Fuzzy Inference for Image Reconstruction from Projections In X-Ray Tomography.

Fuzzy Inference for Image Reconstruction from Projections In X-Ray Tomography.

A.M.T.Gouicem,R.Drai,M.Yahi

Laboratory of Image and Signal Processing (LISP) CSC Research Centre in Welding and NDT

Bp 64 Route de dely Ibrahim Cheraga Algiers Algeria Tel/Fax: +213.21.36.18.50 E-mail: [email protected]

Abstract-

The ill posed problem of image reconstruction was resolved by the Bayesian inference frame work which gives sense to the information known about the inverse problem and aims at smoothing artefact in image. But this method result in a newproblem whichis the edge penalization. It’s the reasons for what we opt to the fuzzy inference penalty functionto preserve edge during the smoothing operation. The proposed algorithm FP-EM does not suffer from the same problem as that of MAP-EM algorithm, and it converges to a low noisy solution.

Keywords: Computed Tomography; Non Destructive Testing; Bayesian Inference ; Fuzzy Inference.

1. Introduction

Recent approaches in Image Reconstruction estimate parameters from projections by defining the solution as a minimizer of appropriate regularized criteria by an iterative method like maximum a posterior MAP which can be interpreted as Bayesian estimation based on iterated conditional mode [10]. This iterative algorithm converges slowly to a local solution and is sensitive to noise artifacts so that the reconstruction begins to degrade when the number of iterations reaches a certain value.

1.1

Image reconstruction Theoretical background

Image reconstruction in x-ray tomography consists in determining an object f(x, y) from its data measures called

projections figure 1:

³

I

I

,

) , ( )

, (

Lr

dl y x f r

p

(1)

We discrete the equation (1) to obtain

p= R.f +n. (2) Where dl is the element of projection ray support, f the discrete value of the object, f(x, y) the pixel value of the image, p values of the projection data p(r, Φ), n vector to represent the modeling and measurement errors(noise)

For the ill-posed problem f=argmin(J(f)) is the quasi solution with minimization of the criteria J(f) chosen to be a least squared in algebraic resolution or likelihood function in the probabilistic estimation. If we add constraints to the criteria as the smoothing, we introduce the constraint violation as a heavy penalization in the criteria [5]:

J(f)=LH+P(f) with P(f)=λ*U(f) (3)

The solution is then defined as the minimizer of this compound criterion. Where LH is a likelihood function l(f) equal to the conditional probability prob(p/f) to observe the vector p when the emission or transmission is the image f (Poisson variable) [10]. U(f) is the priori information that means: what the reconstructed image must been? Which have to be chosen appropriately to reflect our prior knowledge on the noise and on the image.

(4)

Iterative methods based on likelihood function are very interesting but the problem is the accumulation of the noise in the estimated solution. MAP method resolve this problem by including a penalty function to decrease noisy accumulation in the reconstructed image. This method is based on Bayes theorem to combine the Poisson likelihood function with the image prior information. Our contribution takes the prior information as fuzzy penalty and optimizes the new criteria with gradient or genetic algorithm. The most used prior is GIBBS prior defined by [6]:

(5) Where U (f) is the potential function defined by the MARKOV field:

(6)

The derivative of J (f) is taken equal to zero to find the image f (from green formulation) [5]:

(7) The value of the prior in the denominator of MAP makes difference with the ML EM, this correction is

sometimes negligible and its computation is a waste of time. We use the intelligent reasoning to just know if the value is small or large. We will express it with fuzzy model in the method FP EM. If we take J(f) as a fitness function to optimize by the gradient and we will expressed the Value of the prior term in MAP criteria with fuzzy model in the FP CG method.

(8) 1.2 Fuzzy inference

) ) (

( f Ce U f

p r E

)

( )

0 ( )

( k f_b^k

fj N b wjb k

fj k U fj f f f

U ¦ 

w

Ÿ w

aij ni m

j

k fj aij pi n

i

k fj U fj aij

k fj k

fj ¦

¦ w ¦ w

1

1 ( ) 1

1

E

)

)

! ln(

1 ) 1( 1 ln(

)

( m fj pi

j rij pi

fj n

i m j rij f

l ¦ ¦ ¦

k

f i U

j p f m j rij pi

fj n

i m j rij f

J ¦ ¦ ¦ )ln( !)) ( )

ln( 1 1( 1

)

( E

Fuzzy logic is all about the relative importance of precision and the help to decision similar to human reasoning.

Many problems are not known or complex that a mathematical model is difficult to obtain. For this problem, verbal model is simple and standard mathematical are not suitable to incorporate verbal models. This is exac tly where fuzzy linguistic modeling can be applied. The term fuzzy modeling incorporate several kinds of models that uses various concepts from the fuzzy sets theory [3] .Typically fuzzy sets are used for partitioning the continues domains of the problem variables into a small number of overlapping regions labeled with linguistic terms like small, large … etc. A fuzzy model describes the system by establishing relations between these labels.

These relations can be expressed in form of (if… then) rules. Each rule maps a fuzzy region from the premise space to another fuzzy region in the consequence space. The universally Approximation theorem [3] indicates simply that, if we chose properly the number of rules and the parameters of the fuzzy model, we can approximate any non linear function with the precision choice ,the membership functions used are Gaussians figure.3.

2. Applied methods

2.1 Implementation of fuzzy inference The penalized solution is of the form:

^ log ( / ) `

min

ˆ arg J f p

f

f

Where J(f)=Q(f) +λ.U(f) (9)

See Figure.2 where f(i,j) is the central pixel value and others are neighbors pixel value in 3*3 windows.

Q(f) is the log-likelihood function and U(f) is the penalty term. In Bayesian terms, J(f) represents theposterior probability and U(f) is the prior term resulting from the priori probability . A commonly used Bayesian prior is the Gibbs distribution [6] equation (5).

2.1.1 Fuzzy potential function

The purpose of this paragraph is to compute a penalized-likelihood estimate by replacing the function U(f) in Eq.

(9) by a fuzzy penalized function. The reconstruction method (called FP GA) is then obtained by maximizing the

criteria(9).U(f) a prior law of penalty, choice a fuzzy distribution defined as first derivative for the pixel value (i,j) N

f f G f

U _{N ij ij} &

, 1

) ,

( (10)

Where

f

_i,j1 represents the neighborhood pixel in the direction N(North). To computes an edge E along a direction three derivatives (which are perpendicular to this direction in the 3*3 windows) must be calculated and summed.

E^k (i,j) G^k (i,j)G^k (i1,j)G^k (i1,j) (11)

To express the degree of smallest of the fuzzy derivative EN in one direction, we use the fuzzy sets small instead of the hard computations. Now we must found the fuzzy model of the Gibbs prior function, in two steps edge detection and penalization.

2.1.1.1 Detection steps

If the difference G_N between the pixel gray levels is large, then there is an edge, else the region is smooth and the edge value E_N is small.

2.1.1.2 Fuzzification

Conditions to edge detection are:

-To detect (compute) an edge in the direction N three derivatives GN are used.

-The values of derivatives are large in a direction, if there is an edge.

-If two of the three derivatives are small, there isn’t an edge in this direction (rule of 2/3).

-The linguistics terms small and large are defined by the membership functions which define the degree of belonging of the linguistic variable GN ,EN to the fuzzy sets small and large figure.3.

°¯

°®

­ d d

0

) max(

) min(

)] min(

) max(

)

[ min( ²

,

E E

E E for

E E E

G

PE (12)

-To compute the value of the fuzzy variable edge detection EW in a direction W we use the FIS (fuzzy inference system) corresponding to EDGE W.

2.1.1.3 Rules base

If

G

_W^k

( i , j )

is small and

G

_W^k

( i 1 , j )

is small then

E

_W^k

( i , j )

is small If

G

_W^k

( i , j )

is small and

G

_W^k

( i 1 , j )

is small then

E

_W^k

( i , j )

is small If

G

_W^k

( i 1 , j )

is small and

G

_W^k

( i 1 , j )

is small then

E

_W^k

( i , j )

is small

If

G

_W^k

( i , j )

is large and

G

_W^k

( i 1 , j )

is large and

G

_W^k

( i 1 , j )

is large then

E

_W^k

( i , j )

is large (13) Similarly the edge detection in the eighties directions {E, NE, N, NW, W, SW, S, SE} is calculated see figure.2.

2.1.1.2 Penalization steps

The second step is the penalization of pixels for which we haven’t detected an edge (smoothing region).Eight fuzzy rules of the FIS penalty-w are used to indicate the correction in penalization of the eight directions.

If

E

_F^k

( i , j )

is small then C^k(i,j)=

E

_F^k

( i , j )

else

C

^k

( i , j )

=0 (14) N

j i

C^k &

) ,

( is the correction at site (i, j) due to the adjacent pixel in the direction N. The total fuzzy correction is:

C i j C i j N

N k k

T

&

) , 8 (

) 1 ,

( ¦

(15)

Replacing the error term (15) in (8) we obtain the fuzzy criteria to optimize by GA to solve the inverse problem.

2.2.5 Optimization methods Various objective functions that have been evaluated include: Root-Mean- Squared error (RMSE) and Mean-Squared error (MSE). The fitness function F chosen for maximization is based on the error between the computed and measured projections for θ m views of projections as follows:

) ( ) ( )

(f L f U f

J E

F E (16) (17)

(18) CT :Value of prior term in MAP criteria we will express it with fuzzy model in the fuzzy inference method by taking J(f) as a fitness function to optimize. We take as optimization method the conjugate gradient technique.

2.3 Phantom description

We used a 256×256 pixel image of integrated circuit obtained from the tomographe Fein-focus to evaluate the method in term of quality of the reconstruction and robustness to the Poisson or Gaussian noise presence. The size of the final reconstructed image was 256×256 pixels.

3. Simulation Results

A similar comparison was made for projection data simple and degraded by noise level for MSE= 0.005 dB figure.5.Images has been reconstructed with different algorithms: the maximum likelihood-expectation

maximization algorithm (ML-EM), the Maximum a Posterior reconstruction algorithm with fuzzy Prior FP EM.

The performance of the algorithm was evaluated for objective criteria. We measured the convergence rate by computing the mean square error (MSE) between the simulated noiseless activity distribution and the image estimate as a function of the iteration number k. This expresses the dispersion between the reconstructed image and the original image. The MSE resulting for the ML EM, FP EM are plotted in figure.6.The number of iteration choice are comprised between 10 and 100 to remove the accumulation of noise. We saw the stability of the FP EM algorithms in comparison with the ML EM algorithm, this one converge to a noisy solution when the number of iteration increases. The FP EM is more expansive in terms of computation time.

4. Conclusion

The algorithm exploits the fuzzy reasoning to regularize in term of penalties the noise in the image, which is

k

f i U

j p m f j rij pi

fj n

i m j rij f

l ¦ ¦ ¦ )ln( !)) ( )

ln( 1 1( 1

)

( E

)

, (

* ) ( )

, ( )

( k f CT i j

fj U j T i k C fj k U fj w Ÿ

w E

References

[1] Mondal PP,Rajan K, "Iterative image reconstruction for emission tomography using fuzzy potential", IEEE trans Image and signal process.2004. 0-7803-8700-7.

[3] Zadeh LA, "Fuzzy sets. Information and Control",1965. 8:338-353.

[5] Green P J, "Bayesian reconstruction from emission tomography data using a modified EM algorithm", IEEE Trans. on Med. Img., March, 1990, vol.9, No.1.

[6] Zhou Z, Leahy R M and J. Qi , "Approximate maximum likelihood hyper parameter estimation for Gibbs prior", IEEE Trans. on Img. proc, June, 1997,Vol.6, No.6, pp.844-861.

[7] Ville D V, Nachtegael M, Weken D V, Kerre E E , Philips W and Lemahieu I, "Noise reduction by fuzzy image filtering, IEEE Trans Fuzzy Syst ", Aug.2003.vol 11:NO.429-435.

[10] Shep L A and Vardi Y, "Maximum likelihood estimation for emission tomography", IEEE trans.med.ima.

1982.MI.1.pp.113-121.

Figure 1-projection measures in x-ray tomography.

Figure 2- 3 × 3 Neighborhood of a central pixel (i,j), showing the directional derivative along ˆW . f(x,y)

θ

P(s,θ) Detector

X-rays Source

E(j) W

NE NW

Sw SE

f(i-1,j-1) f(i-1,j) f(i-1,j+1) f(i,j-1)

f(i+1,j-1) f(i+1,j) f(i+1,j+1) f(i,j+1) f(i,j)

S(i)

N

Figure 3- Gaussian membership function.

ML EM MAP EM FP EM

Figure 5- Images reconstructed with different algorithms.

Figure 6- MSE Plots.

G, E

Original image Map EM FP EM