Lecture Notes in Computer Science
Decision of Insert Position of a Scale Free Network
Download 12.42 Mb. Pdf ko'rish
|
- Bu sahifa navigatsiya:
- Fig. 4.
- Second”
- 5.2 Evaluation of MAM with Scale Free Network in Both Stages
- Intensity Gradient Self-organizing Map for Cerebral Cortex Reconstruction
- 3.1 The Image Intensity Gradient
- 3.2 The Intensity Gradient Self-organizing Map (IGSOM) Model
|
5.1 Decision of Insert Position of a Scale Free Network The MAM has a two-stage recall process. Introducing a scale free network can be inserted to three locations (first stage, second stage, and both stages). Effective insert position of the scale free network is determined by investigating their noise tolerance. Fig.4 shows noise tolerance of the MAM introducing the scale free network in the first stage, the second stage, and both stages. Here, threshold value is ‘1’. 0 20
60 80 100 0 2 4 6 8 10 Noise Rate[%] P er fe ct R ec al l R at e[ %] Normal First
Second Both
Fig. 4. Difference of Noise Tolerance of the MAM with Different Inserting Position of a Scale Free Network: “Normal “means a noise tolerance of fully connected MAM, “First” means introducing the scale free network in the first stage, “Second” in the second stage, and “Both” in both stages. Table 1. Noise Tolerance and Size of Memory Matrices Normal
First Stage
Second Stage
Both Stage
Average of Recall Rate[%] 30.7 34.9 34.5 34.8 * Noise
Tolerance ― 4%↑ 4%↑ 4%↑ Size of Memory Matrices
20000
10291
10291
578
* As the base, Noise tolerance is defined in case of under 10% one. Effectiveness of Scale Free Network to the Performance Improvement of a MAM 363 Next, the effect on the size of the memory matrices is investigated. Table 1. shows improvement of noise tolerance and size reduction of the memory matrices of the MAM with different inserting position of the scale free network. From results, there is a little difference in improvement on noise tolerance. But noise tolerance is superior to the normal MAM. The ordinary MAM has two fully connected networks. However, the MAM introducing scale free network in the both stages has two low connected networks.
Size of memory matrices is drastically reduced by introducing a scale free network in both stages. 5.2 Evaluation of MAM with Scale Free Network in Both Stages A construction of the scale free network is controlled by threshold value. Therefore we examine the effect that a difference of a threshold value gives improvements for the MAM employing the scale free network in both stages. Table 2. shows the noise tolerance and the size of the memory matrices of the MAM which is introduced the scale free network in both stages for different threshold values.
Threshold θ 2 1 0.1 0.01
Average of Recall Rate[%] 34.2
34.8 33.7
36.5 Size of Memory Matrices 202 578
1334 2128
* As the base, Noise tolerance is defined in case of under 10% one.
As table 2. shows, noise tolerance of each threshold value is not difference. As a result, threshold value has no effect on the noise tolerance. 6 Conclusion In this paper, we proposed a new MAM model using scale free network. The performance of the proposed model was confirmed by the autoassociation experiments for alphabet patterns. Inserting position and threshold value are determined by simulations. As the results, in the proposed model the perfect recall rate is improved about 4% and the size of the memory matrices is drastically reduced 97% with compared to the ordinary fully connected MAM. To design distribution of weight for improvement of noise tolerance and to use more experimental evidence in the various conditions are future works.
Excellence Program Animals and Robots (PI:T.Yamakawa)" granted in 2003 to Department of Brain Science and Engineering, (Graduate School of Life Science and Systems Engineering), Kyushu Institute of Technology by Japan Ministry of Education, Culture, Sports, Science and Technology. 364 T. Saeki and T. Miki References 1. Masuda, N., Miwa, H., Konno, N.: Phys.Rev. E 70(036124) (2004) 2. Barabási, A.-L., Albert, R.: Science 286, 509 (1999) 3. Albert, R., Barab´asi, A.-L.: Rev. Mod. Phys. 74, 47 (2002) 4. Watts, D., Strogatz, S.: Nature 393 (1998) 5. Ritter, G.X., Wilson, J.N., Davidson, J.L.: Comput, Vision Graphics Image Processing 49(3) (1990) 6. Ritter, G.X., Sussner, P., Diaz-de-Leon, J.L.: IEEE Trans.Neural Networks 9(2) (1998) 7. Won, Y., Gader, P.D.: Proceedings of the 1995 IEEE International Conference on Neural Networks, Perth, Australia, November 1995, vol. 4 (1995) 8. Won, Y., Gader, P.D., Coffield, P.: IEEE Trans. Neural Networks 8(5) (1997) 9. Davidson, J.L.: Image Algebra and Morphological Image Processing III. In: Proceedings of SPIE, San Diego, CA, July 1992, vol. 1769 (1992) 10. Davidson, J.L., Hummer, F.: IEEE Systems Signal Processing 12(2) (1993) M. Ishikawa et al. (Eds.): ICONIP 2007, Part I, LNCS 4984, pp. 365–373, 2008. © Springer-Verlag Berlin Heidelberg 2008 Intensity Gradient Self-organizing Map for Cerebral Cortex Reconstruction Cheng-Hung Chuang 1 , Jiun-Wei Liou 2,3, *, Philip E. Cheng 2 2 ,
and Cheng-Yuan Liou 3
1 Dept. of Computer Science and Information Eng., Asia University, Taichung, Taiwan 2 Institute of Statistical Science, Academia Sinica, Taipei, Taiwan 3 Dept. of Computer Science and Information Eng., National Taiwan Univ., Taipei, Taiwan needgem@stat.sinica.edu.tw
model based on the image intensity gradient for the reconstruction of cerebral cortex from MR images. The cerebral cortex reconstruction is important for many brain science or medicine related researches. However, it is difficult to extract deep cortical folds. In our method, we apply the SOM model based on the image intensity gradient to deform the easily extracted white matter surface and extract the cortical surface. The intensity gradient vectors are calculated according to the intensities of image data. Thus the proper cortical surface can be extracted from the image information itself but not artificial features. The simulations on T1-weighted MR images show that the proposed method is robust to reconstruct the cerebral cortex.
deformable surface models, active surface models, gradient vector field (GVF). 1 Introduction Recently, due to advanced magnetic resonance imaging (MRI) techniques, MR images, which reveal high spatial resolution and soft-tissue contrast, have great potential to be used in the analysis of cognitive neuroscience, diseases (e.g., epilepsy, schizophrenia, Alzheimer's disease, etc.), and researches into anatomical structures of human brains in vivo. Modern anatomical MRI studies on human brains have been concentrated on the cerebral cortex, which is a thin and folded layer of gray matter on the cerebral surface and contains dense neurons to control high cortical functions like language and information processing. Many studies have shown that the cortical thickness decreases or changes in association with neurodegenerative diseases and psychiatric disorders, e.g., schizophrenia, Alzheimer’s disease [1], and autism [2]. The cerebral cortex reconstruction nowadays is an on-going research field that can help other researches like brain mapping to explore human brain. Generally speaking, there are three major brain tissues which can be approximately partitioned and segmented in human brains, i.e., gray matter (GM), white matter (WM),
* Corresponding author.
366 C.-H. Chuang et al. and cerebral spinal fluid (CSF). The cerebral cortex is the thin and folded layer between GM/WM and GM/CSF interfaces [3, 4]. The cortex reconstruction means to extract GM/WM and GM/CSF boundaries and rebuild its surface. A lot of methods in the literature have been proposed to solve this problem [1–5]. These methods can be roughly categorized into stochastic and morphological types. The methods using a stochastic model [1, 4] employ labeled cortical mantle distance maps or intensity distance histograms related to the GM/WM interface so that the extraction of GM/CSF interface is needless. On the other hand, the morphological methods [2, 3, 5] apply the dilation of GM/WM interface to extract the accurate GM/CSF boundary surface since the obvious GM/WM interface is easily extracted. In the latter, a cortex is usually regarded as a double surface structure [3]. The exterior surface following the interior surface is deformed with some constraints to find out the GM/CSF interface. The deformable surface model is a powerful method to reconstruct the cortical surface. However, it is difficult to deform the surface with correct topology inside the highly folded and buried cortex with image noise. In [5], a deformable surface model based on the gradient vector flow (GVF) [6] is proposed to overcome this problem. The model using an energy minimizing function, a weighted combination of internal and external energy, is basically a three-dimensional (3-D) snake model. The internal energy controls the continuity force of surface itself, while the external energy governs the attraction force like image gradients that leads the surface. Nevertheless, the computational cost of minimizing the energy function becomes larger as number of points on the surface increases. In [7], the self-organizing map (SOM) model [8- 10] plus a layered distance map (LDM) is applied to deform the GM/WM interface to find out the GM/CSF interface in segmented MR images. The layered distance map is calculated according to the extracted WM surface and segmented GM. Unfortunately, the distance values inside the sulci are usually symmetric that does not match the real cortical thickness. In this paper, we propose an advanced method in which the SOM model based on the image intensity gradient is applied to deform the GM/WM interface to find out the cortical surface in segmented MR images. In the simulation and experiment, a two- dimensional (2-D) T1-weighted MR image and 3-D T1-weighted MRI data are used for test. In contrast with the results of previous method [7], our newly studies on T1- weighted MRI data show that the proposed method gains more precise results to reconstruct cerebral cortex. 2 The Problem In the cerebral cortex, there are many narrow and deep fissures called sulci. These concave parts sometimes contain invisible or unrecognizable CSF, which makes the reconstruction of cerebral cortex laborious. To easily understand the problem, a T1-weighted MR image shown in Fig. 1(a) is illustrated. This image shows a sulcus structure, i.e. the region of interest (ROI). Its segmented image is shown in Fig. 1(b), where the white color region represents WM, the gray color region indicates GM, and the black color region is CSF, other tissues, and background. It is easy to extract the boundaries of the segmented GM, as shown in Fig. 1(c). Now, the ROI image is enlarged and shown in Fig. 1(d). The ROI image is equalized to clearly display the Intensity Gradient Self-organizing Map for Cerebral Cortex Reconstruction 367 ideal boundary, as shown in Fig. 1(e). The extracted and ideal boundaries between Fig. 1(c) and Fig. 1(f) are obvious different and there should exist an interval within the sulcus. If the inner (WM) boundary line is deformed outward to extract the outer (GM) boundary according to the segmentation results, it will probably fail to catch the interval due to fewer extractable features inside the sulcus.
(a) (b) (c)
(d) (e) (f) Fig. 1. Illustration of a missing interval within a sulcus in a T1-weighted MR image: (a) the raw image and the ROI, (b) its segmented image and the ROI, (c) extracted boundary of ROI, (d) magnified ROI, (e) equalized ROI image with ideal boundary, (f) ideal boundary of ROI.
Since it is difficult to partition tissues inside sulci, the ideal GM surface is also hard to be extracted. One popular way is to dilate the GM/WM interface to extract the GM/CSF boundaries because the GM/WM boundaries are obvious and can be easily extracted. In our former method, this kind of dilation strategy is assisted by a layered distance map (LDM) [7]. The LDM is formed by calculating the connected voxels layer by layer from inner to outer surfaces within the segmented GM. Therefore, if the sulci are full of GM, distance values of LDM inside the sulci are symmetric and the detected interval is always in the center. To overcome this problem, the image intensity is employed in the new method. In a T1-weighted MR image, the intensity of WM, GM, and CSF is high, middle, and low, respectively. The GVF [6] is applied to obtain the intensity gradient flow between these three tissues. However, the intensity inhomogeneity should be removed prior to GVF processing. Fortunately,
368 C.-H. Chuang et al. Su et al. [11] proposed a wavelet-based bias correction method which is also independent of tissue classes and computationally simple. 3.1 The Image Intensity Gradient In Fig. 2, the principle of boundary extraction inside the sulci by using GVF is demonstrated. The figures show two types of sulcus profile, where curves represent the dynamic intensity, i.e., the horizontal axis means positions and the vertical one indicates intensity values. GM, WM, and CSF are segmented and arrows denote gradient vectors. In Fig. 2(b), an interval should exist but it cannot be detected by the segmentation. The cortical thickness a is not equal to b, i.e. the structure is asymmetric and the LDM cannot find the ideal boundary. However, from the direction of gradient vectors, the most possible boundary, the long dash line in Fig. 2(b), can be easily extracted.
(a) (b)
Fig. 2. Illustration of boundary extraction inside the sulci by using GVF: (a) the sulcus profile with a clear interval, (b) the sulcus profile with an obscure interval The intensity GVF is computed from the bias-corrected MR images defined as I(x,
GVF field is defined as the vector field V(x, y, z)=[u(x, y, z), v(x, y, z), w(x, y, z)] that minimizes the energy functional [6] 2 2 2 V f V f dxdydz ε μ = ∇ + ∇ − ∇ ∫∫∫
, (1)
where f represents the 3-D negative intensity map, i.e. f =–I , μ is a weighting parameter, and ▽ is the gradient operator, i.e. ( , , )
x y z ∂ ∂ ∂ ∂ ∂ ∂ ∇ =
.
(2) In Eq. (1), the first term controls the smoothing function while the second term governs the maintaining capability. The formulation of computing V by iteratively minimizing ε is given in [6]. The final iterative solution is 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2 ( )( ) ( )( ) ( )( ) n n n n x x y z n n n n y x y z n n n n z x y z u u u u f f f f v v v v f f f f w w w w f f f f μ μ μ + + + ⎧ − = ∇ − − + + ⎪ − = ∇
− − + + ⎨ ⎪ − = ∇
− − + + ⎩ , (3) Intensity Gradient Self-organizing Map for Cerebral Cortex Reconstruction 369 where
▽ 2 is the Laplacian operator, n is the iteration number, and f x , f y , and f z are
partial derivatives with respect to x, y, z, respectively. The initial conditions are set with
0 0 0 0 ( , , ) ( , , ) x y z V u v w f f f f = = ∇ = (4)
Basically, the main goal of this iterative process is to diffuse gradient properties to reduce image noise and form the vector flow all over the volume.
The SOM model, which is a nonlinear, ordered, and smooth function, is an effective algorithm for the mapping between the neuron model and input data sets. In our applications, it is desired to deform the GM/WM interface to find out the GM/CSF interface by the SOM model. The boundary voxels of GM/WM are defined as the neuron data set. The SOM model driven by intensity gradient, called IGSOM, is applied to move the neuron data within the segmented GM. Finally, the GM/CSF interface is reconstructed by the converged neuron data set. The proposed IGSOM model is described in the following. First, it is needed to initialize deformable mesh network M. Here we use the isosurface of GM/WM interface. The mesh network M is a deformable surface inside the segmented GM. The segmented GM is defined as a volume set G. The best matching function of SOM is modified and rewritten as a point function P(.), i.e. ( ) ( ),
and
, j j j j j P m m V m m G m M = + ∈ ∈
(5) where m j is the randomly selected mesh network node, j is the index of mesh node, and V(m
. The update function of SOM denotes ( 1)
( ) ( , )[ (
) ( )],
j j j j m k m k k H D k P m m k α ′ ′ ′ + = + −
(6) where
j m ′ including j m represents all neighbors around j m , k = 0, 1, 2…is the iteration number, ( )
[0,1) k α ∈ is the learning rate, and H is the neighborhood function which decreases when the distance metric D and iteration k increase. The update function is iteratively proceeded until the average variation of input data is less than a threshold value. The Gaussian function is usually applied to be the smoothing kernel, i.e. 2 ( , ) exp 2 ( ) D H D k k σ ⎛ ⎞ = − ⎜ ⎟ ⎝ ⎠
(7) where ( ) k σ is the standard deviation, i.e. the width of the smoothing kernel. The distance metric ( , )
D D j j ′ ≡ defines the distance from j m ′ along the surface to j m . When j ′ is equal to j, i.e. j j m m ′ = , D is zero and the neighborhood function H has the maximal value 1. |
