EVALUATION OF MOVING LEAST SQUARES AS A TECHNIQUE FOR NON- RIGID MEDICAL IMAGE REGISTRATION. Vijayalakshmi Sathyanarayanan. Thesis

Transcription

1 EVALUATION OF MOVING LEAST SQUARES AS A TECHNIQUE FOR NON- RIGID MEDICAL IMAGE REGISTRATION By Vjayalakshm Sathyanarayanan Thess Submtted to the Faculty of the Graduate School of Vanderblt Unversty n partal fulfllment of the requrements for the degree of MASTER OF SCIENCE n Electrcal Engneerng December, 2008 Nashvlle, Tennessee Approved: Professor Robert E. Bodenhemer Professor Benot M. Dawant

2 ACKNOWLEDGEMENTS I would lke to thank my advsor Dr. Robert E. Bodenhemer for gvng me the opportunty to work wth hm and also for gudng me through the course of my thess work. I would also lke to thank Dr. Benot Dawant for helpng me shape my work and gvng valuable suggestons whch helped complete ths thess. Last but not the least, I would lke to express my thanks to my famly especally my sster who has been wth me n each step that I took, for ther contnuous encouragement whch was essental for me to progress.

3 TABLE OF CONTENTS Page ACKNOWLEDGEMENTS... LIST OF TABLES...v LIST OF FIGURE... v Chapter I. INTRODUCTION...1 II. BACKGROUND...4 II.1. Non-Rgd Regstratons...4 II.2. As-Rgd-As-Possble Transformatons...5 II.3. Thn-plate Splne Transformatons...7 III. THEORY...9 III.1. Movng Least Squares transformatons...9 III D transformaton usng Movng Least Squares...10 III D transformaton usng Movng Least Squares...12 III.2. Thn-plate Splne Transformatons...14 III D Thn-plate Splne transformaton...15 III D Thn-plate Splne transformaton...16 III.3. Computatonal aspects of MLS and TPS algorthms...17 IV. EVALUATION OF TWO DIMENSIONAL REGISTRATIONS USING MLS AND TPS ALGORITHMS...18 IV.1. 2D CT Images of whole body mce...18 IV.2. 3D MR Images of human bran...25 V. EVALUATION OF THREE DIMENSIONAL REGISTRATIONS USING MLS AND TPS ALGORITHMS 31 V.1. Methods...31 V.2. Results...33

4 VI. CONCLUSION AND FUTURE WORK...37 REFERENCES...40 v

5 LIST OF FIGURES Fgure Page 1. Regstraton of 2D CT mce. (a) Source mage wth 35 control ponts. (b) Target mage wth 35 control ponts Zoomed secton of the source showng spacng n the spne. The MLS technque does not cause the odd stretchng Regstraton of 2D CT mce. (a) Source mage wth 25 control ponts. (b) Target mage wth 35 control ponts Regstraton of 2D CT mce. (a) Source mage wth 25 control ponts. (b) Target mage wth 10 control ponts Schematc representaton of 2D regstraton of MR bran mages Regstraton of 2D MR mages of the bran. (a) Source mage (b) Target mage wth 19 control ponts. (c) Regstraton Color coded TRE maps supermposed on bran regstraton results of (a) MLS and (b) TPS. The MLS mage Regstraton of 3D CT volumes of mce. (a),(b): 4 slces from source and target volumes (c) 4 slces from (a) 4 slces from source volume. (b) 4 slces from target volume. (c) 4 slces from MLS regstered result supermposed on...36 v

6 LIST OF TABLES Table Page 1. Computatonal tme for MLS and TPS algorthms n seconds Mean, standard devaton and T-test results of TRE for ponts A, B and C n Fgure 3 for MLS and TPS technques Mean and standard devaton of dfference (TPS-MLS) n TRE for MLS and TPS technques..28 v

7 CHAPTER I INTRODUCTION Image regstraton s the process of algnng two mages such that t s possble to determne correspondng ponts n them. Image regstraton 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-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. A detaled classfcaton can be found n Sonka and Ftzpatrck [2000]. Functons used to map the source mage to the target mage are called transformaton functons. Image regstraton s classfed as rgd or non-rgd based on the transformaton functon used. The applcaton determnes whether rgd or non-rgd regstraton s the most approprate technque. Rgd regstratons are those n whch the dstances between all ponts reman constant before and after the regstraton. Ths type of mage regstraton s popular snce t nvolves a rotaton and translaton alone to acheve the mappng between two mages. Rgd regstratons have the advantage of beng smple and easy n the sense that t s possble to predct how the transformaton wll perform. The most common applcaton for rgd regstratons s to regster mages obtaned from same subject over a short duraton of tme. 1

8 Non-rgd regstratons are those n whch dstance between all ponts do not necessarly reman constant after the regstraton. Unlke rgd regstratons, non-rgd regstratons nvolve more complex computatons lke local stretchng and scalng to map the two mages. Applcatons requrng some local scalng and stretchng cannot use rgd regstratons and hence use non-rgd regstratons. An example of such an applcaton would be regsterng mages obtaned from two subjects where anatomcal dfferences exst or to montor dsease progresson [Freeborough and Fox 1998]. Ths thess focuses on non-rgd regstratons, whch are stll an area of ongong research. The man drawback of non-rgd regstraton s the unpredctable nature of the deformaton. It s not possble to exactly specfy the mappng of each pont n the source mage to the target mage. It s also possble that such regstratons can cause certan regons, whch should reman rgd for underlyng physcal reasons, to deform due to scalng or shearng. 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. Such transformatons are called as-rgd-aspossble. Fortunately such technques exst, and ths thess focuses on a one method called the Movng Least Squares (MLS) transformaton. Ths technque s relatvely recent and s better know for surface reconstructon [Kollur et al. 2005] and n computer graphcs for mage deformaton and morphng [Schaefer et al. 2006] but has not been prevously appled to medcal mages prevously. The goal then, of ths thess s to apply the MLS technque to two and three dmensonal medcal mages where rgd transformatons are not sutable and compare the results to smlar non-rgd regstraton technques. MLS s a pont-based technque n 2

9 whch two mages are algned based on feature ponts extracted from them, and therefore we compare t to another popular pont-based non-rgd method, Thn-plate Splne (TPS) transformaton [Booksten 1989; Goshtasby 1988]. Thus, ths thess assesses the applcablty of the MLS technque as a general pont based non-rgd regstraton technque and compares t to another wdely used technque both qualtatvely and quanttatvely. Ths thess s organzed as follows. Chapter II dscusses the related work n nonrgd regstratons and the MLS and TPS technques. Chapter III descrbes the theory behnd the Movng Least Squares and the Thn-plate Splne technques n detal for 2- and 3- dmensonal data. Chapters IV and V descrbe the method and the results for regsterng 2- and 3- D data usng the two technques. We conclude wth a comparson of the two methods based on the results and future possbltes of usng the MLS method n Chapter VI. 3

10 CHAPTER II BACKGROUND Ths chapter descrbes prevous work n non-rgd mage regstraton. We also descrbe related work n as-rgd-as-possble transformatons as well as work n medcal mage regstraton usng the thn-plate splnes transformatons. II.1 Non-rgd Image regstraton Medcal mage regstraton s a very popular area of mage processng wth applcatons rangng from montorng dsease progresson [Freeborough and Fox 1998], buldng medcal atlases [Toga and Thompson 2000], to mage guded surgery [Sauer 2005] to name a few. Mantz and Vergever [1998] provde a survey of the recent publcaton n the feld of medcal mage regstraton. A detaled theory on the varous aspects of medcal mage regstraton ncludng algorthms ther valdaton and applcatons can be found n Ftzpatrck and Sonka [2000]. More detals about the feature selecton and correspondence, transformaton functons and evaluaton methods for 2-D and 3-D mage regstraton can be found n Goshtasby [2005]. Non-rgd regstraton n 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. Each algorthm works well under certan constrants and condtons but may not do so under a dfferent set of condtons. Crum [2004] dscusses the varous types of non-rgd regstraton 4

11 algorthms, ther concepts, applcatons and lmtatons. The paper also provdes lterature references n the area of non-rgd regstratons. The problem faced n non-rgd regstratons s the fact that there exsts no unversal soluton for the mappng problem [Hajnal et al. 2001]. Another drawback for these regstratons s the nablty to valdate results due to the lack of a reference standard to compute the exact errors obtaned n the regstraton process [Hajnal et al. 2001]. The man problem that non-rgd regstratons pose s that t s not possble to actually predct the deformaton feld [Crum et al. 2004]. It s true that t s possble to map the control ponts exactly n the source and target mages/volume usng dfferent methods [Schaefer et al. 2006; Goshtasby 1988; Booksten 1999]. However t s not possble to predct where the other ponts n the source mage/volume map onto n the target mage/volume. Ths s because non-rgd regstraton nvolve more than a rotaton and a translaton as requred for rgd regstratons [Sonka and Ftzpatrck 2000; Crum et al. 2004]. Ths thess focuses on evaluatng a non-rgd regstraton technque whch allows the deformaton feld to be as-rgd-as-possble whle stll performng local stretchng deformatons. The followng secton descrbes related work n as-rgd-aspossble shape transformatons. II.2 As-rgd-as-possble Transformatons The concept of as-rgd-as-possble transformatons was frst ntroduced by Alexa et al. [2000]. Ths paper presents an object space morphng technque that blends the nterors of objects as a smooth blendng. Ths technque nvolves trangulatng the object 5

12 before applyng the morphng algorthm. 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. [2005] s also based on as-rgd-aspossble transformatons. Ths technque also nvolves trangulatng the source object and solvng a lnear system of equatons wth the number of equaton equal to the number of vertces n the trangular mesh. The non-rgd regstraton technque evaluated n ths thess, the Movng Least Squares transformaton s descrbed n [Schaefer et al. 2006]. Ths paper bulds on the technque descrbed by Igarash but ams at achevng faster deformatons. Ths paper proposes the transformaton of objects by usng lnear movng least squares. 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 asrgd-as-possble wth mnmum non-lnear shearng and non-unform scalng. Also, the system s smple to solve n case of 2D mages as t results n a smple lnear 2x2 system. The MLS technque can also be appled to 3D volumes and ths extenson though causes the system to lose ts quadratc nature seen n the 2D case does not nvolve a lot of complex computatons. Chapter III contans the detaled theory behnd the MLS transformaton for both 2D and 3D applcatons. The resultng transformatons n both applcatons are smooth and nvolve the least amount of shearng and non-unform scalng. Ths property of the MLS algorthm make 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. Usng the MLS technque 6

13 ensures that local deformatons are acheved whle retanng the overall rgdty of the object. II.3 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. The detaled theory behnd the TPS transformaton for 2D and 3D applcatons can be found n Chapter III. The remanng part of ths secton descrbes some of the work done n medcal mage regstraton usng the TPS transformaton. Thn-plate Splnes have been used n remote sensng for mappng mages. Goshtasby [1988] descrbes how the thn-plate splnes can be used n mage deformaton for remote sensng mages. Hs book on 2D and 3D regstratons also descrbes the TPS transformaton functon and t applcatons [Goshtasby 2005]. 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 [Crum et al 2004]. The work by Booksten [1999] 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. The TPS algorthm can be used to obtan an elastc transformaton to map the source mage to the target mage. Ths technque has been descrbed by Rohr et al. [2003]. Buldng 7

14 medcal atlases s one mportant applcaton of mage regstraton. The TPS algorthm can be used n constructng medcal atlases and s descrbed n detal by Park et al. [2003]. The TPS technque has also been used wth ntensty as the control feature to obtan consstent regstratons and s descrbed n the work by Johnson and Chrstensen [2001]. Rohr [2001] 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 2D and 3D MR mages. Ths thess 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-rgdas-possble nature. Ths could enable the MLS technque to be used n applcatons lke constructng medcal atlases, studyng the progresson of dseases, post-operatve shft of bran and other such applcatons where the TPS technque has been used so far. 8

15 CHAPTER IIII THEORY Ths chapter dscusses the theory behnd the movng least squares and thn-plate splne technques. III.1 Movng Least Squares Transformaton The movng least squares technque was used n computer graphcs ntally for surface reconstructon [Kollur et. al 2005]. It can also be used for the deformaton of 2D and 3D objects [Schaefer et. al 2006; Cuno et. al 2007; Zhu and Gortler 2007]. The deformaton s carred out by selectng control ponts on the source mage. The ponts to whch these control ponts must be mapped to n the target mage are chosen as the deformed ponts. Ths presents the deformaton as a regstraton problem. A transformaton functon must be computed to map each pont n the source mage to the correspondng pont n the target mage. The prncple of the MLS technque s to mnmze the least squares error functon obtaned durng ths transformaton process. A 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 most 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-aspossble transformatons. Affne transformatons are those whch preserve the parallel 9

16 nature of lnes n the mage and also produce non-unform scalng and skewng, Smlarty transformatons are a part of the affne transformatons but wth unform scalng. As-rgd-as-possble transformatons are those whch are capable of producng local deformatons whle mantanng a global rgdty of the mage. Ths thess focuses on the as-rgd-as-possble transformatons. The transformaton functon s smooth and nterpolates the control ponts. The followng two sectons descrbe the 2D and 3D transformatons usng MLS. III.1.1 2D Transformatons usng Movng Least Squares Gven a set of control ponts on the source and target mages, the MLS technque computes the transformaton lv ( x) that best mnmzes the least squares error: 2 lv( p) q (3.1) 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. 2 wl v( p) q (3.2) The weghtng functon w s of the form w = p 1 2 v α (3.3) 10

17 where v s the pont of evaluaton n the mage and α s a parameter of the weghtng functon whose value decdes f the weghts computed are small or large. The weghtng functon w 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 transformaton functon nterpolates. p, the weght approaches nfnty and the The transformaton functon can be solved as a smple lnear transformaton matrx, M and a translaton vector, T as v ( ) l x = xm + T (3.4) 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 (3.5) where p and q are the weghted centrods of the control ponts gven by p = wp w and q = wq w (3.6) The transformaton functon can now be calculated as v ( ) ( ) l x = x p M + q (3.7) The least squares problem can be wrtten as 2 w ˆ ˆ pm q (3.8) where pˆ = p p and qˆ = q q (3.9) 11

18 The transformaton matrx for the as-rgd-as-possble 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 2 the transformaton matrx must satsfy the condton M M = M M = λ I, where λ s some constant and M 1 and M 2 are the columns of M and are vectors of sze 2x1. M 1 and M 2 have the relatonshp M 2 1 = M such that ( x, y) ( y, ) = x. For rgdty condton to be satsfed M M = I, the scalng constant 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 1 pˆ T M = T w ( ˆ ˆ q q μ ˆ ) (3.10) r p where μ 2 2 T ˆˆ ˆˆ r = wq p + wq p T removes any scalng and thus produces an asrgd-as-possble transformaton. The detaled dervaton for the as-rgd-as-possble transformatons can be seen n Schaefer et. al III.1.2 3D Transformatons usng MLS Ths secton dscusses the extenson of the MLS technque to 3D volumes. The dervaton for the transformaton matrx nvolves more complex computatons than the 2D dervatons. The ncluson of the thrd dmenson causes the system to no longer be quadratc n nature though t s stll possble to obtan closed form solutons. The soluton nvolves fndng the best transformaton mnmzng the weghted least squares problem 12

19 ( ) wl p q v 2 and fndng the optmal axs of rotaton and the angular parameters. The procedure for selectng ponts s the same as n the 2D case but the ponts need to be selected n a volume thus ncludng a thrd coordnate. Control pont sets P and Q are selected from the source and target volumes. In order to obtan as-rgd-as-possble transformatons, t s necessary that the transformaton l ( ) v ( ) v x s of the form l x = Mx+ T (3.11) where M s the transformaton matrx and T s the translaton vector. The translaton vector can be computed usng partal dervatves as seen n secton III.1 n equaton (3.5) as T = q p M, where p and q are the weghted centrods gven by equaton (3.6). The least squares error problem can now be wrtten as w pm ˆ qˆ 2 as seen before n equaton (3.8). Expandng ths summaton gves the followng of whch the second and thrd terms are constants. T 2 2 wqmp ˆ ˆ w q w p ˆ (3.12) Thus t can be seen that the mnmzaton s acheved when M can maxmze the T term wqmp ˆ. The transformaton matrx M can be defned as a rotaton of an angle around an axs. Let θ be the angle of rotaton and e be the axs of rotaton. Applyng such a rotaton to a vector v gves e, ( ) ( ) T T T T T M θ v = e ev + cosθ I e e v + snθ e 0 e v 0 z e y e e x z e y 0 x T (3.13) 13

20 Now the problem of maxmzng equaton 3.14 becomes fndng the best axs of rotaton e and angle of rotatonθ maxmzng cosθ Trace( ) ( θ) 1 cos T T M + eme + snθ Ve (3.14) where T R = wq pˆ and V = wqˆ pˆ (3.15) The soluton for the axs of rotaton and angle of rotaton can be computed by applyng Kuhn-Tucker optmalty condtons to obtan an egenvalue problem. Solvng ths egenvalue problem provdes the axs of rotaton e as the egenvector of the matrx T T M M cv V + + where c s the root of a fourth degree polynomal [Cuno et. al 2007] havng closed form soluton. The angle of rotaton can be computed once the axs s found as cosθ 2 1 u 2 u = and snθ u = (3.16) 1 + u where e s the unt vector of u. The detaled dervaton for the rotaton axs and angle of rotaton can be found n Cuno et al. [2007]. III.2 Thn-Plate Splne Transformatons The thn-plate splnes are a part of the splne famly wth radal bass functons as the nterpolatng functon. The equaton for TPS contans an affne part as well as a nonaffne part wth the nterpolatng functon. Ths secton dscusses the equatons requred to deform 2D and 3D objects usng TPS. 14

21 III.2.1 2D Thn-plate Splne Transformatons The 2D thn-plate splne nterpolatng a set of control ponts s gven by 2 x = a + a x+ a y+ Fr ln r (3.17) N = y = b + bx+ b y+ Gr ln r (3.18) N = ( ) ( ) 2 2 where r = x x + y y, N represents the number of control ponts and x and y th represent the coordnates of the control pont and I ranges from 1 to N. The two equatons specfed above have N +3 unknowns each. The soluton to obtan the deformaton feld s smple. N equatons can be formed by substtutng the coordnates of the control ponts nto the equatons (3.17) and (3.18). Apart from these there are three constrants used for solvng the transformaton. N = 1 N = 1 N = 1 F = 0 xf = 0 yf = 0 (3.19) Thus there are now N +3 equatons to fnd out the N +3 unknowns. The last three constrants ensure that the surface beng deformed does not rotate under the nfluence of the control ponts. Wth the help of these equatons t s possble to solve for the transformaton whch s then appled to deform the source mage. It should be noted here that the transformaton maps the control ponts n the source mage exactly onto the control ponts n the target mage. However the other ponts n the mage do not map exactly producng errors that become more pronounced n regons away from the control 15

22 ponts. TPS have been observed to produce more errors whle mappng mages havng local geometrc dfferences. The symmetrc nature of the logarthmc functon causes the errors to be mnmal n cases where the fducal ponts are dstrbuted symmetrcally whle asymmetrcally dstrbuted ponts produce more errors n the transformaton. III.2.2 3D Thn-Plate Splnes TPS can be extended to three dmensonal volumes as well. In ths case the transformaton requres the computaton of three components: x, y and z to accommodate the three dmensons. The transformaton n ths case s denoted as follows: N x = a + a x+ a y+ a z+ Fr (3.20) = N y = b + bx+ b y+ b z+ Gr (3.21) = 1 z' = c + c x+ c y+ c z+ Fr (3.22) N = 1 Where r = ( x x ) + ( y y ) + ( z z ) Ths equaton has N +4 unknowns. As n the case of 2D splnes N equatons can be obtaned by substtutng the control ponts n the above equaton whle 4 more equatons can be obtaned from the constrants whch ensure the mage does not rotate under the nfluence of the loads. The constrants are as follows: 16

23 N = 1 N = 1 N = 1 N = 1 F = 0 xf = 0 yf = 0 zf = 0 (3.23) Thus wth N +4 equatons for N +4 unknowns t s possble to solve for the deformaton feld to be appled to the mage. III.3 Computatonal Aspects for MLS and TPS algorthms: We mplemented both MLS and TPS n MATLAB. The MLS algorthm nvolves fndng the transformaton at each pont n the mage or volume and thus has a longer computaton tme than for TPS as seen n table 1. 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. Table 1. Computatonal tme for MLS and TPS algorthms n seconds for 128x128 mage. No. of Ponts Tme for transformaton n seconds MLS TPS

24 CHAPTER IV TWO DIMENSIONAL REGISTRATION USING MLS AND TPS TRANSFORMATIONS Ths chapter presents the method and results for evaluaton of two dmensonal mages usng the MLS and TPS technques. We use two dmensonal Computed Tomography (CT) mages and two dmensonal Magnetc Resonance mages for performng the regstraton. IV.1 2D CT Images We frst used whole body CT mages of mce to perform regstratons usng the two technques. CT mages show the bone structures more promnently than the soft tssue n the body. Snce bones are rgd parts of the skeleton, CT mages provde an excellent data set to evaluate the two technques. The followng sectons dscuss the datasets, the method for selectng the control ponts and the results obtaned for the MLS and TPS technques. IV.1.1 Method The CT mages used for regstraton were obtaned from two whole body CT volumes of two dfferent mce. The volumes were of sze 512x512x512 wth a voxel resoluton of 0.2x0.2x0.2 mm 3. A slce showng maxmum detal was selected from each of the two volumes. These two slces were then used n the regstraton process. The head, 18

25 spne and the legs were seen promnently n these two slces. Ths can be seen n Fgure 4.1(a) and (b). Selectng the control ponts s an mportant part of the regstraton process. We selected control ponts manually on the source mage. These ponts were pcked on the head, spne and legs due to ther promnence n the CT mage. Smlarly ponts correspondng to these ponts or homologous ponts, whch have same relatve poston on the target mage wth respect to the control pont on the source mage, were selected manually. Ths set of control ponts on the source and target mages was used n the regstraton process usng the MLS and TPS technques. Fgure 4.1 (a) and (b) shows one such set of homologous ponts on the source and target mages. IV.1.2 Results We evaluated the performance of the two regstraton technques based on the number of ponts used for regstraton and the placement of the ponts. Fgure 4.1 (a) and (b) show the source and target mages wth homologous control ponts placed on the bony structures of the head, spne and legs. Fgure 4.1 (c) and (d) show the result of the regstraton usng MLS and TPS technques. Fgure 4.2 shows a zoomed n secton of the spne of the source mage of the mouse to show the already exstng spacng n the spne. From the results we observed that the MLS technque produces a qualtatvely better result than the TPS technque. The MLS regstered result does not show the unnatural stretchng seen n the spne. We also observed that the resultng transformaton for the MLS technque mantans the overall rgdty of the mage. Ths was attrbuted to the as- 19

26 Fgure 4.1: Regstraton of 2D CT mce. (a) Source mage wth 35 control ponts. (b) Target mage wth 35 control ponts. (c) Regstraton usng MLS method. (d) Regstraton usng TPS method. The MLS method performs qualtatvely better than the TPS method. rgd-as-possble nature of the MLS algorthm. The ablty to mantan the rgdty constrant by lmtng the non-unform scalng produces a deformaton feld whch retans the overall rgdty of the mage. 20

27 Fgure 4.2. Zoomed secton of the source showng spacng n the spne. The MLS technque does not cause the odd stretchng of spne seen n the TPS technque as seen n Fgure 1(d). Fgure 4.3: Regstraton of 2D CT mce. (a) Source mage wth 25 control ponts. (b) Target mage wth 35 control ponts. (c) Regstraton usng MLS method. (d) Regstraton usng TPS method. The MLS method performs qualtatvely better than the TPS method. 21

28 Fgure 4.4: Regstraton of 2D CT mce. (a) Source mage wth 10 control ponts. (b) Target mage wth 10 control ponts. (c) Regstraton usng MLS method. (d) Regstraton usng TPS method. The MLS method shows least stretchng and scalng compared to the TPS method. To evaluate the performance of the two technques based on the placement of control ponts, we selected the control ponts only on the spne leavng out the head and the leg. Fgure 4.3 (a) and (b) show the par of source and target mages wth homologous 22

29 control ponts placed only on the spne. The regstraton result usng MLS and TPS technques can be seen n Fgure 4.3 (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 was not seen to cause any non-unform scalng or stretchng and also produces an as-rgd-as-possble transformaton. The TPS transformaton was seen to stretch the mage unnaturally. In ths case too, we observed the 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. To show that MLS can be used as a better technque for non-rgd regstraton whle mantanng the overall rgdty of the mage we evaluated the two technques 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 4.4 (a) and (b) show the source and target mages wth 10 control ponts placed on the head and the spne. The results of the two regstraton technques can be seen n the Fgure 4.4 (c) and (d). 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 the body of the mce. 23

30 IV.2. 2D MR human bran Regsterng MR bran mages from two dfferent patents requres non-rgd regstraton due to the presence of anatomcal dfference n the bran structures. We evaluated the performance of the two technques by regsterng two correspondng depth two dmensonal Magnetc Resonance (MR) mage slces of the human bran. MR mages provde more detal and greater contrast between the soft tssues n the body than CT mages. The two slces used were 256x256 mages of the sagttal vew of the bran of two subjects. Such mage from two subjects presents a lot of anatomcal varatons and thus provdes a good dataset for non-rgd regstraton. IV.2.1 Method In case of bran mages t s dffcult to dentfy homologous control ponts n two mages from dfferent subjects. 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. We started wth two slces S and T from two dfferent patents. The two slces were regstered non-rgdly usng the Adaptve Bases Algorthm (ABA) [Rohde et al. 2003] an ntensty based method for regsterng two mages or volumes. 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. Ths new mage T was used as the target mage n our regstraton process. S and T were regstered usng the MLS and TPS technques. Fgure 4.5 shows the dagrammatc representaton of the process we followed to obtan the source and target mages. 24

31 Fgure 4.5: Schematc representaton of 2D regstraton of MR bran mages IV.2.2 Results We evaluated the performance of the two technques qualtatvely and quanttatvely. Fgure 4.6 (a) and (b) show the source and target mages wth homologous control ponts. We placed the control ponts on the skull, the tp of the nose, mouth and few control ponts were placed on the nternal structures of the bran. The control ponts on the source mage are shown n lght blue color whle the control ponts on the target mage are shown n dark blue color. The results of the MLS and TPS technques can be seen n Fgure 4.6 (c) and (d). Vsually both the methods look smlar due to the ntrcate 25

32 anatomy of the bran and t was noted that both methods perform acceptable transformatons. Snce qualtatve comparsons were not suffcent to evaluate the performance we performed quanttatve evaluatons as well. We computed the target regstraton error (TRE) 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 [Sonka and Ftzpatrck 2000; Hajnal et al. 2001]. A lower TRE thus ndcates a better method. We performed 25 regstraton trals wth 25 sets of control ponts. Ths was carred out to observe the error obtaned n dfferent regstratons. To compute the TRE, we selected three ponts A, B and C at random on the nternal structures of the bran. We then computed the TRE at these ponts A, B and C, shown on the source mage, 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 ABA we were able to compare the resultng TREs for the two methods. Table IV.1 shows the mean TREs for each of the ponts A, B and C. It was observed that the average TRE was lesser for the MLS method compared to the TPS method. The standard devaton of the error can also be seen n Table IV.1 and shows a lower value for the MLS method. We also performed a t-test at 95% confdence level to check f the means were statstcally dfferent. It was found that pont A dd not have a sgnfcant dfference but ponts B and C had statstcally dfferent means. 26

33 Fgure 4.6. Regstraton of 2D MR mages of the bran. (a) Source mage (b) Target mage wth 19 control ponts. (c) Regstraton usng MLS method. (d) Regstraton usng TPS method. Ponts A, B and C are used n computng the target regstraton error for the two methods. 27

34 Table IV.1. Mean, standard devaton and T-test results of TRE for ponts A, B and C n Fgure 3 for MLS and TPS technques over 25 regstraton trals. The t-test analyss shows that the mean s sgnfcantly dfferent for ponts B and C. Pont A(green*) B(red*) C(yellow*) MLS Mean Std. Dev TPS Mean Std. Dev Dfferent? (t-test, 5%) N Y Y We also computed the mean and standard devaton of the dfference n the TRE for the MLS and TPS methods.e., (TRE TPS - TRE MLS ) as seen n Table IV.2. Ths was done to observe f the overall TRE was lower for the MLS method. To gve and dea of the spatal varaton of the TREs over these trals, we computed the mean TRE for the entre mage and these values are presented as a color-scaled map onto the bran mage as Table IV.2. Mean and standard devaton of dfference (TPS-MLS) n TRE for MLS and TPS technques. Pont A(green*) B(red*) C(yellow*) Mean Std. Dev

35 Fgure 4.7. Color coded TRE maps supermposed on bran regstraton results of (a) MLS and (b) TPS. The MLS mage has lower maxmum and average TRE over the entre mage. shown n Fgure 4.7. The color-scaled map shows that the TPS method has a larger error value for the target regstraton error whch s also confrmed by the computaton of the TRE values. Ths can be seen clearly n base regon of the head. The TPS regstered result has a darker shade of red n ths regon ndcatng a hgher TRE compared to the correspondng regon of the MLS regstered result whch has a lghter red shade. Whle regsterng the 2D bran mages, we used ABA [Rohde et al., 2003] for dentfyng the homologous control ponts. Ths opens the dscusson as to whether systematc errors are present n the quanttatve analyss. The adaptve bases algorthm uses compactly supported radal bass functons for regsterng two mages. The TPS transformaton also uses radal bass functons as ts nterpolatng functon. It s thus possble some systematc 29

36 errors exst due to ths use of radal bass functons but we have not performed any specfc tests to confrm the presence or absence of any bas. In concluson, we found that the MLS transformaton produced better results than the TPS technque both qualtatvely and quanttatvely. We also found that placng control ponts only n certan regons of the mage does not cause the remanng regons to be non-unformly deformed n the case of the MLS transformaton whle the locaton of control ponts s more mportant n case of the TPS transformaton. The MLS transformaton technque was found to work well for both CT and MR mages. 30

37 CHAPTER V EVALUATION OF THREE DIMENSIONAL REGISTRATION USING MLS AND TPS ALGORITHMS Ths chapter descrbes the method and results for performng 3D regstratons usng the MLS and TPS methods. V.1 Method We used two whole body CT volumes obtaned from two dfferent mce for the source and target volumes. The volumes were 512x512x512 n sze wth 02.x0.2x0.2 mm 3 voxel resoluton. To reduce computaton tme we down-sampled these volumes to 128x128x128 n sze. CT volumes obtaned from dfferent mce provdes a good dataset for non rgd regstraton due to the presence of anatomcal dfference n the structures of the mce. We performed the evaluaton of the two methods frst on fve slces and then on ten slces obtaned from the source and target volumes. To obtan the slces we selected fve consecutve slces from the source volume, whch showed maxmum detal. We selected slces n whch the head, spne and legs were most vsble. Ths selecton process was repeated for the target volumes to obtan the fve slces. Control ponts for the regstraton process were selected manually on all of the fve slces of the source volume. CT volumes show the bony structures n the body more promnently than soft tssues. Thus we selected control ponts on the bony structures present n head and spne regons of the fve slces n the source volume. Smlarly the homologous control ponts on the 31

38 fve slces n the target volume were also selected manually on the bony structures n the head and spne regons, thus gvng a set of homologous control ponts for the regstraton to be performed. Fgure 5.1 (a) and (b) show the four slces of the source and target volumes. Fgure 5.1 (c) shows the result of regsterng the two volumes usng the MLS technque and Fgure 5.1 (d) shows the result of regstraton usng the TPS technque. The results are dscussed n detal n the secton V.2. For our next evaluaton we selected ten slces from the source and target volumes showng maxmum detals. In ths case we segmented the bony structures from the ten slces by settng an ntensty threshold value. Ths made the control pont selecton easer and seemed a natural modalty for control pont selecton. As n the prevous case control ponts were pcked manually on the source volume across the ten slces. The control ponts were placed n the bone structures of the head and spne. The correspondng homologous control ponts on the target volume were also pcked manually gvng us a set of control pont for the source and target volumes. Regstraton was performed usng the MLS and TPS technques. Fgure 5.2 (a) and (b) shows four slces from the source and target volumes respectvely. Fgure 5.2 (c) shows the result of the MLS regstraton method supermposed on to the four slces from the target volume and Fgure 5.2 (d) shows the results of the TPS regstraton method supermposed on to the four slces of the target volume. The supermposed results gve a better vsual comparson of the two technques. The results are dscussed n detal below. 32

39 V.2 Results Fgure 5.1 (c) and (d) show the result of regstraton usng the MLS and TPS technques. We found that the MLS algorthm performs better than the TPS method qualtatvely. Comparng the slce one n Fgure 5.1 (c) and (d), t was observed that the MLS transformaton does not show any non-unform stretchng whle the TPS transformaton shows stretchng near the head of the mouse as well as n the regon near the end of the spne and tal. Ths stretchng was observed n all the slces. Observng slce one of Fgure 5.1(d) showed that head of the mouse to be more stretched than n the other slces. Ths unnatural stretchng s not seen n the slces of the MLS regstered volume. Due to the as-rgd-as-possble nature of transformaton of the MLS method the overall rgdty s mantaned whle local deformaton s also acheved. It was observed that the MLS technque maps the homologous ponts exactly and algns the source volume as closely as possble to the target volume whle retanng the rgdty at the same tme. The TPS method though maps the homologous ponts exactly too but does not have control over the rgdty of the object and causes a lot of stretchng. Fgure 5.2 (c) and Fgure (d) show four slces from the result of the MLS and TPS algorthms along wth four slces from the target volume. It was observed that the MLS method outperforms the TPS method n ths case too. It was easer to compare the two methods when only the bony structures were used n the regstraton. The MLS regstraton result was found to be more n algnment wth the target volume. Ths was observed especally n the spne area. Compare slces 2, 3 and 4 from Fgure 5.2 (c) to Fgure 5.2 (d). The TPS versons n 5.2 (d) are stretched n opposte drectons unacceptably because ths s a bony structure and should not deform. Lookng at slces 33

40 two, three and four n Fgure 5.2 (c) and (d), the regon near the end of the spne showed a stretchng, that looks lke the spne was stretched n opposte drectons, n the TPS regstered volume. The MLS transformaton was observed to map the source volume to the target volume wthout ths unnatural stretchng. The head regons n both the results were found to algn equally well. Control ponts were places only on the head and spne regons and hence the leg was not controlled n ether method. Quanttatve comparsons were not carred out as t was dffcult to obtan the exact homologous ponts n the course and target volumes. Any error n the homologous ponts would result n an ncorrect value of the target regstraton error. Based on the results obtaned we concluded that the MLS algorthm outperforms the TPS algorthm qualtatvely. The MLS technque s able to preserve the overall rgdty of the object whle stll performng local non-rgd deformatons. The TPS algorthm was found to cause unwanted stretchng. Ths s unacceptable n cases where the rgdty of the object needs to be mantaned especally areas such as bones whch need to mantan ther rgd structures. 34

41 (a) (b) (c) (d) Fgure 5.1 Regstraton of 3D CT volumes of mce. (a),(b): 4 slces from source and target volumes (c) 4 slces from MLS regstered result volume. (d) 4 slces from TPS regstered volume. The MLS technque performs better than the TPS technque by mnmzng the non-unform scalng and stretchng. 35

42 (a) (b) (c) (d) Fgure 5.2 (a) 4 slces from source volume. (b) 4 slces from target volume. (c) 4 slces from MLS regstered result supermposed on 4 slces of target volume. (d) 4 slces from TPS regstered result supermposed on 4 slces of target volume. The MLS method clearly outperforms the TPS technque whch shows a lot of stretchng n the spne regon. Control ponts were placed on the head and spne (thus the leg was not controlled for n ether method. 36

43 CHAPTER VI CONCLUSION AND FUTURE WORK We have compared two nterpolatng non-rgd regstraton technques, Movng Least Squares (MLS) and Thn-Plate Splnes (TPS). Both quanttatvely and qualtatvely we have found that the MLS technque outperforms the TPS technque for a gven number of control ponts. 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. 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 non-deformed, whle producng local deformatons. Chapters VI present the results of evaluaton of the two non-rgd regstraton technques for 2D CT mages of whole body mce and MR mages of human bran. The results show that the MLS algorthm performs qualtatvely better than the TPS algorthm. The overall rgdty of the mage s mantaned better wth the MLS method. Ths can be seen clearly n the example of the 2D CT mce mages. The bone structures n the head and spne of the mouse n the source mage s deformed to match the target mage whle 37

44 the other regons of the mage reman mnmally affected thus mantanng the global rgdty whle performng local deformatons. In case of the TPS method, the results show unwanted stretchng and shearng n the mage. Ths unwanted stretchng s caused due to the effect of the control ponts on the overall deformaton feld. Each pont has a global effect on the transformaton and thus affects the overall rgdty of the mage. The results for the regstraton of 2D MR human bran mages show that both methods perform equally well. It s dffcult to judge the better method vsually due to the numerous structures n the bran. A quanttatve evaluaton by comparson of the target regstraton error shows that the MLS method has lower error values. The average of the target regstraton errors over 25 trals also show that the MLS algorthm performs better. The t-test results for the mean of the TRE values show that the mean s sgnfcantly dfferent for the MLS method. Chapter V presents and dscusses the results for the 3D regstraton usng the MLS and TPS methods. Agan the results show that MLS algorthm outperforms the TPS algorthm qualtatvely mantanng the rgdty as much as possble but stll producng non-rgd deformatons to map the source and target volumes. 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-aspossble 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 38

45 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/volume 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 better regstraton results. In the future we would lke to evaluate the performance of the MLS algorthm by modfyng the weght parameter. The deformaton feld can be controlled by varyng the value of alpha n the weght functon. We would also lke to examne the results by usng dfferent dstance functons, other than the Eucldean dstance used here, 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. It would be nterestng to how much the control ponts can be perturbed whle stll producng a good transformaton. Another way to optmze the MLS algorthm would be to automate the process of control pont selecton. Ths work would be on the lne of Chu [Chu et al. 2003]. Automatng the process would result n a better algnment of the regstered result and target volume n case of 3D applcatons. Smlarly ncludng an teratve approach would also result n optmzng the movng least squares algorthm. 39

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