论文研究-Extraction of Medical Image EdgeBased on HMT Model and B-Spine Wavelet.pdf

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基于HMT模型和B样条小波的医学图像边缘提取,谷召伟,王波,图像的噪声分为内部噪声和外部噪声,它们在提取医学图像的边缘特征时有很大影响,因此本文使用隐Markov可夫树(HMT)模型对图像进行降�
国科技论文在 http://www.paper.edu.cn (3)Perturb X to obtain a neighboring Design Vector(X, +1) (4) Evaluate E(X +1)using a simulation model ()If E(X +1)<E(X),X, +1 is the new current solution 6)If E(X; +1)> E(X), then accept X, +1 as the new current solution with a probability e a'n where△=E(X,+1)-E(X) (7)Reduce the system temperature according to the cooling schedule ( 8) Terminate the algorithm Simulated annealing can deal with highly nonlinear models, chaotic and noisy data and many constraints. It is a robust and general technique. Its main advantages over other local search methods are its flexibility and its ability to approach global optimality L Figure 1:(a) The medical image with internal noise and its histogram(b)The transform result with the information measure and its histogram 3 Using hmt modcl to remove the cxtcrnal noisc Wavelet coefficients of real-world images generally possess a non-Gaussian distribution, with a lot of small coefficients and few large ones, due to the sparse properties of wavelet decompositions The Hidden Markov Tree(HMt)model developed by Crouse et al. [4] is often described as a quad-tree structured probabilistic graph that captures the statistical properties of the wavelet transforms of images. It is based on a hidden markov model(hmm which exploits hidden states. The non-Gaussian behavior of the coefficients is modeled as a Gaussian mixture with two components o Hiddens statc Sunle 脏 Figure 2: (a)2-D CHMT(c=4)model(b) Diagram of a Hidden Markov Tree in a quad-tree. White dots represent hidden states with arrows as parent-child dependencies, black dots the wavelet coefficients whose conditional distribution depends on the nature of the hidden state The hidden states of the hmm are the large (l)or small(s )nature of the coefficients. It determines the conditional distribution of the associated coefficient. Since the coefficient nature tends to propagate across 中国武技记文在线 http:/www.paper.edu.cn cales, the Hidden Markov Tree materializes the cross-scale (parent-child) link between hidden states. A template HMT is depicted in Figure 2. The parameters of the HMT model are estimated using an Expectation Maximization(Em)algorithm. We refer to [4, 5] for details on the implementation of Hidden Markov Trees Suppose so denotes small and sl denotes big coefficients, a state transferring matrix corresponds to very state of parents transferring to children 10 P=1-P. 5→0 1-P 10 Parameter E(X)E(X +De/A=E(X+D)-E(X is the probability of when the known parents ate big or small and the wavelet coefficients are small or big, so it is called continuance probability. P. and are update probability Figure 1:(a)The medical image before denoising with the new method and its histogram(b) The medical image after denoising with the new method and its histogram Table 1 Comparison of Results for Different Methods METHOD The new MGB HMT method method method RESULTS Output PSNR(dB)94.05889689955 nput PSNR Ir (dB) 79960 The method in this paper improved the existent methods. In our experiments, the internal noise is decreased obviously, and the visual effect of the image is also improved 4 Edge extraction with B-spine wavelet 4.1 The algorith Because of the existence of the external noise of the medical image, when extracting the image edge, the external noise must be removed. Take the one dimensional signal with noise for example Set g(r)as the low-pass filter, y(x)as the result signal through the filter, then y(x)=f(x)*g(x (4) y(x)is believed to be a signal without noise and it is close to the original signal f(x). Smoothing operation will remove many edge features from the original signal, so v(x)can't close enough to the original signal. When the smooth measure of the original signal is unknown, if the original signal is set to be differentiable if onl 山国技论文在线 http://wwwpapcr.ddu.cn g(x) is p-differentiable If g(r)is Gaussian function, due to its infinite differentiable, y(x)is infinite differentiable. Canny operator takes the first derivative of the gaussian function, from above discusses, the result v(x)will be overly smoothed. But the B-spline is not the same, its derivative can be decided by its exponent number. The exponent number can be set according to the noise So we can approach the initial signal. We can prove that the quadratic B-spline wavelet is the optimal edge extraction operator [81 4.2 B-spline wavelet constructing In the reference ofthe lower mistake detecting and higher precision, the optimal edge extraction wavelet should be odd symmetrical compact support wavelet [9]. A compact support quadratic B-spline wavelet can be designed using the B-spline function [10] (x)-8(x+1)(x+1)-4(x+)2(x+)+6x2(x)-4(x-2)(x-)+(x-1)2(x-1) In which set y(x) to be an odd symmetrical compact support wavelet, as same as the Canny operator, which can be thought to be first derivative of an even symmetrical smooth function B(x) in which ((x) is an even symmetrical smooth function 6(x)=[(x+1)(x+1)-4(x+)t (6)is the optimal wavelet basis function according to the lower errors and the higher precision criterion We can confirm the corresponding scaling function (x 9(x)=[(x+)2u(x+)-3(x+)u(x+)+3(x-)u( u(x According to the symmetrical characteristic(xdx=o j du <oo. is the Fourier transform of y(x), so y(x)is a wavelet. More over, the corresponding scaling functions (x) and y(x)can be confirmed. They are shown in Fig4 According to mallat's fast algorithm it needs not to calculate the wavelet transform and the algorithm can be achieved over the recursive calculation of the two filters got by the two-scale relations V(D,iEZ is a Multi Resolution analysis(MRA)of L, and the two-scale relation are q()=∑h"(x-k)分q2wy)=H(w)9"( 10) The calculation of the frequency response of filter H When m is odd H()= (21)2 sin c+(2w) (11) sinc"(w) 山国武技记文在线 http://www.paper.edu.cn (a) (b) Figure 4:(a) Constructed wavelet(b)Corresponding scaling function(c) Smooth function In which sin c(w) m+ 1 k≤ →h +h (12) When m is even H(w)=e2[ (13) m+1 1≤k h={2 k (14) othei The relation of wavelet m and scaling function n g, op(x (2) aisin (15) G(w) SIn Here the finite pulse response is not at the integer point, if shift y at 1/ 2, then the high-pass filter can be g (16) g; k≠0.k≠1 Till now the two filters for the wavelet transform are obtained The recursive formula is f(x)-∑hS21f(2x-k) 2(x)=∑g521/(2x-k) j=1,2,…J-1 (17 Sc f(r)is the original sample. If the original signal has n discrete samples, the complexity of the calculation is o(Nlog 2N 中国武技记又在线 http://www.paper.edu.cn The algorithm can easily be extended to two dimensions, so the two dimension wavelet transform can be obtained. Set 6(x, X y (18 6(x,y) V=(x,y)=,y Correspondingly w. f(r, y) (f*B,)(x,y) f*O)(x,y) (19) f(x,y)」 (f*B0(x,y) In scale s, the mode of the grad vector is M,f(x,y)=IWf(x,y)P+IWf(,D)P2 And the grad phase angle is f(x, y) f(x, y) For every scale parameter s, every point(x; y)in the two dimensions, in the direction of A f(x, y),if M/(r, y) is the local maximum, (x, y) is the sharp variation point of /*6(r, y), namely the sharp variation point of f(x, y) 5 Results of experiments 丝乡 Figure 5: The result with different operators (a)loG(b)Canny (c)our methoc Figure 6: Figure of merit, Q againsL SNR From Fig. 5 and Fig. 6 we can see the results with the methods in the citing literatures and our method According to the simulation results, we can see that our algorithm can extract the edge characteristics in the 国利技论又在线 http://www.paper.edu.cn condition of keeping the better details. So it can provide the more precise image information for the latter works, e.g. image registration and image fusion 6 Conclusion Pe. The traditional edge extraction algorithms are not perfect when the medical image contains noises. For the internal noise, the information measure algorithm combined with SA algorithm does very good. For the external noise we use wavelet domain HMT model to denoise. At last we design a new B-spline wavelet to extract the edges. According to the Canny optimal criterion, we prove that this algorithm can provide the best compromise between noise rejection and accurate edge localization. References [1 Mark Hedley, Hong Yan, Dov Rosenfeld, An improved Algorithm for 2-D Translational Motion artifact Correction, IEEE Transactions on Medical Imaging, vol 10, no 4, 1991, pp. 548-553 [2 J. Fan,D. K. Y. Yau, A. K. Elmagarmid, and W. G Aref, Automatic Image Segmentation by Integrating Color-edge Extraction and Seeded Region Growing, IEEE transactions on image processing, vol. 10, 2001, pp [3] Mitra Basu, Gaussian-based edge detection methods-A survey, IEEE trans. vol 32, no. 3, August 2002, pp. 252 260 14 A Khalil, A Aggoun and A Elmabrouk, Eage detection using Local Histogram Analysis", Proc. of SPIE, vol 5150,2003.pp.2141-2151 [5] Kazuya turuta, Zonghuang Yang, Yoshifumi Nishio, Akio Ushida, On Two Types of Network Topologies of Small-World Cellular Neural Network, RISP International Workshop on Nonlinear Circuit and Signal Processing (NCSP04).Mar,2004,pp.113-116. [6 N. Muranaka, S Kudoh, T. Ashida and M. Tokumaru, On the Mutltiple-Valued Image Contour Extraction Method Using Laplacian Filter, IEICE Transaction on Information and Systems, vol. J85-D-IL, no 10, Oct. 2002, pp 1503-1512 [7 KIrkpatrick, GelaLl, and Vecchi. "Optimization by Simulated Annealing", Science, vol. 220, nO 4598, May 1983 671-680 8 M. Hanmandlu, J. Scc, S. Vasikarla. Fuzzy edge detector usingentropy optimization. 2004 International Conference on Information Technology: Coding and Compuling. Las Vegas, Nevada, USA vOl. 1, 2004, pp 19 Wang Zhengyao, Digital image cdge extraction, Xi'an, Jiaotong University degree paper 2003 [10] M. Unser, A. Aldroubi, M. Eden, B-Spline Signal Processing: Part I--Theory, IEEE Transactions on Signal Processing. vol 41, no. 2, February 1993, pp. 821-833

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