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MODIFIED MINIMUM CROSS-ENTROPY ALGORITHM FOR PET

IMAGE RECONSTRUCTION USING TOTAL VARIATION

REGULARIZATION

Hongqing Zhu, Huazhong Shu, Limin Luo, Jian Zhou

Department of Biological Science and Medical Engineering, Southeast University, Nangjing 210096, China

ABSTRACT

The basic mathematical problem behind PET is an in-

verse problem. Due to the inherent ill-posedness of this

problem,thereconstructedimages usuallyhavenoiseand

edge artifacts. How to decrease the noisy effect whilepre-

serving the edges is an open problem. In this paper, we

propose a new minimum cross-entropy (MXE) image re-

construction method for PET based on the total variation

(TV)normconstraint. Here, TVispresentedasaregular-

ization function in MXE based reconstruction algorithm.

The use of TV is due to the fact that it can effectively re-

duce the noise in 2D images while preserving edges. The

experimental results show that the proposed method is

more effective than the common regularized MXE algo-

rithm, especially for noisy projection data.

1. INTRODUCTION

Positron emission tomography (PET) image recon-

struction using statistical methods can provide more

accurate system modeling, statistical models, and

physicalconstraints than the conventionalfiltered back

projection (FBP) method. Unfortunately, This prob-

lem is usually ill-posed and regularization techniques

are needed to produce reasonable reconstructions. The

regularization of the inverse problem requires smooth-

ing homogeneous areas of the object without degrad-

ing the edges, which are important attributes of the im-

age. The location of these edges is unknown and they

mustbe preservedwhenthe objectis reconstructedand

regularized. In the past decades, there are abundant

literatures that reported the edge-preserved regulariza-

tion methods, but most of them rely on information

from a local neighborhood to determine the presence

of edge: the penalty assigned to each pixel or clique of

pixels depending solely on pixel values within a small

fixed neighborhood[1, 2, 3]. In this paper, a minimum

cross-entropy reconstruction algorithm based on the

total variation norm (TV-MXE) constraint is proposed.

The cross-entropy measures the degrees of dissimilar-

ity between two images. Recently, some regularization

methods based on the minimum cross-entropy (MXE)

havebeenusedbymanyauthors[2,4]. Themethodde-

scribed here uses the total variation (TV) as a prior to

regularizethe MXEalgorithm. Theuseof TVin a con-

strained minimization problem was first proposed by

0-7803-8665-5/04/$20.00 c

?

2004 IEEE

Rudin et al [5]. Since then, the TV norm minimization

has been successfully used in many image processing

applications [6, 7]. But they are mainly focused on im-

age restoration [8], smoothing [9] and removing noise

[5]. Here, the motivationfor us to utilize the TV is that

it can suppress effectively the noise while capturing

the sharp edges without oscillation, since both noise

and edges contribute to high energy component. Un-

like most edge-preserving regularization methods, in

the proposedTV-MXE method,the TV functionis cal-

culated globally, and there is no need to rotate clique

patterns to match edges. To demonstrate the effective-

ness of the proposed method, some results of the ap-

plication of TV-MXE reconstruction to simulated data

are presented.

2. METHODOLOGY

In 1982, Shepp and Vardi [10] introduced a Pois-

son model to the PET reconstruction process. In that

model, the emissions are modeled as a spatial inhomo-

geneous Poisson process with unknown intensity. The

maximum likelihood (ML) method was then used to

estimate the value of unknown parameters. The ML

estimate of the activity

imizing the log-likelihood function of the measured

emission data

??????????? is obtained by max-

??????????? , which is given by

???

where

pixel

EM algorithm starting with a strictly positive vector

is given by:

???????

?

???

?

? ?

?!?"???$#%?&??'?(

?

?

?)?

?*?"?+???

?

',(

?

?&?.-!?0/

(1)

?

?*? is the probability that a positron emitted at

1 will be detected by detector pair

2 . Thus, the

?

??3,46587:9

?

?

?

3,469

?

;

?

?

?!?

?

?

?

?*?"?&?

;=<

?

?

<

?

3,469

<

(2)

ML-EM iterative algorithm converges towards a unbi-

ased estimate of the emission image, but the main dis-

advantage is that the reconstructed image is too noisy

to be useful.

2.1. The cross-entropy algorithm

The goal of iterative image reconstruction in emis-

sion tomography may be considered as minimizing a

S3.4-16BioCAS2004

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given distance measure between the measured pho-

tons and the forward projections of the image esti-

mate. This distance function may be the cross-entropy

or Kullback-Leiber (KL) distance between

[4]. The KL distance between

?

and

>??

? and

>?? is defined as

@A?

where

i.e.,

?CB.>??D?????

?

?

?

?

',(E?

?

?

?

?

',(

?

>??8?

?

?

?

?

#

?

>??8?

?

/

(3)

?

>??8?

? denotes the

2 th component of vector

>?? ,

?

>??8?

?

?)?

?F?

?*?

?

?

(4)

Sincethefirstandthethirdtermsoftheright-handof

Eq. (3) are constant, the maximum likelihood problem

of Eq. (1) is equivalen to minimizing the KL distance

of Eq. (3).

2.2. Total variation norm

The TV norm was first described by Rudin et al [5] as

an iterative method for removing noise while preserv-

ing edges in noisy and blurred images. The TV norm

for two-dimensional case is defined as:

GIHJ?LK

where

energy function using a two-point difference in 2D as

[7]:

????MONQPR

K

PS??TS??Q?)MONQU

KWV

X

#

KWV

Y

S??TS??

(5)

K

X

?Z

Z

X

K,

K

Y

?Z

Z

Y

K. Pania evaluated the

[

?

?

?]\?

U

?LK

?

5D7

\?

?

K

?^\???

V

#

?_K

?]\?

587

?

K

?]\???

V

#%`

or less

V

(6)

The parameter

of the expected maximum value of

of

derivative of Eq. (6) is calculated as follows:

` is equal to approximately

acb

K. Too large value

` will smoothout the edgesin the image. The partial

d

[

d

K

?]\?

?

K

?]\?

?

K

?]e

7

\?

f

?_K

?^\?

?

K

?^e

7

\ ???

V

#

?LK

?]e

7

\?

587

?

K

?]e

7

\?"?

V

#%`

V

#

K

?]\?

?

K

?]\?ge

7

f

?_K

?

587

\?ge

7

?

K

?]\?ge

7

?

V

#

?LK

?^\?

?

K

?]\?ge

7

?

V

#%`

V

?

K

?

5D7

\?h#

K

?^\?

587

??i

K

?]\?

f

?_K

?

587

\?

?

K

?]\?

?

V

#

?LK

?^\?

5D7

?

K

?^\?

?

V

#%`

V

(7)

2.3. TV-MXE algorithm

A number of edge-preserving smoothing filters are

available for the constructionof the penalty term in the

MXEreconstruction[2, 4]. Themethoddescribedhere

is a new MXE method with two terms: a KL distance

term and a TV penalty term for regularization. The

estimated image

j? is given by

j ?k?)l?mon?prq,(

X

??s?t??

?u?.?

(8)

where

s&t??

?W? is the new cost function given by

s?t??

?W?C?

@v?

??Bo>??8??#xw

[

(9)

where

regularizationterm.

stituting Eq. (3) into Eq. (9), the first partial derivative

of

for a given pixel

w

is a weight parameter, which influences the

w is chosenbytrial anderror. Sub-

s&t

??? is given as follows:

d

s?ty?

?W?

d

?

?

???

?

?

?

?&?

?

?!?

?

>??8?

?

#

?

?*????#xw

d

[

d

?

?

(10)

According to the Kuhn-Tucher conditions, the fixel

point iteration for solving this problem is as follows:

?

3,4g587z9

?

?

?

3,4g9

?

;

?

?

?*?

?

?

?

?

?*?"?&?

;

<

?

?

<

?

3,469

<

?

w

d

[

d

?

?

/

(11)

If we let

(11) is omitted, one can see that Eq. (11) turns out

to be ML-EM iterative algorithm. This is due to the

fact that minimizing the cross-entropy is equivalent to

maximizing the log-likelihood function. The TV norm

has good property that it doesn’t penalize edges in the

image. In other words, it only filters the image and the

edges are not smoothed. This is an attractive property

for those applications of PET where the shape of spe-

cial objective (e.g. tumors) should be preserved.

w??|{ , that is, the TV regularization in Eq.

3. SIMULATION

In order to test the effectiveness of the proposed meth-

ods, we compared the reconstructed images of TV-

MXE and MXE using noiseless projections as well as

the noisy projections. The MXE algorithm we used

is described in [4]. In our simulation experiments, we

used a

sibility of the proposed method. The relative activities

of the elements are shown in Figure 1. The projection

space is assumed to be 139 bins and 180 anglar views

evenly spaced over

age is set to a size of

projection data was generated by adding to the origi-

nal projection a uniform field of random coincidences

reflecting a scan ofof the total count. The re-

constructed images using both MXE and TV-MXE al-

gorithms are shown in Figures 2 and 3. The iteration

numberis 40 for each algorithm. We can see from Fig-

ure 3 that the smoothing effect is much more obvious

withincreasingofparameter

smoothes the image while preserving the edges. TV-

MXE reconstruction significantly improves the qual-

ity of the image over images obtained using the MXE

reconstruction, especially for noisy projections case.

We also compare these algorithms in terms of the root

mean square error (RMS) between the simulated activ-

ity distribution and image estimate versus the iteration

number. The RMS is given by

}?~}&~ pixels thorax phantom to test the fea-

a??{& . The final reconstructed im-

}&~
}?~ pixel matrices. The noisy

i&

b

w . TVregularizationterm

and

?

?

denote the value of the simulated

?

?

?_00

?

D

??/?

(12)

where

activity image and the reconstructed image at pixel

?W:.

?

?

o¡:¢

?

1 ,

S3.4-17

Page 3

Fig. 1. A simulated emission thorax phantom

Fig. 2. Projections without noise,

column reconstructed by MXE algorithm. The even

column reconstructed by TV-MXE algorithm.The first

row with

`
??{?£ {&{¤a . The odd

wv?F{¤£

. The second row with

wv?

i, .

respectively. Variance over a region of interest is de-

fined as:

¥T¦?§6¨]¦

u©ª

?

?

?

oC

?L

?L

?

?

(13)

where

image. Figures 4 and 5 display the RMS and Variance

values of reconstructed images using MXE and TV-

MXE, respectively. The results show that the RMS and

Variance values using TV-MXE decrease more rapidly

than those obtained with RMS even if the projections

with statistics noise. The new method TV-MXE is also

evaluated using profiles through the reconstructed and

the original thorax phantom. The line plots of the

row of the reconstructed image obtained from MXE

and TV-MXE are shown in Figure 6. (a) and (b), re-

spectively. The results show that the reconstructed im-

agesusingTV-MXEis moreclose tothe originalphan-

tom than those by MXE.

?

:. is the average gray level of test reference

«?¬]®

4. CONCLUSIONS

In this paper, an effective method for solving the prob-

lem of edge-preserving regularization in PET image

reconstruction has been proposed. The proposed TV-

MXE method was based on minimizing an objec-

Fig. 3. Projections with noise,

column reconstructed by MXE algorithm. The even

column reconstructed by TV-MXE algorithm.The first

row with

`x?¯{?£{&{¤a . The odd

w??F{?£

. The second row with

wv?

i, .

tive function, which is a weighted sum of two parts.

The first part consists of the cross-entropy measuring

the dissimilarity between the measured emission data

and forward projection. The second part is TV norm

smoothing the image while preserving the edges. The

distinguished feature of the TV norm based minimiza-

tion is that it does not introduce over shoots and rip-

ples at edges. Reconstructions of simulated PET data

were performed using the MXE and TV-MXE algo-

rithms. The experimental studies clearly show that the

TV-MXE method yields the reconstructedimages with

better contrast and resolution than the MXE algorithm,

which indicates that the penalty based on TV norm

could be helpful for improving the image quality.

5. ACKNOWLEDGMENT

This work was supported by National Basic Research

Program of China under grant No. 2003CB716102.

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2020

2525

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4040

0.20.2

0.250.25

0.30.3

0.350.35

0.40.4

0.450.45

IterationsIterations

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(b)

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