2. Existing work about dissymmetry. 3. The computation of dissymmetry maps Symmetry, chirality and mid-plane

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1 Statstcal Analyss of Normal and Abnormal Dssymmetry n Volumetrc Medcal Images Jean-Phlppe Thron, Sylvan Prma, Gérard Subsol EPIDAURE Project, INRIA 2004, route des Lucoles, BP Sopha Antpols Cedex, France [thron][sprma][subsol]@sopha.nra.fr Nel Roberts MARIARC. Unv. of Lverpool P.O Box 147. Lverpool L69 3 BX, UNITED KINGDOM nel@lverpool.ac.uk Abstract We present a general method to study the dssymmetry of anatomcal structures such as the human bran. Our method reles on the estmate of 3D dssymmetry felds, the use of 3D vector feld operators, and T 2 statstcs to compute sgnfcance maps. We also present a fully automated mplementaton of ths method whch reles manly on the ntensve use of a 3D non-rgd nter-patent matchng tool. Such a tool s appled successvely between the mages and ther symmetrc versons wth respect to an arbtrary plane, both to realgn the mages wth respect to the md-plane of the subject and to compute a dense 3D dssymmetry map. Inter-patent matchng s also used to fuse the data of a populaton of subjects. We then descrbe three man applcaton felds: the study of the normal dssymmetry wthn a gven populaton, the comparson of the dssymmetry between two populatons, and the detecton of the sgnfcant abnormal dssymmetres of a patent wth respect to a reference populaton. Fnally, we present prelmnary results llustratng these three applcatons for the case of the human bran. Keywords: Asymmetry, Dssymmetry, Bran, Medcal Image Processng, Handedness 1. Introducton The Bauplan or organzatonal scheme of many anmal speces s based on blateral symmetry. Some organs appear n pars n the body, symmetrcal wth respect to the md-plane (lmbs, eyes, ears, antennas, etc.). Other organs are placed near the md-plane and are also approxmately symmetrcal (nose, tal, etc.). Such a symmetry s rather general for the human head, ncludng the bran and ts two also Focus Imagng, Sopha Antpols, France hemspheres, but not for organs such as the lver, that has no correspondng symmetrcal structure and s asymmetrcal. Symmetrcal anatomcal structures, or pared structures, are sometmes also dssymmetrcal 1 whch means that they are roughly symmetrcal but each of the two organs n a par can present a specalzaton and therefore a slghtly dfferent morphology. For the human bran, some normal morphologcal [6] and functonal (descrbed n the early work of Paul Broca) asymmetres are well known. However, due to the lack of precse morphometrc tools t s stll a controversal queston to know to what extent functonal asymmetres translate nto measurable morphologcal dssymmetres. The quantfcaton of abnormal dssymmetry can also be a powerful tool to detect abnormaltes. Ths s an alternatve to the comparson of an ndvdual to the average and standard devaton values measured n a populaton of normal specmens. Sometmes, the nter-ndvdual varatons n the normal populaton are so hgh (for example, bran ventrcle volume varatons) that they prevent a clear detecton of abnormaltes. In that case, comparng the relatve dssymmetry measures of a patent to a populaton can gve more relevant nformaton than comparng absolute szes (for example, comparng the rato of the volumes of the two lateral ventrcles nstead of comparng drectly the absolute ventrcle volumes). However, the normal and abnormal components of the dssymmetry must stll be dentfed n order to detect and quantfy the abnormalty tself. Hence populaton studes are also strongly needed n the analyss of the dssymmetry n a sngle patent to fnd statstcally sgnfcant relatve dfferences rather than absolute dfferences. In the followng, we present a new method to evaluate the normal and abnormal dssymmetry of symmetrcal organs such as the human bran. Our method allows for the 1 Dssymmetry (Merram-Webster) means a defcency of the symmetry whereas asymmetry means a lack of symmetry ( a- = wthout). We make ths mportant semantc dstncton n ths paper.

2 automatc detecton of the md-plane n the 3D mages and the realgnment of the mage wth respect to a fxed drecton. We show how to compute and fuse the dssymmetry nformaton of a populaton and also how to determne the regons whch are sgnfcantly dssymmetrcal (.e., wth respect to perfect symmetry). Then we show that the dssymmetry feld of several populatons can be measured and compared, and that the regons wth sgnfcantly dfferent dssymmetry can be outlned. Lastly, we present expermental results for a varety of questons such as the normal dssymmetry of human brans, the comparson of brans of left and rght handed people, or the comparson of a patent presentng a focal aphasa wth a normal populaton. 2. Exstng work about dssymmetry Asymmetry and dssymmetry have already been extensvely studed n the medcal feld (see for example [8] for a revew of some of these studes). There s for example a number of works dealng wth abnormal dssymmetry of the human bran n the case of schzophrena [9]. Here, we wll not dscuss the medcal outcome of these studes but the geometrcal aspects n these methods. In general, the defnton of homologous features between both sdes of the anatomcal structure s generally performed manually, whch s a tme consumng and tedous task, and creates a senstvty wth respect to the operator. Moreover, the geometrc representatons are often only based on the lengths, or wdths, of anatomcal structures vewed n projecton, such as the ventrcles lengths n the case of ar encephalographc studes [13]. In other cases, structures are studed ndependently slce by slce n MR mages or n cryogenc sectons, wth the underlyng assumpton that the slces are exactly perpendcular to the md-plane, and that there s no dfference n symmetry accordng to the axal drecton. Our method s new n that we attempt to use the geometrc nformaton present n the entre 3D mage. We apply 3D elastc matchng to match both sdes of the object and 3D vector feld analyss technques to perform the statstcal analyss. Ths s dfferent from methods where segmentaton tools are used ndependently to process both sdes of the object before the two sets of shapes are fnally compared. In our method, segmentaton s optonally used and only at the end of the process n order to present or synthesze the dssymmetry nformaton nto a few number of parameters (for example volume varaton measures of organs [19]). There are some smlartes between our work and [16]. In the latter, the md-plane s automatcally detected from the bran mages usng 2D snakes whch are propagatng through the slces to obtan a set of 2D md-lnes. A 3D plane s ftted to the set of md-lnes by a least squares technque and the 3D mage s realgned wth respect to t. The cortcal surface s also extracted usng a propagaton of 2D snakes, and the perpendcular dstance from the md-plane to the cortcal surface s measured for both sdes, leadng to a pctoral representaton of the md-plane, colored wth dssymmetry values. Fnally, the nformaton of several subjects are fused, usng a surface to surface matchng technque based on the cortcal surface. Our method s comparable to to [16] n that the mdplane s computed frst, then the mages are realgned and non-rgd matchng s used to perform nter-patents data fuson. However, several other aspects are very dfferent: we are usng a volumetrc matchng technque nstead of surface segmentaton and surface matchng. In partcular, we determne the symmetry plane by a least squares fttng from features matched n both object sdes nstead of tryng to detect the nter-hemspherc fssure of a bran. Our symmetry plane has therefore a dfferent defnton, much less senstve to the flatness of the nter-hemspherc fssure and, n fact, not at all specfc to bran mages. Another aspect s that our dssymmetry map s defned everywhere n the 3D volume (3D mage) whereas t s only defned n the md-plane n [16] (2D mage). Accordngly, nter-patents data fuson s really volumetrc, allowng for local analyss of the dfferences. Lastly, we wll see that we are able to ndcate effects such as relatve local expansons or atrophes, whereas only bran wdth dfferences can be measured n [16]. 3. The computaton of dssymmetry maps 3.1. Symmetry, chralty and md-plane Chralty s assocated wth symmetry: more precsely, two chral objects are symmetrcal wth respect to a plane but up to a rgd transform. For example, two hands are chral whch means that after a proper rgd placement (.e., by jonng them), they are approxmately symmetrcal wth respect to a plane. Such anatomcal structures have no symmetry plane per se, but we wll see that ther dssymmetry can be studed anyway, thanks to the 3D deformaton feld obtaned between the mage of one structure and a symmetrc verson of the correspondng chral structure. For some other structures, such as the bran, we can reasonably assume the exstence of a symmetry plane that we call the md-plane. As we wll see, ths constrant can be taken nto account explctly n the matchng process that determnes the correspondence between the two sdes of a symmetrcal object. Besdes, the mage of a symmetrcal object can be realgned, that s, the md-plane of ths object can be placed accordng to a gven arbtrary plane. 2

3 3.2. Automatc realgnment of a symmetrcal object Our realgnment method s based on the extensve use of non-rgd matchng tools developed to perform 3D nterpatent matchng. Examples of such tools can be found n [11, 4, 18]. For a gven mage I 1, we assume that the object s roughly symmetrcal and that a drecton approxmately perpendcular to the md-plane s known, whch s a reasonable assumpton for medcal mages such as 3D bran mages. (x; y; z) beng the prncpal axes of the 3D space, we assume for example that ths s the x axs. I1 P p K F1,2 p K(I1) K P p p" S P P It can be demonstrated (see proof n the annex) that P s gong through the barycenter G of the two sets of ponts fp ; p 00 g and that ts normal n s the egenvector assocated to the smallest egenvalue of the followng matrx I: I =X (p, G)(p 00, G)> (3) In partcular, ths plane P s not the plane whch nterpolates fp ; p 00 g. If we note R = S K, thenr s an affne rotaton whose rotaton axs s the ntersecton of planes P and P 0. Determnng S (3 parameters) s therefore equvalent to evaluatng the affne rotaton 2 R, havng a rotaton axs n P 0, that mnmzes the least squares dstance between fp g and fp 0 g or, n other words, that maxmzes the smlarty between I 1 and K(I 1 ). Ths gves other practcal ways to evaluate S: for example, R can be evaluated drectly by usng mutual nformaton mnmzaton technques (see [22, 15]) adapted to affne rotatons wth axes n P 0 and appled between I 1 and K(I 1 ). The symmetry plane P s the mdplane of the object n I 1. Fgure 1. Basc prncpal of the md-plane determnaton: the mage I 1 s transformed nto a symmetrcal mage K(I 1 ) wth respect to an arbtrary plane P 0. Then a pont to pont correspondance F 1;2 s computed between both mages, leadng to pars (p ;p 0 ). Applyng K to the p 0 ponts gves couples (p ;p 00 ) of correspondng ponts wth respect to the md-plane P, whch are used to compute the fnal symmetry S or equvalently the md-plane P tself. The frst step s to choose an arbtrary plane P 0 n the orgnal mage to compute a chral mage K(I 1 ) (see fgure 1). If t x s the number of voxels n the x drecton, ths plane can be: P 0 : x = t x =2 (1) A non-rgd technque, appled between I 1 and K(I 1 ), gves couples f(p ;p 0 )g of correspondng ponts (fp g I 1 and fp 0g K(I 1)). The couples f(p ;p 00 ))g where p 00 = K(p 0 ) represents therefore correspondng ponts between both sdes of the object (for example between the bran hemspheres). The second step s to compute a symmetry S, characterzed by ts plane P (whch means 3 parameters), that mnmzes a crteron C: C =X (S(p 00 ), p ) 2 =X (S K(p 0 ), p ) 2 (2) Fgure 2. Ths mage presents the result of the applcaton of our automatc 3D realgnment tool. On the left: coronal and axal vew on the same patent of the orgnal mage I 1.Onthe rght: the chral mage K(I 1 ).Onthemddle: the realgned mage R,1=2 (I 1 ). The vertcal whte lne s plane P 0 : x = t x =2. We can then demonstrate that R 1=2, the affne rotaton havng the same axs than R but half the rotaton angle, s 2 An affne rotaton s defned by 5 parameters: the rotaton axs, whch s a 3D lne (4 parameters) and a rotaton angle. The constrant that ths rotaton axs has to be wthn a gven plane P 0 reduces the number of free parameters to 3 only, exactly as t s the case for the symmetry S. 3

4 a rgd transform whose nverse (R,1=2 ) can be used to realgn the md-plan wth the arbtrary plane P 0 (see proof n the annex, and also fgure 3). An example of realgnment of a real mage s shown n fgure 2. f R s evaluated drectly (mage based mnmzaton technques), R 1=2 s convenently determned by decomposng R nto a translaton t and a vectoral rotaton represented by ts rotaton vector r. The rotaton n R 1=2 s then r=2 and the translaton s (r=2 +Id),1 (t). f the symmetry plane P s evaluated drectly (correspondng ponts technques), t s more convenent to determne R 1=2 from the ntersecton of the symmetry planes P of S and P 0 of K, and also from the angle between these two planes. Then a re-samplng method such as tr-lnear nterpolaton can be appled to transform mage I 1 nto the realgned mage I 0 1 = R,1=2 (I 1 ). P P I1 R -1 2 K K R 1 2 K(I1) Fgure 3. The transform R,1=2,whereR = S K and R 1=2 R 1=2 = R can be used to realgn the md-plane P wth the arbtrary plane P Dssymmetry feld computaton A practcal feature of most non-rgd nter-patent matchng technques s that the fnal result s senstve to the orgnal relatve poston of the two objects to match. To reduce ths dstance n the case of a symmetrcal object, we propose to compute the dssymmetry feld by applyng the non-rgd matchng technque between the realgned mage I1 0 and ts chral verson I 0 2 = K(I 0 1 ) nstead of drectly between I 1 and K(I 1 ). If the objects to compare are chral but not symmetrcal (hands for example) and maged separately (I lef t for the left hand and I rght for the rght hand), we propose to compute frst the non-rgd correspondence between I lef t and K(I rght ). From these correspondng ponts, we deduce a rgd transform R by a conventonal least squares method (usng for example a quaternon representaton of the rotatons or a rotaton vector representaton and Kalman flterng to reject outlers, see [2]). At last we re-sample one of the two mages wth R: I 0 1 = R,1 (I lef t ) s made supermposable to I 0 2 = K(I rght) or, more symmetrcally, we can re-sample both left and rght mages I 0 1 = R,1=2 (I lef t ) and I 0 2 = R1=2 (K(I rght )). IfR :(r; t) then we have stll: R 1=2 :(r=2; (r=2 +Id),1 (t)) (4) and R 1=2 R 1=2 = R. We note that we have exactly the same formulaton as n the case of a symmetrcal object, except that R s no longer constraned to be an affne rotaton but s a general rgd dsplacement (6 parameters). After the realgnment step, and for both cases (symmetrcal or smply chral), a dssymmetry feld s computed between I 0 1 and I Implementaton For our experments and for both the realgnment and the dssymmetry feld computaton, we have used a non-rgd matchng method based on demons (see [18]), whose output s a dense 3D deformaton feld F 1;2 between the two mages, that s, for each voxel p :(x; y; z) n mage I1,we 0 have three offsets (d x ;d y ;d z ) whch gve the correspondng pont p 0 : (x + d x;y + d y ;z + d z ) n K(I 0 ). A nce feature of ths algorthm s that t provdes a bjectve deformaton feld n the sense that t also computes an nverse deformaton feld F 2;1,whereF 2;1 F 1;2 s very close to dentty (not exactly equal because we are processng dscrete vector felds). 4. The analyss of dssymmetry felds We now dscuss multple ways to perform the analyss of dssymmetry felds. We frst concentrate on the type of nformaton whch can be obtaned from a sngle patent s mage and then on how statstcal analyses can be performed wth respect to one or several populatons The case of a sngle specmen Several dfferent vector feld operators can be appled to a dssymmetry feld n order to obtan a 3D scalar mage, whch can then be vsualzed. A smple one s the norm of the vector feld jjf jj, whch emphaszes ndstnctly many types of dssymmetry, dsplaced structures as well as shearngs, expansons, or atrophes. To be more specfc, dedcated operators can be used (see for example [7], [5]): for expansons or atrophes, we found n the case of temporal evoluton studes of lesons that an nterestng operator s jjf jjdv(f ), that s, the norm tmes the dvergence of the vector feld (see [19]). The dea s that the norm characterzes the magntude of the deformaton, whch holds also for large translatons, whle the dvergence characterzes ts radal aspect whch can also be mportant n nosy regons. The feature hgh dvergence, hgh magntude s very characterstc of atrophes or expansons due, 4

5 for example, to lesons or cancer growths. Our operator gves a very hgh response to such phenomena. We have also developed very precse stereologc methods to evaluate quanttatvely the volume varaton, agan for tme seres (also n [19]), that can be appled almost drectly to the case of dssymmetry feld analyss to evaluate quanttatvely the relatve szes of symmetrcal structures. Fgure 4 presents the result of the dssymmetry feld and jjf jjdv(f ) operator appled to a real patent. We have selected the regon of the temporal lobes for the dsplay (but the dssymmetry feld s really 3D) because ths regon s very dssymmetrcal n ths subject (a young rght handed healthy man) and, as we wll see, n the majorty of the subjects. In the jjf jjdv(f ) mage, whte represents an expanson, whch means a larger structure whle black represents a smaller structure and grey a symmetrcal structure. Hence the subject has a rght temporal lobe (on the left n the mage) larger than hs left temporal lobe, whch s a known normal dssymmetry n ths populaton (see for example [3]). Ths doesn t mean that some sub-structures of the temporal lobe are not larger on the left than on the rght, as t was sad prevously (see also [6]), but the total volume seems to be larger on the rght sde. Also, the dssymmetry seems to be located manly n the whte matter. However, as stated n the ntroducton, one has to establsh precsely the normal and abnormal components of the dssymmetry n order to provde a useful dagnoss. Ths means comparng a subject to a reference populaton. For example, we wll see later on that the dssymmetry of the temporal lobes that we observe for ths partcular subject s confrmed to be normal thanks to a comparson wth a database of 10 rght handed healthy men Inter-patent fuson Agan, non-rgd nter-patent matchng s used to perform data fuson between dfferent subjects, usng the same scheme as presented n [20]. A reference specmen s mage I r s chosen and realgned, and the deformaton felds F ;r from all the realgned mages I of specmens to the reference mage I r are computed. The dssymmetry felds of the realgned mages are then computed and an operator may be appled to ths feld before the result s projected onto the reference specmen s mage (see fgure 5). Once more, we have used the non-rgd matchng method descrbed n [18] to fuse the dfferent specmens, and we have studed ether the averaged dssymmetry vector feld or the averaged result of the jjf jjdv(f ) operator. The frst and second order statstcal parameters are computed for each voxel of the reference mage, usng the projected values of the whole populaton (mean and varance for jjf jjdv(f ), or mean and covarance matrx for F ). Fnally, ndvdual specmens or other populaton spec- Fgure 4. Ths fgure llustrates the dssymmetry feld computaton (left, norm of the feld) and the applcaton of the jjf jjdv(f ) operator (rght) on the realgned mage of a real subject (mddle). Note that the dssymmetry feld s a 3D mage (here, only a coronal and an axal secton of the same 3D mage are presented). Reference subject Realgnment Reference mage Subject 1 Realgnment Dssymetry feld computaton F dv(f) Averagedssymmetry Data fuson Subject n Sgnfcance map Fgure 5. Fuson of the data: the mages of all patents are realgned, and the dssymmetry feld and norm-tmes-dvergence operator are computed. Then all the dssymmetry maps are projected onto the realgn mage of a reference specmen. Lastly, the dssymmetry feld and the sgnfcance map can be compared pont by pont to the reference mage n order to determne whch anatomcal structure s sgnfcantly dssymmetrcal. mens can be projected onto the reference patent and compared wth the reference populaton statstcs to determne sgnfcant dfferences. 5

6 4.3. Statstcal maps and statstcal tests Dfferent types of questons can be addressed, leadng to dfferent statstcal maps and tests. The prncpal questons are: what regons n a gven populaton are sgnfcantly dssymmetrcal? what regons n a gven populaton have a sgnfcantly dfferent dssymmetry than smlar regons n another gven populaton? what regons of a gven specmen are sgnfcantly dfferent from the normal dssymmetry of a populaton? In each case, a probablty map can be computed va the applcaton of the nverse of the Fsher-Snedecor or F- functon to a Mahalanobs dstance or T 2 -value (see for example [1, 21]). We consder here the multvarate case where the samples are random vectors of dmenson p and are supposed to have a Gaussan dstrbuton. When dealng wth 3D dssymmetry felds, we have p =3and f the jjf jjdv(f ) operator s used then p =1(unvarate case) A sgnfcant dssymmetry The frst queston s typcal of pure anatomcal studes. The am can be for example to desgnate the regons of the bran n a populaton of rght handed young, healthy males whch are sgnfcantly dssymmetrcal (wth respect to perfect symmetry,.e., a null dssymmetry feld). Ths s a classcal multvarate analyss test. For a gven voxel x n the reference mage, the random vector for a specmen projected n x beng x, the average on the populaton of n specmens beng and the covarance matrx beng, we have: = npn 1 x =1 (5) = n,1pn 1 (x =1, ) > (x, ) The probablty of beng wrong n sayng that the populaton has a mean dfferent from 0 =0(.e. s dfferent from a perfect symmetry) s called the -value and s gven by the followng formula: = F p;n,p (n, p),1 (n, 1)p T 2 ( 0 ) (6) where: T 2 ( 0 )=n(, 0 ) >,1 (, 0 ) (7) The closer the -value s to zero, the more sgnfcant the dssymmetry. The -values computed for all voxels can be represented n a 3D mage. Settng a threshold 0 n ths mage (for example 0 =0:001) s equvalent to performng an Hotellng s test, that s, to determnng the voxels where T 2 s such that: T 2 >T 2 0 = (n, 1)p (n, p) F p;n,p( 0 ) (8) 0 s called the sgnfcance level of the test. It s, however, unfortunate to reduce the nformaton to only a bnary mage. To have a more pctoral representaton of the map of -values, we propose to dsplay the followng values: = 0 = f > 0, =1otherwse. The output s a 3D mage coded wth floatng pont values where the ntensty s between 0 and 1, and saturated ( = 1) when the dssymmetry s hghly sgnfcant. The Hotellng s test s then smply to determne the voxels =1 n such an mage. In the case of expanson/contracton, the sgn of the dvergence can be used to provde an addtonal nformaton on the nature of the dssymmetry, to lead to an mage wth 2 [0; 1], where = 0 (black) means sgnfcantly smaller (wth a sgnfcance level 0 ), = 0:5 (gray) means undetectable dssymmetry and =1:0 (whte) means sgnfcantly larger (wth a sgnfcance level 0 ) Sgnfcant dssymmetry dfferences between populatons Our second queston s typcal of pathologcal studes where, other parameters beng controlled, a populaton of n 1 pathologcal or atypcal subjects fx 1; g wth mean 1 s compared to a populaton of n 2 controls fx 2; g wth mean 2. It can be for example a populaton of rght handed schzophrenc males wth a populaton of rght handed healthy males. The probablty of beng wrong n sayng that the two populatons have a dfferent mean s: = F p;n1+n2,p,1,1 n1 + n 2, p, 1 (n 1 + n 2, 2)p T 2 ( 1 ; 2 ) (9) where: T n 1n 2 2 = (n 1 + n (x 2, x 1 ) > S,1 (x 2, x 1 ); (10) 2 ) = 1 n1+n2,2 [Pn1 =1 (x 1;, 1 ) > (x 1;, 1 ) + Pn2 (x =1 2;, 2 ) > (x 2;, 2 )] (11) We note however that ths formula s vald only under the hypothess that the varances of the two populatons of subjects (whch are a-pror unknown) are exactly the same, whch s not always true, especally wth respect to groups of dseased patents. What statstcs can tell us s that for a reasonable number of samples n both populatons, ths assumpton s no longer needed. However, for reduced sets of samples and wthout the varance equalty hypothess, more complcated formulae have to be used (agan, see [1]) Sgnfcant atypcal dssymmetry Our thrd queston can be used for ndvdual dagnoss. A typcal queston mght be to detect automatcally a bran tumor as beng a regon sgnfcantly more dssymmetrcal than the same regon n a normal populaton. The probablty of beng wrong n sayng that a value x 0 s sgnfcantly 6

7 dfferent from a populaton havng a mean and a covarance matrx s a smplfcaton of the precedng formula for n 2 =1: F p;n,p n, p,1 (n, 1)p T 2 () (12) where: T 2 () = n n +1 (, x 0) >,1 (, x 0 ) (13) Partal conclusons about dssymmetry maps Many combnatons are possble, such as: s my patent closertoagroupofepleptcsthantoagroupofcontrols... The computatons whch have been presented here are vald for voxel to voxel uncorrelated measurements, whch unfortunately s not the case n practce because the deformaton feld between the drect and chral mages s regularzed. However, the spatal coherency can be used to derve more robust statstcal parameters, as proposed by the theory of random felds and mplemented n the SPM method n the case of the analyss of functonal mages (for example, see [10]). We have not yet nvestgated the possbltes of such a method n the case of dssymmetry studes; ths appears to be an nterestng perspectve to explore. of non-rgd nter-patent matchng, or [19] for tests about volume varaton quantfcaton). To valdate the dssymmetry feld computaton, we have performed the followng experment: startng from the 3D mage I 1 of a real patent, we have smulated an artfcal expanson (a mass effect) at a known locaton and of a known radus n the rght hemsphere of the bran (mage D 1 ). We have then computed the dssymmetry feld F, jjf jj and jjf jjdv(f ) of D 1 (see fgure 6), and compared t wth the dssymmetry feld obtaned drectly wth I 1. The deformaton can be seen vsually by comparng I 1 and D 1, but wth D 1 alone, t s hard to determne vsually the nature and ampltude of the deformaton. In the jjf jj mages, there s a large regon where the norm of the dssymmetry vectors are very hgh. It s therefore easy to detect that there s somethng unusual gong on wth respect to perfect symmetry, but t s however very dffcult to determne the cause of the dssymmetry, that s the focus of expanson. Ths focus can be emphaszed only by usng a vector feld operator approprate to expanson/atrophy. In the mage presentng jjf jjdv(f ), the expanson translates nto a roughly sphercal shaped whte regon n the rght hemsphere (on the left n the mage), centered on the focus of expanson. Of course, t also translates nto a symmetrcal sphercal dark regon n the left hemsphere. The sgnal s less obvous n the outer boundary of the bran because t s corrupted by the natural dssymmetry of these regons. The aperture problem, whch states that deformaton are easer to detect n drectons perpendcular to nterfaces (such as grey/whte matter) than n parallel drectons explans also why perfect sphercal shapes are not retreved. In fgure 7, we present the subtracton between I 1 mages and D 1 mages to emphasze the effects created by the expanson only. Agan, only the jjf jjdv(f ) provdes a clear sgnal wth respect to the localzaton and extenson of the expanson. We hope to be able to emphasze wth ths technque the effect of a growng tumor such as a globlastoma, whch s very dffcult to segment because of ts dffuson wthn the tssue. Fgure 6. Syntheszed expanson n 3D wthn the bran of an healthy subject (upper: wthout expanson, lower: wth expanson). Mddle: ntensty mages, left: jjf jj and rght jjf jjdv(f ). The crossbars represent the focal pont of the expanson. 5. Synthetc experments We have performed varous tests to valdate the dfferent modules used n ths method (see [20] for a frst valdaton Fgure 7. Subtracton of mages I 1 and mages D 1 to emphasze the deformaton effects only. 7

8 6. Some prelmnary expermental results The followng results are very prelmnary. In partcular, they are not valdated medcal studes but are presented here only to llustrate the potental applcatons of our method. Much more work and strong collaboratons wth anatomsts, along wth much larger mage databases are needed to lead to medcally sgnfcant results A populaton of healthy rght handed males A frst experment s on a populaton of 3D MR scans of ten dfferent healthy subjects, all of them beng rght handed males. These subjects have been selected for the medcal study descrbed n [14]. Ther handedness was ascertaned usng the Ednburgh Handedness Inventory (short form) whch s a 10 tem questonnare gvng a lateralty quotent percentage from,100 (left handed) to 100 (rght handed). Other metrcs of handedness are descrbed n [12]. The 10 subjects rated a mnmum of 87 wth respect to ths scale. We have realgned automatcally all the mages wth respect to the md-plane, computed ther dssymmetry maps, appled the norm-tmes-dvergence operator and fused all the nformaton n the frame of an eleventh subject s mage (rght handed ratng 100) also realgned, exactly as t was descrbed n secton and summarzed n fgure 5. The results are presented n fgure 8 for coronal and axal cross sectons and at the level of the temporal lobe only. The left mages present the reference subject. The mages n the mddle present the average of jjf jjdv(f ) for the 10 subjects. The sgnfcance map (the rght mages) present the loc whch are sgnfcantly dssymmetrcal (respectvely larger:whte or smaller:black). We have normalzed the mage of the sgnfcance map wth a sgnfcance level 0 =0:001 (that s, pure whte or pure black means 0:001). A mask has been appled to keep only the data at the level of the reference subject s bran. Ths experment confrms that the dssymmetry map presented n fgure 4 for a sngle subject s representatve of a normal dssymmetry, whch means a larger rght temporal lobe (on the left n the mages) n normal subjects Left handed versus rght handed We now llustrate what was presented n secton 4.3.2: the comparson of two populatons. We have compared the average of 10 rght handed healthy males (handedness score 87) wth the average of 3 left handed healthy males (handedness score,57). The results are presented n fgure 9, wth the averages of jjf jjdv(f )jj for the rght and left handed groups (left and mddle mages), and the sgnfcance map normalzed usng the same sgnfcance level 0 as n fgure 8 (the rght mages). The results appear less conclusve than for normal dssymmetry. In partcular, determnng dscrmnant features of left handed versus rght handed subjects s far from beng evdent. A careful exploraton of the 3D data and more experments wth a larger set of rght and left handed subjects are probably needed to lead to defntve conclusons A patent wth focal aphasa Fgure 8. Sgnfcant dssymmetry for a populaton of 10 healthy rght handed males. Left: reference patent; mddle: average of the 10 jjf jjdv(f )jj maps; rght: sgnfcance map for 0 = 0:001. The rght temporal lobe (on the left n the mage) s ndeed sgnfcantly larger than the left one for normal anatomy. 8 We now study a patent presentng a focal aphasa. In the mage of ths patent (fgure 10), we note an obvous dssymmetry of the ventrcles [17]. The am of ths experment s to retreve ths dssymmetry thanks to the sgnfcance map and accordng to the methodology presented n secton Fgure 10 presents coronal and axal vews of the focal aphasc subjects. On the left s the orgnal mage, n the mddle s the jjf jjdv(f ) dssymmetry map, and on the rght s the sgnfcance map wth respect to the populaton of 10 healthy rght handed males, projected back on the focal aphasc s mage. The obvous dssymmetry of the bran ventrcles s retreved and correctly localzed n the sgnfcance map, wth very hgh magntude sgnfcance values (usng the same sgnfcance level as n fgure 8).

9 Fgure 9. Dssymmetres between a populaton of 10 rght handed males and a populaton of 3 left handed males. Left: average of the 10 jjf jjdv(f )jj rght handed maps; mddle: average of 3 left handed; rght: sgnfcance map for 0 = 0:001; the same sgnfcance level as for fgure 8. These prelmnary results are not conclusve wth respect to a sgnfcant morphologcal dfference between left and rght handed people. Larger datasets are probably requested before drawng any concluson. 7. Concluson We have presented a general method to study the dssymmetry of symmetrcal organs, such as the human bran, usng 3D dssymmetry felds, 3D vector feld operators, and the computaton of 3D sgnfcance maps. The man feature of our method s that we are dealng wth dense volumetrc representatons of the dssymmetry. We have also proposed and tested a fully automated mplementaton of ths method, relyng manly on 3D non-rgd nter-patent matchng tools appled between the mages and symmetrc mages wth respect to an arbtrary plane. A by-product of ths s an unsupervsed method to realgn automatcally symmetrcal structures wth respect to ther md-plane. We have also descrbed three man applcaton felds, whch are the study of the normal dssymmetry n a gven populaton, the comparson of the dssymmetry between two populatons, and the detecton of the sgnfcant abnormal dssymmetres of a patent wth respect to a reference populaton. Fnally, we have presented prelmnary results for the case of human bran. These must be nvestgated n depth, wth the careful support of anatomsts and for much larger databases to enable us to draw conclusve medcal results. Fgure 10. Sgnfcant abnormal dssymmetry of a dseased patent (focal aphasa), wth respect to the populaton of 10 rght handed normal males. Left: focal aphasc s mage; mddle: jjf jjdv(f )jj for the aphasc; rght: sgnfcance map for 0 =0:001, wth respect to the 10 rght handed people. Note the sgnfcant dssymmetry at the level of the ventrcles, whch s correctly localzed. Acknowledgment Many thanks to Mrs Janet Bertot for a careful proofreadng of ths paper. Ths study also relates to the EC Bomed II project BIOMORPH where we ntend to use ths method to study the dssymmetry of schzophrenc patents. 8. Annex 8.1. Symmetry plane computaton We want to mnmze: C = P (S(q ), p ) 2, wth S(q )=q, 2((q, p) > n)n and where p s a pont n the symmetry plane and n the unt normal vector to the plane. By dfferentatng C wth respect to p,weget: dc X dp =4 (2p, q, p ) > nn > (14) whch demonstrates that the barycenter G = 1 np belongs to the symmetry plane. We get: C =X (q +p ) 2 (q, p ) 2 + 4[(q, G) > n][(p, G) > n] (15) X whch s mnmzed when the followng expresson s mnmzed: n > [(p, G)(q, G) > ]n (16) 9

10 whch means than n s the egenvector assocated to the smallest egenvalue of I, where: I =X 8.2. Realgnment of a 3D mage (p, G)(q, G) > (17) We want to demonstrate that R,1=2 (I 1 ) s an mage where the md-plane s P 0 : x = p x =2. Ths s equvalent to demonstratng that after the applcaton of R,1=2,themage R,1=2 (I 1 ) and the mage R,1=2 (S(I 1 )) are symmetrcal wth respect to P 0,thats: K R,1=2 (I 1 )=R,1=2 S(I 1 ) (18) To demonstrate ths, we shall note that S and K are planar symmetres therefore we have K K = Id and S S = Id, hence: R,1 =(S K),1 = K,1 S,1 = K S ) K = R,1 S (19) Furthermore, as R,1=2 s an affne rotaton wth a rotaton axs n P 0, K R,1=2 s also a planar symmetry, hence: K R,1=2 =(K R,1=2 ),1 = R 1=2 K,1 = R 1=2 K (20) Replacng K wth R,1 S n equaton 20 (on the rght) gves the desred relaton: K R,1=2 = R 1=2 R,1 S = R,1=2 S (21) References [1] T. Anderson. Introducton to Multvarate Statstcal Analyss. John Wley & Sons, Inc., New York, USA, [2] N. Ayache. Artfcal Vson for Moble Robots: Stereo Vson and Multsensory Percepton. MIT Press, Cambrdge, Massachussetts, [3] R. Blder, H. Wu, B. Bogerts, G. Degreef, M. Ashtar, J. Alvr, P. Snyder, and J. Leberman. Absence of regonal hemspherc volume asymmetres n frstepsode schzophrena. Amercan Journal of Psychatry, 151(10): , October [4] G. E. Chrstensen, S. C. Josh, and M. I. Mller. Volumetrc Transformaton of Bran Anatomy. IEEE Transactons on Medcal Imagng, 16(6): , Dec [5] L. Collns, T. Peters, and A. Evans. An Automated 3D nonlnear mage deformaton procedure for Determnaton of Gross Morphometrc Varablty n Human Bran. In R. A. Robb, edtor, VBC, volume 2359 of SPIE, pages , Rochester (Mnnesota) (USA), Oct [6] T. Crow. Schzophrena as an anomaly of cerebral asymmetry. In K. Maurer, edtor, Imagng of the Bran n Psychatry and Related Felds, pages Sprnger-Verlag, Berln Hedelberg, [7] Davatzkos, C. and Vallant, M. and Resnck, S. and Prnce, J.L. and Letovsky, S. and Bryan, R.N. Morphologcal Analyss of Bran Structure Usng Spatal Normalzaton. In Höhne, K. H. and Kkns, R., edtor, VBC, volume 1131 of Lecture Notes n Computer Scence, pages , Hamburg (Germany), sep Sprnger. [8] R. Davdson and K. Hugdahl. Bran Asymmetry. A Bradford Book, [9] L. DeLs, M. Sakuma, M. Kushner, D. Fner, A. Hoff, and T. Crow. Anomalous cerebral asymmetry and language processng n schzophrena. Schzophrena Bulletn, 23(2): , [10] K. Frston, C. Frth, P. Lddle, R. Dolan, A. Lammertsma, and R. Frackowak. The relatonshp between global and local changes n PET scans. Journal of cerebral blood flow and metabolsm, 10: , [11] J. C. Gee, M. Revch, and R. Bajcsy. Elastcally deformng 3D atlas to match anatomcal bran mages. Journal of Computer Asssted Tomography, 17(2): , March [12] M. Holder. Hand Preference Questonnares: One Gets What One Asks For. M. phl. thess, Department of Anthropology, Rutgers Unversty, New Brunswck, New Jersey, U.S.A., Electronc verson: prmate/forms/hand.html. [13] R. Hunter, M. Jones, and F. Cooper. Modfed lumbar ar encephalography n the nvestgaton of long stay psychatrc patents. J. Neurol. Sc., 6: , [14] C. Mackay, N. Roberts, A. Mayes, J. Downes, J. Foster, and D. Mann. An exploratory study of the relatonshp between face recognton memory and the volume of medal temporal lobe structures n healthy young males. Behavoural Neurology, [15] F. Maes, A. Collgnon, V. D., G. Marchal, and P. Suetens. Multmodalty Image Regstraton by Maxmzaton of Mutual Informaton. IEEE Transactons on Medcal Imagng, 16(2): , Apr [16] P. Maras, R. Gullemaud, M. Sakuma, A. Zsserman, and M. Brady. Vsualsng Cerebral Asymmetry. In Höhne, K.H. and Kkns, R., edtor, Vsualzaton n Bomedcal Computng, volume 1131 of Lecture Notes n Computer Scence, pages , Hamburg (Germany), Sept Sprnger. [17] G. Subsol, N. Roberts, M. Doran, and J. P. Thron. Automatc Analyss of Cerebral Atrophy. Magnetc Resonance Imagng, 15(8): , [18] J.-P. Thron. Non-rgd matchng usng demons. In Computer Vson and Pattern Recognton, CVPR 96, San Francsco, Calforna USA, June Electronc verson: [19] J.-P. Thron and G. Calmon. Measurng leson growth from 3D medcal mages. In Nonrgd and Artculated Moton Workshop (NAM 97), Puerto Rco, June IEEE. Electronc verson : [20] J.-P. Thron, G. Subsol, and D. Dean. Cross valdaton of three nter-patents matchng methods. In VBC, volume 1131 of Lecture Notes n Computer Scence, pages , Hamburg, Germany, September [21] P. Thompson and A. Toga. Detecton, vsualsaton and anmaton of abnormal anatomc structure wth a deformable probablstc bran atlas based on random vector feld transformatons. Medcal Image Analyss (MEDIA), 1(4): , September [22] P. Vola and W. M. I. Wells. Algnment by maxmzaton of mutual nformaton. In Ffth Int. Conf. on Computer Vson, ICCV 95, pages 16 23, Cambrdge, Massachussetts, June IEEE. 10