Model-Based Multi-Object Segmentation via Distribution Matching

Transcription

1 Model-Based Mult-Object Segmentaton va Dstrbuton Matchng Danel Freedman 1, Rchard J. Radke 2, Tao Zhang 1, Yongwon Jeong 2, and George T.Y. Chen 3 1 Computer Scence Department, Rensselaer Polytechnc Insttute, Troy, NY Electrcal, Computer, and Systems Engneerng Department, Rensselaer Polytechnc Insttute, Troy, NY Radaton Oncology Department, Massachusetts General Hosptal, Boston, MA Abstract A new algorthm for the segmentaton of objects from 3D mages usng deformable models s presented. Ths algorthm reles on learned shape and appearance models for the objects of nterest. The man nnovaton over smlar approaches s that there s no need to compute a pxelwse correspondence between the model and the mage; nstead, probablty dstrbutons are compared. Ths allows for a faster, more prncpled algorthm. Furthermore, the algorthm s not senstve to the form of the shape model, makng t qute flexble. Results of the algorthm are shown for the segmentaton of the prostate and bladder from medcal mages. Keywords: deformable segmentaton, prostate segmentaton, shape and appearance model, medcal mage segmentaton. 1. Introducton The segmentaton of objects from 3D mages usng deformable models s both an mportant and nterestng problem n computer vson. It s mportant because of ts natural applcaton n the medcal arena; for example, segmentaton of tumors from CT or MRI mages can be crtcal n the treatment of cancer. On the other hand, t s nterestng because of the algorthmc challenges nherent n extractng deformable objects from real-world 3D mages. In the context of medcal magery, the key segmentaton-related challenges are the followng: Challenge #1: The objects of nterest are often dffuse and lack strong edges. Challenge #2: There are often many objects, both of nterest and not of nterest, wthn a small volume. Challenge #3: Many objects have farly smlar ntensty profles. Typcally, ths effect cannot be removed by smple pre-processng such as hstogram equalzaton. Challenge #4: Many of the objects are of roughly the same shape. For example, the prostate and bladder are both somewhat deformed spheres. The algorthm presented n ths paper uses learned models for both the shape and appearance of objects to acheve segmentaton; learnng both types of nformaton s the only reasonable way to deal wth all 4 challenges. Our algorthm s certanly not the frst algorthm to combne shape and appearance. However, exstng algorthms that use both shape and appearance models (such as [5]) requre a pxelwse correspondence between the model and the mage; ths correspondence s often very dffcult to compute, and can be extremely tme-consumng. Instead, our algorthm characterzes a model object by (a) ts shape and (b) a probablty dstrbuton of the ntenstes (or colours, textures) of the pxels wthn ts nteror. As a result, comparng a partcular model object to the mage s as smple as comparng two probablty dstrbutons. The algorthm allows the shape to evolve untl the optmal match s found. Furthemore, unlke several exstng algorthms, our algorthm does not requre a partcular form for the shape model; any parametrc shape model can be used. We used a smple PCA-based model n our experments, as ths s suffcent to acheve good results; however, a more sophstcated shape model can be used f the applcaton demands t. The remander of the paper s organzed as follows. Secton 2 revews the exstng lterature on segmentaton of 3D objects usng deformable models. Secton 3 s the heart of the paper; t derves the equatons that comprse the segmentaton algorthm. Secton 4 demonstrates the algorthm s effectveness n a medcal applcaton: segmentaton of the bladder and prostate from medcal magery. Fnally, Secton 5 concludes.

2 2. Pror Work Exstng algorthms for the segmentaton of 3D objects usng deformable models can be categorzed based on the type of learned nformaton they use. No learned nformaton. The man exemplar of ths type of approach s the tradtonal actve contour or snake [11]. More recent work (e.g. [1, 2, 13, 15]) has focused on geometrc curve evoluton, combned wth level sets [16], to allow for both topologcal changes to the object and greater numercal stablty. Standard actve contour methods that seek edges tend to have dffculty when the objects to be segmented are blurry or not sharply delneated from the background (such as the prostate n Secton 4 below). Extensons such as [4] try to segment on the bass of appearance by evolvng the surface based on smple crtera related to the ntenstes (or colours) of the pxels n ts nteror; these methods acheve greater accuracy and robustness at the cost of a major reducton n speed. In general, algorthms that do not use learned nformaton are constraned n terms of what they can segment; they wll have dffcultes wth each of the four challenges posed above. Learned shape models. Some researchers have augmented a level-set actve contour segmentaton algorthm wth a PCA term that bases the curve evoluton towards shapes that are judged to be more lkely based on the tranng set [14, 19]. Cremers et al. ncorporated a more sophstcated (non-pca) shape model [6]. Segmentaton of 3-D medcal mages has also been accomplshed by the coarse-to-fne deformaton of a shape-based medal representaton model, or m-rep [18, 21]. Algorthms that possess only a learned shape model wll typcally fall prey to challenges 3 and 4. Learned shape methods have also been extended to allow for smple (non-learned) models of appearance [22, 23]; for example, the ntenstes wthn the segmented areas may be forced to have means or varances sgnfcantly dfferent than the background. Learned appearance models. One verson of ths type of segmentaton nvolves the non-parametrc warpng of a target surface to a deformable atlas [10]. Contours from the atlas can then be transferred onto the target volume. Ths type of method wll have trouble wth challenges 2, 3, and 4, and can also be slow n hgh dmensons. Some actve contour models [17] assume that one has a probablstc characterzaton of appearance that s learned beforehand. Fnally, approaches related to the current paper, but for whch shape models are ether absent or a mnor component, have been used by the authors n prevous work [7, 26]. Other researchers [9] have used a smlar approach. Learned shape and appearance models. There are a varety of methods that model the shape and appearance of an object usng prncpal component analyss (PCA). The standard-bearer for such methods s the actve shape and appearance model of Cootes et al. [5], whch has been successfully appled to the three-dmensonal segmentaton of medcal volumes, ncludng magnetc resonance mages of the bran, heart, and artcular cartlage [8, 12, 25]. The man drawback of actve shape and appearance model has already been mentoned n Secton 1; they requre the computaton of a pxelwse correspondence between the model and the mage. We wll say more about ths n Secton The Segmentaton Algorthm In ths secton, we descrbe the heart of the algorthm: the procedure for fttng a combned shape-appearance model to an mage. Ths optmal fttng of the model results n the segmentaton of the mage. The basc dea s as follows. The shape s gven by a descrpton of the surface, or multple surfaces n the case of mult-object segmentaton. The appearance s descrbed by a probablty dstrbuton of some photometrc varable nsde the object of nterest, or multple dstrbutons n the case of mult-object segmentaton. A shape-appearance par s then gven by (surface, dstrbuton), and ths par s consdered suffcent to characterze an object for the sake of segmentaton. The learned model s a low-dmensonal manfold n the space of such pars. To verfy how well any partcular shape-appearance par matches the mage, we compute the emprcal dstrbuton of the photometrc varable nsde the shape wthn the mage; ths dstrbuton s then compared to the appearance model. We therefore evolve the shape of an object (or multple objects) untl the emprcal dstrbuton(s) best matches the model dstrbuton(s). In the remander of ths secton, we flesh out these deas Termnology In ths secton, we descrbe some of the notaton needed to defne the problem rgorously. In the case of sngle-object segmentaton, a model-nstance s descrbed by a (surface, dstrbuton) par. The dstrbuton s taken over some photometrc varable; n the experments we perform, ths varable s grayscale ntensty, however, t may also be colour or texture. Gven that the mage s dscrete-valued, we wll assume a probablty mass functon over the ntensty. However, all of the analyss below can easly be transferred to the case of probablty densty functons (whch mght be more relevant n the case of textures). In the case of mult-object segmentaton, a model-nstance wll be specfed by J (surface, dstrbuton) pars, one for each object. We assume each surface s a topologcal sphere, and may therefore be wrtten S : S 2 R 3. For convenence, we wll denote a pont on the surface usng a parametrzaton of S 2 as S(u); however, the partcular

3 parametrzaton chosen s unmportant. Let us denote the mage by I : R 3 {1,..., n}; the mage s pecewse constant, where the peces correspond to voxels (whch have postve volume). We denote the probablty dstrbuton by q = (q 1,..., q n ), where q = prob(i(x) = ); of course q 0 and q = 1. Thus, a model-nstance s gven by (S( ), q). The shape-appearance model s a lowdmensonal manfold n the space of such model-nstances; a d-dmensonal model s parametrzed by β R d, and we wll wrte (S( ; β), q(β)) (sometmes condensng (S( ; β) to S(β)). A partcular form for the shape and appearance model (S(β), q(β)) s dscussed n Secton 4; n the subsequent dervaton, the partcular form s unmportant. The goal s to fnd the partcular model-nstance,.e. the partcular β, for whch the model best matches the mage. In the followng secton, we descrbe a natural crteron for scorng such matches Segmentaton Crteron Gven a surface S, let p S be the dstrbuton (probablty mass functon) of ntenstes lyng nsde the surface S. Ths can be formally defned as follows. Let V be the volume nsde of S; that s, let S = V. In ths case, p S δ(i(x), )dx x V = x V dx (1) where δ(, j) = 1 f = j and 0 otherwse. We wll refer to p S as the emprcal dstrbuton correspondng the surface S. The goal of segmentaton s to fnd a regon n the mage that s most lke the model. That s, we would lke to fnd a model shape S(β) whose emprcal dstrbuton p S(β) most closely matches ts model dstrbuton q(β). In other words, the segmentaton can be posed as mn K(p S(β), q(β)) β where K s some sutable measure of dssmlarty between probablty dstrbutons. There are several obvous canddates for K from nformaton theory. We choose the Kullback-Lebler dvergence, K(p, q) = n =1 p log p q (2) Note a key feature of ths segmentaton algorthm: unlke other jont shape-appearance model-based algorthms, there s no need to fnd a correspondence between the pxels of the model and those of the mage. For example, n [5] fndng the correspondence would nvolve repeated 3D mage warpngs, the most tme-consumng and least rgorous part of that algorthm. In our case, by contrast, pxels are not compared drectly; nstead, dstrbutons are compared. Whle some nformaton s obvously lost n performng dstrbuton comparsons nstead of pxel-wse comparsons, we show that n relevant experments (n medcal mages) ths loss of nformaton does not adversely affect performance. Of course, dstrbuton comparson measures may be computed consderably faster than pxelwse correspondences. When we wsh to segment multple objects at once, our model s gven by J object descrptors {(S j ( ; β), q j (β))} J j=1, and the goal s then to solve mn β J K(p Sj(β), q j (β)) j=1 Note that there s a sngle parameter vector β that controls all of the objects; ths captures the noton that the objects shapes and appearances may be nterrelated. Although a more general verson of ths crteron mght be a weghted sum of Kullback-Lebler dvergences, we have found the unweghted crteron works well n practce Optmzaton of the Crteron We wsh to mnmze K(β) K(p S(β), q(β)) n the case of sngle object segmentaton. (We wll only deal wth the sngle object case n ths secton; the mult-object case follows straghtforwardly.) We wll solve for a local mnmum of the crteron va gradent descent,.e. From (2), we have that K β = n =1 dβ dt = K β where we have shortened p S(β) (1), so that p S(β) = [( 1 + log p ) p q β p ] q q β x V (β) p β = 1 V (3) to p and q (β) to q. From δ(i(x), )dx x V (β) dx ( ) N β p V β N S(β) V (β) In order to compute N / β and V/ β, we need to be able to determne dervatves of the form ψ/ β, where ψ = ν(x)dx. The varatonal dervatve of ψ wth x V (β) respect to the surface S (where S = V ) s gven by = ν(u)n(u), where n(u) s the normal to the surface δψ δs

4 at the pont S(u) (see, for example, [3]). It can be shown by a sort of generalzed chan rule that ψ β = ν(u) S (u; β)n(u; β)du u S β 2 where S/ β s a d 3 matrx (d = dm(β)). To smplfy future computatons, we ntroduce the notaton Γ(u; β) = 1 S (u; β)n(u; β) V (β) β so that ψ β = V (β)γ(u; β)ν(u)du u S 2 We have, therefore, that 1 N V β = 1 V V β = Γ(u; β)δ(i(s(u; β)), )du u S 2 Γ(u; β)du u S 2 Some smplfcaton gves [ K β = n ( Γ(u; β) δ(i(s(u; β)), ) 1 + log p ) ] du S q 2 =1 ( ) [ ] n Γ(u; β)du p + p log p n p q S q 2 =1 q =1 β = Γ(u; β)du + Γ(u; β) log p I(S(u;β)) du S 2 S q 2 I(S(u;β)) n p q Γ(u; β)du K(β) Γ(u; β)du S 2 S q 2 β =1 whch fnally yelds ( K β = Γ(u; β) log p ) I(S(u;β)) K(β) du S q 2 I(S(u;β)) n =1 p q q β (4) We wll use (4) to fnd a mnmum of the Kullback- Lebler dvergence wth respect to the model parameters, va gradent descent. In general, we cannot compute the ntegral n (4) analytcally; we must resort to a fnte element method. Ths s relatvely straghtforward, gven that the surface representaton we use s that of a mesh (smplcal complex). For any trangle of the mesh, the normal s fxed; furthermore, the trangles are chosen to be small enough so that nether S/ β nor I vares much over the trangle. As a result, we can approxmate the ntegral n equaton (4) by Γ(u t ; β) t T ( log p I(S(u t;β)) K(β) q I(S(ut;β)) where T s the set of trangles n the mesh, u t s a representatve pont on the trangle t (typcally the centrod), and a t s the area of the trangle t. ) a t 4. Applcaton: Prostate Radotherapy In ths secton, we present the results of our segmentaton algorthm appled to an mportant real-world problem n medcal magng: automatc contourng of organs from volumetrc computed tomography (CT) magery. A key advance n cancer treatment wth radaton has been the ntroducton of a new technology known as ntensty modulated radotherapy (IMRT). Ths s a computer-controlled method of delverng radaton usng several beams from dfferent angles that can precsely rradate a 3D target of nterest (e.g. a tumor) whle smultaneously avodng nearby radaton-senstve organs. The bottleneck n 3D IMRT systems s the sgnfcant amount of tme and human nterventon requred to delneate the tumor and nearby structures on each scan. A radaton oncologst can often take mnutes to outlne all of the structures of nterest n every axal CT slce. Performng ths operaton each of the tmes a patent s treated s tedous. Usng an uncompled MATLAB mplementaton on a modest machne (1.67 GHz AMD machne wth 448 MB RAM), our algorthm performs the same procedure n fve mnutes. Here, our focus s on prostate cancer, the second leadng cause of cancer death for Amercan men. The state of the art n computer vson algorthms appled to prostate segmentaton s consderably less advanced than for other stes. In the best hosptals across the country, the status quo s stll manual contourng. Several vson approaches have been presented for prostate segmentaton from ultrasound; one of the most effectve s an actve-contour-based method proposed by Shen, Zhan and Davatzkos [20]. The work most comparable to the algorthm descrbed here was recently proposed by Tsa et al. [22, 23] and appled to 3D MRI mages of the prostate. These algorthms do not use learned appearance models; the dsadvantage of usng only learned shape models has already been dscussed n Secton Learnng the Shape Model In order to mplement (4), we must have a shapeappearance model, (S(β), q(β)), deally learned from tranng data. Our tranng set conssted of 17 sets of 512x512x90 CT mages of the male pelvs from dfferent patents, n each of whch a radaton physcst had outlned the prostate and bladder. Our testng set conssted of 3 dfferent mage sets not n the tranng database. There s a substantal amount of shape varablty n ths nter-patent tranng data set. To capture ths varablty, we used a lnear shape model obtaned from prncpal component analyss, a common technque n the model-based segmentaton lterature; see for example [5]. The gen-

5 eral scheme s as follows: the surface of each object s represented as a mesh, and specfed by a vector whose elements are the x-, y-, and z-coordnates of each of the vertces. The vectors representng the organs are then stacked nto a combned vector. There s one such combned vector for each tranng mage, and PCA s performed on these combned vectors. In our case, ponts from homologous structures were automatcally extracted and put nto correspondence usng a varatonal-mplct-surface-based algorthm [24] that resamples every set of contours to have the same number of slces and the same number of ponts equally spaced around each contour. We resampled each object to have 20 slces wth 20 ponts on each contour, and constructed a 10-mode PCA model usng the 800 ponts from each set n the tranng database. We emphasze that f necessary, ths smple model could be replaced wth a more sophstcated shape model lke those dscussed n Secton 2, but the segmentaton equatons would reman the same Learnng the Appearance Model To form an appearance model q(β) from the tranng data, we could perform PCA on tranng hstograms. In fact, the vectors representng the hstograms could be appended to those representng the shapes, whch would yeld the desred jont model. However, there are two major problems wth ths approach. Frst, PCA on hstograms does not preserve the property that hstograms are postve and sum to 1. Second, a lnear combnaton of tranng hstograms often produces new hstograms very unlke any tranng example. Instead, we employ a dfferent approach, based on the dea that there wll be some overlap (perhaps small) between the ntal guess of the object s poston and ts true poston. Our goal should be to extract that secton of the ntal object volume whch overlaps wth the true object, and to form our model densty solely based on ths. Of course, t s not obvous how to extract ths overlappng volume. The followng heurstc s extremely successful n practce. For a gven test mage, the volume correspondng to the ntal object poston s dvded nto blocks; denote the set of blocks B = {b j }. For each block, we compute the hstogram, h(b j ); we then determne how smlar a partcular block s to the model by measurng ts closeness to each of the tranng hstograms, {q tran } (there s one such tranng hstogram for each tranng mage we receve). In partcular, we compute K j = mn K(h(b j ), q tran ) If such a value s low, then we know that the Kullback- Lebler dstance between the block s hstograms and at least one of the tranng hstograms s small; as a result, the block s lkely to belong to the true object. We can then rank order the blocks by ther K j values, and choose only the fracton α of the blocks wth the lowest K j values. These good blocks are then deemed to be part of the true object, and the model densty q can be computed as the hstogram of the unon of these blocks. Note hat α must be chosen to be less than the fracton of the ntal volume that overlaps the true volume; whle ths fracton s not known a pror, α = 0.25 produced good results n practce. In fact, the model densty as computed usng ths algorthm s often almost ndstngushable from the densty correspondng to the true poston of the object Results We ntalzed the model usng β = 0 n (3), that s, at the poston of the mean shape, and allowed t to converge. Note that ths s n contrast to other algorthms where rough manual placement s requred to guarantee convergence to the correct result. The algorthm ran successfully on three test mages; however, for space reasons, the result for only one of the test patents s pctured n Fgure 1. The model and segmentaton algorthms are fully three-dmensonal, but the results n Fgure 1 are represented as 2-D slces to ease vsualzaton. One can apprecate the dffculty of the segmentaton problem n ths context: the CT mages have relatvely low contrast n the area of nterest, and the prostate, and bladder have gray-scale ntenstes that are very smlar to each other as well as to structures n many other regons. 15 teratons of a dscrete verson of equaton (3) were requred; the correspondng runnng tme was just over 4 mnutes on a 1.67 GHz AMD machne wth 448 MB RAM. Therefore, we conclude that our segmentaton algorthm has substantal promse for the problem of rapd, automatc contourng. 5. Conclusons and Future Work We have demonstrated a segmentaton algorthm that matches a learned model of shape and appearance to an mage by comparng emprcal and model probablty dstrbutons. The algorthm produces good results for dffcult 3D medcal mages. Future work wll focus on coarse-tofne methods of samplng the model space, to ensure that the algorthm s not trapped n an ncorrect optmum. We also plan to nvestgate more sophstcated shape models that better ft our data. We wll shortly obtan a corpus of tranng data that ncludes CT volumes and contours that vary along both nterpatent and ntra-patent axes. That s, for each of many patents we wll have many examples of that patent s bodly state. Usng ths data, we plan to learn a model of nter-

6 Fgure 1. Segmentaton results for slces 28, 34, 42, and 48 of patent The top row shows the ntal boundares for the segmentaton (correspondng to the mean shape). The bottom row shows the segmentaton result at convergence (red) versus the hand-drawn ground-truth contours suppled by a radaton physcst (blue). The top contour s the bladder and the bottom contour s the prostate. Note that both organs are not vsble n every slce. and ntra-patent varaton that s more sutable for mageguded therapy applcatons than the nter-patent model descrbed here. We expect ntra-patent data to have much less nherent varablty than nter-patent data, and hence for our segmentaton algorthms to perform more accurately. Acknowledgments Ths work was supported n part by the US Natonal Scence Foundaton, under the award IIS , and by CenSSIS, the NSF Center for Subsurface Sensng and Imagng Systems, under the award EEC References [1] V. Caselles, F. Catte, T. Coll, and F. Dbos. A geometrc model for actve contours n mage processng. Numer. Math., 66:1 31, [2] V. Caselles, R. Kmmel, and G. Sapro. On geodesc actve contours. Int. J. Comput. Vs., 22(1):61 79, [3] A. Chakraborty, L. Stab, and J. Duncan. Deformable boundary fndng n medcal mages by ntegratng gradent and regon nformaton. IEEE Trans. Medcal Imagng, 15(6): , [4] T. Chan and L. Vese. Actve contours wthout edges. IEEE Trans. Image Proc., 10(2): , [5] T. Cootes and C. Taylor. Statstcal models of appearance for medcal mage analyss and computer vson. In Proc. SPIE Medcal Imagng, [6] D. Cremers, T. Kohlberger, and C. Schnorr. Nonlnear shape statstcs n mumford-shah based segmentaton. In Proc. European Conf. Computer Vson, pages , [7] D. Freedman and T. Zhang. Actve contours for trackng dstrbutons. IEEE Trans. Image Processng, Accepted, to appear. [8] A. Hll, A. Thornham, and C. Taylor. Model-based nterpretaton of 3-D medcal mages. In Proceedngs of 4th Brtsh Machne Vson Conference, pages , September [9] S. Jehan-Besson, M. Barlaud, G. Aubert, and O. Faugeras. Shape gradents for hstogram segmentaton usng actve contours. In Proc. Int. Conf. Computer Vson, pages , Nce, [10] S. Josh. Large Deformaton Dffeomorphsms and Gaussan Random Felds for Statstcal Characterzaton of Bran Submanfolds. PhD thess, Washngton Unversty, [11] M. Kass, A. Wtkn, and D. Terzopoulos. Snakes: actve contour models. In Proc. Int. Conf. Computer Vson, London, June [12] A. Kelemen, G. Szekely, and G. Gerg. Elastc modelbased segmentaton of 3-D neuroradologcal data sets. IEEE Transactons on Medcal Imagng, 18(10): , October [13] S. Kchenassamy, A. Kumar, P. Olver, and A. Tannenbaum. Gradent flows and geometrc actve contour models. In Proc. Int. Conf. Computer Vson, pages , Cambrdge, [14] M. Leventon, E. Grmson, and O. Faugeras. Statstcal shape nfluence n geodesc actve contours. In Proceedngs of CVPR 2000, [15] R. Mallad, J. Sethan, and B. Vemur. Shape modellng wth front propagaton: A level set approach. IEEE Trans. Pattern Anal. Machne Intell., 17: , 1995.

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