ti. ). For the algorithms that do not need to predict for the unobserved data, there are mainly two ways to model this p
A Discriminative Method For Semi-Automated Tumorous Tissues Segmentation of MR Brain Images Yangqiu Song, Changshui Zhang, Jianguo Lee and Fei Wang State Key Laboratory of Intelligent Technology and Systems, Department of Automation, Tsinghua University, Beijing, China, 100084
Summary Model and Solution
Numerical Comparison
The prior: We use P (yN ) as the data-dependent prior instead of P (XN ). T K−1y yN 1 N P (yN ) = exp{− } (2) Z 2 where: K−1 = ∇∇yN (− log P (yN )) = ∆ (3)
For numerical comparison, we pick up the following methods: 1) Supervised method SVM [4]. 2) Unsupervised method spcectral clustering (SC) which uses the nystr¨om method [5]. 3) The interactive method GVF snake algorithm [2]. 4) SSGPI which is proposed in this paper.
The problem of tumorous tissues segmentation of MR brain images: • Tumorous tissues vary in size, shape and location. • They are also accompanied with edema, hemorrhage, necrosis and cystic components. • The boundaries of the tumorous tissues may be blurry. • There are a great many pixels (such as 256 × 256 × 124) for 3D MR images. • Segmentation is high computational complexity and large memory requirements. Methods of Segmentation
∆ = I − S is called normalized graph Laplacian in spectral graph theory, and the prior P (yN ) defines a Gaussian random field (GRF) on the graph. This is a Gaussian process, since for any finite selection of points, P (yN ) is Gaussian. The likelihood: 1−λ I{t6=0} + λI{t=0} P (t|y) = 1 + exp(−ty)
We regard the segmentation problem as a classification problem. Supervised and unsupervised methods: • Supervised methods require the scrupulous labeling work by doctors or experts, which is time consuming and costly. • Unsupervised methods are difficult to produce good result fully automatically. When the tumorous tissues are very small in the image, the unsupervised methods will attempt to segment the normal tissues, such as: gray matter, white matter and the cerebrospinal fluid. Our semi-supervised inductive methods: • Our approach uses the labeled data in one image and a subset of unlabeled data to classify the remains. • It can segment 3D data by sampling the unlabeled data from 3D images rather than by segmenting 2D images sequently. • This method is related to semi-automated segmentation and interactive segmentation.
1 M DR SV M 0.2744 SC 0.3383 GV F S 0.2898 SSGP I 0.2400
Slice52 F DR 0.0492 0.1432 0.2087 0.0150
3 M DR 0.2333 0.1914 0.5373 0.2594
Slice81 F DR 0.0460 4.9087 0.0294 0.0047
3D Segmentation Results
(5)
For computing P (tN +1|tN ) = RPrediction: P (tN +1|yN +1)P (yN +1|tN )dyN +1, we need to calculate the posterior: Ψ(yN , yN +1) = − log P (tN +1|yN +1) − log P (yN +1)
2 M DR 0.2433 0.2394 0.0981 0.1836
(4)
which is named as Extended Bernoulli Model (EBM). Training: By computing the mode of posterior P (yN |tN ) as the estimate of yN , which is the negative logarithm of P (yN |tN ) = P (yN, tN )/P (tN ), we define the function of yN as: Ψ(yN ) = − log P (tN |yN ) − log P (yN )
Slice60 F DR 0.0084 0.1192 0.0361 0.0023
The left columns of each figure are the original T1 weighted images. The middle columns are the hand-guided segmentation results. The right columns are the results of SSGPI.
(6)
which is minimized only with respect to yN +1.
Semi-Supervised Classification MRF and GRF
Most of the algorithms in Bayesian framework focus on the joint distribution P (xi, ti). For the algorithms that do not need to predict for the unobserved data, there are mainly two ways to model this probability: a) : b) :
P (XN , tN ) = P (XN |tN )P (tN ) P (XN , tN ) = P (tN |XN )P (XN )
A semi-supervised classification algorithm uses only partially labeled data to find a boundary between classes [1].
Figure 5: Patient 1, slices 60, 50.
(1)
This will lead to the following models:
Figure 3: A toy problem of semi-supervised classification. Only two points are labeled. Our algorithm can get a correct classification boundary.
Figure 1: Graphical Model of MRF.
Figure 6: Patient 2, slices 52, 44.
2D Segmentation Results
We compare the segmentation results with the methods: SVM, spectral clustering, graph cut and active contour.
Figure 7: Patient 3, slices 81, 93. References:
Figure 2: Graphical Model of GRF. Instead of using the direct process xi → ti, we use a latent variable yi to generate the process xi → yi → ti. yi = y(xi) is a function of xi, and yN = (y1, y2, ..., yN )T is the latent variable vector of input data.
Figure 4: “Patient 1” Segmentation Results. a) T1 weighted image. b) T2 weighted image. c) PD weighted image. d) Handmin. e) Handmid. f) Handmax. g) GVF Snake [2]. h) Lazy snapping [3]. i) SVM [4]. j) Spectral clustering [5]. k) Unpost-processed SSGPI. l) Post-processed SSGPI.
[1] X. Zhu (2005): Semi-Supervised Learning Literature Survey. Technical Report. [2] C. Xu and J. L. Prince: Snakes, Shapes and Gradient Vector Flow. IEEE Trans. on Image Proc., Vol. 7(3), pp. 359–369, 1998. [3] Y. Li, J. Sun, C. Tang and H. Shum: Lazy Snapping. SIGGRAPH, pp. 303-308, 2004. [4] Q. WU, W. Dou, Y. Chen and J. Constans: Fuzzy Segementaion of Cerebral Tumorous Tissues in MR Images via Support Vector Machine and Fuzzy Clustering. IFSA. [5] C. Fowlkes, S. Belongie, F. Chung and J. Malik: Spectral Grouping Using The Nystr¨om Method. IEEE Trans. on PAMI, Vol. 26(2), pp. 214-225, 2004.
