Particle Swarm Optimization Of Fuzzy Penalty For 3D Image Reconstruction In X-Ray Tomography
A.M.T.Gouicem
1, M.Yahi
1, A.Taleb-Ahmed
2, R.Drai
11
Laboratory of Image and Signal Processing (LISP)
CSC Research Centre in Welding and NDT Algiers, Algeriamedthrali@yahoo.fr
2 LAMIH UMR CNRS UVHC 8201, Valenciennes, France
Abstract
Engineers last year's works only on the 2D image data, to perceivedefects in the CT images.
This was a handicap facing the challenge of determining the 3D exact defect form. This paper presents a method for 3D image reconstruction, the most interesting in non destructive testing (NDT) especially due to its application in industrial imaging. We propose a new combined approach using particle swarm optimization (PSO) and fuzzy inference penalty, which will be helpful to elevate the hard inverse problem of 3D computed tomography.
Introduction
Image reconstruction in 3D x-ray tomography consists in determining an object f(x,y,z) from its projections p[1]
.A 2-D image represents the projection of a 3-D scene onto a plane, The procedure for detecting edges in 3-D closely follows that in 2-D. But edge contours in 2-D become edge surfaces in 3-D. To avoid detection of noisy edges, it is important that the image is smoothed with a Gaussian before carrying out functional approximation.
f(i,j) is the central voxel value and others are 26 neighbors voxel value in 3*3*3 windows.
Figure 1: 3D Image representation.
Problem Statement
X-ray computed tomography determines an object function
f(x,y)from measures known as projections p.
where (r;θ) : are the relative coordinates of the object.
For the ill posed problem f = arg min {-logJ (f/p)} is the quasi solution with minimization of the critiria J(f) chosen to be likelihood estimation.
J(f)=L(f)+P(f)
Where L(f) is a likelihood function l(f).
The objects will be a 3D images defects of our research center in welding and NDT.
Applied Methods
Due to the ill-posedness of the inverse problem of image reconstruction in 3D x-ray tomography, the quasi solution f(x,y,z) with minimization of the criteria J(f) chosen to be a likelihood function in the probabilistic estimation is:
k
f i U
j p f m j aij pi
fj n
i m j aij f
J
) ln( !)) ( )
1 ln(
1( 1 )
(
r x y z drd d
r p z
y x
f ( , , ) ( , , ) ( cos sin sin )
The term βU (f) is the prior information modeled in our case as a fuzzy penalty. The most used prior is GIBBS prior defined by:
Where U (f) is the potential function defined in the MARKOV field as:
we express U(f) in function of CT Value of the prior term:
where
Results And Discussion
-a- -b-
-c- -d-
Figure 2: Reels 3D Images.
Results will be defused later.
References
Arun, K. S., T. S. Huang, and S. D. Blostein, Least-squares fitting of two 3-D point sets, IEEE Trans.
Pattern Analysis and Machine Intelligence, 9(5):698–700 (1987).
Aubert, G. and L. Blanc-Feraud, Some remarks on the equivalence between 2D and 3D classical snakes and geodesic active contours, Int’l J. Computer Vision, 34(1):19–28 (1999).
Brejl, M. and M. Sonka, Directional 3-D edge detection in anisotropic data: Detector design and per formance assessment, Computer Vision and Image Understanding,
77:84–110 (2000).
)
j , i T( C
* f k) fj ( U ) j , i T( C k) fj ( kU fj
N j i C j
i C
N k k
T
) , 8 (
) 1 ,
(
) ) (
( U f
Ce f
P
) (
) 0 (
)
( k fbk
fj N b wjb k
fj kU fj f f f
U
