Maximum Likelihood Algorithms for Image Reconstruction in Positron

Radionuclides for Oncology – Current Status and Future Aspcets GS Limouris, SK Shukla, HF Bender, HJ Biersack (editors), MEDITERRA Publishers, Athens, 1998 pp. 73-106

Maximum Likelihood Algorithms for Image Reconstruction in Positron Emission Tomography

George Kontaxakis, PhD Ludwig G. Strauss, MD Division of Oncological Diagnostics and Therapy, Medical PET Group - Biological Imaging, German Cancer Research Center (DKFZ), Heidelberg, Germany

Corresponding author: George Kontaxakis, PhD German Cancer Research Center (DKFZ) Division of Oncological Diagnostics & Therapy Medical PET Group - Biological Imaging Im Neuenheimer Feld 280 D-69120 Heidelberg Germany tel. +49-6221-42 2479 fax +49-6221-42 2510 e-mail: [email protected] Web: http://www.dkfz-heidelberg.de/pet/home.htm

Maximum Likelihood Algorithms for Image Reconstruction in Positron Emission Tomography INTRODUCTION The goal in all medical imaging modalities is to get an image of the internal organs of the body, either structural and anatomical (Computerized Tomography, CT, or Magnetic Resonance Imaging, MRI) or functional (Positron Emission Tomography, PET, or Single Photon Emission Tomography, SPET), in a non-invasive way. The main principle of image reconstruction in all computerized tomography modalities is that an object can be accurately reproduced from a set of its projections taken at different angles. In a practical application, as is any CT imaging modality, one can only obtain an estimate of the real image of the object under study. For the faithfulness of the reconstruction in each case there are always questions to be answered, concerning the collection and the preprocessing of the projection data, the numerical implementation of the mathematical reconstruction formulas and the postprocessing of the reconstructed image [1]. PET is a nuclear medicine imaging technique which allows the quantitative evaluation of the distribution of several pharmaceuticals in a target area. PET therefore provides valuable information on the biochemical and biological activity inside a living subject in a non-invasive way, combining techniques applied in nuclear medicine with the precise localization achieved by computerized image reconstruction. The radiopharmaceuticals used in PET studies are chosen to have a desired biological activity, depending on the metabolic activity of the organ under study and are introduced to the subject by injection or inhalation. The most commonly used radionuclides are compounds that constitute or are consumed by the living body, like carbon, nitrogen and oxygen. They are isotopes of biologically significant chemical elements that exist in all living tissues of the body and in almost all nutrients. Therefore the above radionuclides are easily incorporated in the metabolic process and serve as tracers of the metabolic behavior of the body part, which can be studied in vivo. The most common radiopharmaceutical used in PET studies today is fluorodeoxyglucose (FDG), a chemical compound similar to glucose, with the difference that one of the -OH groups has been replaced by F-18. Carbon-11 can also be used as a radiotracer to glucose. The short half-lives of these particles allow so that the subject and the people handling them receive only a low radiation dose. A positron is emitted during the radioactive decay process, annihilates with an electron and as a result a pair of γ-rays is emitted. The two γ-rays fly off in almost opposite directions (according to the momentum conser2

vation laws), penetrate the surrounding tissues and can be recorded outside the subject’s body by scintillation detectors placed on a circular or polygonal detector arrangement, which forms a PET scanner. The event of the annihilation can be identified by the electronic circuits of the scanner, that detect the coincidental arrival of the pair of γ-rays emitted and the location of the event can be identified to occur inside the volume defined between the surfaces of the detectors activated by each one of the rays. After a complete scan, a mathematical algorithm applied by computer corrects the collected data for scatter, attenuation, accidental coincidences, normalizes for the differences in detector efficiencies and reconstructs the spatial distribution of the radioactivity density inside the organ or the system under study. By measuring a coincidence photon, the detector array in a PET system identifies that an annihilation event occurred inside the volume of the detector tube between the surfaces of the pair of detectors that registered the coincidence event. At the end of a PET scan, for each pair of detectors there is a number of coincidence events that have been identified. This information represents the radioactivity in the subject viewed at different angles, when sorted in closely spaced parallel lines. In order to reconstruct the activity density inside the source from its projections gathered along the detector tubes (stripes between the detectors) a reconstruction algorithm has to be applied. The result is a digital image of the source, where the value of each picture element (pixel) is proportional to the activity density inside the source at the area that corresponds to this pixel. This image can be directly displayed on a screen. Further analysis of the data and processing of the produced images can be carried out with the use of a computing system.

y t ray

s r θ

ϕ

x obect to be reconstructed

3

Fig. 1: The position of a point in the plane is specified by (x, y) or (r cosφ, r sinφ). A ray is specified by its perpendicular distance s from the origin and its orientation θ.

Preliminaries in image reconstruction from projections Let f(x,y) represent the spatial distribution of the activity density to be visualized. Two dimensions are used here, since cross-sectional images of the subject’s body are to be reconstructed, although it is now possible to get PET scans in three dimensions. Since the direction of photons is known, the number of counts recorded along the line defined by s (its signed distance from the origin) and θ (its angle with respect to the y-axis), as depicted in

$

Fig.1, is just a measure of the total radionuclide concentration at a point (x,y). By f ( s,θ ) is denoted that function of two variables whose value for any (s,θ) is defined as the line integral of f along the line specified by s and θ:

$

f = ( s, θ ) =

∫

Ω

f ( s cos(θ ) − t sin(θ ), s sin(θ ) + t cos(θ ))dt

(1)

$

where f(x,y) is spatially bounded inside the region Ω. The function f ( s,θ ) is known as the

Radon Transform [2] of the function f(x,y). In the case of PET, the number of counts along the set of detector tubes, constitutes a sampling of the Radon Transform of the activity density function f(x,y). An appropriate inversion or reconstruction algorithm has to be applied in order to recover an approximation of the spatial distribution function. The above formula holds only if the attenuating properties of the medium can be neglected. In the case of PET, an attenuation correction can be made in the data and bring the problem back into the Radon Transform theory [1].

$

The problem of reconstruction now is the following: given f ( s, θ ) , find f(x,y). The solution to

$

the above problem, was published by Johann Radon in 1917

[3],

with an inversion formula

expressing f in terms of f although effort to implementing these formulas in practical problems, like the problem of image reconstruction in CT, was devoted much later. Although the inversion formula can be expressed in abstract mathematics, this is only the beginning for an applied problem

[4].

In almost all cases the physical measurements fail to accurately

define the complete set of line integrals of Eq. (1). In fact, as stated in a theorem by Smith et

al.

[5],

an object of finite extent and completely contained within some unit circle, is uniquely

determined by any infinite set, but by no finite set, of its projections. The lack of the complete set of line integrals leads to inaccuracies and distortions in the resultant reconstruction, because of non-linearities, noise, and insufficient data

[6].

The non-linearities can arise from a 4

non-linear detector process and noise can be the usual statistical uncertainty of the measurement or an interfering component such as scatter. The data can be insufficient because of inadequate sampling or regions of missing data. Unknown attenuation distribution can distort the measurements of a source distribution, resulting in errors in the reconstruction. Several algorithms have been developed in the past two decades and all may be regarded as methods for approximating the inversion of the Radon transform. They can be implemented for the reconstruction in various tomographic modalities and not only for PET. It is important here to mention also that due to many reasons (computer implementation, numerical approximations, etc.) these methods are not all equivalent (e.g., in the Appendix in

[7]

is

shown that algebraic reconstruction techniques (ART) and backprojection methods, are not equivalent). A presentation of many of the transform methods is given in

[1,8].

Most of the

algorithms presented are two-dimensional. The third dimension is usually obtained by appropriate stacking of several adjacent two-dimensional transverse sections. More details on 3-D Imaging are given in [9]. A classification and an overview of the existing image reconstruction algorithms is given in [1],

according to the following scheme: 1. Direct Fourier methods. 2. Signal space convolution and frequency space filtering: • signal space convolution then backprojection; • frequency space filtering then backprojection; • backprojection then signal space convolution; • backprojection then frequency space filtering. 3. Iterative methods. 4. Series methods and orthogonal functions.

These image reconstruction techniques provide solutions to the problem of approximating the inverse Radon Transform. However, the set of variables considered as projections, is usually a set of random variables, whose probability density depends on line integrals. A fundamental assumption of these methods is therefore that the set of measured parameters could represent with high accuracy (as a consequence of the law of large numbers) the means of this process (exact values of the projections). In the case of X-ray tomography, for example, one deals in principle with a large number of counts per projection angle, the above assumption is valid and these methods produce satisfactory results. However, in emission tomography the total number of counts is low, and as a result there is high statistical noise due to detector delay, dead time -- since the coincidence circuits can not process more than one event during the time coincidence window -- and pulse pileup 5

(when different amplifier pulses fall on top of each other and produce a single pulse that distorts the energy information and contributes to counting losses

[10].

This limits the total num-

ber of emission counts detected to about 106 - 107 for a single detector ring, when in transmission tomography the number of counts can be 1015 - 1016 [11]. Other cases exist, especially where short observation times are required, where the number of measured counts can not be considered large enough. Consequently a method for image reconstruction in emission tomography which would incorporate the stochastic nature of the emission process is expected to produce better images than conventional techniques. The next sections describe in detail iterative algorithms for image reconstruction in PET, where the optimization criterion in order to approach optimally the solution of the image reconstruction problem is based on maximum likelihood estimation.

6

ITERATIVE IMAGE RECONSTRUCTION FOR PET BASED ON MAXIMUM LIKELIHOOD ESTIMATION

During the last decade, mathematical models for image reconstruction in PET have been developed that take into account the Poisson nature of positron emissions and the statistical characteristics of the generation of the annihilation events inside the source. From this perspective, the problem of image reconstruction in PET is viewed as a standard statistical estimation problem for incomplete data. The incompleteness in the data is derived from the fact that, although the pair of detectors where the event was registered is known, the origin of the annihilation event (and consequently the point where the positron was produced after radioactive decay of the tracer radionuclide) is not known. In mathematical statistics there is a general iterative method known as the Expectation Maximization (or EM) Algorithm, presented by Dempster et al.

[12]

in its full generality. The

name of the technique stems from the fact that in each iteration there is an expectation step that uses current parameter estimates in order to perform a reconstruction of the unobservable Poisson process, followed by a maximum likelihood step that uses this reconstruction to revise the parameter estimates

[13].

The maximum likelihood approach in image reconstruc-

tion for emission tomography was first introduced by Rockmore and Macovski

[14].

A practical

implementation of the EM method was introduced to the problem of image reconstruction in PET by Shepp and Vardi [16].

[15],

and was extended to transmission CT by Lange and Carson

Recent developments of the application of the EM algorithm in transmission images have

been reported in [17]. The EM algorithm is applied in emission tomography as an iterative technique for computing maximum likelihood estimates of the activity density parameters. In this approach, the measured data are considered to be samples from a set of random variables whose probability density functions are related to the object distribution according to a mathematical model of the data acquisition process. Using the mathematical model, it is possible to calculate the probability that any initial distribution density in the object under study could have produced the observed data. In the set of all possible images, which represent a potential object distribution, the image having the highest such probability is the maximum likelihood estimate of the original object.

7

Description of the EM algorithm for PET As mentioned earlier, there is an array of discrete detector elements on a PET scanner, placed around the detector ring and the pair of photons produced in an annihilation event are detected in coincidence by a pair of detector elements that define a cylindrical volume, or

detector tube [18]. The set of data collected in a PET scan is represented by the vector [y(1),

y(2), ..., y(J)], where y(j) is the total number of coincidences counted in the jth detector tube and J is the total number of detector tubes. If N is the total number of detector elements on the detector ring, then the number of detector tubes is given by J=N (N-1)/2. The measured coincidence events include scattered and accidental coincidences

[10]

and also not all the

events produced inside the source are detected, because of tissue attenuation, photon traveling paths that do not intersect the detector ring(s) and pass undetected, or other inefficiencies due to various reasons. The activity density inside the source (assuming a two-dimensional case), represented by the function f(x,y) has to be estimated using the vector of the measured data y. For purpose of computer implementation and display, the density f is discretized in boxes i=1,2, ..., I, and

$

therefore in each box i there is an unknown count x (i ) , that represents the total number of emissions that occurred in the area of the source covered by the ith box, with mean

$

x(i)=E[ x (i ) ], i=1,2, ...,I. The problem now, is to estimate x(i), or roughly, to guess the true

$

unobserved count x (i ) in each box, from the observed data y(j), j=1,2, ..., J [15].

Derivation of the EM algorithm The mathematical model is based on the assumption that the emissions occur according to a

$

spatial Poisson point process in the region of interest (field of view) in the source. For each box i there is a Poisson distributed random variable x (i ) , with mean x(i), that can be generated independently [15],

$

P(x(i) = m) = e − x(i)

x(i)m m!

(2)

Suppose now that an emission in the ith box is detected in the jth tube with known probability:

a(i,j) = P (event detected in tube j | event emitted in box i)

(3)

8

where a(i,j) ≥ 0. The probability matrix a(i,j) is assumed known from the detector array geometry and other characteristics of the system, described later in this chapter. In this context, the probability of an event in box i to be detected by the scanner is given by: J

a(i) = ∑ a(i, j) ≤ 1

(4)

j =1

This means that some of the photons go undetected. However, it is shown [15] that there is no loss of generality to assume that the equality holds in the above equation, and a(i,j) represents the (conditional) probability that a detected photon emitted in box i is detected in detector tube j. This is equivalent to thinking of the activity density function f(x,y) as the density of emitted counts which are detected. The same assumption holds also for other reconstruction techniques, as the convolution - backprojection (CBP), since there is no clue to include non-detected photons in the measured data. However, if correction techniques for attenuation (lost scatter and photoelectric absorption in the source) are applied, an estimate of the non-detected photons can be included in the measured projections. For both cases, it will be assumed that the equality holds in the above expression without loss of generality. To continue the discussion on the mathematical model, the variables y(j) are independent y ( j ) , where and Poisson, with expectation ~ I

~y(j) = E [ y(j)] = ∑ x(i)a(i, j)

(5)

i =1

Since x(i) are independent Poisson variables, a linear combination of these variables as the one in the above equation is also Poisson distributed. Considering the above, the likelihood of the observed data is: J

L( x ) = P( y| x ) = ∏ e− y(j) j=1

~

~y(j)y(j) y(j)!

(6)

The above likelihood function L(x) expresses the probability under the Poisson probability

model for emission (which according to Shepp

[11]

is a perfect model, except for the need to

discretize the density f(x,y) into pixels and the need to define the probability matrix a(i,j) to give a transition law for an emission in box i to be detected in detector tube j) to observe the

given counts in detector tubes if the true density is x(i). In order to proceed with the maximum likelihood approach, the log-likelihood function of Eq. (6), l(x), is taken and, along with Eq. (5): 9

J

I

I

I

J

j =1

i =1

j =1

l( x ) = log(L( x )) = -∑ ∑ x(i)a(i, j) + ∑ y(j)log( ∑ x(i)a(i, j)) − ∑ log(y(j)!) j=1 i=1

(7)

Taking now first and second derivatives of the log-likelihood function, it can be shown [18] that

$

the matrix of second derivatives is negative semidefinite and that l(x) is concave. Consequently, sufficient conditions for a vector x to be a maximizer of l are the following Kuhn-

Tucker conditions ([19], Theorem 2.19(e)):

J y ( j ) x∃(i )a (i , j ) ∂ l (x)   ∃ 0 =  x (i ) x i ( ) = + ∑ I ∂ x (i )  x∃ j =1  x∃(i ’)a (i ’, j )

(8)

∑ i ’=1

and

$

 ∂l( x )   ∂x(i)  ≤ 0 , if x(i) = 0

$

x

(9)

for each i=1,2,...,I. The formula for the EM algorithm can be now derived by solving Eq. (8):

(a) Start with an initial estimate x(0), satisfying x(0)(i) > 0, i=1,2,...,I. (b) If x(k) denotes the estimate of x at the kth iteration, define a new estimate x(k+1) by: J

x(k +1) (i) = x(k) (i)∑ j =1

y(j)a(i, j) I

∑x i ′=1

(k)

, i=1,2,..,I.

(10)

(i′ )a(i′ , j)

(c) If the required accuracy for the numerical convergence has been achieved, then stop.

The above express the EM algorithm, that can be directly applied to the problem of image reconstruction in PET. Using Eq. (5), Eq. (10) can be rewritten as:

x

(k +1)

J y(j) (i) = x(k) (i)∑ a(i, j) ~ (k) , i=1,2,...,I. y (j) j =1

(11)

y (k) is the reprojection of the estimated image vector x(k) at the kth iteration to the where ~

space of the data vector. In other words, it contains the expected values of the collected data 10

$

if the estimated image vector x(k) was the actual (and unknown) activity density x in the source. The sum in the above expression is the backprojection of the ratio:

y(j) ~y (k) (j) into the image plane with a weight a(i,j). Therefore, the general form of the update process at the (k+1)th iteration is:

x(k +1) (i) = x(k) (i)C (k) (i) i=1,2,...,I

(12)

where: J

C (k) (i) = ∑ j=1

y(j)a(i, j) I

∑x i ′= 1

(k)

(i′ )a(i′ , j)

J y(j) = ∑ a(i, j) ~ (k) y (j) j =1

(13)

is a multiplicative coefficient that updates the value of the ith pixel at the (k+1)th iteration.

General characteristics of the EM algorithm Vardi et al.

[18]

give the following theorem that summarizes the previous discussion and

states the convergence of Eq. (10) to a point of maximum:

THEOREM.

(a) l(x) is concave and hence all its maxima are global maxima. (b) The EM

algorithm converges monotonically in the sense that l(x(k)) < l(x(k+1)) to a global maximum of

l(x). (c) The maximum of l(x) is unique if and only if the grid is such that the J vectors: ( y(j) / y(j))(a( 1, j),a( 2 , j),...,a(I, j) , j=1,2,...,J span EI, the I-dimensional Euclidean space. A proof of the above Theorem, along with other remarks and further discussion can be found in

[18].

Proof of the convergence of the EM algorithms is also given in

[20].

However it is im-

portant to notice here the following: 1. Given a non-negative initial image x(0) and non-negative a(i,j) and y(j) (as in the case of PET), then all images produced by the algorithm are non-negative. This is an important advantage of the method in comparison with backprojection or algebraic techniques, which can result in meaningless negative values for the emission density in certain boxes. 11

Also, the non-negativity is obtained without the use of additional constraints, that complicate the computation work, and is inherent in the formulation of the method. 2. For each image vector x(k) produced by the algorithm the sum of events in the image is equal to the sum of the counts in the data. This property follows immediately from the iterative step formula (10), from which is found that: I

∑x i =1

J

(k)

(i) = ∑ y(j)

(14)

j =1

The last means that the EM algorithm is self-normalizing and the redistribution of the activity in the image cells that occurs after each iteration, is done without any net increase or decrease in the total activity. As noticed by Lewitt and Muehllehner

[21],

from the practical point

of view, the properties of non-negativity and self-normalization are more important than the fact the EM algorithm proceeds in the direction of global maximum in the likelihood function. Another important feature of the EM reconstruction algorithm is that it is based on the exact stochastic model of the projection measurements, unlike the convolution - backprojection or Fourier techniques, that are deterministic and ignore completely the stochastic nature of the data [14,16]. The EM algorithm for emission tomography can provide a physically accurate reconstruction model, since it allows the direct incorporation to it of many physical factors, which, if not accounted for, can introduce errors in the final reconstruction. These factors can be included in the transition matrix a(i,j) [16] and they can be: a) Attenuation correction information can be incorporated in a(i,j) instead of correcting the projection data before the reconstruction [22]. b) Scatter and accidental coincidence corrections. c) Positron range and angulation effects. d) Time-of-flight information [23,24]. e) Information on the time coincidence resolving window for each detector tube and also information on the nature of the specific radionuclide used (with known radioactive decay). f) Normalization for redundant projection sampling (therefore one can correct for oversampling) and detector pair efficiency (since in practice the detection efficiency is not the same for all detector pairs and can be in the range of 80% for most of the radionuclides [10]).

12

g) Variation in spatial resolution (although it is most significant in SPET rather than in PET). A major disadvantage of the EM reconstruction algorithms, as it happens with most of the algebraic and iterative techniques, is the slow convergence rate to an acceptable image and the high computational cost for a practical implementation. However, the use of EM algorithms has been found to produce superior results in comparison with the convolution - backprojection methods

[25,26]

by reducing noise and streak artifacts (better signal-to-noise ratio)

and reducing the expected error in the estimation of radioisotope uptake, especially in low activity areas

[27].

Recent studies

[28,29]

verified the superiority of the EM-MLE techniques in

terms of noise properties and lesion detectability, especially after detailed and systematic studies of the image frequency spectra, bias and variance

[30],

which set several criteria and

indicators as to what can be considered „superior“ in maximum likelihood and other statistically based image reconstruction techniques as opposed to filtered CBP schemes. In addition, the projection data are not required to be equally spaced for EM-MLE reconstructions, and an incomplete set of projection data can still be utilized

[31].

Especially in the case of

SPET, the EM algorithm was found to provide better image quantification

[32]

and better defi-

nition of the object boundaries than filtered-backprojection (FBP) techniques. Monte Carlo preliminary studies of the EM-MLE reconstruction method for SPECT have been presented by Floyd et al. [33]. On the other hand, there are still studies

[34]

claiming that maximum likelihood techniques do

not present significant improvement in overall imaging performance compared to the filtered backprojection. Others

[35]

claim that with adequate post-processing of FBP images the arti-

facts associated with this reconstruction technique could be reduced and improve the image quality to levels very close to the image quality offered by the EM algorithm by using simpler computational means. The above remarks can be reasoned in a way, since it is true that EM techniques are highly dependent on the image structure and the activity levels, and also it is common to go over several tens of iterations to start getting quantitative improvements. In addition, the images produced in emission tomography with the EM algorithm have been observed to become more noisy and to have large distortions near edges (edge artifact

[36])

and the images converge towards the maximum-likelihood estimate

as iterations proceed

[37].

It seems that a high

likelihood value does not necessarily mean better image quality and the source of the noise artifact using unconstrained maximum likelihood estimation (as in Eq. (10)) lies in the properties of Poisson processes.

13

Implementations on parallel and dedicated systems Methods to overcome the problem of the high computational demand of the EM algorithm have been proposed, using parallel machines or specialized hardware design. Therefore, images can be produced with the use of specially designed hardware that exploits the structure of this algorithm to realize a high degree of parallelism and pipelining cated VLSI architecture has also been proposed

[39]

[38],

and a dedi-

towards this direction. It is true that the

EM reconstruction algorithm can be easily parallelized, since the update of each pixel in one iteration does not depend on the updated values of the other elements of the image vector. For a vector machine, the following matrix-form expression for one iteration of the ML-EM algorithm can be used [31,40,41]:

y = A Τ ⋅ x (k ) 1. Set ~ 2. Set z ( j ) = y ( j ) ~ y ( j ) , for j=1,2,...,J 3. Set C = A ⋅z 4. Set x ( k +1) (i ) = x ( k ) (i ) ⋅ C ( k ) (i ) , for i=1,2,...,I where A is the probability matrix, with elements a(i,j). In a parallel implementation of the EM algorithm, data parallelism schemes can be used and are of three types

[31]:

The partition-by-box scheme, in which one processing unit (PE) is as-

signed all the computation associated with a box (pixel or voxel) from the image vector in both forward (step 1.) and backward (step 3.) projection steps. In the partition-by-tube scheme one PE is assigned all tasks and data associated with one tube from the data vector in both steps, and the third scheme is the partition-by-tube-and-box, where the partition-bytube is used for step 1. and the partition-by-box is used for step 3. Ideas on parallel implementation of the EM reconstruction technique have been proposed as early as 1985

[42]

and preliminary results on the application of the EM algorithm on parallel

machines, especially in attempts to perform 3-D image reconstruction in PET, have been published since the late ’80s

[43-45].

Parallel algorithms have been tested on systems such as

a HyperCube multiprocessor Intel iPSC/2 message passing system GP1000 [31], on a set of connected INMOS T800 Transputers computers

[49]

[47,48],

[46]

and a BBN Butterfly

mesh-connected parallel

and others like Cray XMP24, Mercury ZIP, NCUBE, Multiple TMS320C30

DSP’s and NCR GAPP

[43].

A recent parallel implementation of the EM algorithm on a linear

array using 8 DSP chips as processing units has been developed

[50,51].

Steps 2. and 4. of

the above scheme are ideal for SIMD (Single-Instruction Multiple-Data) machines and 2.5 sec per iteration have been reported for a 2-D

[52]

and an extended to a 3-D implementation 14

[53]

for a 16,000 processor MasPar/DECCmpp-Sx machine and for a 64×64×64 image grid.

Similar performance has been reported

[54]

using different vector and parallel programming

techniques, on i860 accelerator boards for a MicroVax 3200. The main disadvantage of the above implementations is that parallel programming for these dedicated architectures is highly platform-dependent. Recently, the idea of distributed processing on a cluster of workstations seems to become more and more popular. Sharing the workload between 8 and 16 CM-5 processors and (in separate experiment) the same number of SPARC5 Workstations

[55],

as well as between 8 SPARCstation 10/40 class CPUs

[56]

has

been reported to significantly reduce the reconstruction time for dynamic studies. A parallel implementation of the algorithm on a heterogeneous workstation cluster has been recently presented [57], where the computational load is distributed to the workstations available within the clinical environment. Instead of occupying the CPUs of all the computing systems available and developing interfaces for the distribution of the computational load to the different machines or developing software for highly specialized parallel architectures, a much simpler implementation of the EM algorithm has been presented by Kontaxakis, et al.

[58].

The proposed computing system

is a standard low-cost Pentium PC, running Windows NT and with adequate RAM, which would be dedicated for the iterative reconstruction of PET data. The ability of the Windows NT to address more than one Pentium processors makes this idea even more appealing, since a multiprocessor Pentium platform allows a certain degree of parallelism for the EM reconstruction, without significant increase in cost or major modification of the basic implementation. The reconstruction module on the PC server accepts Java scripts with the reconstruction parameters and automatically initiates the reconstruction of the data, under the implementation scheme presented in the following sections. The completed module performs all corrections to the raw data on the PC and includes routines for the quantification and parametrization of the reconstructed images.

Image quality criteria and stopping rules In order to test an implementation of the EM Algorithm for convergence, the performance of

$

the method is monitored using several criteria related to the quality of the reconstructed images

[59].

In the case of simulation studies, where the activity density x in the source is

known in advance, the RMS (Root Mean Square) error can be used, that expresses a normalized measure of agreement between the estimated and actual image vectors at the kth iteration:

15

I

RMS( x ) =

$ ∑$

∑(x(i) − x(i)) i =1

2

(15)

I

2

(x(i))

i =1

In the case when the EM reconstruction is performed using real data, the RMS criterion cannot be used. In that case, a (deterministic) measure of agreement between the estimated image and the observed data can be used the following residual norm can be used [21]:

J

2

RN ( x) = ∑ (y ( j ) − ~ y ( j))

(16)

j =1

y(j) is given in Eq.(5). Another criterion (probwhere the expression for the reprojection data ~ abilistic) is the likelihood function of Eq. (6) that the EM algorithm is designed to maximize (which is the probability of observing the measured data vector y given the image vector x) or, since it is more convenient, the natural logarithm of the likelihood function, l(x), Eq. (7), which can be written as: J

J

j =1

j =1

~ l( x ) = ∑ [ y(j)log(y(j)) - ~y(j)] − ∑ log(y(j)!)

(17)

Starting with a uniform initial guess x(0) for the image vector, the RMS shows a rapid decrease during the first iterations of the EM algorithm, while elements of the image vector are getting updated towards an estimate with high likelihood. After this rapid convergence phase, the RMS error reaches an optimum point and then starts increasing. This indicates that statistical noise is added to the reconstructed image and other effects (like the edge enhancement artifact) appear to the image pattern, such that after that optimum point, the image quality expressed with the RMS error criterion is deteriorating [60]. This effect can not be seen using the residual norm criterion or by monitoring the log-likelihood function during the iterative process, which monotonically increases towards the maximum likelihood estimate. The fact that the RN(x) function is decreasing as the iterations proceed, is expected since the

y = y , consequently it is exmaximum of the likelihood function of Eq. (6) will occur when ~ y and y will pected that at each iteration the Euclidean distance between the two vectors ~ decrease. However, using the above criteria, it is not possible to tell when the EM iterations have to be stopped during a reconstruction from a real PET scan.

16

Recent developments and modifications of the EM algorithm As already mentioned, the main results from the on-going research on the MLE methods for image reconstruction in emission tomography during the last decade, are that

[61]

(i) the EM

algorithm converges slowly, with a typical number of 30 to 60 iterations needed to get a good image quality and (ii) if one keeps iterating for more likelihood after a certain optimum point the resulting images become noisy: a „checkerboard“ effect

[62]

becomes visible in the image

pattern, along with an edge-enhancement artifact, which make image interpretation practically impossible. However, there is some discussion

[63]

that deteriorated images after many

EM iterations may still provide useful clinical information.

Modified MLE algorithms for accelerated convergence The problem of the slow convergence rate of the EM image reconstruction technique has been addressed by several authors. Accelerated versions of the EM algorithm have been proposed to overcome this disadvantage. Acceleration here means that one tries to reach the maximum of the likelihood function of Eq. (6) faster than with the conventional EM algorithm of Eq. (10). Certainly, the goal of acceleration techniques is to reduce the computational cost and not to change the convergence characteristics of the method

[61],

however

there is no guarantee that with accelerated techniques better images are obtained or if the same or a better image quality can be reached faster. Eq. (12) can be written in additive form [21], as

x (k +1) (i) = x (k) (i) + ∆x (k) (i) ,

i=1,2,...,I, k=1,2,...

(18)

where ∆x can be found from Eq. (11). Then an overrelaxation parameter λ is introduced whose purpose is to accelerate the iterative process:

x (k +1) (i) = x (k) (i) + λ∆x (k) (i),

i=1,2,...,I, k=1,2,...

After careful choice of the overrelaxation parameter it has been found

[21]

(19) that the acceler-

ated algorithm in the additive form can significantly improve the convergence rate of the EM algorithm, preserving the non-negativity and sum-of-counts properties described earlier. However, the convergence of the method and its performance as an ML estimator are under investigation. Kaufman

[41]

investigated the above idea in more detail and proposed several modified tech-

niques to compute the correction factor λ: LINU, LINB and a Conjugate Gradient (CG) approach. However, the non-negativity is not preserved because of the large step sizes and 17

negative pixels have to be reset to small positive values in order to avoid instability and meaningless results. Later Kaufman

[64]

presented a preconditioned conjugate gradient

(PCG) approach, based on a scaled steepest descent methodology and taking care of the non-negativity property, and also a penalized PCG approach, which introduces an additional smoothing term, as it will be discussed in the next section. Metz

[65]

proposed a similar line search technique, where in order to preserve the non-

negativity of the elements in the image vector the step size is bounded at each iteration (bounded line search). These methods were classified by Lange as EMS (Expectation Maximization Search) methods, since they try in heuristic terms to find a one-dimensional direction for the EM step and obtain a new estimate in that direction. This can be seen if Eq. (10) is written as follows:

  J   y(j)a(i, j) x (k +1) (i) = x (k) (i) + x (k) (i) ∑ I − 1  j =1 ∑ x (k) (i′ )a(i′ , j)    i ′=1

(20)

which with the use of Eq. (8) can be written as:

x (k +1) (i) = x (k) (i) + x (k) (i)

∂ l( x(k) ) ∂x(i)

(21)

and in vector notation:

x(k +1) = x(k) + X(k)dl( x(k) )

(22)

where dl( x(k) ) is the gradient of the log-likelihood function (Eq. (7)) and X(k) is a diagonal matrix with the ith element equal to x(k)(i). The overrelaxation multiplier λ of Eq. (19) can be viewed as a direction multiplier for an one-dimensional search towards increasing loglikelihood and places these methods in a class of rescaled gradient algorithms value of λ for which the log-likelihood function l( x Newton-Raphson search algorithm

[20].

(k+1)

[18,20].

The

) is maximum, can be calculated by the

Rajeevan et al.

[66]

recently presented a different

method of acceleration by vector extrapolation of a set of preliminary parameters, which were obtained by EM and EM-search techniques. All these techniques tend to converge faster than the EM algorithm in terms of likelihood, provided that proper selection of the new parameters that are introduced is made. However,

18

the resulting images in terms of image quality presented a higher noise component after the first iterations than the images produced by the EM reconstruction technique [41]. According to a different principle, another approach for acceleration has been presented by Kaufman

[41],

the Grid Refinement method: here the reconstruction starts on a coarse grid

(e.g., on a 16x16 coarse grid for a 64x64 final grid) and after the first iterative steps each element of the grid is divided by 4. This technique reduces the computational cost by a factor of 4 during the first iterations. The same idea led Ranganath et al. [67] to develop the Multigrid EM (MGEM) algorithm for image reconstruction, according to the principle that the lowfrequency image components can be recovered at the first steps using a coarser grid and then the fine details will be recovered later using the final image grid after few more iterations. However, the drawbacks of this method are the need to calculate and use different transition matrices a(i,j) for each grid and the fact that during the recovery of the fine details of the image, the low-frequency components will start entering the deteriorative phase. In addition, the low-frequency pattern will be used to initialize the finer-grid reconstruction and it has been shown

[41,68]

that unless a uniform initial guess for x(0) is used, there is pattern-

dependency in the images produced by the EM method, which will be discussed later in more detail and contributes to the degraded quality of the images produced after a certain number of iterative steps. Hebert et al.

[69]

proposed a different idea of using a polar pixel

representation of the activity distribution in the source during the forward (Eq. (5)) and backward (Eq. (11)) projection operations. In this case, the circular symmetry of the PET scanner and the pixel representation can be exploited in order to reduce the computational cost, however additional calculations are needed to use rectangular-to-polar representations and there are arguments

[41]

[69]

against the use of pixels that do not correspond to equal areas

in the source, as in the case of a polar pixel representation. Tanaka

[70]

proposed a modified approach of the additive scheme of Eq. (19), where the

overrelaxation parameter λ is now a damping matrix λ(i), calculated based on the precision of the correction for each pixel. This Intelligent Iterative Reconstruction (IIR) method has been proposed not to accelerate the convergence of the EM technique (on the contrary, it has been shown that it slows down the convergence speed), but to suppress the noisy correction which leads to the image deterioration. Another reconstruction technique proposed by Tanaka

[71]

is based on a modified EM algo-

rithm, where the correction factor C(i) is amplified by an exponential constant before the pixel update, in order to accelerate the convergence of the algorithm, and is referred to as Filtered Iterative Reconstruction (FIR) algorithm, since the exponentially amplified correction factor is combined with a smoothing backprojector factor in order to avoid overcorrection and 19

enhance the correction for high-frequency components. This is reasoned by the fact that in the EM algorithm low spatial frequencies converge faster than higher ones

[72].

An advantage

of this method, apart from the convergence acceleration, is that it prevents excess enhancement of the high-frequency component when the inevitable appearance of statistical noise in the EM algorithm starts deteriorating the image quality. However, at each step of the FIR method, a backprojection, 2-D smoothing and renormalization (in order to preserve the total number of counts) steps have to be performed which reduce the computational simplicity of the EM algorithm and reduce the acceleration gain obtained by the enhanced correction. In addition, the FIR method shows instability for pixel values close to zero, since it involves divisions by x(i) and further modifications should be introduced. Another method proposed by Ollinger

[73]

is the Iterative Reconstruction-Reprojection (IRR) algorithm, which is

developed as a relaxed version of the EM algorithm, suitable for use with noisy data sets and with data with gaps and missing projections. It was also initially shown to converge faster to a maximum-likelihood estimate than the conventional EM algorithm. Based on the idea of the exponentiation form of acceleration, Llacer et al.

[74]

proposed a

Successive Substitution (SS) method, which proceeds at the kth step according to the following scheme:

x (k +1) (i) = x (k) (i)( C (k) (i)) , n

i=1,2,...,I

(23)

where 1 ≤ n ≤ 2 for real data. In the above expression, a correction factor for randomcoincidence correction has been included in the correction factor C. This technique was shown to accelerate the convergence of the reconstruction over the conventional EM technique with a speed-up factor proportional to n. The method was applied to real and simulated data and was shown to produce results very similar to the ones produced by the EM algorithm. An additional parameter µ though has to be used as a damping factor in the initial iterations, in order to prevent some instabilities of the SS solution. However, a proof of the convergence of the SS-MLE technique is not yet available and a theoretical validation of the method is needed in order to claim universal applicability of the SS-MLE technique. Techniques applying sequential updating schemes show accelerated performance, since the image is updated in steps involving only parts of the pixels or the elements of the data vector. The space-alternating generalized expectation-maximization (SAGE) algorithm

[75]

uses

the backprojection of the image estimate expressed by Eq. (5) to calculate the correction factors for the update of one or a selected subgroup of image pixels during one iteration. In that way, an acceleration over the ML-EM is achieved using statistical considerations and therefore monotonic increases in the objective function of Eq. (6) are guaranteed.

20

Recently, Hudson and Larkin

[76]

presented a method to accelerate the EM algorithm using

ordered subsets (OS-EM) of the projection data. In that technique, the vector of measured data y is divided in subsets, either non-overlapping or cumulative subsets. At each iteration of the EM update (10), the time-consuming reprojection of Eq. (5) and the summation in the updating coefficient C(i) in Eq. (13), are calculated only for the detector tubes that are included in the selected subset. Then, the backprojection scheme of the pixel update is performed using these correcting factors. This method creates a new image estimate at a fraction of the time required by the conventional EM procedure, where all pixels are updated using the whole set of projection data. The authors of

[76]

claim that the OS-EM method can

produce acceleration (in terms of image quality expressed as a root mean square error) of the order of the number of subsets used. However, this method does not converge to a maximum likelihood solution, except for the case of noise-free data. The OS-EM can be also applied with ordered subsets of increasing population until the whole data vector represents a single subset and consequently coincides at that point with the standard EM reconstruction procedure, with the advantage that the computation time is greatly reduced.

Bayesian and other approaches to the image deterioration problem Acceleration methods for the MLE image reconstruction techniques in emission tomography can provide faster convergence towards high-likelihood estimates, however do not guarantee a better image quality than the standard EM algorithm. Various proposals have been presented on modified MLE techniques, which incorporate a priori information to characterize the source distribution and the data noise. There is also some discussion

[77]

on the en-

hancement of the convergence rate of the reconstruction with the use of Bayesian priors. The fundamental idea of the Bayesian approaches in image reconstruction is the following [78]:

In the derivation of the EM algorithm, one tries to maximize the probability:

P(coincidence data | image vector) (or P(y|x)) to observe the measured data given a current estimate of the activity distribution in the source. According to Bayes’ theorem, one can write:

P(image|data) =

P(data|image) P(image) P(data)

(24)

In the above equation is expressed the conditional probability that the image is true given the set of measured data in terms of the usually computed P(data|image), a normalization constant P(data) and the prior probability P(image). Now, the most probable image, given the set of measured data can be obtained by maximizing the right side of the above equation, 21

called a posteriori probability distribution of the image vector

[79].

Fundamentally, this ap-

proach makes use of some „reasonable“ a priori information in order to prevent the image deterioration that occurs when maximizing P(data|image) alone in an unconstrained reconstruction, as in the EM algorithm. This a priori information is incorporated into the P(image) term of the above expression and represents an a priori estimation of how the resulting image is expected to be

[80,81].

The Bayesian reconstruction methods allow the incorporation of

prior information (such as smoothness constraints or partial specified topological information) and therefore further reduce the noise sensitivity. As an example, Geman and McClure

[82,83]

proposed the use of Gibbs priors which penalize

large deviations between the estimates of the image vector for neighboring pixels, except for edges (where the concept of „line sites“ was introduced

[84]

as indicator of neighbor pixels

belonging to the same region or not). With this approach a noisy image pattern is smoothed to produce more appealing images, since the expected visual outcome of a PET scan is expected to be a smooth image with no large variations of the isotope density. Other authors proposed the use of Gaussian priors [63],

Poisson and Gamma priors [85], and the Good’s Roughness prior [43] in order to model the

property of local continuity of the images. The Gibbs priors involve an energy function

  U ( x ) = exp − β ∑ wn,mV(x(n) − x(m))  [ n ,m ]∈Ω 

(25)

where β is a positive constant which controls the degree of smoothing introduced by the prior, wn,m is a weighing positive coefficient and Ω is a set of pixels, with n and m defining a neighborhood. The important feature in the above expression is the potential function V(x), usually assumed to be even, continuously differentiable and strictly convex

[86].

In

[77]

a list of

recommended potential functions is given, along with their properties. The log-energy function now u(x) = log(V(x)) is added to the log-likelihood function l(x) as a penalty function which expresses any prior knowledge on the smoothness or other property of the estimate x and produces a log-posterior function l(x)+ u(x) [62]. The function l(x)+ u(x) represents now the likelihood of the a posteriori probability distribution of Eq. (24). Taking the logarithms in Eq. (24) and keeping the notation x for the image vector and y for the data vector:

22

logP( x| y ) = logP( y| x ) + logP( x ) = l( x ) + u( x )

(26)

The first term of the right side in the above equation is the log-likelihood function of Eq. (7) and u(x) is a new term given by Eq. (25):

u( x ) = − β

∑w

[ n ,m ] ∈Ω

V( x )

(27)

n,m

Maximization of the (26) by taking partials with respect to each x(i) and setting the result equal to zero (as the standard EM technique of Eq. (10) was obtained), represents the basis of several regularization techniques proposed to solve the problem of the image deterioration in the EM algorithm by assuming a certain degree of „smoothness“ of the resulting images as an a priori information. As an example, it is shown by Green

[87]

that if one substitutes ( ∂u / ∂x(i) )( x ) with

( ∂u / ∂x(i) )( x(k) ) of the current estimates during each iterative step, an iterative pixel updating scheme can be produced, which, in analogy to the EMS notation of Eq. (20), can be expressed as:

x (k+1) (i) = x (k) (i) +

x (k) (i) ∂ l( x(k) ) + u( x(k) )] ∂u (k) ∂x(i) [ 1− (x ) ∂x(i)

which represents the One-Step-Late (OSL) algorithm proposed by Green Lange

[77].

[87]

(28)

and modified by

Some restrictions are applied to the partial derivatives of u(x) in order to avoid

possible negative values in the denominator. By selecting β=0 in Eq. (25) one returns to the standard EM algorithm. Hebert and Leahy

[44]

presented a generalized EM (GEM) algorithm

for a 3-D Bayesian reconstruction using similar Gibbs priors. A method for choosing the parameters for the Gibbs prior is given by Johnson et al.

[88]

along with a procedure to define

neighborhood systems in order to optimize the image deblurring. Several other techniques using the penalized maximum likelihood approach derived by maximizing the a posteriori probability distribution of Eq. (26) have been reported: the Maximum A Posteriori Probability EM algorithm (MAP-EM) proposed by Levitan and Herman

[62],

where a multivariate Gaussian a priori probability distribution is assumed for the image vector; the Fast Maximum A Posteriori with Entropy (FMAPE) algorithm by Nunez and Llacer

[78]

which uses an entropy prior with an adjustable „contrast parameter“ and the SS exponentiation method described earlier; the Bayesian Image Processing (BIP) method by Liang and Hart

[89]

and the Entropy Image Processing method by Liang

[90],

which make use of entropy 23

analysis on the strength correlations of the image pixels; the MLE-PF (PF for postfiltering) introduced by Llacer et al.

[74],

where the MLE procedure is first let iterate for an undeter-

mined number of steps and then the image is postfiltered with an adequate two-dimensional Gaussian kernel in order to suppress noise; the modification of the EM algorithm recently presented by de Pierro

[91]

with proof of convergence for general concave penalizations. In

these methods, one deals with a relaxed EM procedure, often referred to as a generalized EM (GEM) algorithm, since no closed form expression can be derived in principle for the maximization step. A sensor fusion method proposed by Chen et al.

[92]

incorporates prior information from cor-

related CT and MR images to the PET reconstruction. This method was further developed by Ouyang et al.

[93],

where spatially correlated images from high-resolution modalities were

used as priors in order to get higher quality multimodal Bayesian images from PET scans of the same subjects. The Iterative Conditional Average (ICA) method was used to perform the Bayesian reconstructions, a method proposed by Chen et al.

[46]

and applies Gaussian pri-

ors. A similar MAP Bayesian technique that uses registered anatomical MR images as prior information has been presented by Gindi et al.

[94].

Several difficulties are associated with

this kind of approaches, such as alignment and registration, boundary estimation, inherent differences of the imaging modalities used, etc. Other regularization approaches have been presented, which combine several of the abovementioned techniques, such as the MAP-PCG method proposed by Mumcuo ÷lu et al.

[95],

where a conjugate gradient algorithm is implemented to calculate maximum a posteriori estimates of the source distribution, with additional penalty functions (Markov random field priors) to ensure non-negativity and diagonal forms of preconditioners to enhance the convergence rate. Use of linear filters has also been reported

[96],

like the Shepp-Logan filter, the

Butterworth filter, the Gaussian filter, the Hann filter, the Parzen filter and the Lagrange filter (see refs. in

[96])

in order to remove the noise and edge artifacts associated with the MLE

approach. One of the most popular edge and noise-artifact suppression technique is the use of sieves [97,98].

The idea in this approach is to set some constraints in the selection of the possible

estimates x of the image distribution, to be chosen from a subset of all possible non-negative realizations, called the sieve. As an alternative, it is possible to obtain instead of an estimate of the image vector, an estimate of its filtered version, by using a resolution kernel. Snyder et

al. [37] actually proposed the use of both techniques (sieves and resolution kernel) in order to obtain optimum results in avoiding the image quality deterioration and the edge-overshoot. The method of sieves also uses the concept of the a priori belief of smoothness of the image estimate, however it is offering a rather broad selection of reconstruction methods because 24

of the arbitrary degree of smoothness

[99]

which could be selected to impose on the choice of

the sieve kernel. A recently presented Bayesian method

[100]

based on the One-Step-Late (OSL) type algo-

rithms, makes use of a median root prior (MRP) for the prior function P(x). According to this technique an additional weighting multiplicative factor is added to the pixel update scheme:

x

( k +1)

(i ) = M ( k ) (i ) ⋅ C ( k ) (i ) ⋅ x ( k ) (i )

(29)

where:

M (i ) =

1 x (i ) − med ( x, i ) 1+ β med ( x, i )

(30)

med(x,i) is the median of image pixels over a neighborhood at pixel i. The bayesian term

β represents the prior weight and ranges from 0 to 1. The MRP method does not preserve the total number of measured counts in the image vector, in contrary to the EM algorithm and does not converge to a maximum likelihood estimate. However it shows efficient performance in removing the noisy patterns without blurring the locally monotonous structures In all the above techniques, convergence is difficult (if not impossible) to be proven, however they are accepted since they were shown to produce accurate images under simulation experiments and during real data reconstructions. In anycase, each method introduces its set of parameters (such as the β term of the energy function, the penalty weight parameter γ of the MAP-EM algorithm, etc.), which seem to be of crucial importance in the resulting images and have to be selected with care. Often this selection is done empirically and is highly taskdependent, and for that reason someone might question the universal applicability of these methods in the context of image reconstruction for emission tomography. However, all the above techniques prove the flexibility of the EM method to adjust to a specific medical problem involving PET and should always be evaluated on a task-dependent basis

[63].

Image feasibility and stopping rules In the techniques described in the previous sections, the problems of slow convergence and image deterioration in the EM reconstruction were discussed and several techniques that have been proposed to remedy these situations were presented. The regularization methods can be used to prevent the statistical noise from the unconstrained MLE reconstruction from 25

breaking down the image quality, however they do not, in general, produce images better than the ones produced by the standard EM approach at a certain (optimum) point [62]. Since the EM algorithm is an iterative method, stopping the algorithm at a certain point for which the image quality is „good enough“ or (ideally) the optimum that one can obtain, could also be a solution to the problem of image deterioration. As an example, the MAP-EM method proposed by Levitan and Herman

[62]

is shown to produce the same image quality in

terms of RMS error (during phantom experiments) as the one obtained at the optimum of the standard EM reconstruction, with the difference being that the MAP-EM keeps producing similar images at every following iteration. Consequently, an adequate stopping criterion of the EM reconstruction would be just as good of a solution as a modified (and more implementation-wise complicated and task-dependent) regularization approach. The main difficulty to that problem is that in every MLE technique the likelihood function is monotonically increasing after each iteration and does not provide any direct information in the quality of the reconstructed images in terms of noise and edge artifacts. Therefore, the asymptotic behavior of the EM algorithm is not of practical interest and the iterative process has to be stopped at a point when the result of the reconstruction is satisfying enough. The issue of stopping the EM algorithm has been studied extensively in the literature, and quantitative criteria (stopping rules) have existed since 1987

[69,101]

mostly based on statistical

hypothesis testing methods. The feasibility criterion proposed by Llacer and Veklerov

[102]

is

based on the following remark:

„The closer ~ y ( j ) becomes to y ( j ) , the higher the probability that the image generates the projection data. However, if the two get too close for all or almost all projections j, it becomes statistically unlikely that the image could have generated the data. The chance that a source distribution emits a number of γ-rays very close to its mean for all directions where detectors are placed is very small. For instance, the Poisson distribution calls for a standard deviation of y ( j ) from ~ y ( j ) , which is specifically the square root of ~ y ( j ) ’’ [101,103]. Taking the above into account, the following definition for feasible images has been proposed [102]: The image x(1),x(2),...,x(I) is said to be a feasible image with respect to the projection data

y(1), y(2), ..., y(J) if and only if the second moments of y(1), y(2), ..., y(J) are consistent with the Poisson hypothesis, namely:

26

( y( j) − ~ y ( j )) 2 ≈J ∑ ~ y ( j) j =1 J

(31)

And this, since the expected value of the numerator of each term in Eq. (31) is the variance, while that of the denominator is the mean, and for that reason Eq. (31) must be satisfied if the Poisson hypothesis holds. The EM algorithm then has to be stopped when Eq. (31) is satisfied. In the literature, the above formula is used to monitor the progress of the reconstruction as an image quality criterion

[104]

(the residual norm defined in Eq. (16) is included

to that criterion) and as a stopping rule [99,105]. It is usually referred as the χ2/J statistics criterion, where:

1 J ( y( j) − ~ y ( j )) 2 χ /J= ∑ ~ J j =1 y ( j) 2

(32)

with y ( j ) and ~ y ( j ) being the observed and calculated projections and J is the total number of y ( j ) (> 0). The above calls for a

χ2 estimation and, as discussed in [78,106], the upper limit

for χ2 at 99% confidence level should be set at (J + 3.29 J1/2)[107]. A comparison of the FMAPE method with post-filtering, the use of sieves, the application of the feasibility criterion and the post-filtered EM-MLE method is presented by Llacer et al. in [108],

where the insufficiency of the feasibility criterion to produce acceptable images was first

pointed out: the set of feasible images is found to be rather large

[68]

and can even include

noisy and unacceptable images. The feasibility criterion has been found to provide weak results with real data from a PET scanner

[74].

Even in studies with simulation-generated

projection data, the feasibility criterion is highly dependent on the number of counts in the source, apparently to a higher degree than the RMS error figure-of-merit for the image quality, and often under certain circumstances, the feasibility region is never entered during the first hundreds of EM iterations. According to this, the feasibility method does not represent a sufficiently robust criterion to provide us with a stopping rule for the iterative EM reconstruction [74]. Coakley [109] presented a method of stopping the EM-MLE based on the statistical concept of cross-validation. With this technique, the projection data are split in two halves using a thinning technique: either by splitting in two the acquisition time during a PET scan or by randomly assigning each count to one of the two data sets. The EM-MLE is used to reconstruct the first data set and after each iteration, the likelihood of the second data set is calculated with respect to the image estimate produced from the first data set. Initially this likelihood 27

increases, but at some point is peaks and then starts decreasing. At that point the reconstruction of the first data set is halted and the roles of the data sets are reversed: the second data set is reconstructed with the EM-MLE, the likelihood of the first data set is calculated with respect to the image estimates from the second set and iterations are stopped when this likelihood starts decreasing. Then the images produced at the two maxima are added and presented as the final result. The cross-validation method has the advantage of producing a very well defined stopping point, at the maximum of the cross-likelihood, and the addition of the two images contributes to an averaging of the deteriorative noise accumulated into each of the image estimates. An MLE-PF technique can be used at the resulting image, in order to smooth it further

[110].

This

technique has been found to produce good results even with real PET data and can be understood even from the feasibility point of view, since one could think of the cross-likelihood maxima as the points at which the image estimates have acquired all common features and further iterations will accentuate the features related to statistical noise, a feature specific to each data set

[74].

An extension of the method with the use on N data sets has been also

presented [110]. A disadvantage of the cross-validation technique is that it increases the computation time by a factor N, since the EM-MLE has to be performed N times, one for each data set, and there is no evidence that a better image quality than the one at the optimum point (minimum RMS error) of the conventional EM-MLE will be obtained. However, it has been found

[111]

that the stopping point produced with the above method highly depends on

N, for the same data set. This is related to the fact that [112] for a larger number of counts the optimum reconstructed image occurs later in the iterative procedure. Another deficiency of this method is exactly that it is shown to ignore that dependency between the optimum iteration as expressed by the maximum of the cross-validation log likelihood function and the total number of measured coincidence counts [111]. As a remedy to that situation Johnson proposed a modified cross-validation procedure, where a jackknife subsample of the N data sets, in which the total number of counts are divided, is set aside and the MLE-EM reconstruction is performed on a data set which is the sum of the rest N-1 subsets. Then the cross-validation likelihood function is calculated with respect to the jackknife subset (after proper normalization of the image estimates). For N large enough, the jackknife subset will contain only a fraction of the total counts and the reconstruction on the remaining data subsets will be very close to the one of the complete data set. All the proposed stopping methods (feasibility method, cross-validation approach, jackknife techniques, etc.) try to locate the optimally reconstructed and best acceptable image and halt the iterations at that point. It is assumed that the information about the onset of the image 28

deterioration is contained within the EM-MLE reconstruction procedure without the use of any additional constraints or parameters. For the feasibility method, the concept of feasible images had to be introduced, based on the theoretical background of statistical hypothesis testing

[105].

The application of this criterion was based on the fundamental assumption that

in any data set reconstructed with the EM algorithm, a feasible set of images will always be reached, which was later shown to be insufficient for a general application. The crossvalidation approaches are consistent with the feasibility requirements, however it has not yet been proven that the maximum of the cross-validation log-likelihood function provides us with the optimally reconstructed image vector. The development of such a method was initiated by the discovery of the peak in the cross-validation log-likelihood and at a later step a correlation between that point and the best EM-produced image vector has been attempted to be empirically validated. In

[112]

a different approach in the development of a stopping criterion for the iterative EM

algorithm for image reconstruction in PET is presented, which provides with a robust stopping rule for this iterative technique. Through a series of studies using Monte Carlo techniques, it was first identified that the Root Mean Square (RMS) error of the reconstruction shows an initial rapid decreasing phase versus iterations and reaches a minimum just before the excess statistical noise starts deteriorating the reconstructed image quality. This observation has been carefully investigated and was found to be true in all the experimental setups and under all assumptions: the RMS error, in reconstructions where the true activity distribution in the source is known, reaches a minimum at a point that directly corresponds to the most accurately reconstructed image vector. In order to apply this observation in practical cases where the true activity in the source is unknown (and is the quantity one seeks to find via the application of the EM algorithm to the coincidence data vector), one needs to find a quantity which could be calculated in each iteration of a real-data reconstruction, which is somehow in direct relation to the image quality estimator (such as the RMS error) and which can provide a robust stopping criterion for the EM algorithm. This quantity has been searched

[60,113]

in the properties of the updating coef-

ficients in the vector C of the multiplicative correction factors applied to the image vector during each iteration in the EM reconstruction as shown in Eq. (10). A detailed study of the convergence properties of the coefficients C revealed that for a noise-free case (no scattered, random or multiple coincidences) and for an ideal PET scanner model, the minimum value Cmin and other statistical parameters of this vector always reach the same value at the point at which the RMS error reaches its minimum. This value was always reached at that point, independently of the activity density in the source or the image pattern, the image shape and the system configuration. At a subsequent step, a realistic model of a PET scan29

ner and data acquisition was developed as an extention of the previous ideal model and the method has been applied to the EM reconstruction procedure for this case

[114].

The stopping

criterion was shown to apply very well even in that case where all major noise sources have been turned on.

30

Fig. 2: One slice (zoom factor: 1.5, 128x128 image matrix) from a study of a patient with a right-leg necrotic sarcoma. Left: FBP image (0.5 cutoff frequency, ramp filter). Center: OSEM image (4 subsets, 3 iterations). Right: Error-image (quantitative subtraction of the FBP image from the OSEM image).

Examples In cases of low-count short-interval dynamic acquisitions, iterative image reconstruction is clearly advantageous in comparison to filtered backprojection. The strong star-like artifacts produced by FBP are eliminated with the use of the MLEM method and the delineation of the body and other region-of-interest contours is more clear. The example of Fig. 2 shows a slice of a necrotic soft-tissue sarcoma of the right leg of a patient. The images are reconstructed using the filtered-backprojection algorithm (Fig. 2, left) provided by Siemens/CTI for the ECAT EXACT HR+ scanner (0.5 cutoff frequency, ramp filter), and the OSEM method (4 subsets, 3 iterations). Due to the low-count statistics in the sinogram data, there are strong reconstruction artifacts in the FBP image. The iteratively reconstructed image on the other hand (Fig. 2, center) provides a much clearer picture on the activity distribution and even the left leg is much better delineated. The error-image (Fig. 2, right) from the quantitative subtraction of the FBP image from the OSEM image shows that the difference between these two reconstructions lies exactly on the regions-of-interest and contains also the artifacts visible in the FBP image. The efficient implementation of iterative algorithms for PET is therefore expected to provide images of higher diagnostic value by reducing the artifacts that often lead to false-positive results [115]. Iterative image reconstruction may also contribute in eliminating false-negative results: Fig. 3 shows two examples from two colorectal carcinoma studies using

18

FDG. The high tracer

concentration in the bladder produces strong streak artifacts which obscure the presence of 31

a residual colorectal carcinoma and also the body contour in the case of FBP reconstruction (Fig. 3, left). The OSEM reconstruction (3 iterations, 4 subsets, MRP β=0.3) on the other hand clearly delineates the malignancy and the body contours, despite the presence of a high-activity bladder nearby (Fig. 3, center). The error-image (Fig. 3, right) confirms these results: it contains both traces of the streak artifacts and the missing body contour of the FPB image.

Fig. 3: One slice (zoom factor: 1.5, 128x128 image matrix) from two studies of residual colorectal carcinomas. Left: FBP image (0.5 cutoff frequency, ramp filter). Center: OSEM image (4 subsets, 3 iterations). Right: Error-image (quantitative subtraction of the FBP image from the OSEM image). Apart from the strong streak artifacts visible in the FBP images, the high-activity concentration of the bladder obscures the presence of the lesion.

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