Download M10-200 A Projection Access Scheme for Iterative

A Projection Access Scheme for Iterative
Reconstruction Based on the Golden Section
Thomas Köhler
Philips Research Laboratories
Roentgenstrasse 24-26
22335 Hamburg
Germany
Abstract— A new access scheme for projections in iterative
image reconstruction is suggested. It is based on a constant
angular increment. The constant is chosen such that the range of
180◦ is divided by the golden section. It is shown in a simulation
study that the method behaves superior to the random access
scheme and the scheme based on prime number decomposition.
In standard ART [6], the linear system is solved iteratively
by he following procedure:
pl − Alj µj
j
2
= µki + λ
Ali .
(2)
µk+1
i
Alj
j
I. I NTRODUCTION
Iterative reconstruction techniques are widely used in the
area of PET and SPECT because they produce superior image
quality compared with filtered back-projection type algorithms, if the data are very noisy. Typically, these algorithms
operate on subsets of the measured data, e. g. they take one
projection to calculate an update for the intermediate image.
It is known that the ordering of the projection during the
iteration has a large influence on the convergence speed of
iterative image reconstruction algorithms like the algebraic
reconstruction technique (ART) [1]. Several methods have
been proposed and evaluated so far [2]–[4]. Mueller et al. also
evaluated a method with constant angular increment as it will
be used here. However, their choices of 66.0◦ , 69.75◦ , 73.8◦
are not motivated and differ from the choice presented below.
II. M ETHOD
A. Reconstruction Algorithm
The measurement in PET, SPECT, and CT provides (after
proper pre-processing) a set of N line integrals pl through
an continuous object function µ(x), which is the image to be
recovered.
In iterative methods, the image has to be decomposed into
a number of basis functions, and it is then represented by a
finite number of M coefficients. Here, Kaiser-Bessel functions
(blobs) [5] are used as basis functions. The discretization
results in a finite linear system
p = Aµ ,
(1)
where p is an N -dimensional vector of measured line integrals,
µ is an M -dimensional vector of image quantities (e. g.
absorption coefficients for CT), and A is the discrete version of
the integral operator that represents the measurement process.
In the original ART, an update step uses just one line integral. In simultaneous ART (SART) [7], a full projection is
calculated and used for each update step. This improves the
stability of the algorithm significantly. The analysis of the new
projection ordering scheme was performed using SART.
B. Projection Ordering
Traditionally, the design rule for the projection ordering is
the following: At any time, the next considered projection
angle should contain information about the object, which is
as independent of the previously taken projections as possible. This is achieved by maximizing the angular distance of
subsequent projections.
The new method presented here is inspired by the way many
plants position their new leafs [8]. It appears that a lot of plants
use an angle between succeeding leafs of approximately 360◦
times the golden ratio
√
5−1
g=
≈ 0.618 .
(3)
2
The resulting angle of about 222.5◦ might be a result of
an optimization problem: A new leaf should be placed such,
that there is as little overlap with previously grown leafs as
possible in order to avoid unwanted shadowing of previous
leafs. Geometrically, this optimization problem is very similar
to the problem of selecting the best order for projection angles
in iterative reconstruction methods.
In contrast to the botanic counterpart of the solution, the
proper angle between succeeding projection angles can be
chosen as g · 180◦ , since parallel projections, which are 180◦
apart from each other, are already equivalent. Note that there
is another symmetry in this particular problem: Going g · 360◦
counter-clockwise is the same as going (1−g)·360◦ clockwise.
Thus, (1 − g) · 180◦ is also valid choice.
By choosing this angular increment, several key features are
achieved:
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2
10
5
13
TABLE I
F IRST 16 F IBONACCI NUMBERS .
8
7
3
12
11
4
6
9
1
k
1
2
3
4
5
6
7
8
fk
1
1
2
3
5
8
13
21
k
9
10
11
12
13
14
15
16
fk
34
55
89
144
233
377
610
987
2
Fig. 1.
First 13 projections according to the proposed method.
1) Considering at any time the angular distribution of the
projection angles taken so far, it appears that only three
different values occur for the angular gap between these
angles. They have a relative size of 1, g, and g 2 .
2) Each new angle divides one of the largest angular gaps
according to the golden ratio, see Fig. 1, and therefore,
guarantees a maximum of new information to be taken
into account.
3) The sequence is based upon a constant angular increment. Thus, the method is very simple to implement.
4) The sampling pattern is non-periodic because the golden
ratio is irrational. This implies that all measured projections are processed during the iteration. This is a basic
feature required for all sequences to be independent of
the total number of acquired projections.
If this method is applied to a data set with discrete angular
sampling, the proposed angles have to be mapped onto the
discrete samples. This is done by choosing at any time within a
full iteration the nearest angle, which has not been taken so far.
If the profiles are sampled uniformly in angular direction, the
angular increment can be adjusted by adding a small angle to
take this into account. The required adjustment is smallest if N
is a Fibonacci number fj . For example, in the case N = f8 =
21, the adjustment is approximately 0.18◦ . Because of the very
small fraction of this adjustment with respect to the angular
increment of approximately 111.24◦ , the benefits described so
far are still fulfilled, see Fig. 2. The first 16 Fibonacci numbers
are given in Tab. I.
Another minor point is the following: If fi denotes the
ith Fibonacci number, then after fi processed projections, the
angles between processed or measured directions are equal or
they differ only by a factor of the g. This implies an almost
uniform distribution of angles over the full circle. This is
illustrated in Fig. 1 for the case of 13 projections, where the
two values g 5 · 180◦ ≈ 16.2◦ and g 6 · 180◦ ≈ 10.0◦ for the
gaps appear. The next angle of the sequence would partition
one of the g 5 · 180◦ segments into a g 6 · 180◦ and a g 7 · 180◦
segment.
A
1
2
3
B
1
5
2
3
4
C
1
5
2
8
7
3
4
6
D
1
2
10
5
13
8
7
3
12
11
4
6
9
E
1
2
15
7
20
12
4
10 18
5
13
21
8
16
3
11
19
17
6
9
14
F
1
Fig. 2. Access sequence for 21 projections over 180◦ . Note that after the
first two angles, there is a big and a small gap in the angular range (A). The
next angle divides the larger one, resulting in two large gaps and one small
gap (B). The next two angles divide the larger gaps, resulting in 3 large and
2 small gaps (C). In general, after fk angles, there are fk−1 large and fk−2
small gaps and the following fk−1 angles will divide the large gaps.
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resolution by the smoothing of the true image.
Clearly, the prime number decomposition approach suffers
from the large prime factors in the cases with 580 and 610
projections. Only for 640 projections, prime number decomposition provides the same good values for the correlation than
the golden section. For all cases, the new method provides
better results than the random access and better or equal results
than the prime number decomposition.
prime number decomposition
random
golden section
1.8
image resolution / mm
1.6
1.4
1.2
IV. S UMMARY
1
0.8
0.6
0
1
2
3
4
5
6
7
8
9
Fig. 3. Resolution of the reconstructed images as a function of the number
of iterations for the case of 580 projections.
III. R ESULTS
One-dimensional parallel projections of the forbild head
phantom were simulated. Apart from the proposed method,
the random access and the prime number decomposition [1]
were evaluated. Three different numbers of projections were
taken in order to evaluate the performance of the different
methods with respect to this number: 580, 610, and 640. 580
projections were used because this is the number a modern
CT scanner takes within 180◦ . 610 is considered because it is
a Fibonacci number, which is expected to work in particular
well for the proposed method. Finally, 640 contains only small
prime factors, which is expected to work well for the prime
number decomposition.
The images were evaluated after each full iteration. The
correlation method introduced by Liow and Strother [9] was
used for quantitative evaluation. In this method, the correlation
between the reconstructed image and a smoothed version of
the true image is calculated. Gaussian smoothing was used
here, and the size of the Gauss-kernel was alway chosen such
that the correlation with the reconstructed image is maximized.
The smoothing width represents an estimate for the resolution
of the reconstructed image. The correlation serves as a measure
for the image quality.
The resolution obtained by the different access schemes differs only very little. Exemplarily, the resolution as a function
of the iteration number is shown in Fig. 3 for the case of 580
projections.
For the obtained correlation, the results differ significantly
between the different number of projections. Fig. 4 shows
1− the correlation as a function of the number of iterations. Note that the correlation increases until iteration 4
and decreases again. This is a side effect of the additional
smoothing before the correlation is calculated. In all cases, the
resolution improves during the iteration. Once the resolution
reaches a certain limit, the result is influenced by the discrete
representation of the image, which results e. g. in overshoots
and undershoots at sharp edges. This is hidden at lower
The proposed method appears to be very insensitive to the
number of projections. It performs better than the random
access scheme and better or equal to the prime number decomposition. In particular for the case of practical importance
(580), it performs better. Furthermore, it is easier to implement
than the prime number decomposition or other known methods
like the weighted distance approach [3].
Results have been obtained for ART only, but it is anticipated that other iterative reconstruction algorithms like OSML [10] will behave similar.
ACKNOWLEDGMENT
I would like to thank Samuel Matej for providing the code
for the prime number decomposition.
R EFERENCES
[1] G. T. Herman and L. B. Meyer, “Algebraic reconstruction technique can
be made computationally efficient,” IEEE Trans. Med. Imag., vol. 12,
no. 3, pp. 600 – 609, 1993.
[2] H. Guan and R. Gordon, “A projection access order for speedy convergence of ART (algebraic reconstruction technique): A multilevel scheme
for computed tomography,” Phys. Med. Biol., vol. 39, pp. 2005–2022,
1994.
[3] K. Mueller, R. Yagel, and F. Cornhill, “The weighted distance scheme: A
globally optimizing projection ordering method for ART,” IEEE Trans.
Med. Imag., vol. 16, no. 2, pp. 223 – 230, 1997.
[4] H. Guan, R. Gordon, and Y. Zhu, “Combining various projection access
schemes with the algebraic reconstruction technique for low-contrast
detection in computed tomography,” Phys. Med. Biol., vol. 41, pp. 2413
– 2421, 1998.
[5] R. M. Lewitt, “Alternatives to voxels for image representation in iterative
reconstruction algorithms,” Phys. Med. Biol., vol. 37, pp. 705–716, 1992.
[6] R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction
techniques (ART) for three-dimensional electron microscopy and x-ray
photography,” J. Theor. Biol., vol. 29, pp. 471 – 481, 1970.
[7] A. Andersen and A. Kak, “Simultaneous algebraic reconstruction technique (SART): A superior implementation of the ART algorithm,”
Ultrason. Imag., vol. 6, pp. 81 – 94, 1984.
[8] E. Strassburger, Lehrbuch der Botanik, 33rd ed.
Stuttgart: Gustav
Fischer Verlag, 1991.
[9] J.-S. Liow and S. C. Strother, “The convergence of object dependent
resolution in maximum likelihood based tomographic image reconstruction,” Phys. Med. Biol., vol. 38, pp. 55 – 70, 1993.
[10] C. Kamphuis and F. Beekman, “Accelerated Iterative Transmission CT
Reconstruction Using an Ordered Subsets Convex Algorithm,” IEEE
Trans. Med. Imag., vol. 17, no. 6, pp. 1101 – 1105, 1998.
0-7803-8701-5/04/$20.00 (C) 2004 IEEE
580 projections
7e-4
prime number decomposition
random
golden section
1-correlation to smoothed true image
6e-4
5e-4
1e-4
3e-4
2e-4
0
1
2
3
4
5
iteration number
6
7
8
9
610 projections
7e-4
prime number decomposition
random
golden section
1-correlation to smoothed true image
6e-4
5e-4
1e-4
3e-4
2e-4
0
1
2
3
4
5
iteration number
6
7
8
9
640 projections
7e-4
prime number decomposition
random
golden section
1-correlation to smoothed true image
6e-4
5e-4
1e-4
3e-4
2e-4
0
1
2
3
4
5
6
7
8
9
iteration number
Fig. 4. Image quality of different projection access schemes for 580 (top),
610 (middle), and 640 (bottom) projections.
Fig. 5.
Example reconstruction for different access schemes after one
iteration for 580 projections. From top to bottom: Golden section, random,
prime number decomposition.
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Fig. 6.
Example reconstruction for different access schemes after one
iteration for 610 projections. From top to bottom: Golden section, random,
prime number decomposition.
Fig. 7.
Example reconstruction for different access schemes after one
iteration for 640 projections. From top to bottom: Golden section, random,
prime number decomposition.
0-7803-8701-5/04/$20.00 (C) 2004 IEEE