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MAXIMUM LIKELIHOOD – EXPECTATION MAXIMUM RECONSTRUCTION
WITH LIMITED DATASET FOR EMISSION TOMOGRAPHY
A Thesis
Presented to
The Graduate Faculty of The University of Akron
In Partial Fulfillment
of the Requirements for the Degree
Master of Science
Rahul Patel
May, 2007
MAXIMUM LIKELIHOOD – EXPECTATION MAXIMUM RECONSTRUCTION
WITH LIMITED DATASET FOR EMISSION TOMOGRAPHY
Rahul Patel
Thesis
Approved:
Accepted:
______________________________
Advisor
Dr. Dale Mugler
______________________________
Department Chair
Dr. Daniel Sheffer
______________________________
Co-Advisor
Dr. Anthony Passalaqua
______________________________
Dean of the College
Dr. George K. Haritos
______________________________
Committee Member
Dr. Daniel Sheffer
______________________________
Dean of the Graduate School
Dr. George R. Newcome
______________________________
Date
ii
ABSTRACT
Medical Imaging provides a non-invasive technique to look at the structural and
functional information of internal organs and structures. One of the most widely used
Medical Imaging techniques is emission tomography, in which a radioisotope is given to
a patient. Measurement of the radioactive distribution throughout the patient gives the
physiological and patho-physiological information about the patient [2]. The two types of
emission tomography are Positron Emission Tomography (PET) and Single Photon
Emission Tomography (SPECT). The image is reconstructed using the data acquired, also
known as the projection data, by a mathematical technique known as Filtered Back
Projection (FBP).
In emission tomography, it is difficult to locate the exact location from which an
emission originated. If the detectors are placed apart, the probability of scatter being
detected as signal decreases. The main risk with this method is limited projection data
due to the limited number of total number of detectors. The main purpose of this study is
to verify whether or not we can reconstruct images with a limited dataset. This method
will provide us a better estimate of the exact location of the radioactivity. Traditional
reconstruction algorithms like FBP can not reconstruct an image with a limited dataset.
Hence we work with an iterative algorithm, Maximum Likelihood – Expectation
iii
Maximum (ML-EM). This research uses the most commonly used Shepp – Logan head
phantom.
To test whether we can reconstruct the image using a limited dataset without any
statistical difference, we use the Chi-Square Goodness of Fit test. Since it is an iterative
approach, we also look at the line profile of the reconstructed images with a different
number of iterations.
The primary conclusion drawn from this testing was that no statistically significant
differences exist between the images reconstructed from a limited dataset and the original
image. We have proven that by using Modified ML-EM algorithm, we can reconstruct
the image with limited dataset.
iv
ACKNOWLEDGEMENTS
I would like to begin by thanking my parents for giving me the courage and
strength I needed to complete my goals. Dr. Anthony Passalaqua: thank you for your
guidance and valuable insights in the project. Dr. Dale Mugler, and Dr. Dan Sheffer:
thank you for your advice and selfless concern for students. Finally thanks go out to my
friends Manisha Shah, Ashish Jagtiani, Anand Parikh, Nikhil Shrirao, Rupesh Sawant
and Saket Kharshikar for their valuable support.
v
TABLE OF CONTENTS
Page
LIST OF TABLES……………………………………………………………………..…ix
LIST OF FIGURES………………………………………………………………………xi
CHAPTER
I.
INTRODUCTION ………………………………………..………………………1
II.
LITERATURE REVIEW………………………………………………....………3
III.
2.1
Basic Principles of PET...………………………………………....….…..3
2.2
Basic Principles of SPECT…………………………………....………….6
2.3
Radon Transform and Filtered Backprojection…………….………..…....8
2.4
Iterative Reconstruction Algorithms...………………………..…….……12
2.5
Practical Issues…………………………………………………………...14
METHOD…………………………………….....………………….….….……..16
3.1
Mathematics of ML-EM Algorithm.……………………….….…….…...16
3.2
Modified ML-EM Flowchart.……...……………………….….…….…..19
3.3
System Matrix………………………………………………………........23
3.4
Hypothesis and Statistical Testing……………………………………….25
3.5
Advantages and Disadvantages of Ml-EM Reconstruction Algorithm.....26
vi
IV.
V.
RESULTS………………………………………………………….…………….30
4.1
FBP Reconstruction with limited dataset………………………………..30
4.2
Modified ML-EM Reconstruction on image..………….….…………….33
4.3
Effect of the number of iterations on the reconstruction………..……….35
4.4
Effect of the number of bins skipped on the reconstruction...….………..40
4.5
Reconstruction with a real SPECT image………………………….…….44
4.6
Reconstruction with a faulty detector…...……………………………….45
CONCLUSION………………………………………………………...……….48
BIBLIOGRAPHY………………………………………………………………………50
vii
LIST OF TABLES
Table
4.1
Page
Detector Skipped and Resulting Chi-Square Error………..……………..40
viii
LIST OF FIGURES
Figure
Page
2.1.1
Positron emission and annihilation……………….…………………...….4
2.1.2
Conceptual diagram of coincidence detection in a PET system………….5
2.1.3
Types of coincidence in PET……………………………………………..6
2.2.1
Gamma Camera…………………………………………………………..7
2.3.1
Object under reconstruction and its sinogram…………………………....8
2.3.2
Shepp-Logan phantom and its RadonTransform…………………………9
2.3.3
Demonstration of FBP Algorithm…………………………...…………...12
4.1.1
Original Shepp-Logan Head Phantom and FBP Reconstructed image
with ½ dataset and 60 angles……….. ……………..……..……………..30
4.1.2
FBP Reconstructed image with ½ dataset and 1800 angles……………..31
4.1.3
FBP Reconstructed image with ¼ dataset and 1800 angles……………..31
4.1.4
FBP Reconstructed image with ¼ dataset and linear interpolation
(a) 1800 angles (b) 360 angles (c) 180 angles d) 60 angles……………..32
4.2.1
Modified ML-EM Algorithm (a) Original Shepp-Logan phantom
(b) Reconstructed image (c) Sinogram……………..……..……………..33
4.3.1
Chi-Square Goodness of Fit for Modified ML-EM algorithm…………..36
4.3.2
Reconstruction images at (a) 5 iterations (b) 15 iterations
(c) 35 iterations (d) 50 iterations (e) 75 iterations (f) 100 iterations…….37
4.3.3
Line profile values at (a) 5 iterations (b) 15 iterations
(c) 35 iterations (d) 50 iterations (e) 75 iterations (f) 100 iterations…….38
ix
4.4.1
Plot of number of detectors skipped V/S Chi-Square Error……………..41
4.4.2
Reconstructed images and their sinograms with
(a) no detectors skipped (b) ½ detectors (c) 1/3 detectors
(d) 1/5 detectors (e) 1/10 detectors……………………………………....41
4.5.1
Reconstruction with a real SPECT image (a) original image
(b) no detectors skipped (c) ½ detectors (d) 1/3 detectors
(e) 1/5 detectors (f) 1/10 detectors…………………………………….....44
4.6.1
Ring artifact in the image due to faulty detector…………………………46
4.6.2
Sinogram which shows that detector 16 is bad…………………………..46
4.6.3
Image reconstructed using ML-EM Recon Algorithm…………………..47
x
CHAPTER I
INTRODUCTION
Recent advances in computer technology have brought a new revolution in the
field of medical imaging. Now it is easy to acquire data and perform mathematical
operations to reconstruct the data and provide emphasis on the required anatomical
details.
Medical Imaging provides a non-invasive technique to look at the functional and
structural information of internal organs and structure [2]. There are many different types
of modern medical imaging techniques, including Computed Tomography (CT),
Magnetic Resonance Imaging (MRI) and Positron Emission Tomography (PET) to name
a few [10]. We will focus on two types of emission tomography: Positron Emission
Tomography (PET) and Single Photon Emission Tomography (SPECT). The basic
principles of both techniques are outlined in Chapter 2.
Traditionally, mathematical algorithms are used to reconstruct the image. The
most famous reconstruction technique that relies heavily on the Fourier transform is
Filtered Back Projection [10]. The basic principles of Filtered Back Projection are
presented in Chapter 2. These conventional techniques are undoubtedly the most widely
used algorithms for image reconstruction [10]. In the early 1980’s, new methods of image
reconstruction in emission tomography emerged, which could overcome some of the
1
shortcoming of the conventional methods. These methods could take into account
physical characteristics of emission tomography, the Poisson nature of photons, and
many such other factors leading to better image reconstruction.
We will concentrate on one of these iterative reconstruction techniques called
Maximum Likelihood Expectation Maximum (ML-EM) reconstruction. The mathematics
of this method is provided in Chapter 3. It has been observed that ML-EM reconstruction
produces better images than filtered back projection algorithms.
The main reason ML-EM has not been widely accepted is because the algorithm
is computation intensive. We present a new method, which we call the “Modified MLEM”, which will reconstruct the image with a partial data set. In our method we place the
detectors apart, hence decrease the number of detectors in the system, yet reconstruct the
image without any artifacts and the same image sharpness. This new technique may not
only potentially reduce the noise in the system due to photon scattering but will also
reduce the cost of the system. The detailed discussion of this new technique is mentioned
in Chapter 3. Images reconstructed are presented in Chapter 4.
Our analysis suggests that the Modified ML-EM reconstruction algorithm
produces an image with the same spatial and contrast resolution, with a possibility of
reducing the scatter photons and with a better reconstructed image. An outline of possible
future work is given in Chapter 5.
2
CHAPTER II
LITERATURE REVIEW
The goal of Emission Computer Tomography is to give accurate quantitative
measurement of radioactivity distribution throughout the patient to extract physiological
and patho-physiological information. It has been shown that quantitative measurement of
Fluoro Deoxy Glucose (FDG) uptake in tumor is useful for grading diseases (Strauss and
Conti 1991) [22]. Accurate myocardial blood flow has helped to identify triple vessel
coronary artery disease and FDG uptake is useful in studies of cerebral metabolism [22].
In this section we present some of the basic concepts in SPECT and PET that may
help us to understand the reconstruction method better. We also present the standard
reconstruction algorithm, the Filtered Back Projection Algorithm, which is based on
direct inversion of the Radon Transform.
2.1 Basic Principles of PET
A proton-rich isotope, such as Carbon-11, Fluorine-18, Oxygen-15 or Nitrogen13, is given to the subject of a PET study and is placed in the field of view of a number of
detectors. The radionuclide decays to produce a positron, neutron and neutrino. The
3
proton travels a short distance (~1 mm) within the body and gives up its kinetic energy
due to interaction with the human tissue [22]. The positron annihilates with an electron to
produce two 511 keV photons, traveling in opposite directions.
Reference: http://depts.washington.edu/nucmed/IRL/pet_intro/
Figure 2.1.1: Positron emission and annihilation
These photons may be detected by the detectors surrounding the subject. The
detector electronics are linked to detect an emission due to the same annihilation. When a
photon is registered at a detector, it generates a time pulse. The pulses are registered and
called coincident only if the two detection events fall within a small time window
generated by the coincidence circuitry [10]. These coincidence events are stored in arrays
corresponding to projections through the patient which later contribute to the image. The
conceptual diagram is shown in figure 2.1.2.
4
Reference: http://depts.washington.edu/nucmed/IRL/pet_intro/
Figure 2.1.2: Conceptual diagram of coincidence detection in a PET system.
The coincidence events fall into 4 categories: true, scattered, random and multiple
[22]. Pictorial views of the first three events are shown in figure 2.1.3. For a true
coincidence to occur, both the photons occurring due to an annihilation event are detected
within the coincidence time window. A scattered coincidence is one in which one of the
detected photons has undergone a Compton scattering event, resulting in a wrong Line of
Response (LOR) [22]. Random coincidence occurs when two photons not arising from
the same annihilation are detected within the coincidence time window of the system.
Random coincidence is proportional to the rate of single events measured [22]. Multiple
coincidence occurs when more than two photons are detected in different detectors within
the coincidence resolving time. In this event it is difficult to detect the LOR and the event
is rejected. Scattered and Random coincidences contribute to the statistical noise in the
image.
5
Reference: http://depts.washington.edu/nucmed/IRL/pet_intro/
Figure 2.1.3: Types of Coincidence in PET
2.2 Basic Principle of SPECT
SPECT is Single Positron Emission Tomography. As the name suggests, single
photon emissions are the source of emission rather than X-ray transmissions. Similar to
PET, in SPECT the patient is injected with a harmless tracer chemical which will emit
gamma rays, which also have short half lifes [10]. Unlike PET, in SPECT single photons
are emitted. Note that in PET, the detectors are situated in a ring, and the nature of the
positron annihilations combined with the shape of the PET scanner itself provides a selfcollimation. In SPECT, however, the detectors are set up in a strip, so the collimator is
necessary to localize the site from which the photon was emitted [23].
6
Reference: http://www.physics.ubc.ca/~mirg/home/tutorial/hardware.html
Figure 2.2.1: Gamma Camera
Once a tracer is injected, the gamma rays, which have the functional information,
need to be detected. These gamma rays are detected using a gamma camera. The
components making up the gamma camera are collimator, detector crystals,
photomultipler tube array, position logistics controller and data analysis computer. The
collimator allows only rays which are traveling in a straight direction to pass through.
They are made of gamma rays absorbing material such as lead or tungsten [23]. The
scintillation detector detects the gamma rays incident on it. For better image quality the
detection efficiency of the scintillation detector should be very high. The gamma rays
interact with the crystal by means of the Photoelectric effect or Compton scattering. The
released electrons interact with the crystal lattice to produce light, in a process known as
scintillation. The amount of signal given by the scintillation detector is then increased by
passing through a Photomultiplier tube (PMT) [23]. At the end of PMT is a photocathode
7
which emits electrons proportional to the light photons. These electrons are then
multiplied using a series of dynodes. The position circuit helps in locating the exact
position of each scintillation event. The raw data is then passed to a computer to
reconstruct an image using a reconstruction algorithm.
2.3 Radon Transform and Filtered Back Projection
After acquiring the projection data, PET and SPECT reconstruction is done the
same way. Traditionally the most common reconstruction algorithm used is the Filtered
Back Projection algorithm. Gamma rays are released within the tissue. The line integrals
represent the total amount of tracer in the tissues along the line [10].
Reference: http://depts.washington.edu/uwmip/Week_4/tomo_06_ho.pdf
Figure 2.3.1: Object under reconstruction and its sinogram
8
The Radon transform of a distribution is always referred to as the sinogram of the
image. Figure 2.3.1 shows how a sinogram is obtained. We have placed two tracers inside
the object. As the gamma camera moves around the object we obtain the projections at
different bins. All the projection data stacked up together gives us a sinogram.
The Radon transform and its inverse provide the mathematical basics for the
reconstruction of tomographic images.
The Radon Transform of a distribution f(x,y) is given by [24]
p (r ,  )   f ( x, y ) ( x cos   y sin   r )dxdy
where  is a delta function
x, y – coordinates of a point in spatial domain
r,  - coordinates of the point in polar domain
The most commonly used slice image used for reconstruction is the Shepp – Logan
phantom. The phantom and its Radon transform are shown in figure 2.3.2.
Reference: http://www.aapm.org/meetings/99AM/pdf/2806-57576.pdf
Figure 2.3.2: Shepp – Logan phantom and its radon transform.
9
(2.1)
Mathematically the backprojection is defined as

f ( x, y )   p( x cos   y sin  ,  )d
(2.2)
0
Due to the point spread function, the reconstructed image produced is blurred [24].
Filtered Back Projection rectifies this problem by including an additional filtering
step. Hence the algorithm consists of two steps. First, the filtering step concerns the line
integrals that make up the projection of the object. This filter makes the value zero
outside of certain boundaries, leaving behind only the real object to be reconstructed. The
second step is the backprojection algorithm.
If f(x,y) is the unknown image to be reconstructed, the Fourier transform of the
image is give by [10,24]
F (u, v)  



  
f ( x, y )e  2i (ux vy ) dxdy
(2.3)
The Inverse Fourier transform of a point is given by
f ( x, y )  



  
F (u, v)e 2i ( ux vy ) dudv
2 
f ( x, y ) 
  P ( r ,  )e
2ir ( x cos   y sin  )
rdrd
0 0
Now we split the integral into two halves.
 
f ( x, y )    P ( r ,  )e
2ir ( x cos   y sin  )
0 0
 
f ( x, y )    P ( r ,  )e
rdrd 
2 
  P ( r ,  )e

2ir ( x cos   y sin  )
rdrd
0
2ir ( x cos   y sin  )
0 0
 
rdrd    P(r ,   )e 2ir ( x cos(  ) y sin(  )) rdrd
(2.4)
0 0
One of the important properties of polar coordinates is
P(r ,   )  P(r , )
(2.5)
10
Using equation 2.5 in equation 2.4 we get,
 
f ( x, y )    P ( r ,  )e
2ir ( x cos   y sin  )
 0
rdrd    P(r , )e 2ir ( x cos   y sin  ) ( r )drd
0 
0 0
 
f ( x, y )    P(r ,  )e 2ir ( x cos   y sin  ) r drd
0 

f ( x, y )   p 1 ( x cos   y sin  ,  )d
(2.6)
0
where p ( ,  ) 
1

 P ( r ,  )e
2ir ( )
r drd

p 1 ( ,  )  p ( ,  ) * b( )
(2.7)
where b is a filter of magnitude |r|.
Equation 2.6 shows that f(x,y) can be obtained by backprojection p 1 ( ,  ) which is the
filtered sinogram [24]. The filter has a magnitude of |r|. The filtered backprojection is the
most widely used reconstruction algorithm.
11
Original Image
Filtered Backprojection Image
Forward
Back
Projection
Projection
Ramp Filter
Sinogram
Filtered Sinogram
Reference: http://www.aapm.org/meetings/99AM/pdf/2806-57576.pdf
Figure 2.3.3: Demonstration of FBP Algorithm
2.4 Iterative Reconstruction Algorithms
Iterative Reconstruction algorithms are based on a mathematical model of the
physics of PET or SPECT. For mathematical computation we express a true trace
distribution f(x,y), which is reconstructed as an image of LxL pixels, by a vector  (b) (b
12
= 1,2,…,B, B = LxL). If we include N angles and M samples per projection, we represent
the projected data by a 1-D vector n*(d) (d = 1, 2,…, D, D = MxN). The model that
relates the above two factors is given by [2, 5, 6]
B
n * (d )   p (b, d ) (b)
(2.8)
b 1
where p(b,d) is the probability of the emission b to be detected by detector bin d so that
p(b,d) ≥ 0 [2, 5]. These probabilities form a BxD matrix, which is commonly known as
the system matrix. Thus the probability of an emission b being detected is given by
D
p (b)   p (b, d )  1
(2.9)
d 1
In equation (2.8), if the system matrix p(b,d) and the projection data n*(d) are available,
we are left with a set of linear equations for  (b) . A typical SPECT image is of size
128x128 image (with 128 bins x 60 angles) making direct inversion for solving the linear
equations difficult. Direct inversion is relatively slow and needs a lot of memory to store
variables. To avoid numerical instabilities, an iterative approach is used.
In the iterative approach we start off with an initial estimate of the image 0 (b) .
The forward projection is then calculated from the intermediate estimate k (b) , then
multiplying with the probability matrix p(b,d), to give a corresponding projection data
represented by nk(d). [2, 5, 6]
B
n k (d )   p(b, d )k (b)
j 1
This calculation is then compared, by division, with the true measured projection data
n*(d). From this comparison we derive the correction term by backprojection and
13
updating the image to k 1 (b) [2, 3]. The next update of the forward projection nk+1(d) is
closer to the true projection data n*(d). All iterative reconstruction algorithms use the
same approach, but they differ in the way that the correction term is derived and the new
update is calculated.
Iterative reconstruction algorithms can again be divided into two categories. The
first class contains conventional mathematical iterative methods like Algebraic
Reconstruction Technique (ART) and Simultaneous Iterative Reconstruction Techniques
(SIRT) [2, 3]. The second class contains an iterative statistical method, which generally
maximizes a likelihood function. These statistical methods may include some a-priori
information [2, 3]. One of the most commonly used pieces of a-priori information is the
positive value constraint, since the activity distribution can never be negative. The best
known example of a statistical algorithm is the ML-EM algorithm.
2.5 Practical issues
In practical applications we do not always get a perfect sinogram and a perfect
reconstructed image. Some of the practical problems are listed below.
1) Insufficient angular sampling
Angular sampling is directly proportional to the scan time. Hence to reduce the
scan time sometimes we reduce the angular samples in the system.
2) Incomplete or missing data set
Metal implants within the patient block the radiation, leading to missing data in
its shadow.
14
3) Motion artifacts
Patient motions, such as respiration or heart beat, make it harder to reconstruct the
image.
4) Noise
Photon detection depends on the efficiency of the system. To get a better image
with less noise we can increase the radiation dose or the exposure time.
5) Limited detector size
Due to practical limitations the size of the detector is limited. Finite detector size
limits the spatial resolution of the acquired data.
15
CHAPTER III
ML-EM RECONSTRUCTION ALGORITHM
One of the important issues in SPECT and PET is quantitative accuracy of
radionuclide distribution and concentration. Image reconstruction falls broadly into two
categories [3]. The single step Fourier transform methods are the most widely used
methods due to its short reconstruction time. The other category, algebraic reconstruction
techniques may take a longer time to reconstruct but may incorporate stochastic
variations, thus yielding more accurate results.
3.1 Mathematics of ML-EM Algorithm
In the ML-EM algorithm, the projection data collected plays an important role. In
a SPECT scanner, the size of the projection data depends on both the number of detectors
in the camera strip and the number of angles. If the camera has b number of detectors
and we measure at a angles, then the total number of counts is the projection data is J =
a*b. For ease of calculations, this vector is generally represented as a column vector.
In PET, there is a ring of detectors around the patient which measures the
annihilation event. As discussed in Chapter I, an event is recorded only if the two events
16
occur within a time window. If the detector ring has N detectors, the number of counts in
projection data is given by J = N(N-1)/2 [10].
For computer reconstruction, the image to be reconstructed is digitized into a
matrix x with nx rows and ny columns. Again for computational purposes, we represent
the image as a column vector I with I = nx*ny elements. Physicists have proved that such
emissions follow a Poisson model. So the unknown total number of emission events in
the ith pixel, xˆ (i ) , represents a Poisson random variable, with mean x (i ) .
We know that; the system matrix represents the probability distribution of the
projection data. Hence elements of the system matrix p(i,j) represents the probability of
emission i to be detected by detector j. The system matrix will be discussed in detail in
later sections. It is possible to calculate the expected value of the projection data
depending on the system matrix using the formula: [3, 5, 7, 10]
I
nˆ ( j )  E (n(i ))   x(i ). p (i, j )
(3.1)
i 1
The probability of the entire projection dataset is the product of individual counts, so the
likelihood function is given by [3, 5, 7]
J
L( x)  P(n x)  
j 1
e  nˆ ( j ) nˆ ( j ) n ( j )
n( j )!
To simplify the above equation we take the log on both sides:
l ( x)  log( L( x))
J
l ( x)   (log(e  nˆ ( j ) )  log(nˆ ( j ) n ( j ) )  log(n( j )!))
j 1
J
l ( x)   ( nˆ ( j )  n( j ) log(nˆ ( j ))  log(n( j )!))
j 1
17
Substituting equation 3.1 we arrive at the log-likelihood function:
J
I
J
I
J
j 1
i 1
j 1
l ( x)   x(i ) p(i, j )   n( j ) log( x(i ) p(i, j ))   log(n( j )!)
j 1 i 1
It can be shown using the first and second derivatives of the log-likelihood function that
the matrix of second derivatives is negative semidefinite and that l(x) is concave [6]. As a
result, sufficient conditions for a vector x̂ to yield a maximum of l(x) are the KuhnTucker conditions [14]:
0  ( x(i )
J
l ( x)
n( j ) xˆ (i ) p (i, j )
)   xˆ (i )   I
x(i )
j 1
 xˆ (i' ) p(i', j)
i ' 1
and
 l ( x) 

  0  if xˆ (i )  0
 x(i )  xˆ
The algorithm requires an initial estimate x 0 , and using the maximization condition to
iteratively improve the estimate. Researchers have used a variety of initial estimates to
reach the results faster. The main formula for ML-EM algorithm is derived by solving the
above maximization condition for xˆ (i ) .
n( j ) p (i, j )
J
x n 1 (i )  x n (i )
j 1
I
x
n
(i ' ) p (i ' , j )
i '1
This can be written as
x n 1 (i )  x n (i )x n (i )
Hence the sum is really a multiplicative coefficient that corrects the image at every step.
As the number of iterations increase, the x n (i ) term gets closer and closer to 1.
18
3.3 Modified ML-EM Algorithm
Given below is the simplified flowchart for the ML-EM algorithm [9].
Calculate Probability
matrix Gij
Start with initial image
estimate  (b)
Forward Projection:
B
ni   Gi , j kj
j 1
Comparison:
n ' i  n *i / n i
Back Projection:
D
x j   Gi , j n' i
i 1
Normalization:
D
x ' j  x j /  Gi , j
i 1
Update:
kj 1  kj * x' j
19
To better understand the working of the algorithm we consider a 4x4 image.
100
0
0
0
This image can be represented in a column matrix as
100
0 
 
0 
 
0 
Now let us try to reconstruct this image using 4 reconstruction angles and 2 bins. The
total number of projection data points is the product of number of angles and the number
of bins. Hence in our example it is 8. The projection data points are represented below in
yellow. The image is represented in blue.
7
8
6
100
0
1
5
0
0
2
4
3
At each point the value measured is the sum of all the pixel values that the bin can
measure using line integral approach. Hence the projection data for our example is
20
100
0 
 
0 
 
100
n   
0 
 
100
100
 
0 
Probability (G) matrix represents the probability of a pixel b being located at detector d.
This makes the size of the matrix huge; generally BxD (where B represents the image
size and D represents the projection data). Hence the size of the Probability (G) matrix in
our example is 8x4. It is represented as
1
0

0

1
G
0

1
1

0
1
0
1
0
0
1
0
1
0
1
0
1
1
0
1
0
0
1
1

0
1

0
0

1
To reduce the number of iterations we start off with an initial estimate of the image with
individual values equal to the average projection data.
25
25
 
25
 
25
B
We perform forward projection ( ni   Gi , j kj ) on this initial estimate.
j 1
21
50
50
 
50
 
50
n 
50
 
50
50
 
50
The comparison step is done in ML-EM by dividing the original projection data with the
obtained projection data.
2
0 
 
0 
 
2
n   
0 
 
2
2
 
0
D
Backprojection ( x j   Gi , j n' i ) of the data gives us our correction term. This correction
i 1
D
term needs to be normalized ( x' j  x j /  Gi , j ) before we multiply it with the real image.
i 1
2
1 
x   
1 
 
0 
We then multiply this matrix by the original image to get the first estimate of the image.
22
50 
25
1   
25
 
0 
Performing similar iterations as shown above the second image correction terms are
4 / 3
2 / 3


x1 
2 / 3


0 
Hence the second estimate of the image is
66.66
16.66 

2  
16.66 


0

As the number of iteration goes on increasing the estimated image approaches closer and
closer to the real image.
We know that; as the number of iterations goes on increasing, the error goes on
decreasing. But if the number of iterations is too high, due to the Poisson nature of the
data, the image may look a little noisy. Hence many researchers have suggested that 50
iterations give an acceptable result in real SPECT image. In our study too, we will
consider 50 iterations.
3.4 System Matrix
The system matrix is the most important part of the ML-EM algorithm. It contains
the information of the geometry of the imaging system being used. It provides the model
23
of how detectors in the system see the emission event that takes place in the object being
imaged. The ability to add various mathematical models like attenuation correction,
scatter correction, interaction of photons with collimators and detectors, point spread
function and filters make it a more sophisticated model.
The size of the system matrix depends on both the size of the image and the
number of points in the projection dataset. If we have an image with size lxl, then for
computational purpose we represent the image by a column vector  (b) (where b =
1,…,B, B=lxl). As discussed earlier, for PET the projection data is represented by a
column vector n(d) (where d = 1,…,D, D=N*(N-1)/2) where N represents the number of
detectors surrounding the object to be reconstructed. In SPECT the projection data is of
size (na*nb) where na is the number of angles and nb represents the number of bins.
Hence the total size of the system matrix is BxD. In other words the entire image is
represented in each row and the entire projection data is represented by one column of the
system matrix. Thus a fixed row in the system matrix corresponds to a particular detector
at a particular angle.
Since this matrix contains the probability of an emission at a particular point
showing up at a particular detector at a particular angle, many elements within this matrix
are zeros. Very few pixels in this matrix will have non-zero terms. This makes the system
matrix very sparse. Let us consider an image with following parameters:
Image size = 32x32,
Number of angles = 60
Number of bins = 128
24
The size of the system matrix is 1024*7680 (7,864,320 elements). The number of nonzero elements in this matrix is just 173,586 i.e. 2.2073%. In addition to this, if we know
that the image lies within a specific area of the digitized pixel grid, then a mask can be
used. Pixels which lie outside this mask are made zeros which will not contribute to the
final image. This makes the system matrix sparser.
Different techniques are used to calculate the system matrix. The technique we
used solely depends on the geometry of the imaging system. The values are calculated
depending on the area of intersection of the ith pixel and the strip defined by the jth
detector. Various constraints can be added to this matrix.
For our Modified ML-EM algorithm, we add one more constraint to the system
matrix. As discussed earlier, each row in the system matrix contributes to a particular
detector pair. So to remove the detectors from the reconstruction algorithm, we can
replace the value at that point by a zero. It is very important to modify the system matrix
according to the system change or it may introduce artifacts in the image. The most
common artifacts introduced due to improper sampling are the ring artifacts. This will
make the system matrix sparser.
3.6 Hypothesis and Statistical Testing
The null hypothesis of this research state that:
 The original image and image reconstructed by Modified ML-EM method will
demonstrate no statistically significant difference.
25
To test our null hypothesis we will use the Chi-Square Goodness of Fit method. For a
Chi-square goodness of fit computation, the data is divided into k bins and the test
statistic is defined as
k
 2   (Oi  Ei ) 2 / Ei
i 1
where O is the observed result and E is the expected result.
3.6. Advantages and Disadvantages of ML-EM reconstruction algorithm
There are several promising characteristics of the modified ML-EM algorithm
that give it an advantage over conventional techniques such as Radon Transform or
Filtered Back Projection. However there are some drawbacks which may avoid its use for
practical purposes.
Advantages:
1) Modeling of the measurement process
The projection data are Poisson in nature with mean value equal to the line
integrals, as discussed in Chapter 2. Filtered Back Projection works well if the
number of counts is very high, since for higher count statistics, the mean value of the
counts is close to the line integrals [10, 2, 7]. But for low count statistics, the
measured data may have a large variation from the mean. ML-EM can use this
information in the system matrix p(b,d) to give a better image. The ML-EM method
26
can also model the interaction of the photons with the body, the collimators and
detectors. [2]
2) Modeling of image degrading factors
The major problem that SPECT and PET face is the scatter of photons. Various
models have been presented that can be incorporated in the system matrix p(b,d). A
few of the most common methods mentioned in [2] are Scatter Correction done by the
subtraction based method, Point Spread Function Recovery by convolution method,
attenuation correction method, geometrical weighting of the point spread function,
and Monte Carlo simulation.
3) Reduction of Cross Talk
One of the problems with imaging reconstruction algorithms in PET is the cross
talk between detectors. These detector crystals convert photons to a proportional
amount of light energy produced. These crystals are coated with light reflecting
material which forces the light to go in the downward direction. In spite of the light
reflecting coat; some amount of light gets detected by the adjacent detectors, hence
contributing to noise. By placing the detectors apart we reduce the possibility of cross
talk between the detectors.
4) Reduction of cost of the system
By placing the detectors apart, the system requires fewer detectors to cover the
same area. This can reduce the overall cost of the system without affecting the image
quality.
5) Reduction of storage space
27
The projection data acquired by the imaging system is huge. This projection data
may be saved to evaluate the reconstruction algorithm. As the size of the image and
the number of slices goes on increasing, the projection data size also increases. Hence
to reduce the storage space required, the Modified ML-EM algorithm may be used. It
can reduce the space requirement to 1/10th of the original size without affecting the
spatial or contrast resolution of the reconstructed image.
6) Reconstruction with faulty detectors
Sometimes the detectors in the camera strip malfunction. In this case the normal
Filtered Back Projection may introduce ring artifacts in the image. There are special
codes installed in the system which detect faulty detectors. To avoid artifacts due to
this, the data can be interpolated. A better way to solve this problem will be to modify
the system matrix. The corresponding values in the system matrix can be made zero
to give a reconstructed image without artifacts. This point is explained in detail is
section 4.6.
Disadvantages:
1) Noisy Reconstruction with higher iterations
As the number of iterations goes on increasing, the result in real images with
noise initially converges to an acceptable image. For a higher number of iterations,
the likelihood still increases but the resulting image becomes noisy. [2, 10] This is
due to the Poisson nature of the variables. [2, 10] This can be avoided by using a
fixed stopping rule depending on the error. A second approach is to use a smoothing
filter after the image is reconstructed. The most common approach is the second one.
28
2) Longer Calculation Time
Filtered Back Projection is a single step process whereas ML-EM uses an iterative
approach. ML-EM may require from 10-100 iterations depending on the stopping rule
used. Several methods have been suggested to decrease the calculation time in the
ML-EM algorithm. The most common approach is the OS-EM method proposed by
Hudson and Larkin in 1994. [2, 10] In this method, projection data is divided into
subsets and uses only one subset for each iteration. OS-EM has proved to increase the
reconstruction time by a factor of 6. [2, 10]
3) More iterations with fewer detectors
As we decrease the number of detectors, the number of iterations required to
reconstruct an acceptable image goes on increasing. Hence there is always a tradeoff
between the number of detectors skipped and the reconstruction time.
29
CHAPTER IV
RESULTS
4.1 FBP Reconstruction with limited dataset
In this section, we test whether FBP can reconstruct images with a limited dataset.
We use half the number of detectors to reconstruct the image. Figure 4.1.1 shows the
original Shepp- Logan head phantom and the reconstructed image with ½ detectors and
60 angles.
Figure 4.1.1 Original Shepp-Logan Head Phantom and FBP Reconstructed Image with 60
angles
30
To reconstruct the image, we can increase the number of angles to 1800. Figure 4.1.2
shows a reconstructed image with 1800 angles. In practical situations, it is impossible to
increase the number of angles to such a huge number.
Figure 4.1.2 FBP Reconstructed image with ½ dataset and 1800 angles
Now we try to reconstruct the image with ¼ data points and 1800 angles. Figure 4.1.3
shows the reconstructed image. Hence we see that even with such high number of angles
we can not reconstruct the image.
Figure 4.1.3 FBP Reconstructed image with 1/4 dataset and 1800 angles.
31
Now, we try to interpolate the projection data between the original data points and then
reconstruct the image using the modified projection data. Figure 4.1.4 represents
reconstructed images.
(a) 1800 angles
(b) 360 angles
(c) 180 angles
(d) 60 angles
Figure 4.1.4 FBP Reconstructed image with ¼ data points and linear interpolation (a)
1800 angles (b) 360 angles (c) 180 angles (d) 60 angles
The image produced is good with a loss in resolution, which causes blurring at the edges.
32
4.2 Modified ML-EM Reconstruction on image
In this section, we show the results for the Modified ML-EM algorithm. Through
out the study we use a Shepp-Logan head phantom with the following parameters:
1) size = 32x32
2) angles = 60 angles
3) number of bins = 128 bins
4) number of iterations = 50.
These parameters are the most commonly used parameters in the field. Many studies have
suggested that 50 iterations produce an acceptable result in the medical field. Figure 4.2.1
shows a sample reconstructed image.
(a) Original Shepp-Logan phantom
Figure 4.2.1: Modified ML-EM Algorithm: (a) Original Shepp-Logan phantom (b)
Reconstructed image (c) Sinogram
33
(b) Reconstructed image
(c) Sinogram
Figure 4.2.1: Modified ML-EM Algorithm: (a) Original Shepp-Logan phantom
(b) Reconstructed image (c) Sinogram continued
34
4.3 Effect of the number of iterations on the reconstruction
As mentioned earlier, ML-EM is an iterative algorithm. Therefore as the number
of iterations goes on increasing, the error goes on decreasing. To calculate the error, we
use the Chi-Square Goodness of Fit method. For a Chi-square goodness of fit
computation, the data is divided into k bins and the test statistic is defined as
k
 2   (Oi  Ei ) 2 / Ei
i 1
where O is the observed result and E is the expected result.
The number of pixels in the image is 1024 (32x32). So the degree of freedom is 1023.
Assuming a significance level of 0.05,  (20.05,1023)  124 .
Figure 4.3.1 shows the plot of error with respect to the number of iterations. As
the number of iterations increase the likelihood goes on increasing.
35
Figure 4.3.1: Chi-Square Goodness of fit for Modified ML-EM algorithm
Figure 4.3.2 below represents images acquired at different number of iterations.
As the number of iterations increase, the image quality gets better. The images are taken
with the following parameters:
1) size = 32x32
2) angles = 60 angles
3) number of bins = 128 bins
36
(a) 5 iterations
(b) 15 iterations
(c) 35 iterations
(d) 50 iterations
Figure 4.3.2: Reconstructed image at (a) 5 iterations, (b) 15 iterations, (c) 35 iterations,
(d) 50 iterations, (e) 75 iterations and (f) 100 iterations
37
(e) 75 iterations
(f) 100 iterations
Figure 4.3.2: Reconstructed image at (a) 5 iterations, (b) 15 iterations, (c) 35 iterations,
(d) 50 iterations, (e) 75 iterations and (f) 100 iterations continued
Figure 4.3.3 represents the values of a single horizontal line through the center of
the image with respect to number of iterations. The red line represents the reconstructed
image and blue represents the true values. We can see that as the number of iterations
increase, the difference between the two plots goes on decreasing.
(a) 5 iterations
(b) 15 iterations
Figure 4.3.3: Line Profile values at (a) 5 iterations, (b) 15 iterations, (c) 35 iterations, (d)
50 iterations, (e) 75 iterations and (f) 100 iterations
38
(c) 35 iterations
(d) 50 iterations
(e) 75 iterations
(f) 100 iterations
Figure 4.3.3: Line Profile values at (a) 5 iterations, (b) 15 iterations, (c) 35 iterations, (d)
50 iterations, (e) 75 iterations and (f) 100 iterations continued
39
4.4 Effect of the number of bins skipped
In this section, we keep on skipping the detectors, which results in an incomplete data
set. By skipping the data sets we are also decreasing the number of detectors in the
system. Table 1 shows the results with following parameters:
1) image size: 32x32
2) number of angles: 60
3) number of bins: 128
4) number of iterations: 50
The reconstructed images and their respective sinograms are shown in figure 4.4.2. The
results show that as we increase the number of bins skipped, the error goes on increasing.
This error can be reduced by increasing the number of iterations.
Table 1: Detector skipped and resulting Chi-Square error
Data points
Chi-Square
Number
skipped
Error(*10^-3)
1
None
3.8
2
½
3.8
3
1/3
3.9
4
1/5
5.2
5
1/10
6.3
40
Figure 4.4.1: Plot of number of detectors skipped V/S the Chi-Square error
(a) no detectors skipped
Figure 4.4.2: Reconstructed images and their sinograms with (a) no detectors skipped, (b)
½ detectors, (c) 1/3 detectors (d) 1/5 detectors (e) 1/10 detectors
41
(b) ½ detectors
(c) 1/3 detectors
Figure 4.4.2: Reconstructed images and their sinograms with (a) no detectors skipped, (b)
½ detectors, (c) 1/3 detectors, (d) 1/5 detectors and (e) 1/10 detectors continued
42
(d) 1/5 detectors
(e) 1/10 detectors
Figure 4.4.2: Reconstructed images and their sinograms with (a) no detectors skipped, (b)
½ detectors, (c) 1/3 detectors, (d) 1/5 detectors and (e) 1/10 detectors continued
43
4.5
Reconstruction with a real SPECT image
In this section, we reconstruct a single axial SPECT bone scan image through
the pelvis using real sinogram data from a gamma camera. This sinogram was collected
from a Trionix gamma camera. From Figure 4.5.1, we can see that as we skip the
number of bins, the image looks more and more noisy. The original image (a) below is
the filtered back projection reconstruction and the images b-f are reconstructed by
Modified ML-EM Algorithm. As seen in section 4.3, we could have improved this
image quality if we increased the number of iterations.
(a) Original image
(b) none skipped
(c) ½ detectors
(d) 1/3 detectors
Figure 4.5.1: Reconstructed images (a) original image (b) all detectors(c) ½ detectors (d)
1/3 detectors (e) 1/5 detectors (f) 1/10 detectors
44
(e) 1/5 detectors
(f) 1/10 detectors
Figure 4.5.1: Reconstructed images (a) original image (b) all detectors (c) ½ detectors (d)
1/3 detectors (e) 1/5 detectors (f) 1/10 detectors continued
4.6 Reconstruction with a faulty detector
Faulty detectors are a common scenario in real life which led to rings in the final
image making it un-diagnostic. Currently vendors detect these faulty detectors and
interpolate the data losing the resolution. In our method, we can modify the system
matrix to give a image without losing any resolution. Figure 4.6.1 shows an image
reconstructed with a faulty detector. Figure 4.6.2 shows the sinogram of this image. The
line at detector number 16 shows that the detector is faulty. We modified our system
matrix and the reconstructed image is shown in figure 4.6.3.
45
Figure 4.6.1: Ring artifact in the image due to a faulty detector.
Figure 4.6.2: Sinogram which shows that detector 16 is bad.
46
Figure 4.6.3: Image reconstructed using ML-EM Recon Algorithm
47
CHAPTER V
CONCLUSION
In summary, Medical Imaging provides a non-invasive technique to look at the
structural and functional information of internal organs and structures. Measurement of
the radioactive distribution throughout the patient gives the physiological and pathophysiological information about the patient. The most widely used mathematical
technique to reconstruct this image, using the data acquired, is known as Filtered Back
Projection (FBP).
The main purpose of this study is to verify whether or not we can reconstruct
images with a limited dataset. This method will provide us a better estimate of the exact
location of the radioactivity. Traditional reconstruction algorithms like FBP can not
reconstruct an image with a limited dataset. We work with the iterative method, ML-EM,
by modifying the probability matrix (system matrix). Chi-Square Goodness of Fit test is
used to test whether we can reconstruct the image using a limited dataset without any
statistical difference.
Based on the statistical data shown in section 4.4 we fail to reject the null hypothesis.
The modified ML-EM algorithm is a good alternative to the filtered back projection
algorithm and can be used successfully to reconstruct images with limited dataset.
The real SPECT and PET images are generally of size 512x512. All the
reconstruction done in our study was done using Matlab. Due to the huge size of the
48
system matrix, the virtual memory allocation in matlab restricted the size of the
reconstructed images to 64x64.
For future work, we suggest the implementation of the same code in a lower level
language. This would not only allow us to reconstruct images with larger size, but will
also decrease the reconstruction time. Another interesting study would be to develop a
technique that collects limited dataset by using point spread function information and
see its effect on image quality.
49
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