Augmented Likelihood Image Reconstruction with Non-local Prior Image Regularization

Abstract

The presence of high-density objects remains an open problem in medical CT imaging. The recently published Augmented Likelihood Image Reconstruction (ALIR) algorithm has shown to outperform current methods for phantom data and real clinical cases of patients with different kinds of metal implants. A variation of the algorithm with an additional non-local prior image based regularization term is proposed. The prior image should hold anatomical information that are similar to the target image. In every iteration of the ALIR algorithm, a new image is calculated based on the given prior image and a registration step. The resulting image is then used to penalize intensity variations. Reconstruction results show that the regularization step improved the reduction of streaking artifacts.

Full text

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Augmented Likelihood Image Reconstruction with
Non-local Prior Image Regularization
Maik Stille and Thorsten M. Buzug
Abstract—The presence of high-density objects remains an
open problem in medical CT imaging. The recently published
Augmented Likelihood Image Reconstruction (ALIR) algorithm
has shown to outperform current methods for phantom data
and real clinical cases of patients with different kinds of metal
implants. A variation of the algorithm with an additional non-
local prior image based regularization term is proposed. The
prior image should hold anatomical information that are similar
to the target image. In every iteration of the ALIR algorithm,
a new image is calculated based on the given prior image
and a registration step. The resulting image is then used to
penalize intensity variations. Reconstruction results show that the
regularization step improved the reduction of streaking artifacts.
I. INTRODUCTION
Computed tomography (CT) remains one of the key imaging
methods in clinical practice. Image quality of reconstructed
CT images can be reduced by the occurrence of different
artifacts, which are caused by physical phenomena such as
scattering, beam hardening, noise, or total absorption. These
phenomena can be amplified due to high-density objects
such as metal implants or surgical instruments. The resulting
streaking artifacts obstruct the assessment of the anatomy of
the patient and can reduce the diagnostic value of the images
drastically.
In order to reduce metal artifacts a variety of approaches
have been proposed in the last decades [1]–[3]. The re-
cently published Augmented Likelihood Image Reconstruction
(ALIR) algorithm has shown to outperform current methods
for phantom data and real clinical cases of patients with
different kinds of metal implants including hip implants,
knee implants and amalgam fillings [4]. Due to its iterative
reconstruction scheme and the augmented Lagrangian based
optimization the algorithm enables a high degree of flexibility.
We present an ALIR variation with a prior image based non-
local regularization term, which was recently published in [5].
The regularization term penalizes intensity variations between
the image to be reconstructed and a prior image. While the
prior image holds information from an image that looks similar
to the image that is to be reconstructed, the regularization term
forces the reconstruction to keep anatomical information of the
original image while reducing streaking artifacts.
II. ME TH OD S
Given a set of intensity measurements {ni}M
i=1, the nega-
tive log-likelihood function for transmission tomography for
statistical image reconstruction is defined as
l(f) =
M
X
i=1 
−niln(n0) + ni
N
X
j=1
aij fj
+ ln(ni!) + n0exp(−
N
X
j=1
aij fj)

(1)
where f∈RNis a vector that consists of the expected
attenuation coefficients [3]. The number of photons that are
detected in the absence of absorption is denoted by n0, the
total number of projections is denoted by M, and the number
of pixels in the image is denoted by N.
In order to reconstruct an image without the usage of x-
rays that run through a metal object, the set of projections
indices M={1, . . . , M }can be divided into a set of indices
for projections that are not affected by metal, M1, and a set
for projections that are affected by metal, M2. In ALIR the
constant terms in (1) are omitted and only projection indices
of x-rays that are not affected by metal are used, which results
in
ˆ
lΛ(f) = 1
|M1|X
i∈M1

ni
N
X
j=1
aij fj
+n0exp(−
N
X
j=1
aij fj)
.
(2)
Furthermore, the ALIR algorithm works with the assump-
tion that prior knowledge in form of shape and known at-
tenuation coefficients of the metal object is available. This
knowledge can be gained by an exact computer-aided design
(CAD) description of these objects, which can be potentially
provided by manufacturers [6]. However, if an exact model
of the metal object is not available, the proposed algorithm
is able to operate with an approximation of the shape, gained
from a segmentation step, combined with arbitrary attenuation
coefficients. Let b∈RNbe a vector that contains attenuation
coefficients of the implant and Q∈[0,1]N×Na diagonal
matrix with qij = 0 if i6=jthat represents a mask.
In the ALIR algorithm, the prior knowledge of the metal
object is used to introduce the equality constraint
cΛ(f) = µ
2
N
X
j=1 qjj (fj−bj)
N2
−
N
X
j=1
λj
(qjj (fj−bj))
N,
(3)

2
Ground Truth Metal Object Prior Image Difference
Figure 1. The used XCAT phantom. From left to right: the ground truth image of the image that is be reconstructed, the artificial metal object, the prior
image, which is located approx. 2cm proximal to the first image, and the difference between the prior image and the image that is to be reconstructed.
with the multipliers λ∈Rnand µ∈R, which is applied in
order to assign the given attenuation coefficients to the correct
position in the reconstructed image. The objective is formu-
lated as an augmented Lagrangian, which incorporates (3)
directly in
Λζ(f;λ, µ) = ˆ
lΛ(f) + ζcΛ(f) + γ R, (4)
where γis a regularization parameter, Ris a regularization
term and the weighting factor ζ > 0is introduced in order
to control the influence of the constraints in relation to the
log-likelihood function [4].
In the course of the algorithm, projection values
pi=
N
X
j=1
aij fj, i ∈ M2(5)
are replaced by a forward projection of a bilateral filtered
version of interim results and the set of all indices Mis used
in (2) instead of the set M1[4], [7].
For the regularization Rthe previously proposed non-local
prior image regularization is used [5]. The term penalizes
intensity variations to a prior image g∈RNthat should
include similar anatomical information as the image fand
is defined as
R(f, Γ(g, γ)) = v
u
u
t
N
X
x=0
δxfx−1
ωx
Ψx(f, Γ(g, γ))2
,(6)
with
Ψx(f, Γ(g, γ)) =
X
y∈Nx
Γ(gx, γ) exp −||fηx−Γ(g , γ)ηy||p
h2(7)
where Γ(g, γ )is the transformed prior image gwith the trans-
formation parameter γ[5]. Without loss of generality, an affine
transformation is used. The parameters of the transformation
result from the optimization problem
D(f(k),Γ(g, γ )) !
=min (8)
where D:R2N→Rdenotes a distance measure. Problem (8)
is solved in every iteration of the reconstruction algorithm
using the l-BFGS-b algorithm [5], [8]. Within (7) ηxdenotes
a patch window around pixel x,Nxdenotes a search window
around pixel x, and || · ||pdenotes the Minkowski distance of
order p. Furthermore, λ∈ {0,1}Nis a mask with
δx=(0if gx= 0
1if gx6= 0 ,(9)
which forces the regularization to ignore all pixels where the
prior image holds no information.
III. RES ULTS
In order to investigate the performance of the proposed
ALIR algorithm with non-local prior image regularization, a
software phantom was generated using the XCAT software [9].
Two different slices were used that are located around the
pectoral girdle. In the first slice an artificial metal object was
manually added within the left humerus. This slice is used as
the target image f. The second slice is located approximately
2cm proximal to the first slice and is used as the prior image
g. Most importantly, the image gshows anatomical differences
compared to the image fand contains no metal artifacts nor
metal objects as can be seen in figure 1.
For the initialization of the ALIR algorithm a forward-
projection of the ground truth image is calculated and all
projection values that are affected by the artificial metal object
are removed. The description of the metal object in the form
of band Qis gained from a segmentation of the metal object
in the ground truth image. Furthermore, the prior image that is
seen in figure 1 is used for the regularization, which is given
in (6).
After 19 iterations the ALIR algorithm reached conver-
gence. In figure 2 selected interim results are shown together
with the calculated prior images Ψx(f, Γ(g, γ)). In the course
of the reconstruction the each time recalculated prior imaged
shows more and more similarity to the image f. After approx.
13 iterations no changes in the new prior image can be
observed. However, the regularization ignores pixels that do
not hold any information. Therefore, holes in the recalculated
prior image are not inherited into the image f.
In figure 3 the final reconstruction results are given for
the ALIR algorithm, the ALIR algorithm with non-local prior
image regularization and the linear interpolation approach [1].
The amount and severity of streaking artifacts is highest in
the reconstruction result of the linear interpolation approach.
The ALIR algorithm is able to reduce most of the artifacts
and results in a substantially enhanced image. However, the

3
Ground Truth Linear Interpolation ALIR ALIR with Reg.
Figure 3. Reconstruction results. From left to right: the ground truth image, the linear interpolation approach, ALIR and ALIR with additional non-local prior
image regularization.
Iteration 1 Prior Image 1
Iteration 3 Prior Image 3
Iteration 5 Prior Image 5
Iteration 13 Prior Image 13
Figure 2. Interim results and the corresponding prior image of the ALIR
reconstruction with non-local prior image regularization.
regularization step based on a prior-image is further beneficial
for the metal artifact reduction.
In order to confirm the visual impression quantitatively,
the sum of squared differences between the ground truth
and the result of each reconstruction method is calculated.
Table I shows an unambiguous result that the linear interpo-
lation shows clearly the highest error with 1195.9HU. The
ALIR algorithm already shows a much better performance
with 134.4HU. However, an additional non-local prior image
MAR Method SSD [HU]
linear interpolation 1195.9
ALIR 134.4
ALIR with regularization 32.7
Table I
SUM O F SQUAR ED DI FFER EN CES (SSD) O F DI FFER ENT MAR METHODS
CO MPAR ED TO G ROU ND TRU TH . SEE FIGURE 3FOR THE CORRESPONDING
IM AGES .
regularization reduces the error further to a minimal error of
32.7HU.
IV. CONCLUSION
A variation of the Augmented Likelihood Image Recon-
struction algorithm with a non-local prior image based regular-
ization is proposed. In every iteration a transformation between
intermediate results of the ALIR algorithm and the prior image
is found. Based on a non-local approach a new prior image is
calculated that is used to penalize intensity variations between
the recalculated prior and the image that is to be reconstructed.
Reconstruction results show an enhanced artifact reduction
compared to ALIR without regularization and the linear in-
terpolation approach. While incorporating information based
on a prior image that holds similar anatomical structures, the
correct detailed anatomical information of the target image
could be reconstructed.
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