ITERATIVE DISCRETE STEERING
OF THREE-DIMENSIONAL RECONSTRUCTIONS FROM
X-RAY IMAGES WITH LIMITED ANGLE
Anja Frost, Eike Renners
and Michael Hötter
Institut für Innovationstransfer, University of Applied Sciences and Arts, Ricklinger Stadtweg 120, Hannover, Germany
Keywords: Computer Tomography, CT, Discrete Tomography, DT, X-ray, Emission Data, Limited Data Problem,
Three Dimensional Image Reconstruction, Algebraic Reconstruction Technique (ART), Binary Steering,
Evaluation, Probability Calculus, Accuratio.
Abstract: Computer Tomography is aimed to calculate a three dimensional reconstruction of the inside of an object
from series of X-ray images. This calculation corresponds to the solution of a system of linear equations, in
which the equations arise from the measured X-rays and the variables from the voxels of the reconstruction
volume, or more precisely, their density values. Unfortunately, some applications do not supply enough
equations. In that case, the system is underdetermined. The reconstructed object, as only estimated, seems to
be stretched. As there are a few voxels, that are already representing the object true to original, it is possible
to exclude these variables from the system of equations. Then, the number of variables decreases. Ideally,
the system gets solvable. In this paper we concentrate on the detection of all good reconstructed voxels i.e.
we introduce a quality measure, called Accuratio, to evaluate the volume voxel by voxel. In our experi-
mental results we show the reliability of Accuratio by applying it to an iterative reconstruction algorithm. In
each iteration step the whole volume is evaluated, voxels with high Accuratio are excluded and the new
system of equations is reconstructed again. Steadily the reconstructed object becomes “destretched”.
1 INTRODUCTION
First and foremost, Computer Tomography was
introduced for clinical diagnostics. Nowadays, it is
also used for quality assurance in the production and
maintenance of any object. As it generates a three
dimensional reconstruction of the inside of the
object from series of X-ray images, inner structures
such as casting defects or cold soldered connections
become visible. Moreover, exact measurements of
the shape are feasible. But the very use in quality
assurance demands reconstructions that are
absolutely true to original. To calculate such
reconstructions, it is necessary to provide many X-
ray images from different angles of vision. Ideally,
the object is turned through 360° while x-raying.
In some applications it is not possible to turn the
object through 360°. For example, if the shape is
bulky and stops the rotation in the computer
tomography scanner, then the number of X-ray
images is reduced. Mathematically speaking, the
reconstruction problem is underdetermined. The
reconstructed volume can only be estimated. For the
most voxels this estimation differs drastically from
the target. The object seems to be stretched (shown
in Figure 1).
Figure 1: Cross sections through the reconstructed volume
from a series of X-ray images spanning 360° (left image)
and 135° (right image).
An underdetermined system of equations can not
be solved in principle. But including a priori
knowledge that does not arise from the measurement
can improve the reconstruction quality. There are
two types of a priori knowledge; knowledge about
the shape, and knowledge about the materials the x-
rayed object consists of, i.e. the density of each
89
Frost A., Renners E. and Hötter M..
ITERATIVE DISCRETE STEERING OF THREE-DIMENSIONAL RECONSTRUCTIONS FROM X-RAY IMAGES WITH LIMITED ANGLE.
DOI: 10.5220/0003809200890094
In Proceedings of the International Conference on Computer Vision Theory and Applications (VISAPP-2012), pages 89-94
ISBN: 978-989-8565-03-7
Copyright
c
2012 SCITEPRESS (Science and Technology Publications, Lda.)
material. An abundance of approaches exists for
both types.
Most of all solutions that deal with knowlege
about the shape fit a parametric model to the
reconstructed object. They get useful results, as
shown for example by Benameur et al. (2003) or
Sadowsky et al. (2011). Others handle fragmentary
knowledge about the shape demanding smoothness
of the surface or similar, and usually combine
knowledge about the material densities (Varga,
Balázs and Nagy, 2010).
It becomes more difficult to improve the
reconstruction of any object due to the sheer
knowledge about the material densities. Some
approaches achieve passable results for objects that
consist of one material only. Then, the
reconstruction problem is a binary decision between
object and air. But the algorithms get unreliable if
the object is composed of several materials and x-
rayed with limited angle. Herman and Kuba
published a compendium of these works in 2007.
In our approach, we introduce a new quality
measure to evaluate a volume voxel by voxel
including a priori knowledge about the materials, no
matter the number of different materials. By
evaluating, voxels that are representing the target
true to original get detected. In a second step, we
exclude the detected voxels from the system of
equations. This means, the number of variables
decreases. The reconstruction problem, which was
once underdetermined, becomes solvable.
In Section 2.1 we introduce the basic notation
and give an overview about conventional
reconstruction techniques. Then, in Section 2.2 we
enter into the question how to decrease the number
of variables. In Section 3 the evaluation step is
described. We present and discuss our experimental
results in Section 4. A few remarks conclude the
paper.
2 RECONSTRUCTION
TECHNIQUES
2.1 Foundations
Physically, the grey value y
i
of pixel i in an
measured X-ray image is equal to the line integral of
the density x(l) along the ray path l.
= dllxy
i
)(
(1)
To calculate a reconstruction of the x-rayed
object in a quantised grid the equation (1) changes
into equation (2).
=
=
N
n
nnii
xwy
1
(2)
Here, N is the number of voxels in the whole
volume, x
n
represents the density of voxel n, and w
ni
describes the contribution of the n
th
voxel to the i
th
measurement. Figure 2 shows a ray path through the
volume. Voxels with w
ni
0 are marked with a red
outline.
Figure 2: X-ray in a regular grid.
There are several techniques to calculate the
densities x
n
from the measured X-rays y
i
, listed very
comprehensively in (Kak and Slaney, 1988). These
techniques can be divided into two categories: On
the one hand, there are analytical methods, which
involve all X-ray images to generate one
reconstruction. On the other, there are iterative
methods that approach the solution step by step. In
each step a correction for the current reconstruction
is calculated.
One of the most frequently referred iterative
reconstruction algorithms in highly topical works is
ART, first published by Gordon, Bender and
Herman (1970), recently adapted to GPU-based
calculation and sophisticated by Xu and Müller
(2007). As we adapt this technique to our work, we
explain it more detailed.
In ART, the differences y
i
- w
ni
x
n
of all
measurements i are minimised by applying the
Kaczmarz method (Kunze, 2007). This leads to the
correction equation (3) for the density x
j
of a voxel j
and for iteration number k+1.
ji
N
n
nini
N
n
k
nnii
k
j
k
j
w
ww
xwy
xx
+=
=
=
+
1
1
)(
)()1(
(3)
VISAPP 2012 - International Conference on Computer Vision Theory and Applications
90
As long as y
i
w
ni
x
n
, the difference y
i
- w
ni
x
n
is divided by the number of voxels that are part of
the ray i (w
ni
0), and distributed uniformly among
these voxels. In this way, after adding in accordance
with (3), the difference y
i
- w
ni
x
n
is zero. Equation
(2) is met, but for the current ray i only.
2.2 Vary the Number of Variables
In the following, A = {1, 2, 3, ..., N } is the set of all
voxels, strictly speaking their indices. B is a subset
of A and contains all bad reconstructed voxels i.e.
the remaining variables. The equation (2) can be
expressed by (4).
+=
Bn
nni
Bn
nnii
xwxwy
(4)
Based on (4), the ART correction equation for
any voxel j
Β
changes into
ji
Bn
nini
An
k
nnii
k
j
k
j
w
ww
xwy
xx
+=
+
)(
)()1(
(5)
and only for j
B the update is executed.
Now, the difference y
i
- w
ni
x
n
0 is distributed
among the bad reconstructed voxels only, i.e.
exactly the voxels that have produced the difference.
The basic idea of excluding good reconstructed
voxels from the system of equations is already
mentioned by Batenburg and Sijbers (2007). In
contrast to our topic, they work on the problem of
limited data, which yields reconstructions that are
slightly deformed in all directions. In our approach
we use a novel quality measure in the evaluation
step, better fitted to the problem of limited angle i.e.
extremely stretched reconstructions.
3 EVALUATION STEP
3.1 Include A Priori Knowledge
There are many ways of including a priori
knowledge about the materials of which the X-rayed
object consists into its reconstruction. Some present
works use a conventional reconstruction technique
(according to Section 2.1), and afterwards put in a
priori knowledge. In the simplest case, for example
in (Censor 2001), the reconstructed density x
n
is
discretised via thresholding.
Others, such as Kuba, Ruskó, Rodek and Kiss
(2005) extend the cost function with an additive
term which includes a priori knowledge. For our aim
to evaluate the reconstruction, we pursue a related
strategy: For every voxel j the reconstructed density
x
j
is replaced successively with each predefined
material density m
d
. For each inserted material
density and for each X-ray i crossing the selected
voxel the mean square deviation between y
i
and
w
ni
x
n
is calculated. The sum of all mean square
deviations, normalised to the number of rays I, is
called density error f
j
(m
d
) (6).
I
mwxwxwy
mf
I
i
djijji
An
nnii
dj
∑∑
=∈
+
=
1
2
)(
(6)
The higher the support of the measurements for a
material density m
d
is (or, metaphorically speaking,
the more X-rays the material density m
d
prefer), the
smaller will be the density error f
j
(m
d
).
3.2 Probability Modelling
The density errors f
j
(m
d
) already provide a basis for
the selection of the most probable material m
d
(q.v.
Frost and Hötter (2010)). But in order to classify a
voxel, we need a quality measure that is independent
of parameters such as the object size and enables a
quantitative evaluation of the reconstruction quality.
Therefore, we convert the density error f
j
(m
d
) into a
probability p
j
(m
d
): a probability of existence of the
material m
d
at the position j. The conversion takes
three constraints into account: p
j
(f
j
=0) = 1, 0 p
j
(f
j
)
1 and p
j
(f
j
) is strictly monotonic decreasing. The
Gaussian function in (7) fulfils the three constraints.
2
~)(
j
f
jj
efp
π
(7)
The output of (7) still depends on the material
densities and size of the x-rayed object. In case it is
demanded to compare various reconstructions with
different objects, the conversion in (7) has to be
expanded by a fourth constraint; If two distinct
material densities, for example m
d
and m
d+1
, produce
density errors f
j
(m
d
) and f
j
(m
d+1
) that are equal in
value, it is impossible to come to a decision. Both
materials are equiprobable. In this instance the
density error is the squared half difference of both
material densities and p
j
(f
j
= ((m
d+1
m
d
)/2)²) = 0.5.
Considering this fourth constraint, we get the
conversion formula (8).
2
1
2/)((
2ln
)(
+
=
dd
j
mm
f
jj
efp
(8)
ITERATIVE DISCRETE STEERING OF THREE-DIMENSIONAL RECONSTRUCTIONS FROM X-RAY IMAGES
WITH LIMITED ANGLE
91
3.3 Probability Distribution
The probability p
j
(m
d
) of material density m
d
with d
= 1...D and D different materials yields a probability
distribution. For a good reconstructed voxel j the
very density m
d
, which was really existing at the
position j while x-raying, is supported by the most
rays and stands out with a high probability p
j
(m
d
).
For a more inaccurate reconstructed voxel, the rays
do not correspond with the preferred material
density. In this case, less rays support the real
density m
d
. The probability p
j
(m
d
) is lower.
Figure 3 shows exemplarily the probability
distributions of two voxels in a reconstructed
volume. Voxel g is situated inside of the large
sphere and evidently good reconstructed. The
probability distribution holds a distinct maximum at
material density m
1
, which stands for the sphere
material. Voxel b is near to the surface. The
reconstruction in this area is difficult, because some
measurements prefer material density m
1
, and others
favour the air density m
0
. There is no outstanding
maximum in the probability distribution.
Figure 3: Probability distribution of a good reconstructed
voxel g (green coloured) as well as of a bad reconstructed
voxel b (red coloured).
3.4 Quality Measure
Now, for each voxel there is a probability
distribution which expresses the quality of the
reconstruction. For the practical application it is
necessary to handle one numerical value instead of a
function. Hence, we pick out the maximum of
probability distribution max(p
j
), briefly called
Accuratio a
j
, since this maximum already obtains all
the information required for quality determining.
Figure 4 shows a cross section through the Accuratio
volume that corresponds to a 135°-reconstruction.
The higher the Accuratio of a voxel, the lighter the
grey is displayed. It becomes visible that Accuratio
is generally high, except for the falsified areas,
where a low Accuratio predominates.
Figure 4: Cross section through the reconstructed volume
from a series of X-ray images spanning 135° (left image)
and corresponding Accuratio (right image).
Binarising the Accuratio volume, we distinguish
“good” from “bad” reconstructed voxels.
4 EXPERIMENTAL RESULTS
4.1 Algorithm of Discrete Steering
In our experiment we executed the reconstruction
algorithm outlined in Figure 5.
Figure 5: Iteration of Discrete Steering.
At the start, the subset B is equal to A. The whole
volume will be reconstructed. When each X-ray
image was used for reconstruction once, each voxel
is evaluated by our quality measure. Voxels with a
j
> 0.5 are defined as “good” and excluded from the
subset B. Moreover, their reconstructed density x
j
is
replaced by the most likely material m
d
. In this way,
the reconstruction is steered into a discrete solution.
In the next reconstruction step (according to
equation 5), the “good” voxels go into w
ni
x
n
, which
is also called forward projection. But only the “bad”
voxels become reconstructed again.
4.2 Test Objects and Scenes
The experiment deals with one mathematically
defined object (A) and one real work piece made of
aluminium (B). Object A modelled on a circuit
board (Figure 6). Three soldered points (material
VISAPP 2012 - International Conference on Computer Vision Theory and Applications
92
Figure 6: Cross section in xy-plane of test object A.
density m
3
= 2.7) are situated between two plates (m
1
= 0.9). On the top, there is a flat box (m
2
= 1.8).
From this model we generated 29 X-ray images in a
157° rotation around the z-axis. Each image has the
size of 64² pixels. The volume contains 64³ voxels.
Object B we describe as screw-nut. 168 X-ray
images on a scale of 168 x 128 pixels were
measured in a 360° rotation. The reconstruction of
the complete data is shown in Figure 1 (on the left).
Afterwards, we limited the angle range to 135°.
The Figures 7 and 8 demonstrate the influence of
limited angle on conventional ART as well as the
improvements by Discrete Steering. As can be seen,
the results of ART are falsified: In Figure 7 (Object
A) all horizontal surfaces, such as the plates, are
blurred. Hence, the crack between the right soldering
and upper plate vanishes completely. In Figure 8
(Object B) the air gap is invisible, too. Additionally,
the reconstruction volumes do not change
significantly during 8 iterations of ART. At the same
time, in each iteration step of Discrete Steering the
reconstructed objects converge more and more to the
original shape. The first evaluation step already
improves the reconstruction. Here, due to the
inclusion of a priori knowledge, some falsified areas
can be corrected immediately. For example, light
artefacts in the air disappear. In case the falsified
areas can not be corrected, they show low
Accuratios and will be reconstructed again. In the
following evaluation steps, the number of bad
reconstructed voxels decreases. In Figure 7 the
plates get a distinct shape and the crack between the
right soldering and upper plate becomes visible. In
Figure 8 the air gap is detected by low Accuratios.
To sum up, in the end the reconstructed objects
show the rough shape of the original or, at least, an
improvement compared to the results of ART.
In 24 test series with varied limited angles from 129°
to 157° we recorded the number of correctly
discretised voxels while reconstructing. In
comparison to ART (with discretisation via
thresholding), the Discrete Steering Algorithm
performs a faster as well as longer rise, and finally
keeps 98 % correctly discretised voxels on average.
5 CONCLUSIONS
We have presented a technique to generate three
dimensional reconstructions from X-ray images
spanning a limited angle only, i.e. in the first
instance the system of equations is underdetermined.
By detecting and excluding all good reconstructed
voxels from the system of equations, the number of
variables decrease and the system gets solvable.
To distinguish “good” from “bad” reconstructed
voxels, we have applied the new quality measure
Accuratio: For each voxel there is calculated a
probability distribution of a priori known material
density by taking into account all X-rays that are
crossing the selected voxel. The better the
reconstruction is, the more X-rays prefer the same
material and the more the maximum of the
distribution increases. If the maximum exceeds 0.5,
we define the voxel as “good” assuming that the
corresponding material density really existed at the
position of the selected voxel while measuring.
We have shown in our experimental results that
Accuratio is suited to detect “good” and “bad”
voxels. The very first evaluation step marks falsified
areas or correct them reliably. In the following
iterations, the number of bad reconstructed voxels
Figure 7: Cross sections through the reconstruction volume of test object A applying conventional ART (upper row) and
Discrete Steering (lower row). In the results of Discrete Steering, voxels with a
j
0.5 are red marked.
ITERATIVE DISCRETE STEERING OF THREE-DIMENSIONAL RECONSTRUCTIONS FROM X-RAY IMAGES
WITH LIMITED ANGLE
93
Figure 8: Cross sections through the reconstruction volume of test object B applying conventional ART (upper row) and
Discrete Steering (lower row). In the results of Discrete Steering, voxels with a
j
0.5 are red marked.
decreases. The over all quality increases. In the end,
the reconstructed objects are closer to the original,
though a lot of noise appeared. In further works we
want to address the problem by separate noise
suppression.
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