Applicability of Non-Rigid Medical Image Registration using Moving Least Squares

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1 010 Internatonal Journal of Computer Applcatons ( ) Applcablty of Non-Rgd Medcal Image Regstraton usng Movng Least Squares Ms.Hema P Menon Department of Computer Scence and Engneerng Amrta School of Engneerng, Combatore Dr.K.A.Narayanankutty Department of Electroncs and Comm. Engneerng Amrta School of Engneerng, Combatore ABSTRACT A drawback of the non-rgd regstraton s ts unpredctable nature of the deformaton on the target mage. Mappng every pont on mages can cause deformatons even to regons, whch are expected to reman rgd. A non-rgd regstraton s therefore desrable, that produces only local deformatons where needed, whle stll preservng the overall rgdty. Ths work focuses on one such method called the Movng Least Squares (MLS) transformaton and compares the results wth Thn Plate Splnes (TPS). An ntensty based non-rgd regstraton algorthm s appled aprory, f the nput medcal mages are from two dfferent patents n order to facltate for the selecton of homologous control ponts n them. We compare the performance of both the technques by calculatng the Target Regstraton Error (TRE) at certan ponts and results are encouragng. Keywords Movng Least Squares, Non-rgd medcal mage regstraton, Thn Plate Splnes, As-rgd-as-possble transformatons. 1. INTRODUCTION Image regstraton, a processng requrement for combnng mages (.e., mage fuson), s used extensvely n the medcal feld for montorng dsease progresson, mage guded surgery, studyng bran shft after surgery and developng medcal atlases. The process for mage regstraton nvolves two mages namely, the source and the target mages, and fndng the best deformaton feld to algn the two mages. Varous features lke ponts, lne segments and ntenstes can be used as the bass for the regstraton. Functons used to map the source mage to the target mage are called transformaton functons. It s classfed as rgd or non-rgd, based on the transformaton functon used. Unlke rgd regstratons, where the dstances between all ponts reman constant before and after the regstraton, a non-rgd regstraton nvolve more complex computatons lke local stretchng and scalng to map the two mages whch mght generate unpredctable deformatons. It s not possble to exactly specfy the mappng of each pont n the source mage to the target mage. It would thus be good to have a non-rgd regstraton technque that produces local deformatons only where needed whle stll preservng the overall rgdty of the transformaton as much as possble called as-rgd-aspossble methods. Fortunately such technques exst, and ths work focuses on one such method called the MLS transformaton []. Ths technque s relatvely recent and s better known for surface reconstructon and n computer graphcs for mage deformaton and morphng. Here, we are explorng the applcablty of the MLS technque as a pont based non-rgd regstraton technque. Methods of MLS usng affne, smlarty transformatons and rgd transformatons are presented elsewhere [].The results are compared wth TPS method, whch s another wdely used technque both qualtatvely and quanttatvely. The source and target mages consdered for regstraton are of the same modalty. To evaluate the performance of the MLS technque, D MR bran mages from the same patent and dfferent patents were consdered as the nput to the system. If the source and target mages are from the same patent, then homologous control ponts are selected from the respectve mages and are passed to each of the regstraton algorthm to get the transformed source. If the nput mages are from two dfferent patents, an ntensty based non-rgd regstraton algorthm s used for the selecton of homologous control ponts. Then the selected control ponts are appled to each of the pont based technque to obtan the regstered source. Durng ths process we can compare the performance of both the technques by calculatng the TRE at certan ponts n the mage other than the control ponts. Fgure 1: The Image Regstraton process. Steps n Image Regstraton nvolve pre-processng, feature selecton, feature correspondence, determnng and applyng a transformaton, and n many cases re-samplng of the mages [3].. MEDICAL IMAGE REGISTRATION Image regstraton has evolved ndependently n varous research areas, rangng from geo-surveyng to medcal magng, each wth a number of unque 85

2 010 Internatonal Journal of Computer Applcatons ( ) applcatons. Medcal mage analyss, n partcular, has many challengng and useful applcatons. The need for regstraton can arse when mages of a gven pece of anatomy are taken over a perod of tme and need to be compared, as s the case for the study of tumor growth or n functonal magnetc resonance magng(fmri) studes [4, 5]. Regstraton can also be used to characterze normal versus abnormal anatomcal shape varatons. For example, tumors may be regstered to normal or abnormal tumors wthn the same class. Other applcatons nclude labelng and segmentaton, n whch the mage to be labeled/segmented s regstered to a prevously labeled/segmented mage (the atlas).ths atlas regstraton crcumvents the need for explct labelng/segmentaton. The mage that needs to be compared may also be obtaned usng dfferent hardware devces, each hghlghtng a specfc part of the body (these mages are also referred to as mult-modalty mages). Examples of common mult-modalty mages n medcal magng nclude Computed Tomography (CT), X-ray magng, Postron Emsson Tomography (PET) magng, and Nuclear Magnetc Resonance (NMR) Imagng. If these mages are regstered, they can be combned (fused) n a more meanngful way to provde an ntegrated vew. Medcal mage regstraton s a very popular area of mage processng wth applcatons rangng from montorng dsease progresson [6], buldng medcal atlases [5], mage guded surgery etc [4]. A survey of the recent publcatons n the feld of medcal mage regstraton wth algorthms, valdatons and applcatons can be found n [6] and [7]. More detals about the feature selecton and correspondence, transformaton functons and evaluaton methods for -D and 3-D mage regstraton can be found n [8]. 3. NONRIGID IMAGE REGISTRATION 3.1 Normal or Body Text Non-rgd regstraton s an actve area of research n the feld of medcal mage regstraton. Ths actvty s due to the fact that one cannot decde on the best algorthm for all applcatons,.e., no unversal soluton exsts for the mappng problem and lack of a reference standard to compute the exact errors obtaned n the regstraton process [10]. Each algorthm works well under certan constrants and condtons but may not do so under a dfferent set of condtons. Crum et al [9] dscusses the varous types of non-rgd regstraton algorthms, ther concepts, applcatons and lmtatons. The man problem that non-rgd regstratons pose s that t s not possble to actually predct the deformaton feld [9]. The control ponts may be mapped exactly n the source and target mages/volume usng dfferent methods [11], but the accuracy n mappng the other ponts n the source mage/volume map onto n the target mage/volume s unpredctable. Ths s because non-rgd regstraton nvolves more transformatons than just rotaton and translaton as requred for rgd regstratons [9]. Ths work focuses on evaluatng a non-rgd regstraton technque whch allows the deformaton feld to be as-rgd-aspossble whle stll performng local stretchng deformatons. 3. As-rgd-as-possble Transformatons The concept of as-rgd-as-possble transformatons was frst ntroduced by Alexa et al [1] where an object space morphng technque that blends the nterors of objects as a smooth blendng was proposed. Ths morphng was called as-rgd-as-possble because the objects undergo mnmum dstorton durng morphng. The shape manpulaton technque presented by Igarash et al [13] s also based on as-rgd-as-possble transformatons. Frst Page Copyrght Notce 3.3 MLS Transformatons The prncple of the MLS technque s to mnmze the least squares error functon obtaned durng the mappng transformaton process. A set of control ponts are chosen and transformaton functon s obtaned for each pont n the mage and s based on a weght functon ncluded n the least squares error functon at each pont of evaluaton. Ths weght functon ensures that the effect of a control pont s seen mostly n the regons mmedately surroundng t, whle ts effect s less promnent n far off regons. The transformaton matrx of the MLS technque can nclude affne, smlarty and as-rgd-as-possble transformatons and therefore extended here for regstraton. Ths work focuses on the as-rgd-as-possble transformatons whch are capable of producng local deformatons. The transformaton functon s smooth and nterpolates the control ponts. Gven a set of control ponts on the source and target mages, the MLS technque computes the transformaton I v(x) that best mnmzes the least squares error: I( p ) q where p and q are the set of control ponts n the source and target mages respectvely. Ths transformaton however produces a sngle affne transformaton of the entre mage as there s no control over the scalng or shearng n the mage. A weghtng functon ncluded to ths least squares error fxes ths problem and thus produces a dfferent transformaton functon for each pont of evaluaton of the mage. w I( p q The weghtng functon w s of the form w p 1 v where v s the pont of evaluaton n the mage and α s the fall-of-parameter of the weghtng functon whose value decdes f the weghts computed are small or large. The weghtng functon s dependent on the pont of evaluaton and thus produces a dfferent transformaton for each pont of the mage. Hence the method s called Movng Least Squares. We can see that as v approaches the control ponts, 86

3 010 Internatonal Journal of Computer Applcatons ( ) the weght approaches nfnty and the transformaton functon nterpolates. The transformaton functon can be solved as a smple lnear transformaton matrx, M and a translaton vector, T as: I v ( x) xm T The transformaton matrx M can be modfed to nclude affne, smlarty and rgd transformatons. To perform as-rgd-as-possble transformatons, the matrx, M must be constraned to satsfy the condton for rgdty M'M=I. The translaton component can be easly computed by: T q p M where control ponts gven by: are the weghted centrods of the The transformaton functon can now be calculated as: The least squares problem can be wrtten as: where p and pˆ q I v ( x) ( x p) M q p w pˆ M The transformaton matrx for the as-rgd-aspossble transformatons can be obtaned by elmnatng the scalng constant. The soluton s smple and closed form. It can be obtaned easly by a slght modfcaton of the smlarty transformaton for whch the transformaton matrx must satsfy the condton, M 1 M =M 1M =λ I, where λ s some constant and M 1 and M are the columns of M and are vectors of sze x1. M 1 and M have the relatonshp such that, For rgdty condton to be satsfed, M M=I, the scalng constants needs to be removed. By usng partal dervatves wth respect to the free varables n M and substtutng the values back nto the error functon the optmum transformaton functon s obtaned as: r where removes any scalng and thus produces as-rgd-as possble transforms. qˆ w p p and q w p ( x, y) ( y, x) ˆ and q w qˆ pˆ T M M 1 q q w qˆ pˆ w q T w The non-rgd regstraton technque evaluated n ths paper, the MLS transformaton s escrbed n [11]. Though ths paper s buld on technque descrbed by Igarash t ams at achevng faster deformatons. For ths purpose we propose the transformaton of objects by usng lnear MLS. No trangulaton of the nput s needed n ths case and the transformaton can be computed at each pont n the mage. The transformatons acheved usng rgdty constrants are as-rgd-as-possble wth mnmum nonlnear shearng and non-unform scalng, whch s a property of MLS transforms, thus makng t a sutable canddate for applcatons n medcal mage regstraton where rgd regstratons are nsuffcent and non-rgd regstratons are necessary to algn features of the mage. 3.4 Thn Plate Splne Transformatons Thn-plate splnes are a part of the splne famly wth radal bass functons as the nterpolatng functon. They produce smooth and closed form transformatons and have been used extensvely n mage deformaton. Thnplate Splnes have been used n remote sensng for mappng mages. Goshtasby et al [8] descrbes how the thn-plate splnes can be used n mage deformaton for remote sensng mages. Hs book on D and 3D regstratons also descrbes the TPS transformaton functon and t applcatons [14]. Though the TPS produces smooth transformatons, the man dsadvantage of the technque s that each control pont n the mage has a global effect on the transformaton and thus even f one pont s perturbed all other ponts n the mage do not get mapped correctly [9]. The work by Booksten et al [15] also descrbes the theory behnd TPS transformatons and how t can be used n medcal mage processng. Numerous papers have been publshed n the medcal mage processng area usng the TPS algorthm n mage regstraton [16, 17]. The TPS algorthm can be used to obtan an elastc transformaton to map the source mage to the target mage. Rohr et al [18] uses the TPS as a technque to obtan elastc regstraton of bran mages takng nto account the errors at the landmark ponts n the mages. The method proposed n ths paper s applcable to D and 3D MR mages. Ths paper establshes that the MLS transformaton technque can be used as an alternate to the TPS transformaton technque and can perform better due to ts as-rgd-as-possble nature. Ths could enable the MLS technque to be used n applcatons lke constructng medcal atlases, studyng the progresson of dseases, postoperatve shft of bran and other such applcatons where the TPS technque has been used so far. 4. IMPLEMENTATION DETAILS The overall system for the non-rgd regstraton of the medcal mages obtaned from the same patents usng the MLS and TPS technques s shown n fgure 4.1. For each par of the source and target mages frst of all a set of homologous control ponts are extracted. These control ponts from the source and the target are taken as the nput to the Movng least Squares (MLS) and Thn Plate Splne (TPS) algorthms, whch wll fnd out a transformaton functon for each pxel poston n the source mage to 87

4 010 Internatonal Journal of Computer Applcatons ( ) obtan the regstered source. In the case of mages taken from two dfferent patents, t s dffcult to dentfy homologous control ponts. As these ponts play the most mportant role n the regstraton process t becomes necessary to accurately obtan homologous control ponts from the source and target mages. Intally two mages of the same modalty S and T from two dfferent patents were gven as the nput to the system. The two mages were regstered usng an ntensty based method. The deformaton functon thus obtaned was appled to the source mage S to get the new transformed mage T. T has homologous ponts to the source S. The transformed source mage T obtaned by the ntensty based regstraton algorthm was used as the new target mage for the pont based regstraton process. In ths case we can quanttatvely evaluate the performance of the two technques by computng the Target Regstraton Error(TRE) as shown n the followng fgure Intensty Based Non-Rgd Regstraton Algorthm Here an error functon s estmated usng Taylorseres expanson, n order to smplfy the mnmzaton. A more accurate estmate of the actual error functon can be determned usng a Newton-Raphson style teratve scheme. In partcular, on each teraton, the estmated transformaton s appled to the source mage, and a new transformaton s estmated between the newly warped source and target mage. As few as fve teratons greatly mprove the fnal estmate. Secondly, the requred spatal/temporal dervatves have fnte support, thus fundamentally lmtng the amount of moton that can be estmated. A coarse-tofne scheme s adopted n order to contend wth larger motons []. A Gaussans pyramd s bult for both source and target mages and the local affne parameters are estmated at the coarsest level. These parameters are used to warp the source mage n the next level of the pyramd. A new estmate s computed at ths level, and the process repeated through each level of the pyramd. The transformatons at each level of the pyramds are accumulated yeldng a sngle fnal transformaton. Ths multscale approach s crtcal gven the dfferental nature of our measurements, allowng us to regster mages wth larger motons. Fnally the estmaton of the spatal/temporal dervatves s crucal step. Spatal/temporal dervatves of dscretely sampled mages are often computed as dfferences between neghbourng sample values. Such dfferences are typcally poor approxmatons to dervatves and lead to substantal errors. In computng dervatves we employ a set of dervatve flters specfcally desgned for mult-dmensonal dfferentaton [3]. The spatal/temporal dervatves are estmated as follows. The mages are frst pre-fltered n tme(usng the two-tap flter( [ ] ).The dervatve n x s then estmated by pre-flterng the result n y (usng the 3-tap pre-flter [ ] ), followed by dfferentatng n x (usng the dervatve flter[ ]).Smlarly the dervatve n y s estmated by frst pre-flterng the result n x (usng the 3-tap preflter [ ]), followed by dfferentatng n y (usng the dervatve flter [ ]).The dervatve n tme s estmated by frst preflterng n space (n x and y) usng the 3-tap pre-flter [ ]), followed by applyng the two-tap dervatve flter [ ] n tme to the result. These flters sgnfcantly mprove the resultng regstraton. System overvew for the regstraton algorthm s splt nto three sectons as depcted n Fgure 3. It depcts the multscale estmaton of the regstraton map, from a coarse to fne scale. The multscale regstraton algorthm proceeds as follows. The source and target mages at the coarsest scale are regstered to obtan an ntal estmaton of the regstraton map [10]. Ths ntal estmate s used to warp the source mage at the next scale. The warped source mage s then regstered wth ts correspondng target mage. Ths process repeated at each level of the pyramd. A sngle regstraton map s obtaned by accumulatng successve estmated regstraton maps at each scale. Ths multscale regstraton approach allows us to recover large motons as well as small motons. Wthn each scale, the regstraton map s determned n an teratve fashon. After an ntal estmaton of the regstraton parameters the source mage s warped wth the estmated parameters and regstered agan wth the target mage. Durng each of these teratons, successve ntermedate regstraton maps are accumulated to form a sngle regstraton map [4]. The teratons are stopped when the average dsplacement of the estmated moton s less than 0.1 pxels. Wthn each scale, and wthn each accuracy-teraton, a smooth regstraton map s obtaned as follows. Gven source and target mage, an estmate of the regstraton map wthout smoothness s frst obtaned. Fgure 3: System dagram Intensty Based Non Rgd Regstraton Ths ntal estmate s used to bootstrap the nonlnear teratve estmaton of the smooth regstraton 88

5 010 Internatonal Journal of Computer Applcatons ( ) map []. These regstratons are referred to as smoothness-teratons. These three components together form the complete regstraton algorthm. 4.. Control Pont Selecton Control pont selecton s an mportant aspect n pont based mage regstraton technque. For good regstraton results t s necessary to select these control ponts as accurately as possble. Control ponts are selected from the source and target mages by the user. Anatomcally smlar structures from the two mages are usually selected as the control ponts. After the selecton process t s appled to both the regstraton algorthm to obtan the regstered source Target Regstraton Error (TRE) Wthout quanttatve evaluaton, no regstraton method can be accepted for practcal utlzaton. The Target Regstraton Error (TRE) s one such method whch s the dsplacement between two correspondng ponts after regstraton,.e., after one of the ponts has been subjected to the regsterng transformaton. A lower TRE ndcates a better method. The word target n the name of ths error measure s meant to suggest that the error s beng measured at an anatomcal poston that s the target of some nterventon or dagnoss. Such errors would be expected to be more meanngful than errors measured at ponts wth no ntrnsc clncal sgnfcance. Let p represent a pont n the frst mage of a par to be regstered, and q a pont n the second mage. A regstraton method appled to ths par leads to a transformaton T that, wthout loss of generalty, regsters the frst mage to the second. The dfference between the two vectors representng the transformed pont and the correspondng pont gves the target regstraton error. Thus, TRE = T (p) q, whch s nothng but the regstraton error at that pont and s computed as the dsparty n the poston of two correspondng ponts after regstraton. A low TRE thus ndcates a better method. 5. EXPERIMENTAL RESULTS 5.1. Datasets Used Ths secton presents the results for evaluaton of two dmensonal mages usng the MLS and TPS technques. We use two dmensonal Magnetc Resonance mages (MR) of bran from same patents as well as dfferent patents for performng the regstraton. The datasets used are jpeg mages of sze 56x Quanttatve Analyss- Performance Metrcs Qualtatve (Vsual) comparsons are not suffcent to evaluate the performance of any algorthm, quanttatve evaluatons have to be made. For ths purpose the target regstraton error (TRE) was computed at certan ponts n the mage for both the methods. The target regstraton error at any pont, whch s nothng but the regstraton error at that pont, s computed as the dsparty n the poston of two correspondng ponts after regstraton [0,9]. To compute the TRE, three ponts (namely A, B and C) at random was selected on the nternal structures of the bran as shown n the Fgure 4(a & b).the TRE at these ponts A, B and C, shown on the source mage, was computed by fndng the ponts they mapped to on the MLS and TPS regstered mages. Snce the ponts correspondng to A, B and C could be found on the target mage by usng the Intensty based non-rgd regstraton Algorthm comparsons could be made on the resultng TRE s for the two methods. Table 1 shows the resultng TRE values for the two methods. Table 1: Comparson of MLS and TPS Fgure 4(a:d) Shows the Regstraton of D MR Bran Images wth ponts marked wth Yellow, Whte and Blue used to calculate TRE. Fgure 4 (a) Source Image wth 1 Control Ponts Fgure 4 (c) Regstraton -MLS Fgure 4 (b) Target Image wth 1 Control Ponts Fgure 4 (d) Regstraton - TPS 89

6 010 Internatonal Journal of Computer Applcatons ( ) 5.3. Computatonal Aspects for MLS and TPS The MLS and TPS algorthm were mplemented n Matlab. The MLS algorthm nvolves fndng the transformaton at each pont n the mage and thus has a longer computaton tme as seen n Table. One way to reduce the computaton tme would be to decmate the mage or volume usng a grd and apply the deformaton to each vertex of the grd nstead of each pont and nterpolatng the other ponts usng blnear nterpolaton. TPS transformatons are computatonally less tme consumng Observatons If the control ponts are unformly dstrbuted n both the source and target, t s dffcult to vsually analyse the performance of both the algorthms. In ths case both the algorthms wll gve vsually smlar results. Hence the Target Regstraton Error (TRE) s used as a quanttatve metrc to evaluate the performance of both the technques. stretchng caused n the spne regon along wth the stretchng seen near the head regon. Thus t can be sad that non-unform placement of control ponts affects the deformaton feld mnmally n case of the MLS technque than the TPS technque whch showed more stretchng. Fgure 5 (a) Source and Target Image wth 7 Control Ponts Table : Executon Tme for MLS and TPS Algorthm Based on Number of Control Ponts A lower TRE ndcates a better method. Here three ponts other than the control ponts are randomly selected from the source and target. Then the TRE value s calculated. In each case t was found that the TRE for MLS technque s low as compared to TPS. The executon tme for both the methods by varyng the number of control ponts ndcates that TRE perform faster than MLS. Even f the executon tme of TPS s faster compared to MLS, the later wll gve less error durng the regstraton. So we can conclude that MLS technque s quanttatvely better than TPS algorthm Qualtatve Analyss Qualtatve analyss s done based on Human Percepton. To evaluate the performance of the two technques based on the placement of control ponts, the homologous control ponts were selected only on the spne leavng out the other regons as shown n fgure 5 (a) and (b). The regstraton result usng MLS and TPS technques can be seen n fgure 5(c) and (d). It was observed that even though control ponts were not placed n the head regon, the MLS technque produced a qualtatvely better result. Ths technque dd not show any non-unform scalng or stretchng and also produced an as-rgd-as-possble transformaton. The TPS transformaton was seen to stretch the mage unnaturally. In ths case too, we observed the Fgure 5 (b) Regstraton MLS Fgure 5 (c) Regstraton - TPS To show that MLS can be used as a better technque for non-rgd regstraton whle mantanng the overall rgdty of the mage the two technques were also evaluated by usng a lmted number of control ponts. Usng more control ponts wll produce equally good results for the two methods snce ths allows more control over the deformaton feld. Fgure 6 (a) shows the source and target mages wth 3 control ponts placed on the head regon. The results of the two regstraton technques can be seen n the Fgure 6 (b) and (c). A vsual nspecton of the two transformatons showed that the MLS algorthm produces a qualtatvely better mage than the TPS method. The MLS transformaton was observed to map the source to the target mage wth mnmal stretchng, due to ts as-rgd-as-possble transformaton ablty, whle the TPS transformaton was observed to cause more stretchng across shape. If the number of control ponts selected are less or f they are not unformly dstrbuted the Thn Plate Splne (TPS) regstraton algorthm fals. But n both the scenaros MLS technque wll produce vsually good results than TPS. 6. CONCLUSIONS In ths work we have appled an nterpolatng non-rgd regstraton technque.e., Movng Least Squares and compared the results both quanttatvely and qualtatvely wth another pont based technque.e., Thn Plate Splnes. MLS technque was found to be a good 90

7 010 Internatonal Journal of Computer Applcatons ( ) canddate for non-rgd regstraton for a gven number of control ponts than thn plate splnes. A theoretcal ncety of the MLS technque s that the deformaton feld s as rgd as possble, gven the constrant that the control ponts are to be nterpolated. Fgure 6 (a) Source and Target Image wth 3 Pvots Fgure 6 (b) Regstered Source- ML Fgure 6 (c) Regstered Source -TPS Ths can offer advantages over other non-rgd regstraton methods, partcularly those where bony structures should be mnmally deformed durng regstraton. Non-rgd regstratons often produce unwanted stretchng n the mages and the unpredctable nature of the deformaton feld poses a major drawback whch makes an algorthm wth the ablty to produce as-rgd-as-possble transformatons attractve. The as-rgd-as-possble nature of the MLS technque thus makes t a sutable canddate for non-rgd regstratons as t provdes a transformaton that mantans the rgdty of structures that need to reman nondeformed, whle producng local deformatons. The overall rgdty of the mage s mantaned better wth the MLS method. Ths can be seen clearly n the example of the D MR bran mages taken from the same patents. Certan structures of the bran n the source mage s deformed to match the target mage whle the other regons of the mage reman mnmally affected thus mantanng the global rgdty whle performng local deformatons. The results show that there are no unwanted stretchng and shearng n the mage. Ths s because the effect of the control ponts on each pxel locaton s weghted by a weght functon. The results for the regstraton of D MR human bran mages of dfferent patents s used for qualtatve comparson. A quanttatve evaluaton s done by comparson of the target regstraton error (TRE).It shows that the MLS method has lower error values. Selectng the control ponts forms an mportant aspect of mage regstraton. Thus t becomes necessary to select these control ponts as accurately as possble. Any error n ths placng of ponts affects the fnal regstraton result. Ths s more pronounced n case of the TPS transformaton as each pont has a global effect on the deformaton feld. From the results obtaned for the MLS transformaton we observed that the as-rgd-as-possble nature of transformaton produces an acceptable deformaton even f one or two control ponts are not placed accurately. Ths s because of the weghtng functon on the least squares error functon. Ths weghtng functon ensures that the effect of the control pont n regons far away from t s less affected. The MLS transformaton nvolves computng the transformaton at each pont n the mage or volume. Ths leads to a longer computaton tme though t produces good results. One way to reduce the computaton tme would be to decmate the mage wth a grd and computng the transformaton only at the grd vertces and nterpolatng the other pont n the mage usng an nterpolaton technque lke blnear nterpolaton. The MLS method can be used for producng even better regstraton results. In the future we would lke to evaluate the performance of the MLS algorthm by modfyng the weght parameter. Study can also be made by examnng the results by usng dfferent dstance functons to see the varaton of rgdty n the deformaton. Another area of nterest would be to study the effect of perturbaton of the control ponts. Research can be carred out n future to fnd out ways for optmzng the MLS algorthm by automatng the process of control pont selecton. 7. REFERENCES [1] J. Mantz and M. Vergever, A survey of medcal mage regstraton, n Medcal Image Analyss, pp. 1 36, March [] R. Kollur, Provably good movng least squares, n Proc. of Internatonal Conference on Informaton Processng n Medcal Imagng, pp , 005. [3] H.Chu and A. Rangarajan, A new pont-matchng algorthm for non-rgd regstraton, n IEEE Internatonal Conference on Computer Vson and Image Understandng, pp , 003. [4] F. Sauer, Image regstraton: Enablng technology for mage guded surgery and therapy, n IEEE Transactons on Engneerng n Medcne and Bology Socety, pp , 005. [5] A.W. Toga and P.M. Thompson, Handbook of medcal magng, 1 st ed., 000. [6] P. Freeborough and N. Fox, Modelng bran deformatons n alzhemer dsease by flud regstraton of seral 3d mr mages, Journal of Computer Asssted Tomogra- phy, vol., pp , [7] M. Sonka and M. Ftzpatrck, Handbook of medcal magng, Medcal Image Processng and Analyss, SPIE Press Monograph, vol., pp ,

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