=Paper=
{{Paper
|id=None
|storemode=property
|title=SPECT Reconstruction with a Non-linear Transformed Attenuation Prototype
|pdfUrl=https://ceur-ws.org/Vol-715/bvm2011_85.pdf
|volume=Vol-715
}}
==SPECT Reconstruction with a Non-linear Transformed Attenuation Prototype==
https://ceur-ws.org/Vol-715/bvm2011_85.pdf
SPECT Reconstruction with a Non-linear
Transformed Attenuation Prototype
Sven Barendt, Jan Modersitzki
Institut of Mathematics and Image Computing, University of Lübeck
sven.barendt@mic.uni-luebeck.de
Abstract. This work deals with the single photon emission computed
tomography (SPECT) reconstruction process. As a SPECT measure-
ment also depends on unknown attenuation properties of the tissue, such
a process is challenging. Furthermore, the given attenuation may not be
a good approximation to the true attenuation field. Reasons are reposi-
tioning or movement of the patient such as relaxation during scan time or
even breathing. We propose a novel model for an attenuation correction
in SPECT reconstruction, which is a natural extension of an idea of Nat-
terer in that way, as the linear transformation of a so-called attenuation
prototype is enhanced to an arbitrary transformation. We present nu-
merical results for a non-linear spline transformation model which clearly
indicate the superiority of the proposed reconstruction model compared
to the case of no motion correction and the correction with a linear
transformation model.
1 Introduction
Single photon emission computed tomography (SPECT) is a nuclear medicine
imaging technique which can provide in vivo 3D functional information of tis-
sue. More precisely, functional information corresponds to the density of an
administered radio-pharmaceutical which has to be reconstructed from a set of
projections also known as sinogram. The reconstruction process is challenging
as the sinogram also depends on unknown attenuation properties of the tissue.
A commonly used simplification is to assume that the attenuation field is given
and to solve only for the density [1, 2]. Practically, the attenuation informa-
tion is supplied by an extra measurement such as a computed tomography (CT)
scan. Drawbacks of such an approach are that it requires a costly additional scan
which adds stress to the patient. Furthermore, the given attenuation may not be
a good approximation to the true attenuation field. Reasons are repositioning or
movement of the patient such as relaxation during scan time or even breathing
or heart beat.
Because of these limitations, recent approaches aim to simultaneously recon-
struct the density and attenuation [3, 4, 5, 6, 7, 8, 9, 2]. However, a fundamental
problem is that the combined reconstruction is generally ill-posed [4, 9]. In order
to circumvent ill-posedness of the combined reconstruction problem, Chang used
a simple model for the unknown attenuation field, which is essentially based on a
Non-linear SPECT Reconstruction 415
piecewise constant function [10]. However, the attenuation model is inadequate
to cover complex patient anatomy. Dicken addressed ill-posedness by introducing
non-linear Tikhonov regularization for both, density and attenuation [9]. How-
ever, limited success has been reported in the literature [2]. Natterer phrased
so-called consistency conditions on the range of the projection operator [4], which
then constrain the set of feasible attenuations. This approach was then used in
combination with a linearly transformed prototype for attenuation [6] and a
piecewise constant attenuation field [2].
This paper pursues the work of Natterer [6] as it removes the limitations to
an affine linear transformation model and enables essentially arbitrary transfor-
mations. In addition, we present a continuous mathematical framework for the
combined reconstruction model and follow the so-called discretize then optimize
paradigm to compute a numerical solution.
This paper is organized a follows. A novel reconstruction model is presented
in Section 2 and two numerical examples which clearly demonstrate superiority
of the non-linear approach are shown in Section 3. Finally, in Section 4, results
are discussed and an outlook is given.
2 Materials and Methods
In this section we introduce a novel model for the simultaneous reconstruction
of the tracer density f and the attenuation field µ, where we assume that the
unknowns f, µ ∈ L2 (R2 , R) are compactly supported on a domain Ω ⊂ R2 . As
common, the forward projection process is modeled by g = P[f, µ] with the
projection operator P (attenuated ray transform) and the measurement denoted
by g, e.g., [11]. Our reconstruction model is to find a minimizer (f, µ) of the
energy functional
J [f, µ] = ∥g − P[f, µ]∥L2 + αR[µ] (1)
where α denotes a positive regularization parameter and R a regularizer, which
is discussed in detail later.
Natterer used a similar approach with the constraint µ(x) = ν(ℓ(x)), where ℓ
denotes an affine linear spatial transformation and ν denotes a given prototype
of the expected attenuation [6]. Thus, the set of feasible µ consists only of affine
linear transformations of the prototype. Moreover, regularization is skipped and
consistency conditions are used to determine the affine linear transformation.
The new idea is to extend the class of feasible transformation and to replace
the consistency assumption by regularization. To simplify our presentation, we
restrict our model to free-form transformations, i.e. the transformation can be
expanded in terms of basis functions Bi (e.g. monomials of degree less equal
to one and B-splines of a certain order) and coefficients ci . Moreover, we also
limit our discussion to plain Tikhonov regularization of the coefficient vector.
However, it is noticeable that our model is far more general and allows for
non-parametric transformations and arbitrary regularization. With the above
416 Barendt & Modersitzki
specification, we have
{ ( ∑ ) }
R[µ] = ∥c∥22 where µ ∈ M := µ(c; x) = ν x + i ci Bi (x) , c ∈ Rn (2)
The mathematical model is to minimize J 1 subject to µ ∈ M 2. As an ana-
lytical solution to this problem is unknown, we employ numerical optimization
techniques. In particular, we follow the discretize then optimize paradigm and a
straightforward cell-centered discretization of the spatial domain. To eliminate
the constraint, we use a reduction approach and the unknown of the reduced
model are the density values f = (fi )m i=1 on cell-centers and the coefficients
c ∈ Rn . The optimization is performed with a generalized Gauss-Newton scheme
and an Armijo Linesearch, e.g. [12] for details.
3 Results
The scheme has been implemented using MATLAB R 2010a. In order to have
a gold standard for comparison and validation, we use synthetic data generated
using the XCAT phantom [13]. To be more precise, from a 4D simulation of a
human torso 2D slices are taken. That is, an average activity image f0 and two
attenuation images µ0 and ν at different time steps are chosen, such that µ0 and
ν differ with respect to respiratory motion.
The activity and the attenuation images are calculated by the XCAT software
based on a 140 KeV photon energy. The two time steps related to the attenuation
images assume a respiration period of 5 seconds and are simulated at 0 seconds
and 2.5 seconds. This corresponds to a 0 % and 96 % inhale, according to the
default respiration of the XCAT phantom. The maximal motion of the chest is
parameterizable and choosen to be 1.2 cm. According to the XCAT phantom a
movement of 1.2 cm simulates a normal breathing activity. For the sake of testing
(a) f0 (b) ν (c) µ0 (d) g
Fig. 1. One of the two test cases are exemplarily shown above. The averaged density
of radioactivity over a period of 5 seconds is shown on the left (f0 ). The next two
images (ν and µ0 ) show the attenuation at two different time steps. As respiration
movement is taken into account, these images differ in a non-linear movement. The
image on the right (g) is the simulated SPECT projection (sinogram) which is formally
the application of the attenuated ray transform on f0 and µ0 . The task is to reconstruct
f0 and µ0 , given the sinogram g and ν.
Non-linear SPECT Reconstruction 417
Table 1. Relative error between the gold standard (Fig. 1) without any motion cor-
rection, linear, and the proposed non-linear motion correction.
err(f ) = ∥f∥f−f 0 ∥2
0 ∥2
err(µ) = ∥µ−µ 0 ∥2
∥µ0 ∥2
chest movement no correction linear non-linear no correction linear non-linear
1.2 cm 48.43% 31.16% 18.84% 29.13% 32.36% 17.12%
2.4 cm 81.48% 36.50% 18.19% 39.65% 47.06% 23.31%
of the proposed motion correction in SPECT reconstruction, the comparatively
larger non-linear movement of the chest with a 2.4 cm a.-p. movement is choosen
as a second test case. In Figure 1 the latter test case is shown exemplarily. In
the following the reconstruction results for the test case presented in Figure 1
are illustrated in Figure 2 in comparison with the gold standard provided by
the XCAT phantom. In Table 1 a relative error between the gold standard and
different reconstructions is shown.
(a) no correction (b) linear (c) non-linear
Fig. 2. The absolute value of the difference between the gold standard and different
reconstructions. Rows (a) and (b) are related to the density of radioactivity f and the
attenuation µ, respectively. For instance, the image in the lower left corner illustrates
the absolute difference of the attenuation prototype ν and the gold standard µ0 .
418 Barendt & Modersitzki
4 Discussion
The proposed model for a motion correction of SPECT reconstruction is a nat-
ural extension of the idea of Natterer as the linear transformation model for
the attenuation prototype ν is enhanced to an arbitrary transformation. We
present results for a non-linear spline transformation model which clearly indi-
cate the superiority of the proposed reconstruction model compared to the case
of no motion correction and the correction with a linear transformation model
(Tab. 1).
Due to the nature of the addressed SPECT reconstruction problem, the type
and weight of the regularization is a crucial issue. In particular, an improved
regularization model is under current investigation.
As the current implementation deals with 2D reconstructions of the density
of radioactivity and the attenuation, future work involves the implementation of
a 3D reconstruction. Furthermore it is planned to explore different transforma-
tion models, as well as different regularizations. Because of the promising and
superior results of the previous section it would be interesting, to validate the
proposed reconstruction model in a clinical setting.
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