tomographic image reconstruction for partially …
TRANSCRIPT
TOMOGRAPHIC IMAGE RECONSTRUCTION
FOR PARTIALLY-KNOWN SYSTEMS AND IMAGE SEQUENCES
BY
JOVAN G. BRANKOV
Submitted in partial fulfillment of therequirements for the degree of
Master of Science in Electrical Engineeringin the Graduate College of theIllinois Institute of Technology
Approved Adviser
Chicago, IllinoisDecember 1999
ii
iii
ACKNOWLEDGMENT
First I want to express my thanks to my parents for their full support throughout
my education and to my sister for her encouragement and advice.
I want to thank my adviser Dr. Miles Wernick for introducing me in this filed, for
all his help and for believing in me.
My sincere thanks to Dr. Nikolas Galatsanos and Dr. Yang Yongyi for valuable
discussions.
I also wish to acknowledge the National Institute of Health (NS 35273) for their
financial support of the project.
Special thanks for all my friends, especially to Ana, for encouraging me and
sharing with me the both the sadness and the joy.
J. G. B.
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TABLE OF CONTENTS
Page
ACKNOWLEDGMENT....................................................................................................iii
LIST OF FIGURES.............................................................................................................v
LIST OF TABLES ............................................................................................................vii
CHAPTER
I. INTRODUCTION ...........................................................................................1
1.1. Tomography by SPECT and PET ......................................................21.2. PET and SPECT Studies based on Image Sequences ........................41.3. Spatially Adaptive Temporal Smoothing for PET and SPECT
Image Sequences ...............................................................................71.4. Image Reconstruction with Partially-Known Blur.............................81.5. Arrangement of Thesis .......................................................................9
II. IMAGE RECONSTRUCTION UNDER A PARTIALLY-KNOWN BLURMODEL .........................................................................................................11
2.1. Theory ..............................................................................................112.2. Simulation Results............................................................................152.3. Discussion ........................................................................................23
III. SPATIALLY ADAPTIVE TEMPORAL SMOOTHING FOR GATEDSPECT AND DYNAMIC PET......................................................................25
3.1. Temporal Smoothing using the KL Transformation ........................253.2. Clustering .........................................................................................283.3. Simulations Results ..........................................................................29
IV. CONCLUSIONS AND FURTHER RESEARCH.........................................50
4.1. Summary ..........................................................................................50
APPENDIX
A.....................................................................................................................51
B.....................................................................................................................55
C.....................................................................................................................58
BIBLIOGRAPHY .............................................................................................................61
v
LIST OF FIGURES
Figure Page
1. Single Photon Emission Computed Tomography (SPECT) uses a Gamma Camerawith a Collimator to Measure the Radiotracer Distribution......................................3
2. Coincidence Detection. True coincidence (solid line), singles and randomcoincidence (dashed line), and scatter coincidence (broken line) [1] .......................5
3. Gating by Electrocardiograph .........................................................................................7
4. Original Image...............................................................................................................16
5. Assumed and Actual Point Spread Functions ...............................................................16
6. Mean Square Error Vs. λ for fixed σ2∆a and σ2
n=10, M=10 ........................................17
7. Mean Square Error for Hot Spots Vs. λ for fixed σ2∆a and σ2
n=10, M=10 ..................18
8. Mean Square Error for Cold Spots Vs. λ for fixed σ2∆a and σ2
n=10, M=10 ................19
9. Original Image...............................................................................................................19
10. Image Reconstructed from Blurred Noisy Sinogram using FilteredBack-Projection. MSE1=5428.68............................................................................20
11. Image Reconstructed without Modeling PSF Uncertainty using:λ=0.013 and σ2
n=10. MSE1=3642.81 .....................................................................20
12. Image Reconstructed with Model of PSF Uncertainty using:λ=0.013 σ2
∆a=1.3e-5 and σ2n=10. MSE1=1187.23 ................................................21
13. Original Image.............................................................................................................21
14. Image Reconstructed from Blurred Noisy Sinogram using FilteredBack-Projection with a Ramp Filter. MSE1=1021.1...............................................22
15. Image Reconstructed without Modeling PSF Uncertainty using:λ=0.013 and σ2
n=10. MSE1=9825.8 .......................................................................22
16. Image Reconstructed with Model of PSF Uncertainty using:λ=0.013 σ2
∆A =1.3e-5 and σ2n=10. MSE1=315.23 .................................................23
vi
Figure Page
17. k-means Clustering ......................................................................................................29
18. Four Compartment Kinetic Model ..............................................................................31
19. Blood Sampled Data and the Fitted Curve..................................................................35
20. Zubal Brain Phantom, from Right to Left: slice No. 50, Sagittal view, Coronal view, and Transversal view, respectively ................................................36
21. Transversal view of the Brain Regions in Slice No. 50 ..............................................37
22. Time Curves for Thalamus and Occipital Cortex obtained byDifferent Spatial Smoothing in Sinogram Domain Methods..................................39
23. Difference between Estimated Time Curves...............................................................40
24. Square Difference Estimate.........................................................................................40
25. Image Samples ............................................................................................................41
26. Regions in Slice No. 70, Frame No. 23.......................................................................42
27. Original Time Curves for Lesion, Lung and Heart .....................................................43
28. Estimated Time Curves for the Small Lesion .............................................................44
29. Difference between the Estimated Curves and the True One......................................44
30. Image Example............................................................................................................45
31. Upper Part of gMCAT which contains a Heart and marked Slice No. 70 ..................46
32. Slice No.70 and ROI ...................................................................................................47
33. Estimated time curves of ROI without Preprocessing.................................................48
34. Estimated Time Curves of ROI with Preprocessing by KL smoothing ......................48
35. Estimated Time Curves of ROI with Preprocessing by KL/Clustering ......................49
vii
LIST OF TABLES
Table Page
1. Properties of Common Isotopes Used in PET...............................................................33
2. Rate Coefficient Values.................................................................................................33
3. Blood Samples...............................................................................................................34
4. Blood Curve Fitting Parameters ....................................................................................34
5. Regions/Thalamus Ratio ...............................................................................................37
viii
ABSTRACT
Tomographic imaging, such as that used in medicine, relies on a step known as
image reconstruction to compute the image from the measured data. The problem of
image reconstruction can be a challenging one because of noise and blur that corrupt the
data.
In this thesis we describe two new methods for use in image reconstruction: a
method for image reconstruction when the imaging system properties are only partially
known; and a spatially adaptive technique for temporal smoothing in image sequence
reconstruction.
In image reconstruction, it is usually assumed that the system matrix describing
the behavior of the imaging system is known exactly, although this is not usually the case
in the practice. In the first part of the thesis, we investigate the potential benefit of
modeling the system matrix as the sum of a known part and an unknown random part that
accounts for errors. Using some simplifying assumptions, we develop a penalized
weighted least squares reconstruction algorithm. Our experiments indicate that this
approach can, indeed, lead to significant improvements in the reconstructed image, both
visually and quantitatively.
In the second part of the thesis, a new method is proposed for reconstruction of
image sequences. In this method, between-image temporal correlations are exploited in
order to improve image quality. Pixels are clustered according to their temporal behavior,
ix
then smoothed using a temporal Karhunen-Loève transformation of the data. This method
allows for spatially-adaptive filtering of image sequences.
Experimental results are shown that demonstrate the potential improvements in
image quality obtainable by both techniques.
1
CHAPTER I
INTRODUCTION
Nuclear medicine refers to a collection of medical imaging methods that can
capture functional, as well as structural, information. A nuclear medicine imaging study
begins by administering to the patient a small amount of a radioactive material, called a
radiotracer, either through injection or inhalation. The radiotracer is an analog of a
biologically active substance of known physiological properties, which is labeled with a
radioactive isotope. Its behavior in the body is the same, or similar, to its naturally
occurring counterpart, but it can be imaged because, as the radioisotope decays, it
produces gamma-ray emissions that can be measured by detectors placed outside the
body. The measured emission data are then transformed mathematically, in a process
known as image reconstruction, to obtain images of the spatial (and sometimes temporal)
distribution of the radiotracer concentration in the body.
In this thesis, two new methods for image reconstruction are described and
evaluated. The new methods attempt to improve on existing techniques through better
modeling of the imaging process, and of the images themselves. The first method is based
on a proposed improvement to the conventional linear model of the imaging system, in
which the inevitable errors in modeling are explicitly accounted for in the mathematical
description. The second method improves on the reconstruction of image sequences by
better representing their statistical properties.
Before describing the methods, we begin with a brief explanation of nuclear
2
medicine imaging techniques, specifically single photon emission tomography (SPECT)
and positron emission tomography (PET), which are the focus of this thesis.
1.1. Tomography by SPECT and PET
In the context of nuclear medicine, tomography is an imaging approach that
creates images that reflect the concentration of radiotracer at each point in the body, as
distinguished from planar imaging which only produces projections (similar to line
integrals) of the distribution.
SPECT imaging uses radiotracers that emit single photons; PET uses radiotracers
that emit a positron, which undergoes mutual annihilation with a neighboring electron,
and produces two photons.
1.1.1. Single Photon Emission Tomography (SPECT). In SPECT, a gamma
camera is used to detect the emitted photons (gamma rays) (see Figure 1). Localization of
the photon source is made possible by using a collimator, which is a thick perforated
metal sheet placed in front of the detectors. The aim of the collimator is to allow only
parallel rays to reach the detectors. In practice, the collimation is not perfect and a cone
of rays is allowed to pass (this is viewed as a form of blur). With the collimator in place,
the gamma camera measures (neglecting the blur) a set of line integrals of the radiotracer
distribution along lines parallel to the collimator holes (perpendicular to the camera face).
Multiple images are obtained by the gamma camera (from different angles) from which
the two-dimensional (2D) or three-dimensional (3D) radiotracer concentration can be
reconstructed.
3
A significant drawback of SPECT is its low sensitivity which is due to the fact
that the collimator identifies parallel rays by discarding all other rays. Since the amount
of radiotracer that can be injected into a patient is limited, the images obtained by SPECT
suffer from low signal-to-noise ratio. But SPECT is widely used because it is inexpensive
and does not require a cyclotron to produce the radiotracer.
Figure 1. Single Photon Emission Computed Tomography (SPECT) uses a GammaCamera with a Collimator to Measure the Radiotracer Distribution.
Because of the finite size of the collimator holes each detected photon can be
projected back to a cone of possible origins. This makes the point spread function (PSF)
of the gamma camera depth-dependent (the width of the cone increases with distance
from the camera). Image quality in SPECT is also limited by scatter and by the difficulty
of correcting for the non-uniform attenuation of gamma rays by the body.
4
1.1.2. Positron Emission Tomography (PET). PET differs from SPECT
primarily in the way that the direction of the gamma rays is determined. In SPECT, the
collimator acts as a physical sieve that aims to allow through only rays traveling in a
certain direction. In PET, an electronic collimation scheme is used that leads to better
sensitivity. The PET system (Figure 2) consists of a ring of detectors. In PET, because a
positron-emitting radiotracer is used, pairs of gamma rays are emitted, which travel in
nearly opposite directions. If two different detectors sense the two photons at roughly the
same time, the photons are assumed to have been created by a positron annihilation event
somewhere along the line connecting the two participating detectors.
The causes of blur in PET are different from those in SPECT. The principal
blurring effects are: 1) finite detector size, 2) positron range (the positron travels some
distance before emitting the photons that reveal its position), 3) angulation error (the
photon pairs do not travel in exactly opposite directions, and 4) scatter (the photons may
be deflected causing a misleading indication of their point of origin, see Figure 2).
An additional source of image degradation is the contribution of accidental
coincidences, also known as randoms. Randoms are false events caused when two
photons emitted from separate events happen to be detected at the same instant of time.
The imaging system cannot discriminate these accidental coincidences from true ones.
1.2. PET and SPECT Studies based on Image Sequences
To study physiological processes in the body, PET and SPECT studies frequently
are based on a sequence of images depicting changes in the radiotracer distribution with
5
time. We will refer to one image in a sequence as a frame.
Figure 2. Coincidence Detection. True coincidence (solid line), singles and randomcoincidence (dashed line), and scatter coincidence (broken line) [1]*
There are two ways to acquire an image sequence in PET and SPECT. In a
dynamic study, image frames are acquired sequentially as in a video or movie, usually
over a period of 10 minutes to 2 hours. A dynamic study usually seeks to determine the
way in which the radiotracer interacts with the body as a function of time, and thereby
learn something about the tissue. The organs imaged in a dynamic study are assumed to
be motionless; the temporal variations of the image are due only to the dynamics of
* Corresponding to numbered references in the bibliography.
6
radiotracer concentration. From a dynamic study, one usually measures the time variation
of radiotracer concentration in a region of interest (ROI). A graph of radiotracer
concentration (activity) as a function of time is known as a time-activity curve.
A gated study is used to image organs that move in a periodic fashion, namely the
heart and the organs of the torso that move because of respiration. Though respiratory
gating is receiving increasing increasing interest recently, the primary use of gating is for
imaging the heart.
A gated image sequence is actually a loop, with each cycle of the loop
representing one cycle of the periodic organ motion. Thus, one cycle of a gated cardiac
sequence depicts one cardiac cycle (one heartbeat).
A gated image sequence is obtained by synchronizing the data acquisition process
to the cardiac rhythm, as measured by electrocardiography (see Figure 3). Each frame of
a gated sequence is actually the sum many time intervals. For example, frame 1 of the
sequence is obtained by summing images obtained the short time interval following the
beginning of many different cardiac cycles, frame 2 is obtained by summing all of the
second time intervals, etc. Typically, a gated cardiac image sequence consists of just
eight frames showing one period of the cardiac cycle, synthesized from hundreds of
actual heartbeats.
The advantage of gating is that it reduces the motion blur that would be present if
the heart were imaged in a dynamic study. This allows the clinician to observe subtleties
of wall motion, and small defects that would not be visible otherwise.
The cost of both dynamic and gated image acquisitions is that the number of
7
counts in each frame is less than what would be obtained by integrating counts over the
entire imaging time. Hence, the noise in each frame is higher than in a static image. The
additional noise calls for special image reconstruction and processing methods, which are
the subject of the second part of this thesis.
Fram
e 1
Fram
e N
Fram
e N
-1
Fram
e N
Fram
e 2
Fram
e N
-1
Fram
e 1
Fram
e 2
Figure 3. Gating by Electrocardiograph
1.3. Spatially Adaptive Temporal Smoothing for PET and SPECT Image Sequences
To alleviate the problem of noise in PET and SPECT image sequences we
develop a spatially adaptive temporal smoothing method. Temporal smoothing takes
advantage of the fact that the signal (desired) part of the observations is characterized by
strong between-frame correlations, whereas the noise is entirely uncorrelated.
Review of Previous Work. Temporal smoothing as means for improving the
quality of image sequences has been studied for many years, but is enjoying increasing
interest in the nuclear medicine research community in recent years. Smoothing across
8
frames of related images (such as those in a sequence) is a well-known technique in
image processing known as multichannel image processing (see, for example [2]). In the
field of nuclear medicine, a temporal Wiener filter was proposed in [3]. Principal
component analysis [4] was used in [5] to smooth PET data along their time axis. In [6]
this idea was expanded to show that spatial resolution recovery could be achieved by first
applying temporal smoothing to reduce noise. Fully four-dimensional reconstruction
methods have been proposed recently that have temporal smoothing built into the
algorithm [7, 8].
The method in [6] is limited by its use of space-invariant statistical descriptions of
the temporal correlations in the data. In [9] and [10] methods were proposed, using
known tissue properties, to improve on the method in [6]. In this work we propose an
alternative, data-driven approach to overcoming the limitation of the method in [6].
1.4. Image Reconstruction with Partially-Known Blur
Tomographic imaging systems, such as those used in PET, make use of
algorithms that reconstruct images from measured projection data. The system matrix that
describes the behavior of such systems is usually not known exactly, either because it is
object dependent or because of errors in its measurement or modeling. In this work, we
examine the problem of image reconstruction for the case in which the system matrix is
not known exactly. In our analysis we model the system matrix as the sum of a
deterministic part, which reflects an assumed model, and a random part, which accounts
for any difference between the assumed model and the true model.
9
1.4.1. Review of Previous Work. The approach we propose is based on the
same concept as the method of total least squares (TLS) [11]. The TLS principle has been
employed before in image restoration [12] and image reconstruction for optical
tomography [13]. The approach of assuming errors in the system model is related to the
problem of blind deconvolution, but there the system is assumed to be entirely unknown
except perhaps for some constraints or priors.
In [14] and [15], using a model that incorporates errors in the system matrix, the
problem of sinogram restoration using a maximum a posteriori (MAP) formulation was
addressed by members of our group. In this thesis, we estimate the image directly from
the noisy and blurred data using a penalized weighted least squares (PWLS) [16]
approach.
1.5. Arrangement of Thesis
In Chapter II we address the problem of image reconstruction under a model of
partially-known blur. A description of the imaging model, a derivation of the penalized
weighted least squares (PWLS) cost functional, and an explanation of the optimization
procedure are given in Section 2.1. In Section 2.2 computer simulation results are
presented and a discussion of these results is given in Section 2.3.
In Chapter III a new approach to spatially temporal smoothing for gated SPECT is
described. The theoretical background is given in Section 3.1. This is followed by a
description of the k-means clustering method for identification of data elements sharing
similar time-activity curves in Section 3.2. The new method is tested for three possible
10
applications: a kinetic study of the brain is considered in Section 3.3.1, a lesion imaging
problem investigated in Section 3.3.2 and a gated SPECT study is examined in Section
3.3.3.
Final conclusions and future directions for research are given in Chapter IV.
11
CHAPTER II
IMAGE RECONSTRUCTION UNDER A PARTIALLY-KNOWN BLUR MODEL
2.1. Theory
2.1.1. Imaging Model. An idealized model of tomographic data is the Radon
transform:
s f x y x y dxdy( , ) ( , ) ( cos sin )ρ θ δ θ θ ρ= + --�
�
-�
� II , - < < , 0� � � �ρ θ π , (2-1)
which is a set of line integrals through the object f x y( , ) . This can be written in discrete
form as:
g Pf= , (2-2)
in which g and f are lexicographically ordered representations of the ideal projection
data (sinogram) and the source image, respectively, and P is a matrix of weights
describing the contributions of each volume element to the data.
In positron emission tomography (PET), the system matrix must also express a
blurring effect that can often be approximated as depth-independent. Thus, the blurred
and noisy sinogram g can be modeled by:
E[ ]g APf= , (2-3)
where A is a blurring operator, and E[ ]¼ represents the expected value over realizations
of the noise. In this preliminary study, for tractability of the solution, we will assume that
A is circulant, and hence represents a blur that is space-invariant.
12
We will depart from the traditional model in (2-3) by introducing uncertainty in
the observation model as follows:
E[ | ]g A (A A)PfD D= + . (2-4)
In this model, the blurring operator is represented as the sum of a deterministic
component A and a stochastic component DA , which are circulant matrices composed
from blurring kernels a and Da , respectively. The expectation in (2-4) is conditional on
the realization of the random model error DA .
2.1.2. Penalized Weighted Least Squares (PWLS) Cost Functional. We use
the following PWLS cost functional as our reconstruction optimality criterion:
J ( ) ( ) ( )f g APf C g APf Qfg= - - +-T 1 2λ , (2-5)
in which Cg is the covariance matrix of g , λ is a positive unknown parameter that
controls the smoothness of the image, and Q is a circulant Laplacian high-pass operator.
Assuming the noise to be white and signal-independent, and by using the
commutative property of convolution, (for details see Appendix A or [17]), we can write
Cg as:
C SS Ig a n= +σ σD2 2T , (2-6)
where S is a circulant matrix, the rows of which consist of the elements of sinogram s
arranged in lexicographic order, σDa2 is the variance of the error in the blur point spread
function (PSF) and σ n2 is the additive noise variance. Note that the first term in Cg
13
reflects a signal-dependent noise induced by the system model error.
Because A and S are circulant, we can rewrite the PWLS functional in terms of
quantities in the DFT domain as follows:
J f Qfa n
1 6 = -+
���
��� +=
Ê12
2 2 20
2
N
G(i) A(i)S(i)
S(i)i
N-1
σ σλ
D
, (2-7)
where A i1 6, S i1 6, and G i1 6 are DFT coefficients of the blurring kernel a , the sinogram
estimate s, and the observed sinogram g , respectively.
2.1.3. Conjugate Gradient optimization. To find the desired image, we
minimize the PWLS cost functional, J f1 6 , by employing the modified conjugate gradient
method suggested by [18] and quadratic interpolation for the line-search procedure [19].
At each iteration, we calculate the gradient of J f1 6 which is given by:
grf
fa n
nn
J=��
=-
+
����
* *
=Ê1 6 1 63 82
2 2 2 20N
A (i)P (i) G(i) A(i)S(i)
S(i)
n
i
N-1 Re
σ σD
--
+
�
��� +
*σ
σ σλD
D
a
a n
TQ Qf2 2
2 2 22 2
G(i) A(i)S(i) Re P (i)S (i)
S(i)
n
4 9, (2-8)
where P in 1 6 is the ith coefficient of the DFT of the nth column of the projection matrix P ,
for details see Appendix B or [17].
To evaluate the benefit of accounting for error in the system matrix, we performed
experiments with and without the error term. When the error term is omitted (i.e., when
σDa2 0= ) the PWLS cost functional is quadratic; when the error term is present, J f1 6 is
14
nonconvex. When applying the conventional error-free model (Eq.(2-3)), we initialize the
optimization procedure with a filtered-backprojection (FBP) reconstruction, i.e.,
$f s01 6 ; @= FBP . (2-9)
When applying the model that includes error (Eq.(2-4)) we initialize with the
result obtained by the error-free model, i.e.
$ arg minf ff
0
02
1 6 1 6J L==
Jaσ D
(2-10)
In both cases we begin with:
d gr0 01 6 1 6= - , (2-11)
β 0 01 6 = . (2-12)
The nth iteration is described by:
grf
fn
n
n
J1 61 6
1 64 9
=�
�
$
$ , (2-13)
β n
n n n
n n1 6
1 6 1 6 1 6
1 6 1 6=-�
!
"
$##
-
- -max ,0
1
1 1
gr gr gr
gr grT, (2-14)
d gr dn n n n1 6 1 6 1 6 1 6= - + -β 1 , (2-15)
α α αα
nn nJ
otherwise
1 61 6 1 64 94 9= + ¼ > >%
&K'K
> >-
arg min $ , .
,.0 5 0
4
05 0
10
f d(2-16)
15
$ $f f dn n n n+ = +11 6 1 6 1 6 1 6α (2-17)
We stop the iteration when both of the following conditions are met:
J Jn n$ $f f1 6 1 64 9 4 94 9- <+11D (2-18)
maxi
in
inf f1 6 1 64 9- <+1
2D (2-19)
where, in the experiments we choose D1 0 001= . and D2 0 0005= . .
After completion of the minimization procedure we apply a non-negativity
constraint, i.e.,
$ ,
,f
f f
otherwiseii i=
�%&'0
0. (2-20)
The proposed conjugate gradient algorithm converges within about 100 iterations
$f for a 32 32� image. Each iteration requires about four seconds on a Pentium Pro
200MHz computer.
2.2. Simulation Results
The phantom shown in Figure 4. was used to test the proposed algorithm. The
phantom is divided into four regions: upper background, cold spots, lower background,
and hot spots. The intensities of the pixels in these regions are 10, 5, 5, and 10,
respectively.
The observed sinogram was simulated by degrading the source sinogram using a
Gaussian shaped PSF with full width at half maximum (FWHM) equal to 2.355 pixels
16
and additive white noise with σ n2 10= . For reconstruction we assumed a Gaussian PSF
with FWHM of 3.33 pixels. Both PSFs are shown in Figure 5.
5 10 15 20 25 30
5
10
15
20
25
30
0
1
2
3
4
5
6
7
8
9
10
Figure 4. Original Image
o- A ctu a l P S F+ - A ssu m ed P S F
-5 -3 -1 1 3 5 0
0 .1
0 .2
0 .3
0 .4
2 4 6-4 -2 0-6
Figure 5. Assumed and Actual Point Spread Functions
To evaluate the results we used two figures of merit: spatial mean square error
(MSE) of the pixel intensity, and an ensemble MSE for region of interest (ROI) intensity
estimates.
17
The spatial MSE is defined as:
MSE f f1 = E1
N-�
! "$#
$ 2(2-21)
where ¼ represents the Euclidean norm, $f is the reconstructed image, f is the source
image, N is the number of pixels in the image, and E ¼ is an average over an ensemble of
noise realizations.
In Figure 6 we show MSE1 as a function of λ for different values of the PSF
noise variance σDa2 assumed by the reconstruction algorithm. This figure shows that, for
every value of λ , we obtain better results when we take into account the PSF uncertainty
(σDa2 0� ) than when we ignore it (σDa
2 0= ).
MSE1 Vs. λ
0
2000
4000
6000
8000
10000
12000
0 0.005 0.01 0.015 0.02
σ2∆a =0
σ2∆a =1Ε−6
σ2∆a =5Ε−6
σ2∆a =9Ε−6
σ2∆a =1.3Ε−5
σ2∆a =1.6Ε−5
σ2∆a =2Ε−5
Figure 6. Mean Square Error Vs. λ for fixed σ2∆a and σ2
n=10, M=10
A frequent task in the application of PET is measurement of the average intensity
within a region of interest (ROI) of the image. To quantify the accuracy of the
reconstructed images for this task we define:
18
MSE2
2= E θ θ-�
! "$#
$ , (2-22)
where θ is the true value and $θ is estimated value of ROI. Figure 7 and Figure 8 show
MSE2 as a function of λ . Like Figure 6, Figure 7 and Figure 8 show that, for all values
of λ , it is better to account for PSF error than to ignore it.
Figure 9 shows the source image used to obtain the results in Figures 6-8 with
plots of intensity profiles along two lines through the image. Figures 10-12 show images
reconstructed by FBP, PWLS without accounting for PSF error and PWLS with
accounting for PSF error, respectively.
MSE2 for hot spots Vs. λ
0
50
100
150
200
250
300
350
400
0 0.005 0.01 0.015 0.02
σ2∆a =0
σ2∆a =1Ε−6
σ2∆a =5Ε−6
σ2∆a =9Ε−6
σ2∆a =1.3Ε−5
σ2∆a =1.6Ε−5
σ2∆a =2Ε−5
Figure 7. Mean Square Error for Hot Spots Vs. λ for fixed σ2∆a and σ2
n=10,M=10
19
MSE2 for cold spots Vs. λ
0
50
100
150
200
0 0.005 0.01 0.015 0.02
σ2∆a =0
σ2∆a =1Ε−6
σ2∆a =5Ε−6
σ2∆a =9Ε−6
σ2∆a =1.3Ε−5
σ2∆a =1.6Ε−5
σ2∆a =2Ε−5
Figure 8. Mean Square Error for Cold Spots Vs. λ for fixed σ2∆a and σ2
n=10,M=10
0 5 10 15 20 25 30-2
0
2
4
6
8
10
12
14
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5
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Figure 9. Original Image
20
5 10 15 20 25 30
5
10
15
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25
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1
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8
0 5 10 15 20 25 30-2
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Figure 10. Image Reconstructed from Blurred Noisy Sinogram using FilteredBack-Projection. MSE1=5428.68
0 5 10 15 20 25 30-2
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5
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25
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0
2
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16
Figure 11. Image Reconstructed without Modeling PSF Uncertainty using:λ=0.013 and σ2
n=10. MSE1=3642.81
21
5 10 15 20 25 30
5
10
15
20
25
30
0
2
4
6
8
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12
0 5 10 15 20 25 30-2
0
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14
0 5 10 15 20 25 30-2
0
2
4
6
8
10
12
14
Figure 12. Image Reconstructed with Model of PSF Uncertainty using: λ=0.013σ2
∆a=1.3e-5 and σ2n=10. MSE1=1187.23
The image in Figure 12 appears to exhibit the least variability within uniform
regions. This is explained by the experiments shown in Figure 13-16 in which a uniform
phantom with a small hot spot was used as the source object.
0 5 10 15 20 25 30-2
0
2
4
6
8
10
12
14
0 5 10 15 20 25 30-2
0
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Figure 13. Original Image
22
5 10 15 20 25 30-2
0
2
4
6
8
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14
16
18
5 10 15 20 25 30
5
10
15
20
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30
0
1
2
3
4
5
5 10 15 20 25 30-2
0
2
4
6
8
10
12
14
16
18
Figure 14. Image Reconstructed from Blurred Noisy Sinogram using FilteredBack-Projection with a Ramp Filter. MSE1=1021.1
5 10 15 20 25 30-2
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Figure 15. Image Reconstructed without Modeling PSF Uncertainty using:λ=0.013 and σ2
n=10. MSE1=9825.8
23
0 5 10 15 20 25 30-2
0
2
4
6
8
10
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0 5 10 15 20 25 30-2
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301
2
3
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6
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8
9
Figure 16. Image Reconstructed with Model of PSF Uncertainty using: λ=0.013σ2
∆A =1.3e-5 and σ2n=10. MSE1=315.23
In Figure 16 it is apparent that inaccuracy in the PSF model leads to ringing in the
image unless the presence of PSF error is accounted for in the imaging model, as in
Eq.(2-4). This, in turn, leads to the quantitative inaccuracies exhibited in the MSE plots.
2.3. Discussion
In this work we have derived and evaluated a method for incorporating a model of
point-spread function (PSF) uncertainty in the reconstruction of tomographic images. We
derived a penalized weighted least-squares (PWLS) cost functional which we minimize
using a conjugate gradient method.
The studies in this paper are preliminary and designed for proof of concept. To
make the method practical, a strategy for automatic estimation of the algorithm
parameters will need to be developed, and more realistic noise and blur models will be
required.
The results suggest that incorporation of a model of partially-known blur can
potentially improve the quality of reconstructed tomographic images, both visually and
24
quantitatively.
25
CHAPTER III
SPATIALLY ADAPTIVE TEMPORAL SMOOTHING FOR GATED SPECT ANDDYNAMIC PET
3.1. Temporal Smoothing using the KL Transformation
In gated SPECT and dynamic PET the data gathered during one of the recording
time intervals are called a frame. The strong temporal correlation between frames in the
sequence can be exploited in order to achieve better image quality.
A shortcoming of our group’s previous approaches were built on an assumption
that the temporal statistics of all pixels are the same (i.e., the time-activity of every pixel
is a realization of the same random sequence). Our goal in this work is to develop a new,
spatially adaptive approach for applying temporal smoothing in the sinogram domain.
Let us define a N -dimensional random vector representing the values of one pixel
in the image across time:
xT
= x x x N1 2, , ,L . (3-1)
This can be rewritten in terms of a set of N orthonormal basis vectors φ i; @,
i N= 1, ,K
x y= ==Êφ i i
i
N
y1
F , (3-2)
where:
F = φ φ φ1 2 L N T, (3-3)
26
and the coordinates (transform coefficients) are given by:
yT
= y y y N1 2, , ,L . (3-4)
The Karhunen-Loève (KL) or principal component (PC) basis vectors φ i; @ are the
eigenvectors of Cx , the covariance matrix of the random vector x , i.e.,
Cxφ φi i i= λ , (3-5)
where λi are the eigenvalues.
We can express this as a linear transformation (the KL transform) with
transformation matrix A=ΦT , which diagonalizes Cx and decorrelates x :
y Ax= . (3-6)
A key property of the KL transform is that this choice of basis vectors minimizes
MSE of the representation of x obtained by leaving out terms from the expansion in (3-2).
In other words, it optimally compacts information about x into first terms.
By assuming a set of points with in the same distribution, (same correlation
matrix) we can replace true statistic by sample statistic:
LxTxx= . (3-7)
If we have many realizations of the same sample vector x denoted as x j (e.g.
more then one pixel in the image sequence) we can take the average as:
LxTx x= Ê1
K j jj
, (3-8)
27
where K is the number of realization (e.g. pixels with the same statistic).
To improve the sensitivity of the first KL component we normalize the variables
prior to KL decomposition by the transformation:
� = --
x x xxS1
2 1 6 (3-9)
where x is the mean of x and Sx x x= diag Nvar var13 8 3 84 9L .
Here the x and var x j3 8 is obtained from:
x x= Ê1
K jj
(3-10)
var x x xiki i
kK2 7 2 7=
--Ê1
1
2. (3-11)
In the case that data are greatly corrupted by noise one can obtained better results
in estimating the sample statistic by prefiltering the data [20].
For decomposition of a variable in KL components we assumed the same statistic
of the variable in each realization. If we know that some of the data are samples from
different statistic we can do the KL decomposition separately on each data set with
different statistics and get the better results by keeping the same number of KL
components. For purpose of classifying the data from different statistics into different
clusters one can use the k-means / Generalized Lloyd Algorithm (GLA) clustering can be
used.
28
3.2. Clustering
As explained in previous section in order to make a better estimate of the data
statistic (covariance, mean) for data from mixed statistics/distribution we can try to
separate them in several groups/clusters. For dynamic sequences we want to separate
them according to their different time behavior. For that purpose in this work k-mean
algorithm was used [21].
The algorithm consist of:
1. Partitioning data into K initial clusters. In this work we divided the image
sequence in K equally spaced regions.
2. Finding a mean vector of each region/cluster.
3. Making a new cluster assignment that minimizes the distance measurement of
each data vector from each cluster mean. Euclidean distance is used as
distance measurement.
4. If any reassignments took place go back to step 2.
The final assignments of the data usually depends on the first initial guess. In our
experiments, for sinogram data, we did not observe significant differences in final
assignment caused by different initial assignment.
If there exist a region with significantly higher mean/intensity from all the other
regions, a hierarchical clustering is performed. First the data is clustered into two
clusters: the cluster with the high mean and the cluster with the low mean and then a
region of interest is further clustered into required numbers of clusters.
29
First guess
Second iteration
Final partitioning
Calculated Mean
Figure 17. k-means Clustering
3.3. Simulations Results
Our method for spatial smoothing in sinogram domain was tested for three
possible applications: kinetic study of the brain, lesion detection in dynamic PET and in
gated SPECT. All methods were tested in capability to preserve important features (e.g
time curves) for given study.
We compare results of three procedures: no sinogram preprocessing with
Expectation Maximization (EM) reconstruction [22-24], sinogram presmoothing by
lower order approximation using KL transform (KL) [6] with EM reconstruction and a
new one: lower order approximation by KL transform taking into account different
statistics (e.g. time behavior) of the different regions in the sinogram domain
(KL/Clustering) with EM reconstruction.
The simulation procedure was the following: First we made a sinorgam from
30
given model/phantom, then we blurred the sinogram by assumed Gaussian blurring
kernel. The level of 20% randoms and Poisson noise was simulated. Finally sinogram
was corrected for randoms assuming exact value of randoms 20%.
Obtained Sinogram was then prefilterd by some of the presmoothing methods and
reconstructed by EM algorithm assuming the same deblurring kernel as it was for
blurring.
The reference image sequence went through the same procedure, except for
simulating the noise, in order to get a reference image which dependent on number of
reconstruction iterations and reconstruction algorithm in the same fashion as
reconstructed images sequence.
For each simulation the number of retained KL components was two.
3.3.1. Kinetic Study of the Brain. Four-compartment tracer kinetic model
shown in Figure 18. was used to generate the time curves for regions in the brain
containing specific bindings. For nonspecific bindings regions the three compartment
method was used, which was derived from four compartment model with restriction of k3
and k4 to be zero.
31
Cf
Cn
Cb Cp
PlasmaCompartment
Free-ligandCompartment
Specific bindingCompartment
Non-specific bindingCompartment
k1
k2
k3
k4
k5 k6
Figure 18. Four Compartment Kinetic Model
The tracer kinetics was modeled by following equations:
dC t
dtk C t k C t k C t k k k C tf
p b n f
( )( ) ( ) + ( ) - ( ) ( )= + + +1 4 6 2 3 5 , (3-12)
dC t
dtk C t k C tb
f b
( )( ) - ( )= 3 4 , (3-13)
dC t
dtk C t k C tn
f n
( )( ) ( )= -5 6 , (3-14)
C t C t C t C tb n f( ) ( ) ( ) ( )= + + , (3-15)
where C tp 1 6 , denotes the concentration (nCi/ml) of ligand in plasma. C tf 1 6, C tn 1 6 and C tb 1 6
are the radioligand concentrations of free, nonspecifically-bound and specifically-bound
ligand, respectively. C t1 6 stands for the observed radioligand activity measured by PET.
K1 (ml/min) and ki (min-1) i = 2 3 6, , ,L are the rate constants.
It can be shown that the solution for l compartment method can be written as:
32
C t L e C tiR t
pi
ni( ) = ( )
=1
- ©Ê (3-16)
where Li and Ri can be calculated from K1 and ki i = 2 3 6, , ,L with n l= -1 for the l
compartment case.
And finally with the consideration of the decay of radioligand we have:
C t e L e C tdec
t
t
iR t
pi
ni( ) ( )
( )
= ©-
=Ê
12
1
2
1
ln
(3-17)
where t1/2 is the half-life of the isotope. In our studies, the half-life of the [11C]
isotope was 20.4min, see Table 1.
For [11C] Carfetanil the rate coefficients values were obtained from [25], see
Table 2. Those data were taken from real PET study.
The input function (concentration of lingand in plasma) will be assumed to have
the following form.
C tL e t t
c e t tp
pmt
iD t
i
i( )
( ) ( )
( )=
- � �
�
%&K'K
-
=Ê
1 0 0
00
3 (3-18)
in which c L m Di i and t0are constants obtained by fitting a curve trough the measured
concentration of the lingand in the blood plasma.
In this thesis, we used the blood sample values obtained in a PET study conducted
by the Department of Radiology at the University of Chicago shown in Table 3.
33
Table 1. Properties of Common Isotopes Used in PET
Isotope Isotope half-life Maximum Energy(MeV)
Range in water(mm)
18F 109.7 min 0.635 2.3911C 20.4 min 0.96 4.1113N 9.96 min 1.19 5.3915O 2.07 min 1.72 8.2
Table 2. Rate Coefficient Values
Coefficient Three compartment Four compartment
K1 0.174 0.205k2 0.209 0.246k3 - 0.214k4 - 0.100k5 0.027 0.027k6 0.036 0.036
34
Table 3. Blood Samples
Time Concetraction (nCl/ml)
0.08 00.33 1.260.52 9.650.73 8.770.95 6.661.2 4.951.75 3.361.95 3.212.18 2.813.18 2.254.8 1.67.15 1.219.55 1.1215.27 1.0824.12 1.1335.45 1.0844.28 0.9972.63 0.83106.5 0.77
From blood samples we calculated the values of the constants in the input
function C tp ( ) , see Table 4.
Table 4. Blood Curve Fitting Parameters
Parameters Value
C1 0.0129C2 0.1470C3 0.0173D1 0.0075D2 1.3452D3 0.3741L 6.3323m 0.0223t0 0.5200
35
The fitted curve C tp ( ) and blood samples are shown in Figure 19.
0 20 40 60 80 100 120 140 1600
0.02
0.04
0.06
0.08
0.1
0.12Blood sample
Con
cent
ratio
n (n
Ci/m
l)
Time (min)
Fited curveSampled data
Figure 19. Blood Sampled Data and the Fitted Curve
For this study a realistic MRI voxel-based numerical brain phantom developed by
Zubal et al, [26] was used. There are 124 slices in the brain phantom. Each slice is
256x256 array with the pixel size of 1.09 mm. The original phantom was down sampled
to 64x64 at pixel size of 4.36 mm. The slice thickness was 1.4 mm. A single slice in the
middle of the brain was used.
In our study we distinguish six different regions in the brain: Thalamus, Caudate,
Front Cortex, Temporal Cortex, White matter and Occipital Cortex. For each of them the
kinetic model is estimated using the real PET study data.
36
Figure 20. Zubal Brain Phantom, from Right to Left: slice No. 50, Sagittal view,Coronal view, and Transversal view, respectively
37
Temporal Cortex
Caudate
Thalamus
White Matter
Front Cortex
Occipital Cortex
Figure 21. Transversal view of the Brain Regions in Slice No. 50
For our study we assume [11C] Carfetanil µ-selective opiate receptor agonist. The
activity ratio for each brain region/thalamus was made as measured by Frost, et al., [27]
taking an average for two given subjects (see Table 5).
Table 5. Regions/Thalamus Ratio
Brain region Three compartment Four compartment
Thalamus - 1Caudate - 0.87
Front Cortex - 0.805Ant. Temporal +Sup. Temporal
Cortex
- 0.805
White matter - 0.1Occipital Cortex 1 -
The sinogram was simulated assuming a Gausian shape PSF of 8 mm FWHM and
38
the four million counts per slice.
In our study twenty-tree frames were generated.
Sampling times were:
1 2 3 4 5 6 9 12 15 18 21 24 27 30 35 40 45 50 55 60 70 80 90 min
Clustering is preformed in sinogram domain. First we did preliminary clustering
into two groups: Group with high activity level and a group with low activity level. The
fist group, assumed to be a brain, is subdivided further into 5 clusters. The second is
assumed to be the background.
Results are shown for the thalamus and occipital cortex, for 300 iteration in EM
reconstruction.
In Figure 22. results suggest that by assuming the same covariance matrix for all
pixels in the dynamic sequence we can introduce some error into time behavior of some
of the regions of the reconstructed brain images. For purpose of finer comparison of the
obtained time curves we can look at the Figure 23 which represent the difference between
the original time curves and obtained after spatial-temporal smoothing and
reconstruction. It can be seen that without presmoothing, reconstructed time curve
estimation is noisy. By assuming the same covariance matrix for all points in the dynamic
sequence we introduced an error in occipital cortex time curve. By taking into account
the different time behavior (covariance matrix) for different regions we obtained better
results. We can further explore this in Figure 24. For results in Figure 24 we first
subtracted the original from the reconstructed sequence and then we estimated time curve
for different regions. The results can be considered as a summed mean square error of
39
each pixel in the image. It can be seen that with KL/Clustering smoothing we can obtain
better results then with the plain EM reconstruction, and KL smoothing.
In Figure 25 we present image samples from our experiment. The images are the
3th reconstructed image of given data sequence. In the first row we have the original
followed by the image reconstructed without any presmoothing, image smoothed with
KL, and image smoothed by KL/Clustering. All of them were reconstructed with EM
algorithm. In the second row we show the images obtained by subtracting those images
from the original. One can notice the brighter spot in KL difference image, which does
not appear in KL/Cluster difference image.
0 10 20 30 40 50 60 70 80 900
2
4
6
8thalamus
time (min)
Estimation of time curves
0 10 20 30 40 50 60 70 80 900
2
4
6occ cortex
time (min)
originalEMKL EMKL/Clustering EM
Figure 22. Time Curves for Thalamus and Occipital Cortex obtained by DifferentSpatial Smoothing in Sinogram Domain Methods
40
0 10 20 30 40 50 60 70 80 90-0.4
-0.2
0
0.2
0.4thalamus
time (min)
Diference of the true and the estimated time curve
0 10 20 30 40 50 60 70 80 90-1.5
-1
-0.5
0
0.5occ cortex
time (min)
originalEMKL EMKL/Clustering EM
Figure 23. Difference between Estimated Time Curves
0 10 20 30 40 50 60 70 80 900
5
10thalamus
time (min)
Estimation of the time curves of MSE
0 10 20 30 40 50 60 70 80 900
1
2
3
4occ cortex
time (min)
originalEMKL EMKL/Clustering EM
Figure 24. Square Difference Estimate
41
Figure 25. Image Samples
42
3.3.2. Lesion Enhancement. The purpose of the experiment in this section is to
show that there is a possibility to improve time curve of a lesion that will lead to
improvement in detection capability of the lesion as shown in [28] and [9]. The time
activation curves are taken from those articles.
The slice No. 70 from 4D mathematical cardiac-torso MCAT [29] was chosen as
a test subject for our simulation. In this experiment we assumed that the heart beating
does not synchronize the imaging. The time curves used for the heart, lung, small and big
lesion used by our experiment are shown on Figure 27. One can find more detail about
them in [28] and [9].
Heart
Lungs
Large Lesion Small Lesion
0
0.5
1
1.5
2
2.5
Figure 26. Regions in Slice No. 70, Frame No. 23
In our study twenty-tree frames was generated for the time sequence.
The simulated bin size was 5.625mm/pixel. The distant-independent blur is
assumed to be 8mm of FWHM.[6, 30]. The total number of count per slice was four
43
millions.
Clustering is performed in mixed hierarchical-K-mean manner. First the sinogram
was divided in two clusters, high level and lower level (heart and the rest), then the lower
level cluster is divided into five clusters because we assumed that the region of interest is
not in the heart.
The number of iteration of the EM algorithm was 150 iterations.
In Figure 28. one can see improvement in the values of the MSE and in the
visualization of the time behavior of the smaller lesion.
If we take a closer look at the time curves in Figure 29. we can see the difference
between the original curves and the estimated. It can be seen that the last method,
KL\Clustering, has the best performance.
In Figure 30, which is given for visual comparison, it is not so clear as in the case
of the brain study that there is an improvement but the MSE shows us that there is.
0 500 1000 1500 2000 25000
2
4
6
8
10
12
14
Time(min)
Imag
e in
tens
ity
big lesionsmall lesionheartbackground
Figure 27. Original Time Curves for Lesion, Lung and Heart
44
0 500 1000 1500 2000 25001
1.5
2
2.5
3
3.5
4
4.5
5small lesion
Time(min)
Imag
e in
tens
ity
Original MSE=0EM MSE=0.10906KL EM MSE=0.052292KL/Clustering EM MSE=0.014862
Figure 28. Estimated Time Curves for the Small Lesion
0 500 1000 1500 2000 2500-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1small lesion
Time(min)
Original MSE=0EM MSE=0.10906KL EM MSE=0.052292KL/Clustering EM MSE=0.014862
Figure 29. Difference between the Estimated Curves and the True One
45
Figure 30. Image Example
46
3.3.3. Gate SPECT. The 4D gated mathematical cardiac-torso gMCAT (D1.01
version- fixed anatomy, dynamic (beating heart)) phantom [29] was chosen as a test
subject for our simulation. In our simulation we used the upper part of the phantom
(Figure 31.). Sixteen frames were simulated to represent the gated SPECT study. For
getting preliminary results, which are shown here, we assumed that the blur is distance-
independent, which is the first approximation to the reality. The relative activity levels of
the heart, lung, and background were 1.0: 0.16: 0.2. The number of iterations for each
reconstruction was 150.
Figure 31. Upper Part of gMCAT which contains a Heart and marked Slice No. 70
In Figure 32. we present the slice which we used for measurements. On the wall
47
of the heart one can notice a white region of interests (ROI) for which we will find the
time behavior curve.
As shown on Figure 33. - Figure 35. the time curve for ROI without any
preprocessing is noisy as it was expected, but we can obtained better results (e.g. less
MSE) for the case of KL/Clustering.
In case of KL preflittering we can notice that the reconstructed time curve does
not follow neither the original time curve nor the noisy time curve version.
KL/Clustering, as one can see from Figure 35, produced time curves with the
smallest variation around the true time curve. It also follows the noisy time curve, except,
that the level of noise is significantly reduced.
Figure 32. Slice No.70 and ROI
48
0 5 10 15 20 25 30 351.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2Estimate time curve MSE=0.12218
OriginalEstimated
Figure 33. Estimated time curves of ROI without Preprocessing
0 5 10 15 20 25 30 351.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8Estimate time curve MSE=0.033062
OriginalEstimated
Figure 34. Estimated Time Curves of ROI with Preprocessing by KL smoothing
49
0 5 10 15 20 25 30 351.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8Estimate time curve MSE=0.025367
OriginalEstimated
Figure 35. Estimated Time Curves of ROI with Preprocessing by KL/Clustering
50
CHAPTER V
CONCLUSIONS AND FURTHER RESEARCH
4.1. Summary
In first part of this thesis we developed and evaluated one new method for
modeling PET imaging systems with allowance for uncertainty in PSF of the system
response. The preliminary study, in this thesis, show us that there is a possibility of
improvement the quality of an image reconstructed by algorithm that incorporated PSF
uncertainty comparing to algorithm that does not.
In second part we showed that one can obtain better quality of reconstructed
image sequence by incorporating the strong correlation between consecutive frames.
Further, we showed that by incorporating the information about differences in the region
time behavior one can obtained less artifacts produced by algorithm that just exploit the
between frame correlation.
51
APPENDIX A
52
Let us consider our cost functional from Chapter II.
J T( ) ( ) ( )f g APf C g APf Qfg= - - +-1 2λ , (A-1)
in which Cg is the covariance matrix of g , λ is a positive unknown parameter that
controls the smoothness of the image, and Q is a circulant Laplacian high-pass operator.
By definition, Cg is given by:
C g g g gg = - -E E E2 72 7T. (A-2)
In the PWLS approach, the image is treated as deterministic, the system is model
as a sum of deterministic part AP and random error DAP and additive noise n as random
with zero mean. The expected value of the observed sinogram can be written as:
E Eg APf APf n APf= + + =D . (A-3)
Substituting (A-3) into (A-2) we obtain Cg, which is the autocorrelation of the
random part in the system model:
C R APf n APf ng T= = + +E D D1 61 6T . (A-4)
Applying the expectation operator to (A-4) we obtain:
C APf APf nng = +E ED D1 61 6T T . (A-5)
Future by assuming s Pf= one can obtain:
C Ass A Ig n2= +E D DT T σ . (A-6)
where σ n2 is the additive noise variance.
53
The product DAs represents a convolution operation between vectors Da and s .
Using a convolution commutative property, DAs can be rewritten as S aD , where S is a
circulant matrix, the rows of which consist of the elements of sinogram s arranged in
lexicographic order. Therefore,
C S a a S Ig n2= +E D D T T σ (A-7)
Using the commutative properties of circulant matrices we obtain
C SS Ig a n2= +Tσ σD
2 , (A-8)
where σ Da2 represents the covariance of the PSF error since it is modeled to be zero
mean as well.
Now we have:
J T( ) ( ) ( )f g APf SS I g APf Qfa n= - + - +-
σ σ λD2 2 1 2T3 8 , (A-9)
here the matrixes S and A are circulant matrixes.
Using properties of DFT described in Appendix C this can be rewritten as:
J T T( ) ( )f g W WPf W W W W W IWa a s s n= - +�� ��- - - -
-1 2 1 1 2 1
1
L L LDσ σ3 8
¼ - +-( )g W WPf Qfa1 2L λ (A-10)
J T T T( ) * * ( )f g W Wf P W I W W Wg WPf Qfa a s s n a= - + - +- --
-1 1 2 21
1 2L LL LD3 8 3 84 9σ σ λ
J T T( ) * * ( )f g W Pf W WW I WW Wg WPf Qfa a s s n a= - - - - -- + - +1 1 1 2 2 1 1 21 64 9 3 8L LL LDσ σ λ
J T T( ) * * ( )f g W Pf W I Wg WPf Qfa a s s n a= - - -
- + - +1 1 2 2 1 21 64 93 8L LL LDσ σ λ (A-11)
54
and finally by :
b W W b
W b
Wb
T T
N
- -
-
=
=
=
1 1
1
*
, (A-12)
JN
( ) * * * * ( )f W g W Pf I Wg WPf Qfa a s s n a= - + - +-1 2 2 1 2L LL LD3 83 8σ σ λ (A-13)
Now we can rewrite the PWLS functional in terms of quantities in the DFT
domain as follows:
J f Qfa n
1 6 = -
+
���
���+
=Ê1
2
2 2 20
2
N
G(i) A(i)S(i)
S(i)i
N-1
σ σλ
D
, (A-14)
where A i1 6 , S i1 6, and G i1 6 are DFT coefficients of the blurring kernel a , the
sinogram estimate s , and the observed sinogram g , respectively.
55
APPENDIX B
56
From obtained PWLS cost functional:
J f Qfa n
1 6 = -
+
���
���+
=Ê1
2
2 2 20
2
N
G(i) A(i)S(i)
S(i)i
N-1
σ σλ
D
(B-1)
and the gradient definition:
grf
fnn
n
J=��1 6
(B-2)
by knowing the only the S i1 6 is dependent of f one can get:
grf f
a n
nn n=
- ��
- +- �
�-
+
�
�
�����=Ê1
2 2 20N
A i S iG i A i S i
A i S iG i A i S i
S i
* *
i
N-1
1 6 1 62 7 1 6 1 6 1 62 7 1 6 1 63 8 1 6 1 6 1 62 71 6
*
σ σD
--
��
+��
���
���
+
�
�
����+
G i A i S iS i
S iS* i
S i
S i
*1 6 1 6 1 6 1 6 1 6 1 6 1 6
1 64 9
2 2
2 2 22
2
σ
σ σλ
D
D
a
a n
Tf f
Q Qfn n (B-3)
where the derivation is regarding the nth pixel in image
Let us consider Ws, where W is a Fourier transform matrix and s is a sinogram
defined as s=Pf. Here P is a system projection matrix and the f lexicography ordered
image.
57
��
=��
=
--
-
�
!
"
$
#####= ��
WPf
fWP
f
f
WP
Sf
1 6 1 6
1 61 6
1 6
n n
n
n
n
N n
δδ
δ
1
2
M(B-4)
where δ ii
otherwise1 6 = =%&'
1 0
0, this will select only the nth row from P , wich can be
denoted by Pn .
So from (B-4),
��
= =fn
S Pni i i N1 6 1 6, , ,1L , (B-5)
Combining (B-3) and (B-5) one can get the gradient functional of the PWLS cost
functional.
grf
fa n
nn
J=��
=-
+
����
* *
=Ê
1 6 1 63 822 2 2 20N
A (i)P (i) G(i) A(i)S(i)
S(i)
n
i
N-1 Re
σ σD
--
+
�
����+
*σ
σ σλD
D
a
a n
TQ Qf2 2
2 2 22
2G(i) A(i)S(i) Re P (i)S (i)
S(i)
n
4 9, (B-6)
58
APPENDIX C
59
For circulant matrices B composed from element of vector b , one can write
WB Wb= L (C-1)
tr Lb Wb1 6 =
where Lb is a diagonal matrix whose elements on main diagonal Lb k k,1 6 are
related to the discrete Fourrier transform of the vector b. W is the matrix that transforms
data into the DFT domain. Transformation matrix can be written as:
W =
�
!
"
$
######
-
-
- - - -
1 1 1 1
1
1
1
2 1
2 4 2 1
1 2 1 1 1
LLL
M M M O ML
W W W
W W W
W W W
N N NN
N N NN
NN
NN
NN N
1 6
1 6 1 61 6
(C-2)
W eN
j
N=- 2π
(C-3)
Following properties of DFT matrices will be used in derivations:
W W WW I 1 1- -= = , (C-4)
where I is identity matrix.
W NW 1* = - , (C-5)
where ¼1 6* denotes complex conjugate,
W NW 1= -3 8*, (C-6)
W WT = . (C-7)
60
And for matrix B composed from real elements:
B W W
W W
W W
WW
Tb
T
Tb
1 T
b
b
=
=
=
=
-
-
-
-
1
1
1
L
L
L
L
3 81 6 3 8
3 8NN
**
, (C-8)
B WW
W W
B B
b
b
T
T
NN
* ** *
*
=���
���
=
= =
-
-
1
1
3 8 LL . (C-9)
61
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