main loop,
the output of which is fed back to the CG minimizer for the next iteration. Results are shown in figure 3.
