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

)²2/)((

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.

REFERENCES

Benameur, S., Mignotte, M., Labelle, H., De Guise, J.A.,

2003. A hierarchical statistical modeling approach for

the unsupervised 3-D biplanar reconstruction of the

scoliotic spine. In IEEE Transactions of Biomedical

Engineering, 52(12), 2041-2057.

Sadowsky, O., Lee, J., Sutter, E. G., Wall, S. J., Prince, J.

L., Taylor, R.H., 2011. Hybrid cone-beam

tomographic reconstruction: incorporation of prior

anatomical models to compensate for missing data. In

IEEE Transactions on Medical Imaging, 30(1), 69-83.

Varga, L., Balázs, P., Nagy, A., 2010. Projection selection

algorithms for discrete tomography. In Advanced

Concepts for Intelligent Vision Systems, Springer. pp.

390-401.

Herman, G. T., Kuba, A., 2007. Advances in discrete

tomography and its applications, Birkhäuser. Bosten.

Kak, A. C., Slaney, M., 1988. Principles of computerized

tomographic imaging, IEEE Press. New York.

Gordon, R., Bender, R., Herman, G. T., 1970. Algebraic

Reconstruction Techniques (ART) for three-

dimensional electron microscopy and X-ray

photography. In Journal of Theoretical Biology, 29(3),

pp. 471-481. Elsevier.

Xu, F., Müller, K., 2007. Real-time 3D computed

tomographic reconstruction using commodity graphics

hardware. In Physics in Medicine and Biology. 52, pp.

3405-3419. IOP Publishing.

Kunze, H., 2007. Iterative Rekonstruktion in der

medizinischen Bildverarbeitung, Shaker.

Batenburg, K. J., Sijbers, J., 2007. Dart: A Fast Heuristic

Algebraic Reconstruction Algorithm for Discrete

Tomography. In Image Processing, 4th IEEE

Conference on Image Processing. doi: 10.1109/ICIP.

2007.4379972

Kuba, A., Ruskó, L., Rodek, L., Kiss, Z., 2005.

Preliminary studies of discrete tomography in neutron

imaging. In IEEE Transactions on Nuclear Science.

52(1), pp. 380-385. IEEE Press.

Censor, Y., 2001. Binary steering in discrete tomography

reconstruction with sequential and simultaneous

iterative algorithms. In Linear Algebra and its

Applications (Vol. 339(1-3), pp. 111-124). Elsevier.

Frost, A., Hötter, M., 2010. Discrete Steering: Eine

statistisch orientierte Diskretisierung von

dreidimensionalen Rekonstruktionen aus

Röntgenaufnahmen. In Puente León, F. and Heizmann,

M, Forum Bildverarbeitung: [2. - 3. Dezember 2010

in Regensburg] (pp. 365-376). KIT Scientific

Publishing.

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