A novel probabilistic simultaneous segmentation and registration using level set

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
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  ANOVELPROBABILISTICSIMULTANEOUSSEGMENTATIONANDREGISTRATIONUSINGLEVELSET  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 We propose a new shape-based segmentation approach usingthe statistical shape prior and level sets method. The seg-mentation 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 takingin consideration the evolution of a contour represented by alevel set function. The evolution of the front gathers informa-tion from the image intensities and shape prior. The segmen-tation approach is applied in segmenting the vertebral bodiesin CT images. Our results shows that the technique is ac-curate and robust compared with the other alternative in theliterature.  Index Terms —  Simultaneous segmentation and registra-tion, vertebral body (VB). 1. INTRODUCTION LevelsetmethodswerefirstintroducedbyOsherandSethian[1].The level sets method presents several advantages over theparametric active contours. The contours represented by theLevel sets function may break or merge naturally duringthe evolution, and changes are automatically handled. An-other advantage is that the level set contour always remainsa function on a fixed grid, which allows efficient numericalschemes.Level sets method is one of the techniques used in theshape based segmentation which is an important complexproblem in computer vision, computer graphics and medicalimaging. In the shape based segmentation, embedding themodel into the image domain is the key issue and depends onthe registration of the given shape template to the image. Theshape registration problem is formulated such that a trans-formation that moves a point from a given shape to a targetone according to some dissimilarity measure [2] needs to beestimated. An active contour algorithm that can incorporateshape priors was introduced in [3]. Shape priors, in additionto the image gradient, are embedded into the energy functionof the contour. In [4] the distance function is used to im-plicitly represent open/closed shapes (structures). The signeddistance function is used for closed shapes.In this paper, our objective is to help to diagnose andtreat osteoporosis by accurate segmentation. Osteoporosis isa bone disease characterized by a reduction in bone mass, re-sulting in an increased risk of fractures. To diagnose the os-teoporosis accurately, the bone mineral density (BMD) mea-surements of the vertebral bodies (VBs) are required. There-fore, the accurate VB segmentation is an important step toidentify vertebral fractures and to measure BMDs. The verte-brae consists of the VB and spinal processes. In this paper 1 ,we are interested in CT images of the vertebral body. Limitedapproaches have been introduced to tackle the segmentationof spine bones such as in [7, 8].There are difficult segmentation challenges in spine com-puted tomography (CT) images for a good segmentation asshown in Fig 1. Also, exposure levels (X-ray tube amperageand peak kilovoltage), slice thickness, and volume of inter-est (VOI) effect the resolution of CT images. To overcomethose limitations, we propose a new shape based segmenta-tion method using level sets. We use a dissimilarity measurein the shape information which is analytically invariant un-der affine transformation. In some of previous publications(i.e. [2, 4]), registration parameters and weighting parametersof the shape model are usually estimated numerically usinggradient descent. This iterative optimization requires an ap-propriate tuning of the time step in order to guarantee a stableevolution. Also, experiments show that the order of updat-ing the different pose parameters and weighting vectors affectthe resulting segmentation process. In this paper, we use theintrinsic registration as in [5], which solves the common dis-advantages in the statistical dynamic shape information. Insection 2, we present the proposed method. Section 3 ex-plains the experiments, and compares our results with anotheralternative. 1 This work has been supported by Image Analysis, Inc., Columbia, Ken-tucky, USA. 2011 18th IEEE International Conference on Image Processing 978-1-4577-1303-3/11/$26.00 ©2011 IEEE2161  (a) (b) (c) (d) Fig. 1 .  Typical challenges for vertebrae segmentation. (a) Innerboundaries. (b) Osteophytes. (c) Bone degenerative disease. (d)Double boundary. 2. PROPOSED METHOD Intensity models may not be enough to obtain the optimumsegmentation. Hence, we propose a new shape based iterativesegmentation and registration method. In the first step, theMatched filter [9] is employed to detect the VB region auto-matically. This procedure eliminates the user interaction andimproves the segmentation accuracy. We tested the Matchedfilter using  3000  clinical CT images. The VB detection accu-racy is  97 . 6% . For more information, see our work in [10].In the second phase, we initialize the evolving contour onthe VB. Then, an iterative process which simultaneously doesthe segmentation and registration begins. In the segmenta-tion step, we use an improved level sets approach in whicha probabilistic shape model is integrated. To make the shapeprior to be invariant to the transformation, we register it to theevolving contour at each iteration. Our overall segmentationframework is given in Algorithm 1. The next section presentsthe proposed segmentation and registration framework. Algorithm 1  Proposed Framework  Given :Aninputimage( I  ), and80manuallysegmentedtrain-ing images. Objective : Minimizing the energy function ( E  ) to obtain theoptimum segmentation. Training Stage : Aligning training shapes and constructingthe probabilistic shape model. Testing Stage : Simultaneous segmentation and registration. 1 .  Detect the VB region in  I   using the Matched filter. 2 .  Iterative segmentation and registration algorithm to mini-mize the energy function in Eq. 4. 3 .  End the iteration until the energy minimization is saturatedor the iteration number is reached. 2.1. Shape Prior Reconstruction In our experiments, we use 80 training CT images (slices)which are obtained from 10 different patients and differentregions such as lumbar and thoracic bones. We obtain a prob-abilistic shape model we presented in [11]. First, VBs aremanually segmented (under the supervision an expert). Thenthe segmented VBs are aligned using  2 D registration. Thealigned training images are shown in Fig. 2. Finally, a shape Fig. 2 . Obtaining the shape prior image. Training CT slicesof different data sets. The last column shows the shape priorimage with variability region. 51015202500. Distace (Wave)       P     r     o      b     a      b      i      l      i      t     y     ObjectBackground (a) (b) Fig. 3 . (a) A sample projection of the shape prior image.The white color represents the object region,  O , the red con-tourrepresentstheobject/variabilitysurface, C OV  , theyellowwaves represent the iso-surfaces, C d x . (b) The probability of the object and background in the variability region, V  , respectto each iso-surface, C d x .image represented as  P  s  =  O  B   V   is generated. Thewhite color represents the object or the VB ( O ), the black color represents the background ( B  ), and the gray color rep-resents the variability region ( V  ).To model the shape variations in  V  , the distance proba-bilistic model is used. The distance probabilistic model de-scribes the object (and background) in the variability regionas a function of the normal distance  d x  =  min  x − c  (where c  ∈  C OV  ) from a pixel  x  ∈ V   to the VB/variability sur-face C OV  . Each set of pixels located at an equal distance  d  p from C OV   constitutes an iso-surface C d x  for C OV  . To esti-mate the marginal density of the VB, it is assumed that eachiso-surface C d x  is a normally propagated wave from C OV   asshown in Fig. 3(a). The probability of an iso-surface to be anobject decays exponentially as the discrete  d x  increases. TheVBdistance histogramisestimatedasfollows. Thehistogramentity of the object region at distance  d x  is defined as h ( d x  | O ) = M   i =1 K   ℓ =1  x ∈ C dx δ  ( x  ∈ O iℓ )  (1)where the indicator function  δ  ( A )  equals  1  when the condi-tion A istrue, andzerootherwise, M   isthenumberoftrainingdata sets,  K   is the number of CT slices of each data set, and O iℓ  is the VB region. The distance,  d x , is changed until thewhole distance domain in the variability region is covered. Inpixel-wise, this process can be done by obtaining the outer 2011 18th IEEE International Conference on Image Processing 2162  edge of the previous iso-surface. Then, the histogram is mul-tiplied with shape prior value which is defined as follows: π O  = 1 M  |V|  x ∈V  δ  ( x  ∈ O ) .  (2)The distance marginal density of the object region is calcu-lated as P  O ( d x ) =  h ( d x  | O )  π O M  | C d x |  .  (3)The same scenario is repeated to obtain the marginal den-sity of the background. An example of the distance marginaldensities of the object and background region is shown inFig. 3(b). In the next section, we explain our proposed seg-mentation method. 2.2. Segmentation The level set segmentation framework contains the movingfront, denoted by  C  , which is implicitly represented by thezero level of a higher dimensional function,  φ , that is:  C  ( t ) = { x/φ ( x,t ) = 0 } . The energy function of the segmentationcan be written as E   =  E  cv ( φ ) +  αE  shape ( φ )  (4)where α isaconstantwhichcontrolshowmuchwedependonthe probabilistic shape prior. The first energy (data penalty)term is based on the intensity of the testing image. The sec-ond term is based on the shape prior after registering it to theevolving contour to be invariant to the transformation param-eters. 2.2.1. Intensity information The first term is modelled using similar way in [12] as fol-lows: E  cv ( φ ) =   Ω ( I   − u + ) 2 Hφ ( x ) dx +   Ω ( I   − u − ) 2 (1 − Hφ ( x )) dx  +  v   Ω |∇ Hφ ( x ) | dx,  (5)where  φ  represents the signed distance function of the evolv-ing 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 probabilis-tic energy function in the level set method using previouslypresented shape model [11]. To register the shape model tothe evolving contour, we use the similar approach presentedin [5]. Each pixel in the shape prior has two probabilities forbeing  i ) an object and  ii ) a non-object. Our shape prior isembedded in the level sets function in order to obtain moreaccurate segmentation results and extract the spinal processesautomatically. The shape model is registered into the imagedomain by maximizing the probability of pixels inside thecontour belonging to the object space and the pixels outsidethe contour belonging to the non-object space. This approachleads to the following energy function: E  shape ( φ ) =   Ω (1 − P  O ( σ φ d x  +  µ φ )) Hφ ( x ) dx +   Ω (1 − P  B ( σ φ d x  +  µ φ ))(1 − Hφ ( x )) dx.  (6)where the translation and scaling parameters can be estimatedusing a similar method shown in [5] as µ φ  =   Ω xHφ ( x ) dx   Ω Hφ ( x ) dx ,  (7) σ 2 φ  =   Ω ( x − µ ) 2 Hφ ( x ) dx )   Ω Hφ ( x ) dx .  (8) 2.2.3. Gradient Descent flow of   φ The change of the level set function with time using the twoenergy function is calculated by the Euler-Lagrange with thegradient descent as : ∂φ∂t  =  − ∂E ∂φ  =  − ∂E  cv ∂φ  −  ∂E  shape ∂φ .  (9)The gradient of two energy terms is obtained as follows: ∂E  cv ∂φ  =  δ  ( φ )[( I   − u + ) 2 − ( I   − u − ) 2 − vdiv (  ∇ φ |∇ φ | )] ,  (10) ∂E  shape ∂φ  =  δ  ( φ )[ P  O ( σ φ d x  +  µ φ ) − P  B ( σ φ d x  +  µ φ )] .  (11) 3. EXPERIMENTS AND DISCUSSION To assess the accuracy and robustness of our proposed frame-work, we tested it using clinical data sets. The clinical datasets were scanned at  120 kV and  2 . 5 mm slice thickness. Allalgorithms 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. Tocompare the proposed method with other alternative, VBs aresubsequently segmented using the active appearance model(AAM) [13]. Finally, segmentation accuracy is measured foreach method using the ground truths (expert segmentation).To evaluate the results we calculate the percentage segmenta-tion accuracy ( A ) as follows: 2011 18th IEEE International Conference on Image Processing 2163  Table 1 . Accuracy of our VB segmentation on 40 CT images. Our AAM [13] Mean accuracy, % 95.59 86.15Min. accuracy, % 92.92 83.21Max. accuracy, % 99.40 89.15Stand. dev.,% 2.11 2.68 A % = 100 ∗ ( number of correctly segmented voxels ) Total 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 methodand the other alternative are shown. Figure 4 shows the seg-mentation results of the proposed framework. As we show inthe results, the spinal processes which are not required in theBMD measurements are eliminated automatically. We obtainvery high segmentation accuracy using our new probabilisticshape energy function. 4. CONCLUSION In this paper, we have presented a new simultaneous segmen-tation and registration framework. We tested our method onVBs in CT images. The probabilistic shape model has twoadvantages: i) the spinal processes is eliminated, ii) the reg-istration and segmentation errors are refined. We comparedthe results with the AAM method. Experiments on the datasets show that the proposed segmentation approach is moreaccurate and robust than other known alternative. 5. REFERENCES [1] S. Osher and J. A. Sethian, Fronts propogating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formu-lations, J. Comp. Phys. vol. 79, pp. 12-49, 1988.[2] N. Paragios, M. Rousson, and V. Ramesh, Matching DistanceFunctions: A Shape-to-Area Variational Approach for Global-to- Local Registration Proc. Seventh European Conf. ComputerVision 2002.[3] Y. Chen, H.D. Tagare, S. Thiruvenkadam, F. Huang, D. Wil-son, K.S. Gopinath, R.W. Briggs, and E.A. Geiser, Using PriorShapes in Geometric Active Contours in a Variational Frame-work Intl J. Computer Vision vol.50, no. 3, pp.315-328, 2002.[4] X. Huang, N. Paragios, and D. Metaxas, hape Registration inImplicit Spaces Using Information Theory and Free Form De-formations IEEE Trans. Pattern Analysis and Machine Intelli-gence vol. 28, no. 8, pp. 1303- 1318, 2006.[5] D. Cremers, S. J. Osher, S. Soatto, Kernel density estimationand intrinsic alignment for shape priors in level set segmenta-tion, International Journal of Computer Vision, vol. 69(3), pp.335-351, 2006.[6] www.cedarssinai.com Fig. 4 . Segmentation results of some clinical CT images. Theyellow color shows the contour of the segmented region. [7] A. Mastmeyer and K. Engelke and C. Fuchs and W. A. Kalen-der, A hierarchical 3D segmentation method and the definitionof vertebral body coordinate systems for QCT of the lumbarspine 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, identifica-tion, 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 patternrecognition 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 Verte-bral Bones from Volumetric CT Images Using Graph Cuts,ISVC’09, 2009.[11] M. S. Aslan, A. Ali, D. Chen, B. Arnold , A. A. Farag, andP. Xiang, 3D Vertebrae Segmentation Using Graph Cuts WithShape Prior Constraints, Proc. of 2010 IEEE International Con-ference on Image Processing, pp. 2193-2196, 2010.[12] T. F. Chan and L. A. Vese, A level set algorithm for minimiz-ing the Mumford-Shah functional in image processing, IEEEWorkshop Proceeding on and Level Set Methods in ComputerVision, 2001.. pp. 161-168, 2001.[13] T. F. Cootes and G. J. Edwards and C. J. Taylor, Active Ap-pearance Models, IEEE Transactions on Pattern Analysis andMachine Intelligence, vol.23(6), pp.681-685, 2001. 2011 18th IEEE International Conference on Image Processing 2164
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