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ИРКУТСКИЙ ГОСУДАРСТВЕННЫЙ УНИВЕРСИТЕТ ПУТЕЙ СООБЩЕНИЯ
Jiang Xian-gang YAK 614.8
THE TECHNOLOGY RESEARCH ON IMAGE SMOOTH AND MEDICAL ORGAN'S 3D-RECONSTRUCTION BASED ON ANISOTROPIC DIFFUSION
1. Introduction.
The noises existed in the digital image should been eliminated through image mean filtering, median filtering, Gaussian filtering, open operations ,closing operation and so on, but the main effect of these methods are to remove the high-frequency components of the image. Some details of the image existed in the high-frequency part ,which keep the edge characteristics information have been filtered in the same time. So we need a processing method of smoothing homogeneous neighborhood without destroying the image edges .
In recent years ,the application of partial differential equations model in the image processing technology get a significant developments .And the application of the theory actually develop steps of uniformity linear diffusion, non-uniformity linear diffusion, nonlinear diffusion ,anisotropic diffusion and so on.
The filtering method which based on the diffusion equation is a new medical image filtering technology. It is arising from the diffuse phenomena of physics. That is to solve the equation of nonlinear thermal diffusion in which the initial value is the input image. In the diffusion equation ,by introduction of image features, design of appropriate diffusion coefficient ,it control the action of the diffusion equation. It is not only to smooth image but also to preserve and even enhance the image edge feature, It can be used for the coarse image segmentation practically.
In 1990, Perona and Malik put forward the anisotropic diffusion equation which based on the partial differential equations. In the nonlinear scale space, the different diffusion coefficients of images determined by the direction of the gradient.the smooth noise and retaining details of
the performance was advanced, its equation is as following :
^ = dzv[c(|Vu|2 )-Vu],
Here, 'div' is the divergence operator, 'u' is the gray scale of a image's pixel, 'Vu'is the gradient, 't' is the time which expending on process of diffusion caloric in physics ,which is expressed by the iterative steps in designing program ,c is the diffusion coefficient ,which is the non-negative and monotonic decreasing function of the'Vu'. Perona and Malik recommends two forms, as follows :
c(|Vu|2 ) =-U—,
1 ) i+|vu|;/2
c(|Vu|2 ) = exp[-|Vu%].
Through the experiment, taking the fast and stability into account ,the test adopting the diffusion coefficient to this:
c(|Vu|2 ) =-
1+k\ Vul2
Among them, kis a parameter of the diffusion coefficient, which controls the intensity of the diffusion.
Adopting different diffusion coefficient in different direction, so it is named anisotropic diffusion equation. Because the anisotropic diffusion equation apply the gradient of the monotone decreasing function in different directions of the image,in homogeneous (the same organ) region, where the change of value of the gray scale is little ,also is the gradient, it can smooth the noise in homogeneity region effectivly as its diffusion coefficient is big .But in edge of the image, the value of gray scale changes rapidly, and the gradient is larger ,and the diffusion coefficient is smaller, so that the edge of
МЕХАНИКА. ТРАНСПОРТ. МАШИНОСТРОЕНИЕ
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the image information can been retained, in the whole, it can autocontrol its diffusion action.
2. Designing the image smoothing algorithm in anisotropic diffusion.
Considering diffusion algorithm of the gradient in four directions of top, bottom ,left and right, the steps as following :
The first step: Calculating the gradient in the step n iterations of four points ,supposing a point asfn(i, j),so calculating the gradient as following:
Gn (/, j ) = sgrt
[ fn (i, j ) - fn (i, j-1) ]•
•f (i, j ) - fn (i, j-1)]
fn (i, j) --fn (i-1, j)
f (i, j ) - 4 (i "1- j]
The second step : Calculating the diffusion coefficient in the step n iterations of four neighbor points.
r Ï = 1
Cn(i- j)_1 + k\Gn(i- j)|2 '
The third step : Calculating the value of gray scale after the step n iterations.
Un + 1(i,j ) = Un ( i<j ) +
[ Cn ( i,j ) + Cn ( i-1, j ) ]
2
,[Cn ( i,j ) + Cn (i + 1, j )]
2
,[Cn ( i,j ) + Cn (i, j -1)]
2
,[Cn ( i,j ) + Cn (i, j +1)]
2
Un (i-1,j) + Un (i + 1,j ) + Un (i,j -1) + Un (i,j +1)-
4Cn (i,j ) + Cn (i-1, j ) + +Cn (i + 1,j) + Cn (i, j-1) + +Cn (i,j + 1)
2
•Un ( ^ j )
coefficient's parameters k , anisotropic diffusion's smoothing effect will be worse in the homogeneous region. Generally, k is 0.08. With the increasing of step Ai's size, anisotropic diffusion's smoothing effect will be better in the homogeneous region. Generally, At is 0.16.
3. Designing the improved image smoothing algorithm in anisotropic diffusion.
On above ,we introduced the gradient of anisotropic diffusion in four directions, but in practice, We will involves 45 degrees, 135 degrees, 225 degrees and 315 degree's brink during the processing dealing with images. Therefore, we adopt the gradient in eight directions to calculate the intensity of a point's transformed valve. Fig. 1 shown the eight directions' anisotropic diffusion.
Fig.1 Image smoothing sketch in 8 directions anisotropic diffusion.
Correspondingly, the same third step in the process of iterative formula should be changed to:
Un +1(^ j ) = Un (i,j ) + ¿t •
[ Cn ( i j ) + Cn (i-1,j ) ] U ) +[Cn(J,j) + Cn{i + \,j)\
2 n ( ) 2
•U„ „ + Ц ) + [C- 1 +C- ( " ''Я •U. (ij-1) +
Here, At is the size of step, un (i, j)is the value of gray scale in the step n iteration of point (i, j).
The fourth step Judging the iterative process ,if it was finished ,the process would exit ;If not, it will continue to the first step in next loop.
With the number of iterations increasing, anisotropic diffusion's smoothing effect will be better in the homogeneous region. Although the more the number of iterations and the better the results, the processing time for images will extend longer . The number of iterative should be 25 as the best one after a lot of experiments testing. With the increasing of diffusion
[Cn (i,j) + Cn (i,j + 1)]
Un (i,j +1) +
[c. ( i, j ) + Cn (i-1,j +1)]
2
[Cn ( i, j )- ^Cn (i +1,j+1)]
2
[Cn ( i, j )- ЬCn (i -1, j-1)]
2
[Cn ( i, j )- ЬCn (i +1,j+1)]
2
Un (i-1,j +1) +
Un ( i +1, j +1) +
Un (i-1,j-1) +
Un ( i + 1, j-1) -
/2 •Un (i,j )
8Cn (i,j ) + Cn ( i-1, j ) + Cn (i + 1j ) +
+Cn ( i, j -1) + Cn ( i, j + 1) + Cn ( i +1, j + 1) +
+Cn (i-1,j + 1) + Cn (i-1,j-1) +
+Cn (i + 1, j -1)
Under the conditions of
n = 25,k = 0.08,At = 016,the smoothing effect as Fig. 2 showed ,its smoothing effect is better than the four directions anisotropic diffusion which
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ИРКУТСКИЙ ГОСУДАРСТВЕННЫЙ УНИВЕРСИТЕТ ПУТЕЙ СООБЩЕНИЯ
shows in figure 2 in the same conditions of parameters.
4. Designing the image edge enhancement algorithm in anisotropic diffusion.
Anisotropic diffusion smooth the noise effectively while it retain the edge. Under some conditions, we hope that anisotropic diffusion at the same time ,the edge can been enhanced, also the details been focused on. In this paper we choice different edge-enhanced according to the different of the gradient values ,edge will be enhanced in the bigger gradient ,and will have little change in the smaller gradient areas.
Using the following iteration format to get the result of enhanced edge:
u' (i, j) = min (255,max(0, round( u( i, j) + power • [4 ■ •u(i, j) - u(i-1, j) - u(i+1, j) - u(i, j-1) - u(i, j +1)]))).
Here, u'(i, j) is the value of gray scale after
being changed, u(i, j) is the value of gray scale before being changed.
With the enhancement coefficient 'power' increasing ,both edge and noise are enhanced .In this way, while the edge is enhanced ,at the same time the noise was enhanced. Noise gradient becomes much greater and the effect of smoothing becomes worse. Considered all of these ,we can improve the image quality by taking improved eight direction anisotropic diffusion at first, then applying edge enhancement ,only by this procedure we can solve the conflict effectively .With the enhancement coefficient 'power' increasing ,the edge was enhanced but the noise not, smoothing effect is the best in homogeneous region. Fig. 2 (i) show the image processing effect by improved
Now the results of all kinds of image preprocessing results are compared as followings:
'A'." i
(a) original image +FCM (b) gray-opening operation +FCM (c) gray-closing operation +FCM
гШ—4
9
fir i
(d)mean filtering+FCM (e)median filtering+FCM (f) Gaussian fi I ten rig +FCM
1 ._.
'ÊÊÎ HfP V ^J I n
(g) four direction anisotropic (h > i m p rov ed ei ght di rect i on (i )e dge en ha nc ed i m proved ei gh t diffusion +FCM anisotropic diffusion +FCM direction anisotropic diffusion
+FCM
n = 25 , * = 0.08, Д/ = 0.16 и = 25, к = 0.08^ At =0.1
и = 25, fr = 0.08, M = 0,16, power=1.6
Fig.2. Compared image effect in different processing methods.
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eight direction edge enhancement anisotropic diffusion.
5. Compared results in different image Pretreatment methods.
The above results show that the gray-opening operations eliminate the some small bright details, retain the holistic gray-level and big size bright region unchanged , eliminate the small objects,and smooth the edge of the larger objects at the same time; gray-closing operations eliminate the dark details of the image, retain bright region do not be affected relatively, filled the objects' small hollow and smooth its borders at the same time .It does not significantly change their area; Mean and median filtering for noise filtering and edge information without distinction, eliminated a certain level of noise and made the edge become blurred; Gaussian filter is the proliferation of the isotropy, use the templates to the original image convolution operation, noise eliminated, edge blurring, and the overall image is gray; anisotropic diffusion smooth the noise efficiently and maintain the edge's information; eight direction anisotropic diffusion is better than four direction anisotropic diffusion; Eight direction anisotropic diffusion with Edge enhancement not only eliminated the noise in homogeneous region but also strengthened the edge. Edge enhancement of anisotropic diffusion's effect is better, it can adjust the edge enhancement factor to strengthen the edge largely, when the edge enhancement factor is taken as a small negative absolute value , edge will become fuzzy.
After image was smoothed, following process is to separate the different parts of the organs, the aim of taking gray image segmentation is to separate the image's regions into several parts which have some uniform properties by analysis ,extracting and identification. In this paper, K-means clustering and fuzzy-means clustering (FCM) algorithm can gain more accurate and reasonable classification.
By analysis of the experiments of the medical images processing ,we concluded that under normal circumstances, organ and tissue are separated into four categories for bone, muscle, subcutaneous tissues and background preferably. The initial cluster center of gray values would be taken approximately 230,130,50 and 0 as a better choice. Different image preprocessing will produce different effects of 3D reconstruction. Fig.3 is the reconstruction effects of the slice images which was preprocessed in
different way then further processed in 3D reconstruction.
Item (e) and (f) are results of reconstruction after preprocessed every slice in the parameters ofn = 25, k = 0.08, At = 0.20. Comprehensive
analysis, gray-opening operations, mean filter, median filter, Gaussian filter, anisotropic diffusion and improved anisotropic diffusion filter the noise in varying degrees ,but improved anisotropic diffusion in eight direction is the best one. Edge enhancement of anisotropic diffusion filter eliminate a part of noise, but also bring into other noises, this is because the part of edge was strengthened through edge enhancement ,which make the gray value of edge leaping promotion and the boundary of edge become clearer , but perhaps some parts of edge were easily separated into other organs, which will be conducive to the goal of two-dimensional image segmentation and recognition.
The hardware and software environment of the operation of procedures are: Windows 2000 operating system, Delphi 7 development platform, mainframe is PIV 1.81GHz,512MB memory. The following table is time (milliseconds) compared about the consumption of all kinds of image reconstruction after pretreatment.
6. Conclusion.
Experiments show that the 8 direction anisotropic diffusion method takes diffusion effects in 8 directions into account , which is a better one to get the image smoothing effect than the way of 4 direction anisotropic diffusion. The anisotropic diffusion technology should be used flexibly according to special requirements and situations in image smoothing and three-dimension reconstruction. The parameter k of diffusion coefficient C and edge enhancement factor can be adjusted to achieve better reconstruction results than that of the generally used image preprocessing methods. The impurities suspended in spatial objects have been reduced reconstructed objects become smoother . The better filtering method is the pre-condition of significant reduction in the number of triangles of a 3D model. It enable procedures to reduce the demands on hardware configuration, reduce costs, improve the accuracy of reconstruction. It is worth noting that all the existing image filtering and the anisotropic diffusion image smoothing technology focus on 2D planar images. If we consider spatial 26 directions in anisotropic diffusion equation in the image
(a)gray-opening operations (b) mean ЯИег (c) median filter
(d)Gaussian filter (e) four direction anisotropic diffusion (f) improved eight direction
anisotropic diffusion Fig.3 Compared reconstruction effects with different processing methods.
Table 1
Reconstruction consumption time of all kinds of image pretreatments.
Pretreat - ment O/I G/O/O Mean filter Median filter Gaussian filter 4D/A/D I/8D/A/D
N/G 29938 20047 33156 25562 18391 19812 16453
K/G 26484 33078 26891 37547 35453 23438 21641
F/G 34359 32406 25625 22171 16859 29313 27719
O/I for original image.
4D/A/D for four direction anisotropic diffusion.
I/8D/A/D for improved eight direction anisotropic diffusion.
G/O/O for gray-opening operations.
N/G for not grouped by classification.
K/G for grouped by the K-means clustering.
F/G for grouped by Fuzzy-means clustering.
processing technology. Spatial objects will be BIBLIOGRAPHY
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