A B-Snake Model Usng Statstcal and Geometrc Informaton - Applcatons to Medcal Images Yue Wang, Eam Khwang Teoh and Dnggang Shen 2 School of Electrcal and Electronc Engneerng, Nanyang Technologcal Unversty Nanyang Avenue, Sngapore 639798 2 Department of Radology, Johns Hopkns Unversty, Baltmore, MD 2287 Emal: ps263375g@ntuedusg eekteoh@ntuedusg dgshen@cbmvjhuedu ABSTRACT A B-snake model usng statstcs nformaton for segmentng 2D objects from medcal mages s presented n ths paper Based on our prevous research work[], a statstcal model s proposed for our B-snake model, n order to use avalable pror knowledge about the object shape beng studed Ths method allows the deformaton of B-snake to be nfluenced prmarly by the most relable matches Expermental results show that our method s robust and accurate n object contour extracton n medcal mages INTRODUCTION Object segmentaton s a very mportant procedure n mage analyss, computer vson, and medcal magng Many medal mage analyss applcatons, lke the measurement of anatomcal structures, requre pror segmentaton of the organ from the surroundng tssue Our specal nterest s the segmentaton of the ventrcle from magnetc resonance mages (MR) for further study The snake was orgnally developed by M Kass [] It was deformed by the external and the nternal forces From the orgnal phlosophy of snake, an alternatve approach s usng a parametrc B-splne representaton of the curve Such a formulaton of a deformable model allows for the local control and a compact representaton Moreover, ths formulaton has only less number of parameters to control and the smoothness requrement has been mplctly bult nto the model By the way, for the case where the pror knowledge s avalable, current researches make t possble to be embedded nto the snake model In ths paper, we are combned B-snake model wth a statstcal model, n order to get a better segmentaton results The structure of ths paper s arranged as follows In Secton 2, a revew of the exstng B-snake model and statstcal model s presented Secton 3 brefly ntroduces a B-snake model In Secton 4, the statstc model s gven to gude the B-snake deformaton The smulaton results are shown n Secton 5 Ths paper concludes n Secton 6 2 RELATED WORKS Cootes presented a pont dstrbuton model [2] for buldng flexble shape models The shape s represented by a set of labeled ponts The shapes are algned and the devatons from the mean are analyzed usng prncpal component analyss Unfortunately, the labeled ponts have to be chosen manually for each shape n the tranng set, t s very tme consumng Moreover, as the method works by modelng how dfferent labeled ponts tend to move together as the shape vares, f the labelng s ncorrect, wth a partcular pont placed at dfferent stes on each tranng shape, the method wll fal to capture shape varablty As an extenson research work to the pont dstrbuton model, Baumberg proposed a cubc B- splne model [3] for detectng and trackng the walkng pedestrans The control ponts of B-splne are treated exactly the same way as the labeled ponts of pont dstrbuton model [2] Ths method has been appled to a real-tme processng system
Stammberger [4] proposed a B-splne snake algorthm for the segmentaton of the knee jont cartlage from MR mages by usng a mult-resoluton approach As the external forces are generated by mage edges and the dstance transformaton of a standard model, the B-snake may not deform to a desred object, because there s no statstcal nformaton has been ncluded n ths algorthm Máro presented an approach to unsupervsed contour representatons and estmatons by usng B- splne [5] The problem s formulated n a statstcal framework wth the lkelhood functon beng derved from a regon-based mage model However, no any pror knowledge of the shape s used n ths model Wang [6][7][8] presented a B-snake based lane model for lane detecton The external forces n ths model are desgned based on the perspectve relatonshp of lane boundares on the mage plane However, although the results are good n lane detecton, the number of control ponts s fxed to three, ths lmts the capablty to descrbe the complex shape In ther later paper [], a structureadaptve B-snake model wth a strategy of adaptve control pont nserton was proposed for segmentng the complex structures n medcal mages Here, we present a B-snake model usng statstcal nformaton, t s an extenson of our prevous research work [] The detals are gven n the followngs r ( s) = g ( s ), where s The external energy term on r(s) s defned as E( r( Therefore, the total energy functon of the B-snake E( r( EB snake can be defned by ntegratng along the B-snake That s, (3) E B snake = E( r( ds (4) 32 Estmatng B-Snake Parameters by Image Data Based on the ntal locaton of the control ponts, the B-snake would be deformed to the studed object by Mnmum Mean Square Energy Approach (MMSE) wth an adaptve strategy of nsertng control ponts For more detals, please refer to our paper [] Fgure shows a result of usng B-snake for ventrcle extracton from MR mage 3 B-SPLINE SNAKE 3 A Close Cubc B-Snake Model A close cubc B-splne has n + control ponts T { = [ x y ], =,,, n}, and n + connected curve segments { g() s = ( u (), s v () s ), =, 2,, n + } Each curve segment s a lnear combnaton of four cubc polynomals by the parameter s, where s s normalzed between and ( s ) That s, ( ) mod( n+ ) mod( n+ ) g () s = M R() s, =,2,, n +, () ( + ) mod( n+ ) ( + 2) mod( n+ ) where 6 2 2 6 3 2 M () = [ ] 2 2 R s s s s 2 2 2 6 3 6 A B-snake s defned as follows: (2) Fgure B-snake usng 7 control ponts 4 B-SNAKE MODEL USING STATISTICAL INFORMATION In ths secton, we suggest a knowledge-based strategy for B-snake deformaton In order to be able to use statstcal nformaton to gude the B-snake deformaton, the correspondence problem between two shapes should be solved Frst we have to reconstruct B-snake wth a fx number of control ponts, and then fnd the correspond control pont between the tranng sets 4 B-splne Re-Constructon Re-construct the B-splne wth a fx number of control pont s based on a fx rato of whole length for each segment on splne curve Please see Fgure 2 for a example of 4 control ponts of B-snake, t s reconstructed from Fgure whch has 7 control
ponts The method for re-constructon of B-splne s a standard algorthm whch can be found n [] element of the th attrbute vector vs n/ 2 F Here, For our case, B-snake, these feature ponts can be smply replaced by the control ponts of B-snake, as we know B-splne s affne-nvarant The attrbute vector for the th control pont,, s generated by adjacent control ponts Please see the shadow areas n Fgure 3 Some examples of shape algnment are shown n Fgure 4 F Fgure 4 Some algned results of B-snake model Fgure 2 B-splne wth 4 control ponts 42 Shape Algnment Strategy The method used here for determnng the correspondence between sets of data s obtaned from the paper [9] In [9], a shape algnment algorthm s proposed by usng an affne -nvarant feature It s mplemented to a set of feature ponts of pece-wse deformable model These feature ponts are extracted drectly from the sample ponts, whch are evenly dstrbuted along the gven shape For each feature pont, an attrbute vector, whch s calculated by the areas formed by adjacent feature ponts, has been assgned to t As the attrbute vectors are affne-nvarant, shape algnment could be acheved by an error mnmzaton process 43 Mappng the B-Snake to the Space Derved from the Tranng Set In paper [2], a snake deformaton mechansm usng statstcal nformaton to constran the deformable model n the space of allowable (or lkely) confguraton was presented In ths mechansm, the snake model seeks mage boundares wth the smlar shape structure of feature ponts of tranng set rather than only nfluenced by nearby edges To mplement ths algorthm n our B-snake model, the control ponts of B-snake are treated as the feature ponts agan We brefly descrbe the algorthm below Gven a set of the tranng vectors, { S}, the average vector and the covarance matrx can be S mean calculated Then, compute the egenvectors of the covarance matrx, and sort by the sze of ther correspondng egenvalues The M egenvectors correspondng to M hghest egenvalues can be selected as the bass of the shape subspace of the tranng samples Here, we stack these M egenvectors as a matrx H The followng formula s for fttng the control pont vector of the model to the control pont vector of the algned B-snake shape: = T + (5) T2 T3 where T = W T 3 = H, T = H 2 T W, and ( W W H H T ) S mean (6) Fgure 3 Schematc representaton of the concept of the attrbute vector on the th control pont The area of a trangle ] s used as the vsth [ vs] [ ] [ + vs Once obtanng the best control pont vector, we can transform back to update the postons of control ponts n the current B-snake by usng the
affne-transformaton matrx A algn on how to get A, see [2] 44 Complete Algorthm algn For more detals The complete algorthm usng both affne nvarant algnment and geometrcal nformaton s as follows: Get a reference model mod { el, =,,, N} It can be done by the method presented n Secton 3 2 Intalze the B-snake as mod { el =,,, N}, 3 Deform the B-snake by usng MMSE to mnmze external force (Secton 32) If the teraton number exceeds a predefned number or external force among B-snake s below a defne value, go to step 6 A algn 4 Algn the current snake confguraton wth the standard model contour by usng the affnetransformaton matrx calculated from the snake to the model [9] Then, stack the algned snake as a pont vector algn algn algn algn T [ x, y x N, y N ] =,, 5 Map the control pont vector nto the new vector model,, whch s descrbed n nto the new vector usng the statstcal = T T 2 + T 3 Secton 43 Then, transform back nto the orgnal coordnate space of the snake va the algn nverse matrx of A and update the B-snake Go to step 3 6 Stop 5 EXPERIMENTAL RESULT The algorthm presented above has been smulated by Matlab codes and tested to real MR medcal mages In our experment, we used over 5 shapes to form the tranng sets, and the number of control ponts for our B-snake s fxed to 4 Fgure 5 shows some results of our B-snake model The grays are the ntal shapes of B-snake n each mage, whle the brght are the fnal results These results show that our B-snake model approaches to the desred object precsely 6 CONCLUSION We have presented a B-splne snake model usng statstcal nformaton for segmentng 2D complex shapes from the medcal mages The obtaned results have showed that ths model can be used to acheve a more accurate segmentaton and hence a refned model REFERENCES [] M Kass, A Wtkn, and D Terzopoulos, Snakes: Actve Contour Models, n Int J Computer Vson, (4):32-33, 987 [2] TFCootes, CJ Taylor, DH Cooper, and JGraham "Tranng models of shape from sets of examples", n Proc Brtsh Machne Vson Conference, pages 9-8, 992 [3] Baumberg, A M, and Hogg, D C "Learnng flexble models from mage sequences", n European Conference on Computer Vson94, vol, Pg 299-38, 994 [4] Stammberger T, Rudert S, Mchaels M, Reser M, Englmeer KH, "Segmentaton of MR mages wth B- splne snakes: A mult-resoluton approach usng the dstance transformaton for model forces", n CEUR Workshop Proceedngs, vol 2, 998 [5] Máro A T Fgueredo, José M N Letão, and Anl K Jan, "Unsupervsed Contour Representaton and Estmaton Usng B-Splnes and a Mnmum Descrpton Length Crteron," n IEEE Transactons on Image Processng, vol 9, no 6, pp 75-87, June, 2 [6] Yue Wang, Eam Khwang Teoh, and Dnggang Shen, Lane Detecton Usng B-Snake, n IEEE Internatonal Conference on Informaton, Intellgence and Systems (ICIIS 99), Washngton, DC, Nov -3, 999 [7] Yue Wang, Dnggang Shen and Eam Khwang Teoh, A Novel Lane Model for Lane Boundary Detecton, n IAPR Workshop on Machne Vson Applcatons, pp 27-3, 998 [8] Yue Wang, Dnggang Shen and Eam Khwang Teoh, Lane Detecton and Trackng Usng B-Snake, Revsed for IEEE Transactons on Intellgent Transportaton Systems for possble publcaton [9] Horace H S and Dnggang Shen, "An affne-nvarant actve contour model (AI-snake) for model-based segmentaton", Image and Vson Computng 6(2): 35-46, 998 [] Yue Wang, Eam Khwang Teoh and Dnggang Shen, Structure-Adaptve B-Snake for Segmentng Complex Objects, n Internatonal Conference on Image Processng (ICIP 2), pp 769-772, Thessalonk, Greece, Oct 7, 2 [] Rchard H Bartels, John C Beatty and Bran A Barsky, An Introducton to Splnes for Use n Computer Graphcs and Geometrc Modelng Morgan Kaufmann: Los Altos, CA, 987 [2] Dnggang Shen and Chrstos Davatzkos, "An adaptve-focus deformable model usng statstcal and geometrc nformaton", n IEEE Trans on Pattern Analyss and Machne Intellgence (PAMI), 22(8):96-93, August 2
Fgure 5 Some segmentaton results of MR bran mages usng B-snake model