Fast scaled gradient decomposition methods for maximum likelihood

Proceedings of the 25* Annual Intemational Conference of the IEEE EMBS Cancun, Mexico September 17-21,2003

Fast Scaled Gradient Decomposition Met hods for Maximum Likelihood

Transmission Tomography A. R. De Pierro and M. E. B. Yamagishi Department of Applied Mathematics, State University of Campinas, SP, Brazil. Abstract-

11. THENEW ALGORITHM: T-RAMLA

New iterative algorithms are presented for

Maximum Likelihood (ML) and Regularized Maximum Likelihood (MAP) reconstruction in Transmission Tomog-

We aim at finding a solution of the transmission

raphy (CT). The algorithms are natural extensions to CT

tomography ML problem. As in [6], the ML approach

of RAMLA, a well known method for ML reconstruction

looks for solutions maximizing the function

in Emission Computed Tomography (ECT). We show that the new algorithm for ML solutions produces similar, or

m

L(z) =

even better results than EM-like algorithms, but in much

where

ter than other ordered subsets methods. Keywords-

EM algorithm, OS-EM, RAMLA, transmis-

sion tomography.

-diexp(-(ai,Z))

-yi(ai,z),

(1)

i

fewer iterations. Also, its convergence properties are betdi

is the number of photons emitted along the

- th line (blank scan) and yi the number of detected photons. ai i = 1,...,m are the rows of the projection

i

matrix and z the image vector (n pixels). (,) denotes the standard inner product. It is easy to prove that L is

I. INTRODUCTION

a concave function.

Our contribution in this article is a new algorithm for ML reconstruction in CT, that is fast because it uses ordered subsets as [B], it has guaranteed positivity, and shares the convergence properties of RAMLA [l]. Also,

Consider now the partition of the integer interval

M = [l,2,3,...,m]as a unim of p disjoint subsets

si , i = 1,...,p . Let us now decompose L as P

we adapt the new algorithm to the regularized problem. In section I1 we formulate the problem and present the

main algorithm. Section I11 is devoted to the extension

i=l

where

of the main algorithm to the MAP problem. In section

IV we describe the results of simulations comparing the performance of our algorithms with the EM algorithm and ordered subsets versions. Section V is dedicated to

Calculating the gradient of the functions Li(x) gives

some concluding remarks.

Work by M. E. Beleza Yamagishi was supported by FAPESP grant No 96/09186-2and work by A. R. De Pierm by CNPq grant

Using the previous definitions and denoting the starting

No 301699/81 and FAPESP grant No 02/07153-2.

image as d o )the , new algorithm is defined as follows. k

0-7803-7789-3/03/$17.0002003 IEEE

829

the sequence is defined es

and i are the indices for the iterations and subiterations respectively.

=

XFi+l)

(kd) + X k X ( k ” )

Qljdlexp

(- (a’,x(kii)))-Qljgi

(7)

Ak

lESi

X ( ~ I P )=

aljdr exp

(-

(Q’,

))

x ( ~ ~ ~--aljY1 ~ ) (12)

lESi

where j = 1, ...,n, i = 1, ..,p (pixels and subsets respectively) and

x$?*~-’)

for j = 1,...,n e i = 1, ..,, p.. Then we define

xk+’. That is, the second index

goes through the subsets of equations (projections) and the first one is associated to a complete cycle. A t is a

where D k = diag(x:,

sequence of positive relaxation parameters such that A&

+

k-rw

0;

2A&

= fm.

...,xi). The sequence of relaxation

parameters satisfies condition 8. As for RAMLA, convergence results in [l]are still

(8)

k=O

valid once boundedness is assumed (this is always true

We will call the new algorithm T-RAMLA because

for the parameter’s range in applications).

it is a natural extension of RAMLA (from Row Action IV. ;SIMULATIONS

Maximum Likelihood Algorithm) [l]to transmission to-

We performed several experiments comparing T-

mography. The same convergence results presented in [l] apply in this case, once it is assumed that the sequence

RAMLA against T-EM (the EM algorithm for transmis-

is bounded (the proof of this fact will appear elsewhere).

sion tomography ML), as well as its ordered subsets (OS) version. Our goal is to ,show that our algorithm and its extension, respectively for ML and MAP solutions, is

111. THEBAYESIAN APPROACH: T-BSREM.

faster than the EM, attaining higher values of the objec-

We consider in this section the extension of our

tive function, even if coimpared to the OS version.

algorithm for the computation of Maximum ‘A Posteri-

All the simulations were performed using SNARK93

ori’ (MAP) solutions, the Bayesian approach. Instead

[2], a reconstruction software developed by the Medical

of maximizing the likelihood, we aim at maximizing the

Image Processing Group of the University of Pennsylva-

penalized likelihood function,

nia.

T-RAMLA versus T-EM. It G ( 4 = L ( 4 +yR(4,

is well known

(9)

that the EM for the transmission tomography ML prob-

where L(x) is the likelihood function, R(x) is a func-

lem is very slow [4], and many iterations are necessary

tion containing ‘prior’ information and y a real positive

to obtain high likelihoold levels and image quality. We performed several experiments comparing T-

parameter allowing a trade-off between the ‘prior’ information and the consistency with the data.

RAMLA with Ollinger’r; [lo] version of the EM. We used

Using the same notation of T-RAMLA and following

120 views with 231 rays per view. The number of blocks

[3] we define T-BSREM. Given a positive starting point

of projections (subsets) was 16 for T-RAMLA. The se-

xo and considering that

quence of relaxation parameters of our choice was defined by XO = 1.5 and A& = 1.5/k1I4.This sequence was found after some experimentation; it satisfies the convergence

830

conditions and it tends to zero very slowly, but allows rel-

T-RMILA x OSEM

-mm

atively large steps at the beginning. The mean number of photons was 10000000, but only 10 % went detected.

Figure 1 shows the behavior of the likelihood function for a typical case

t/

-1.47 -l.&

-1.43.

-7mo[

1

2

I

I

I

3

4

5

L

6

I

?

I

I

0

9

10

IW -1.49.

-1.5

Figure 2: Likelihood Function for T-RAMLA and

T-OS-EM (very low statistics).

-1.51

I

A similar behavior was observed for RAMLA, when compared with OS-EM in emission tomography. By further reducing the number of views, the images obtained by the EM fastly deteriorate, but the ones obtained by T-RAMLA are still acceptable. This is crucial in Figure 1: Likelihood Function for T-RAMLA &d

PET or SPECT transmission studies, because it can

T-EM.

decrease considerably the exposure for the transmission scan. When we take one view each 30 degrees, after 10 iterations, T-RAMLA still shows an image in spite of the artifacts, while the EM does not generate any image

T-RAMLA attains very high likelihood levels in the first

at all, and negative d u e s show up. The same behav-

5 iterations, but the EM needs at least 60 iterations for

ior is observed when using the ordered subsets version

the same likelihood values.

instead of the EM.

Several experiments showing the

regularization effect of T-BSREM were also performed.

T-RAMLA versus T-OS-EM We also accelV. CONCLUDING REMARKS

erated the EM using ordered subsets as in [8] and the results were similar to those obtained with T-RAMLA

In this article he have introduced new scaled it-

for the same noise level as before. However, when the

erative decomposition methods for ML reconstruction in

photon statistics was drastically decreased (mean lOOOO),

CT. The new methods are faster than the EM algorithm

as well as the number of views (60),T-RAMLA attains

for CT, they have better convergence properties (increas-

higher likelihood values. This is shown in Figure 3.

ing the likelihood, preserving positivity, ensuring bound-

83 1

edness in the case of T-RAMLA) than its ordered subsets versions and they produce less artifacts when applied to problems with small number of views. We are now working on further experiments to opti-

mize the choice of the relaxation parameters and a comparison with other methods. From the theoretical point of view we are trying t o improve the convergenceresults in [l],that are applicable to T-RAMLA and T-BSREM.

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-

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