A B-Snake Model Using Statistical and Geometric Information ...

points. The method for re-construction of B-spline isa st**and**ard algorithm which can be found in [11].element of the ith attribute vector1 ≤ vs ≤ n/ 2.F i. Here,For our case, B-snake, these feature points can besimply replaced by the control points of B-snake, aswe know B-spline is affine-invariant. The attributevector for the ith control point, , is generated byadjacent control points. Please see the shadow areasin Figure 3.Some examples of shape alignment are shown inFigure 4.F iFigure 4 Some aligned results of B-snake model.Figure 2 B-spline with 40 control points4.2. Shape Alignment StrategyThe method used here for determining thecorrespondence between sets of data is obtained fromthe paper [9].In [9], a shape alignment algorithm is proposed byusing an affine -invariant feature. It is implemented toa set of feature points of piece-wise deformablemodel. These feature points are extracted directlyfrom the sample points, which are evenly distributedalong the given shape. For each feature point, anattribute vector, which is calculated by the areasformed by adjacent feature points, has been assignedto it. As the attribute vectors are affine-invariant,shape alignment could be achieved by an errorminimization process.4.3. Mapping the B-**Snake** to the Space Derivedfrom the Training SetIn paper [12], a snake deformation mechanism usingstatistical information to constrain the deformablemodel in the space of allowable (or likely)configuration was presented. In this mechanism, thesnake model seeks image boundaries with the similarshape structure of feature points of training set ratherthan only influenced by nearby edges. To implementthis algorithm in our B-snake model, the controlpoints of B-snake are treated as the feature pointsagain. We briefly describe the algorithm below.Given a set of the training vectors, { S} , the averagevector **and** the covariance matrix can beS meancalculated. Then, compute the eigenvectors of thecovariance matrix, **and** sort by the size of theircorresponding eigenvalues. The M eigenvectorscorresponding to M highest eigenvalues can beselected as the basis of the shape subspace of thetraining samples. Here, we stack theseM eigenvectors as a matrix H .The following formula is for fitting the control point'vector Q of the model to the control point vector Qof the aligned B-snake shape:'Q= T +(5)1⋅ T2⋅ Q T3where T = W1T 3−1=⋅ H,T = H2T ⋅W , **and**− −1( W − W ⋅ H ⋅ H T ) S mean1(6)Figure 3 Schematic representation of the concept ofthe “attribute vector” on the ith control point. Thearea of a triangle Q ] is used as the vsth[ i−vs] Q [ i] Q[ i+vsOnce obtaining the best control point vector'Q , we'can transform Q back to update the positions ofcontrol points in the current B-snake by using the

affine-transformation matrix Aalignon how to get A , see [12].4.4. Complete Algorithmalign. For more detailsThe complete algorithm using both affine invariantalignment **and** geometrical information is as follows:1. Get a reference modelmod{ Qeli, i = 0,1,..., N}. It canbe done by the method presented in Section 3.2. Initialize the B-snake asmod{ Qel i = 0,1, ..., N}.i,3. Deform the B-snake by using MMSE to minimizeexternal force (Section 3.2). If the iterationnumber exceeds a predefined number or externalforce among B-snake is below a define value, goto step 6.Aalign4. Align the current snake configuration with thest**and**ard model contour by using the affinetransformationmatrix calculated from thesnake to the model [9]. Then, stack the alignedsnake as a point vectoralign align align align T[ x0, y0xN, yN]Q = ,...,5. Map the control point vector into the new vectorQmodel,, which is described in'into the new vector Q using the statistical'Q = T1⋅T 2⋅Q+ T 3'Section 4.3. Then, transform Q back into theoriginal coordinate space of the snake via thealigninverse matrix of A **and** update the B-snake.Go to step 3.6. Stop.5. EXPERIMENTAL RESULTThe algorithm presented above has been simulated byMatlab codes **and** tested to real MR medical images.In our experiment, we used over 50 shapes to formthe training sets, **and** the number of control points forour B-snake is fixed to 40.Figure 5 shows some results of our B-snake model.The grays are the initial shapes of B-snake in eachimage, while the bright are the final results. Theseresults show that our B-snake model approaches tothe desired object precisely.6. CONCLUSIONWe have presented a B-spline snake model usingstatistical information for segmenting 2D complexshapes from the medical images. The obtained resultshave showed that this model can be used to achieve amore accurate segmentation **and** hence a refinedmodel.REFERENCES[1] M. Kass, A. Witkin, **and** D. Terzopoulos, “**Snake**s:Active Contour **Model**s,” in Int. J. Computer Vision,1(4):321-331, 1987.[2] T.F.Cootes, C.J. Taylor, D.H. Cooper, **and** J.Graham."Training models of shape from sets of examples", inProc. British Machine Vision Conference, pages 9-18,1992.[3] Baumberg, A. M., **and** Hogg, D. C. "Learning flexiblemodels from image sequences", in EuropeanConference on Computer Vision'94, vol. 1, Pg. 299-308, 1994.[4] Stammberger T, Rudert S, Michaelis M, Reiser M,Englmeier KH, "Segmentation of MR images with B-spline snakes: A multi-resolution approach using thedistance transformation for model forces", in CEURWorkshop Proceedings, vol. 12, 1998.[5] Mário A. T. Figueiredo, José M. N. Leitão, **and** Anil K.Jain, "Unsupervised Contour Representation **and**Estimation **Using** B-Splines **and** a MinimumDescription Length Criterion," in IEEE Transactionson Image Processing, vol. 9, no. 6, pp. 1075-1087,June, 2000.[6] Yue Wang, Eam Khwang Teoh, **and** Dinggang Shen,“Lane Detection **Using** B-**Snake**,” in IEEEInternational Conference on **Information**, Intelligence**and** Systems (ICIIS’99), Washington, DC, Nov. 1-3,1999.[7] Yue Wang, Dinggang Shen **and** Eam Khwang Teoh,“A Novel Lane **Model** for Lane Boundary Detection,”in IAPR Workshop on Machine Vision Applications,pp. 27-30, 1998.[8] Yue Wang, Dinggang Shen **and** Eam Khwang Teoh,“Lane Detection **and** Tracking **Using** B-**Snake**,”Revised for IEEE Transactions on IntelligentTransportation Systems for possible publication.[9] Horace H. S **and** Dinggang Shen, "An affine-invariantactive contour model (AI-snake) for model-basedsegmentation", Image **and** Vision Computing 16(2):135-146, 1998.[10] Yue Wang, Eam Khwang Teoh **and** Dinggang Shen,“Structure-Adaptive B-**Snake** for Segmenting ComplexObjects,” in International Conference on ImageProcessing (ICIP 2001), pp. 769-772, Thessaloniki,Greece, Oct 7 – 10, 2001.[11] Richard H. Bartels, John C. Beatty **and** Brain A.Barsky, An Introduction to Splines for Use inComputer Graphics **and** **Geometric** **Model**ing. MorganKaufmann: Los Altos, CA, 1987.[12] Dinggang Shen **and** Christos Davatzikos, "Anadaptive-focus deformable model using statistical **and**geometric information", in IEEE Trans. on PatternAnalysis **and** Machine Intelligence (PAMI), 22(8):906-913, August 2000.

Figure 5 Some segmentation results of MR brain images using B-snake model