A Novel Probabilistic Simultaneous Segmentation and Registration ...

2011 18th IEEE International Conference on Image Processing

A NOVEL PROBABILISTIC SIMULTANEOUS SEGMENTATION AND REGISTRATION USING LEVEL SET Melih S. Aslan, Eslam Mostafa, Hossam Abdelmunim∗ , Ahmed Shalaby, Aly A. Farag, and Ben Arnold∗∗ Computer Vision and Image Processing Lab., University of Louisville, Louisville, KY, 40299, USA ∗ Computer and Systems Engineering Department, Ain Shams University, Cairo, Egypt ∗∗ Image Analysis, Inc., 1380 Burkesville St., Columbia, KY, 42728, USA ABSTRACT

distance function is used for closed shapes.

We propose a new shape-based segmentation approach using the statistical shape prior and level sets method. The segmentation depends on the image information and shape prior. Training shapes are grouped to form a probabilistic model. The shape model is embedded into the image domain taking in consideration the evolution of a contour represented by a level set function. The evolution of the front gathers information from the image intensities and shape prior. The segmentation approach is applied in segmenting the vertebral bodies in CT images. Our results shows that the technique is accurate and robust compared with the other alternative in the literature. Index Terms— Simultaneous segmentation and registration, vertebral body (VB).

In this paper, our objective is to help to diagnose and treat osteoporosis by accurate segmentation. Osteoporosis is a bone disease characterized by a reduction in bone mass, resulting in an increased risk of fractures. To diagnose the osteoporosis accurately, the bone mineral density (BMD) measurements of the vertebral bodies (VBs) are required. Therefore, the accurate VB segmentation is an important step to identify vertebral fractures and to measure BMDs. The vertebrae consists of the VB and spinal processes. In this paper1 , we are interested in CT images of the vertebral body. Limited approaches have been introduced to tackle the segmentation of spine bones such as in [7, 8].

There are difficult segmentation challenges in spine computed tomography (CT) images for a good segmentation as shown in Fig 1. Also, exposure levels (X-ray tube amperage 1. INTRODUCTION and peak kilovoltage), slice thickness, and volume of interest (VOI) effect the resolution of CT images. To overcome Level set methods were first introduced by Osher and Sethian [1]. those limitations, we propose a new shape based segmentaThe level sets method presents several advantages over the tion method using level sets. We use a dissimilarity measure parametric active contours. The contours represented by the in the shape information which is analytically invariant unLevel sets function may break or merge naturally during der affine transformation. In some of previous publications the evolution, and changes are automatically handled. An(i.e. [2, 4]), registration parameters and weighting parameters other advantage is that the level set contour always remains of the shape model are usually estimated numerically using a function on a fixed grid, which allows efficient numerical gradient descent. This iterative optimization requires an apschemes. propriate tuning of the time step in order to guarantee a stable Level sets method is one of the techniques used in the evolution. Also, experiments show that the order of updatshape based segmentation which is an important complex ing the different pose parameters and weighting vectors affect problem in computer vision, computer graphics and medical the resulting segmentation process. In this paper, we use the imaging. In the shape based segmentation, embedding the intrinsic registration as in [5], which solves the common dismodel into the image domain is the key issue and depends on advantages in the statistical dynamic shape information. In the registration of the given shape template to the image. The section 2, we present the proposed method. Section 3 exshape registration problem is formulated such that a transplains the experiments, and compares our results with another formation that moves a point from a given shape to a target alternative. one according to some dissimilarity measure [2] needs to be estimated. An active contour algorithm that can incorporate shape priors was introduced in [3]. Shape priors, in addition to the image gradient, are embedded into the energy function 1 This work has been supported by Image Analysis, Inc., Columbia, Kenof the contour. In [4] the distance function is used to imtucky, USA. plicitly represent open/closed shapes (structures). The signed

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(a) (b) (c) (d) Fig. 1. Typical challenges for vertebrae segmentation. (a) Inner

boundaries. (b) Osteophytes. (c) Bone degenerative disease. (d) Double boundary.

Fig. 2. Obtaining the shape prior image. Training CT slices of different data sets. The last column shows the shape prior image with variability region. 0.9

Object Background

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2. PROPOSED METHOD Intensity models may not be enough to obtain the optimum segmentation. Hence, we propose a new shape based iterative segmentation and registration method. In the first step, the Matched filter [9] is employed to detect the VB region automatically. This procedure eliminates the user interaction and improves the segmentation accuracy. We tested the Matched filter using 3000 clinical CT images. The VB detection accuracy is 97.6%. For more information, see our work in [10]. In the second phase, we initialize the evolving contour on the VB. Then, an iterative process which simultaneously does the segmentation and registration begins. In the segmentation step, we use an improved level sets approach in which a probabilistic shape model is integrated. To make the shape prior to be invariant to the transformation, we register it to the evolving contour at each iteration. Our overall segmentation framework is given in Algorithm 1. The next section presents the proposed segmentation and registration framework. Algorithm 1 Proposed Framework Given: An input image (I), and 80 manually segmented training images. Objective: Minimizing the energy function (E) to obtain the optimum segmentation. T raining Stage: Aligning training shapes and constructing the probabilistic shape model. T esting Stage: Simultaneous segmentation and registration. 1. Detect the VB region in I using the Matched filter. 2. Iterative segmentation and registration algorithm to minimize the energy function in Eq. 4. 3. End the iteration until the energy minimization is saturated or the iteration number is reached.

Probability

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In our experiments, we use 80 training CT images (slices) which are obtained from 10 different patients and different regions such as lumbar and thoracic bones. We obtain a probabilistic shape model we presented in [11]. First, VBs are manually segmented (under the supervision an expert). Then the segmented VBs are aligned using 2D registration. The aligned training images are shown in Fig. 2. Finally, a shape

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(a)

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Fig. 3. (a) A sample projection of the shape prior image. The white color represents the object region, O, the red contour represents the object/variability surface, COV , the yellow waves represent the iso-surfaces, Cdx . (b) The probability of the object and background in the variability region, V, respect to each iso-surface, Cdx . S S image represented as Ps = O B V is generated. The white color represents the object or the VB (O), the black color represents the background (B), and the gray color represents the variability region (V). To model the shape variations in V, the distance probabilistic model is used. The distance probabilistic model describes the object (and background) in the variability region as a function of the normal distance dx = minkx − ck (where c ∈ COV ) from a pixel x ∈ V to the VB/variability surface COV . Each set of pixels located at an equal distance dp from COV constitutes an iso-surface Cdx for COV . To estimate the marginal density of the VB, it is assumed that each iso-surface Cdx is a normally propagated wave from COV as shown in Fig. 3(a). The probability of an iso-surface to be an object decays exponentially as the discrete dx increases. The VB distance histogram is estimated as follows. The histogram entity of the object region at distance dx is defined as h(dx | O) =

2.1. Shape Prior Reconstruction

0.6

M X K X X

δ(x ∈ Oi` )

(1)

i=1 `=1 x∈Cdx

where the indicator function δ(A) equals 1 when the condition A is true, and zero otherwise, M is the number of training data sets, K is the number of CT slices of each data set, and Oi` is the VB region. The distance, dx , is changed until the whole distance domain in the variability region is covered. In pixel-wise, this process can be done by obtaining the outer


edge of the previous iso-surface. Then, the histogram is multiplied with shape prior value which is defined as follows: πO =

1 X δ(x ∈ O). M |V|

(2)

x∈V

The distance marginal density of the object region is calculated as h(dx | O) πO . (3) PO (dx ) = M |Cdx | The same scenario is repeated to obtain the marginal density of the background. An example of the distance marginal densities of the object and background region is shown in Fig. 3(b). In the next section, we explain our proposed segmentation method. 2.2. Segmentation The level set segmentation framework contains the moving front, denoted by C, which is implicitly represented by the zero level of a higher dimensional function, φ, that is: C(t) = {x/φ(x, t) = 0}. The energy function of the segmentation can be written as E = Ecv (φ) + αEshape (φ)

(4)

where α is a constant which controls how much we depend on the probabilistic shape prior. The first energy (data penalty) term is based on the intensity of the testing image. The second term is based on the shape prior after registering it to the evolving contour to be invariant to the transformation parameters. 2.2.1. Intensity information The first term is modelled using similar way in [12] as follows: Z Ecv (φ) = (I − u+ )2 Hφ(x)dx Ω Z Z 2 |∇Hφ(x)|dx, (5) + (I − u− ) (1 − Hφ(x))dx + v Ω

Ω

where φ represents the signed distance function of the evolving contour, H is the Heaviside step function, and u− and u+ represent the mean intensity in the two regions. 2.2.2. Embedding shape prior information In this paper, our contribution is to propose a new probabilistic energy function in the level set method using previously presented shape model [11]. To register the shape model to the evolving contour, we use the similar approach presented in [5]. Each pixel in the shape prior has two probabilities for being i) an object and ii) a non-object. Our shape prior is

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embedded in the level sets function in order to obtain more accurate segmentation results and extract the spinal processes automatically. The shape model is registered into the image domain by maximizing the probability of pixels inside the contour belonging to the object space and the pixels outside the contour belonging to the non-object space. This approach leads to the following energy function: Z Eshape (φ) = (1 − PO (σφ dx + µφ ))Hφ(x)dx Ω Z + (1 − PB (σφ dx + µφ ))(1 − Hφ(x))dx.

(6)

Ω

where the translation and scaling parameters can be estimated using a similar method shown in [5] as R xHφ(x)dx µφ = RΩ , Hφ(x)dx Ω R (x − µ)2 Hφ(x)dx) 2 . σφ = Ω R Hφ(x)dx Ω

(7) (8)

2.2.3. Gradient Descent flow of φ The change of the level set function with time using the two energy function is calculated by the Euler-Lagrange with the gradient descent as : ∂E ∂Ecv ∂Eshape ∂φ =− =− − . ∂t ∂φ ∂φ ∂φ

(9)

The gradient of two energy terms is obtained as follows: ∇φ ∂Ecv = δ(φ)[(I − u+ )2 − (I − u− )2 − vdiv( )], (10) ∂φ |∇φ| ∂Eshape = δ(φ)[PO (σφ dx + µφ ) − PB (σφ dx + µφ )]. (11) ∂φ 3. EXPERIMENTS AND DISCUSSION To assess the accuracy and robustness of our proposed framework, we tested it using clinical data sets. The clinical data sets were scanned at 120kV and 2.5mm slice thickness. All algorithms are run on a PC 3Ghz AMD Athlon 64 X2 Dual, and 3GB RAM. All implementations are in C++. In this experiment, we use 40 testing CT images. To compare the proposed method with other alternative, VBs are subsequently segmented using the active appearance model (AAM) [13]. Finally, segmentation accuracy is measured for each method using the ground truths (expert segmentation). To evaluate the results we calculate the percentage segmentation accuracy (A) as follows:


Table 1. Accuracy of our VB segmentation on 40 CT images. Mean accuracy, % Min. accuracy, % Max. accuracy, % Stand. dev.,%

Our 95.59 92.92 99.40 2.11

AAM [13] 86.15 83.21 89.15 2.68

100 ∗ (number of correctly segmented voxels) . T otal number of VB voxels (12) The statistical analysis of our method is shown in the Table 1. In this table the results of the proposed segmentation method and the other alternative are shown. Figure 4 shows the segmentation results of the proposed framework. As we show in the results, the spinal processes which are not required in the BMD measurements are eliminated automatically. We obtain very high segmentation accuracy using our new probabilistic shape energy function. A% =

4. CONCLUSION In this paper, we have presented a new simultaneous segmentation and registration framework. We tested our method on VBs in CT images. The probabilistic shape model has two advantages: i) the spinal processes is eliminated, ii) the registration and segmentation errors are refined. We compared the results with the AAM method. Experiments on the data sets show that the proposed segmentation approach is more accurate and robust than other known alternative. 5. REFERENCES [1] S. Osher and J. A. Sethian, Fronts propogating with curvaturedependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys. vol. 79, pp. 12-49, 1988. [2] N. Paragios, M. Rousson, and V. Ramesh, Matching Distance Functions: A Shape-to-Area Variational Approach for Globalto- Local Registration Proc. Seventh European Conf. Computer Vision 2002. [3] Y. Chen, H.D. Tagare, S. Thiruvenkadam, F. Huang, D. Wilson, K.S. Gopinath, R.W. Briggs, and E.A. Geiser, Using Prior Shapes in Geometric Active Contours in a Variational Framework Intl J. Computer Vision vol.50, no. 3, pp.315-328, 2002. [4] X. Huang, N. Paragios, and D. Metaxas, hape Registration in Implicit Spaces Using Information Theory and Free Form Deformations IEEE Trans. Pattern Analysis and Machine Intelligence vol. 28, no. 8, pp. 1303- 1318, 2006. [5] D. Cremers, S. J. Osher, S. Soatto, Kernel density estimation and intrinsic alignment for shape priors in level set segmentation, International Journal of Computer Vision, vol. 69(3), pp. 335-351, 2006. [6] www.cedarssinai.com

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Fig. 4. Segmentation results of some clinical CT images. The yellow color shows the contour of the segmented region. [7] A. Mastmeyer and K. Engelke and C. Fuchs and W. A. Kalender, A hierarchical 3D segmentation method and the definition of vertebral body coordinate systems for QCT of the lumbar spine Medical Image Analysis. vol. 10, no. 4, pp. 560-577, 2006. [8] T. Klinder, J. Ostermann, M. Ehm, A. Franz, R. Kneser, C. Lorenz, Automated model-based vertebra detection, identification, and segmentation in CT images, Medical Image Analysis, vol. 13, pp. 471-482, 2009. [9] B. V. K. V. Kumar, M. Savvides, and C. Xie, Correlation pattern recognition for face recognition, Proceedings of the IEEE, vol. 94, no. 11, pp. 1963-1976, 2006. [10] M. S. Aslan, A. Ali, H. Rara, B. Arnold , A. A. Farag, R. Fahmi, and P. Xiang, A Novel 3D Segmentation of Vertebral Bones from Volumetric CT Images Using Graph Cuts, ISVC’09, 2009. [11] M. S. Aslan, A. Ali, D. Chen, B. Arnold , A. A. Farag, and P. Xiang, 3D Vertebrae Segmentation Using Graph Cuts With Shape Prior Constraints, Proc. of 2010 IEEE International Conference on Image Processing, pp. 2193-2196, 2010. [12] T. F. Chan and L. A. Vese, A level set algorithm for minimizing the Mumford-Shah functional in image processing, IEEE Workshop Proceeding on and Level Set Methods in Computer Vision, 2001.. pp. 161-168, 2001. [13] T. F. Cootes and G. J. Edwards and C. J. Taylor, Active Appearance Models, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.23(6), pp.681-685, 2001.