Efficient automatic correction and segmentation based 3D visualization of magnetic resonance images

Transcription

1 Lousana State Unverst LSU Dgtal Commons LSU Dotoral Dssertatons Graduate Shool 5 Effent automat orreton and segmentaton based 3D vsualzaton of magnet resonane mages Mkhal V. Mlhenko Lousana State Unverst and Agrultural and Mehanal College mmlt@lsu.edu Follow ths and addtonal works at: Part of the Computer Senes Commons Reommended Ctaton Mlhenko Mkhal V. "Effent automat orreton and segmentaton based 3D vsualzaton of magnet resonane mages" 5. LSU Dotoral Dssertatons Ths Dssertaton s brought to ou for free and open aess b the Graduate Shool at LSU Dgtal Commons. t has been aepted for nluson n LSU Dotoral Dssertatons b an authorzed graduate shool edtor of LSU Dgtal Commons. For more nformaton please ontatgradetd@lsu.edu.

2 EFFCENT AUTOMATC CORRECTON AND SEGMENTATON BASED 3D VSUALZATON OF MAGNETC RESONANCE MAGES A Dssertaton Submtted to the Graduate Fault of the Lousana State Unverst and Agrultural and Mehanal College n partal fulfllment of the requrements for the degree of Dotor of Phlosoph n The Department of Computer Sene b Mkhal V. Mlhenko Dploma Mosow State Unverst 999 Deember 5

3 Aknowledgements wsh to thank m advsor Dr. John M. Tler for hs uneasng energet support and preless gudane of m researh. wsh to thank also Dr. Oleg S. Pankh for hs numerous and valuable sentf and tehnal adve nredble responsveness and support that helped to mprove the qualt of ths researh. wsh to thank Dr. Warren Waggenspak Jr. for the pleasure of workng under hs gudane n order to fulfll m mnor requrement. A large part of what know about 3D vsualzaton and modelng learned from hm and large part of the work that performed under hs gudane served the foundaton for ths researh. wsh to thank m ommttee Dr. Baa B. Kark Dr. Clare D. Advokat and Dr. Janhua Chen for ther valuable omments and rtques. wsh to thank all the professors and nstrutors of Department of Mehans and Mathemats Mosow State Unverst whose treless eduatonal effort and eeptonal sentf mert allowed me to obtan the eellent mathematal bakground neessar for dotoral researh. wsh to thank m mother ountr Russa and ts people who provded me wth opportunt to reeve a mathematal eduaton of hghest qualt for free. wsh to thank m parents for thngs we an reeve onl from parents. And wsh to thank m fanée m beloved Jenne for beng m endless soure of nspraton and true knowledge about the world.

4 Table of Contents ACKNOWLEDGEMENTS... ABSTRACT... v NTRODUCTON... CHAPTER MAGNETC RESONANCE NHOMOGENETY CORRECTON Magnet Resonane mage Aquston Proess nhomogenet n MR mages Non-unformt Artfat Model MR nhomogenet Correton Methods MR Non-unformt Correton: Wh Another Method? New Method Dervaton The Algorthm Bas Evaluaton of Dsf Comparson wth Seleted Publshed Methods Dsusson and Conlusons CHAPTER AUTOMATED MEDCAL MAGE VOLUME RENDERNG ntroduton Medal mage Volumes Surfae Renderng Surfae Renderng and sosurfae Representaton Volume Renderng Renderng Based on Boundar Voels of a Segmented Volume The Problem of Effent Renderng Pre-segmentaton of a Medal mage Volume Gravtatonal Shadng Algorthm Evaluaton of Gs.... Dsusson and Conlusons...8 CHAPTER 3 SUMMARY Results Conlusons Future Researh...4 REFERENCES...6 APPENDX: DSF ALGORTHM PSEUDOCODE... VTA...6

5 Abstrat n the reent ears the demand for automated proessng tehnques for dgtal medal mage volumes has nreased substantall. Estng algorthms however stll often requre manual nteraton and newl developed automated tehnques are often ntended for a narrow segment of proessng needs. The goal of ths researh was to develop algorthms sutable for fast and effetve orreton and advaned vsualzaton of dgtal MR mage volumes wth mnmal human operator nteraton. Ths researh has resulted n a number of tehnques for automated proessng of MR mage volumes nludng a novel MR nhomogenet orreton algorthm dervatve surfae fttng dsf automat tssue deteton algorthm atd and a new fast tehnque for nteratve 3D vsualzaton of segmented volumes alled gravtatonal shadng gs. These newl developed algorthms provded a foundaton for the automated MR proessng ppelne norporated nto the UnVewer medal magng software developed n our group and avalable to the publ. Ths allowed the etensve testng and evaluaton of the proposed tehnques. Dsf was ompared wth two prevousl publshed methods on 7 dgtal mage volumes. Dsf demonstrated faster orreton speeds and unform mage qualt mprovement n ths omparson. Dsf was the onl algorthm that dd not remove anatom detal. Gs was ompared wth the prevousl publshed algorthm fsvr and produed renderng qualt mprovement whle preservng real-tme frame-rates. These results show that the automated ppelne desgn prnples used n ths dssertaton provde neessar tools for development of a fast and effetve sstem for the automated orreton and vsualzaton of dgtal MR mage volumes. v

6 ntroduton n the last two deades dgtal magng has been replang onventonal flm n hosptals and other medal nsttutons. Dgtal s beomng a de fato standard for medal mage storage and ommunaton. Currentl all modern medal mage aquston deves produe a dgtal format for ths output. Consequentl the dssemnaton of these dgtal mages has nreased the demand for the omputer-based proessng and vsualzaton of dgtal mage volumes n medne and man new dgtal proessng algorthms emerged. The earler proessng methods were desgned to help lnans manage medal magng data n a new format and transfer establshed analss nto ths new envronment. Whle the elements of automat omputer-based proessng provded new apabltes the orgnal medal methods took a ver onservatve approah to the data handlng whh requred a larger than neessar substantal manual labor omponent. The onstantl fast evoluton of omputer tehnolog has produed enormous volumes of dgtal mages.e. almost all radolog lns proess mages on the order of thousands dal. The urrent trend s to nrease the mage throughput wth the same number of radologsts nreasng ther effen. To obtan ths nreased mage throughput requres man new automated proessng methods. For ths reason man reent medal mage proessng and vsualzaton methods are more automated and tend to redue all the requred human operator nteratons. Our researh ontnues ths trend. We onentrated our effort on magnet resonane aqustons whh are non-nvasve and possess good spatal resoluton and soft tssue ontrast. For these reasons magnet resonane s wdel used n the dagnoss of man dseases e.g. multple sleross atrophes nfarton tumors vsual/hearng dsturbanes traumas et.

7 We addressed two maor problems n the aquston and analss of MR mages: nhomogenet artfat orreton and vsualzaton of nternal strutures. Our goal was to develop a sstem of data orreton and subsequent vsualzaton that requres mnmal operator nteraton n the proessng for hgh volumes of mages. Addtonall suh a sstem should use nepensve generall avalable PC hardware and requre no addtonal thrd-part software lbrares to enourage a broader range of possble applatons. Fnall proessng tmes must requre onl few seonds to be of use n the real-tme lnal applatons. Based on these requrements we developed a new novel real tme dgtal MR mage volume proessng sstem wth two maor modules:. Automat nhomogenet orreton presented n Chapter ;. Automat analss/vsualzaton presented n Chapter.

8 Chapter Magnet Resonane nhomogenet Correton n the frst half of ths hapter we dsuss the magnet resonane mage aquston proess artfats assoated wth t and methods developed for orreton of resultng medal mages. n the seond half we wll present a novel method for automat MR nhomogenet orreton detals of our mplementaton and results of our method s performane evaluaton.. Magnet Resonane mage Aquston Proess An nuleus wth odd atom number.e. odd number of neutrons and protons has a non-zero spn alled ts magnet moment. n a normal state all nule n the tssue are randoml orented and ther net magnetzaton s zero. n the strong eternal magnet feld B however nule start preessng about an as parallel to the dreton of that feld vetor and the tssue beomes magnetzed see Fgure.. The man tssue magnetzaton haratersts are net magnetzaton vetor M algned along B wth the preesson frequen ω. Preesson frequen depends upon the partular tssue tpe and serves for the subsequent dstnton between dfferent tssues. Preesson frequen s determned from the Larmor equaton: ω γb π where γ s a salar onstant. Sne the energ levels of magnetzed nule dffer from the non-eted state eted nule an nterat wth the eternal eletromagnet pulses refer to [] for further detals on underlng phss. The magnet resonane eperment nvolves the applaton of radofrequen spetrum pulses RF pulses to the volume to be maged. After the applaton of an eletromagnet pulse nule gan addtonal magnetzaton f the RF pulse ontans the frequen lose to ther own resonant frequen and the net magnetzaton vetor M 3

9 moves to a transverse plane. Durng the subsequent lapsed tme nule re-emt at the same frequen and M returns to ts orgnal orentaton. Ths proess s alled free nduton dea FD and s llustrated on Fgure.. Fgure.. Magnetzed nule spn vetor omponents. B s an eternal onstant magnet feld ω s the preesson frequen. f a loop of wre the reever s plaed n the transverse plane the plane perpendular to the magnet feld and plane on Fgure. an alternatng eletr urrent s ndued n the reever. Ths urrent s alled the MR sgnal and s haraterzed b ampltude frequen and phase relatve to the phase of the transmtter. T s the tme requred for the z omponent of M to reover 63% of ts orgnal magntude and T s the tme requred for the transverse omponent of M to dea to 37% of ts value mmedatel after the RF pulse. T and T alled FD relaaton tmes are often used as a measure of resultng pel ntenst on mages produed usng T or T weghtng. 4

10 Fgure.. Free nduton dea. a Net magnetzaton response to an RF pulse; b ts Fourer transform. ωnq s Nqust frequen ωnq Total number of data ponts / [*Samplng tme] ωtr s the transmtter referene frequen. To obtan MR mages for a three-dmensonal volume the MR sgnal needs to be loalzed for ever sample pont n the volume. For ths small perturbatons are appled to the man magnet feld B n short tme ntervals. These perturbatons are alled gradent pulses and depend lnearl upon the and z oordnates of the magnet feld. So t s possble to deode the pont poston n spae from the resultng sgnal desrbed b the epanded form of Larmor equaton : r r ω B V γ r where ω s the proton frequen at poston r and V r s the gradent vetor. The ombnaton of gradent pulses RF pulses and data samplng s alled a pulse sequene. Several tehnques haraterzed b spef pulse sequenes for the RF and 5

11 magnet gradent feld arrangements are used to obtan a good spatal resoluton wthn a reasonable tme san. These tehnques have the followng haratersts:. D multsle magng utlzes the etaton of several D sles n the same tme san;. Sequental sle tehnque mplements a sle-b-sle sequental etaton; 3. 3D volume aquston uses double-phase enodng for one-tme etaton of small volumes; 4. Half-aquston/half-Fourer tehnque takes advantage of the ntrns smmetr of the raw data to redue the san tme [].. nhomogenet n MR mages Magnet resonane magng s based on the resonant frequenes and relaaton tmes for dfferent tssues beng dfferent enough to produe ontrast mages. The underlng phsal theor suggests that wthn one unform tssue the MR sgnal would have onl nsgnfant devaton from the mean value that haraterzes ths tssue. The onl nonunformt that the pel ntenstes an have naturall s due to the tssue mrostruture [] [3] although other authors [4] [5] assume the deal pel ntenst varaton wthn a sngle tssue to be zero. Ths model mples the deal ondtons for a MR eperment: unform eternal magnet feld absene of random nose no orrelaton between the sgnals obtaned from neghborng sample ponts et. The atual MR sanners produe ontrast but non-unform mages. MR artfats are lassfed b Brown et al. [] nto three groups aordng to the ause of sgnal msnterpretaton:. Artfats aused b patent moton durng aquston;. Artfats due to measurement tehnque parameters; 3. Artfats generated b sanner or the soure eternal to both patent and sanner. 6

12 Ths lassfaton mples that the soure MR sgnal an be nterpreted to produe orret deptons of anatom detal n all 3 ases. n ths work we are more nterested n the artfats that an be orreted after mage aquston.e. when the do not remove some anatomal nformaton from the mage. Therefore we lmt our onsderaton to artfats of whh the most mportant are the followng:. Statonar gradents where the man magnet feld B nonunformt affets the haratersts of reeved sgnal. Aordng to [] mperfetons n the magnet from manufaturng as well as the presene of metal obets n the vnt of MR sanner that dstort the magnet feld. The RF reever ol ma also be a soure for smooth sgnal varaton [3]. Ths nhomogenet s onstant durng the aquston tme and results n ontnuous gradents n soft tssues. The amount of statonar nonunformt depends upon magnet feld haratersts and an sometmes be sgnfant as an be seen on the lnear profle of the soft tssue on Fgure.3.. Dfferenes n the magnetzaton n adaent tssues ma ntrodue a dstorton nto loal magnet feld near the nterfae between the tssues. 3. Sne the tehnque used for loalzaton of reeved sgnal mplements magng gradents the loal temporar magnet feld nhomogenet ndues proton dephasng. Ths ma result n repeated ntenst flutuatons wthn the same tssue wth a noteable struture. Fgure.4 shows a magnfed area of whte matter wth pel ontrast enhaned to observe the nhomogenet mrostrutures. The statonar gradent s the maor artfat that nterferes wth both human and automat proessng of MR data and t does not neessarl remove anatomal nformaton from the resultng sgnal. For ths reason t s theoretall possble to appl a orreton proedure and redue MR mage non-unformt. n further setons onl statonar gradent non-unformt s dsussed and s alled the MR nhomogenet or non-unformt artfat. 7

13 n the net seton we wll onsder more losel the struture of ths artfat. Fgure.3. nhomogenet wthn a tssue. Poston of the proflng lne wthn the same tssue on the MR mage left ntenst along ths profle rght Fgure.4. Contrast optmzed loal nhomogenet ndued b the gradent-drven edd urrents.3 Non-unformt Artfat Model n man mages MR nhomogenet gradents are not vsble. Ths s due to the ee s hgh apat for aommodaton as well as the fat that our bran often adapts the dstorted mage from the opts of an ee orretng t toward more nterpretable results. For ths 8

14 reason t s hard to desgn an automat orreton method wthout desrbng the nhomogenet artfat b analtal model frst. t follows from that MR bas feld ma be onsdered three-dmensonal for 3D mage volumes. For smplt the dervaton of mathematal foundaton for the orreton methods wll be performed for a two-dmensonal ase; t s mpled that t an be epanded to 3D b addton of a thrd oordnate and approprate hange n notaton. t follows from the MR aquston proess that the observed mage s the produt of the spn denst dstrbuton n the tssue and senstvt profle of the surfae ol [6]. The multplatve nature of MR non-unformt an be desrbed for two-dmensonal mage b the followng equaton: f n 3 where represents atuall obtaned two-dmensonal mage the non-dstorted deal mage f a multplatve bas feld and n an addtve nose. The latter s present n almost an MR mage and should be taken nto aount before an transformaton of model 3 sne t an be the soure of sgnfant omputaton error as dsussed n Seton.6.. The bas feld model 3 s the foundaton of a maort of orreton methods dsussed n the net seton..4 MR nhomogenet Correton Methods Ths seton ontans a general revew of estng MR orreton tehnques. Sne the estng tehnques are numerous and dverse an lassfaton would be formal and nomplete. We do not ntend to form our lassfatons based on an spef formal rule sne we beleve t s more nformatve to vew estng orreton methods as havng natural trends that an be ombned b a ertan qualt. Therefore some methods mentoned below fall nto more than one ategor. 9

15 .4. Phantom Methods The earler methods for orreton of MR non-unformt emerged n the earl 98 s when ommeral MR sanners spread. Brown and Semelka et al. [] menton that nreasng samplng frequen bandwdth of the MR sanner leads to a reduton of non-unformt n observed mages but ths also leads to a lower sgnal-to-nose rato and the loss of anatom nformaton [7]. The more effent earl methods use a phantom mage to ompute f n 3. Phantoms are smple obets flled wth a unform substane whh an be maged usng MR. After aquston of phantom mage the bas feld f for MR sanner S an be etrated usng 3. An eample of ths proedure s desrbed n [7]. One of the dffultes assoated wth ths method s the dffult determnng a phantom that eatl fts the sanner s three-dmensonal feld of vew and separatng the obet from ts bakground when neessar. Due to non-unformt n the phantom mage manual proessng ma be requred see Fgure.5 for llustraton of a phantom mage wth ambguous edge defnton. Furthermore t s not alwas possble to aess the atual sanner where the mages were aqured. Sne the development of broadband omputer networks man lnans and researhes vew medal data remotel n whh ase the applaton of a phantom-based method ma not even be possble. Ths lak of unversal applablt and hgh manual labor requrements led to development of other orreton tehnques that do not requre a prelmnar MR eperment to estmate the bas feld f. These tehnques are often alled post-aquston..4. Regstraton Based Methods The problem of bas estmaton s losel related to segmentaton of MR data. We have alread shown that the estmaton of bas feld n phantom mages requres

16 Fgure.5. Dstorted mage of lndral phantom. Note the loss of edge defnton at the top. segmentaton nto obet and bakground. A smlar approah used on bran mages s based on regstraton. The task of regstraton s to fnd the transformaton between the orgnal bran mage volume and a known model volume for whh the bas feld and tssue dstrbutons are establshed. After the regstraton transformaton operator Θ s found applng ts nverse Θ to orgnal MR mage and omparng ths result wth the model volume allows omputng the bas feld based on 3. The model mage for regstraton an ome from dfferent soures. For nstane La and Fang [8] suggest usng an addtonal low resoluton mage aqured from the same spatal poston as the orgnal mage ths approah resembles phantom methods. The unformt of ths small addtonal mage aqured usng a bod-ol nstead of a surfae-ol pulse sequene La and Fang [8] allows the estmate of the bas feld f of orgnal mage after regstraton. Man other authors as Studholme et al. [9] Chrstensen et al. [] and Collns et al. [] use referene MR ntenst templates for regstraton. The prnple dffult wth regstraton based methods s to determne the operatorθ. Most use an teratve approah for ths and therefore ma requre an etended

17 tme to omplete. The fnal error n Θ depends upon the orretness of the regstraton. Sgnfant non-unformt an nterfere wth the regstraton so these methods work better wth a relatvel small bas feld. Addtonall onstrutng a tssue model requres substantal prelmnar work that ma or ma not be reproduble. Fnall a regstraton based method s applablt s lmted to a spef bod part..4.3 Statstal Methods From a statstal pont of vew the MR mage an be onsdered a mture of several probablt dstrbutons; n ths ase f s onsdered to be a probablt denst funton. n the statstal vew the segmentaton problem onssts of fndng the unknown soft tssue dstrbuton of the deal mage. Then a subsequent analss of 3 would alulate f. Man statstal methods develop two epressons E and E : E for estmatng the bas feld f and E to ompute a tssue dstrbuton map. The alulaton of ether E or E depends on the other. E and E are estmated teratvel usng the Epetaton-Mamzaton EM algorthm desrbed orgnall n Dempster et al. []. Varous authors derved the estmates E and E usng Baesan statsts [] [3] Markov proess theor [4] deonvoluton flter based on Fourer transform [5] and other tehnques. The dffult wth the orreton methods based on the EM algorthm s that E and E must be ntalzed for the frst teraton. Therefore some pror knowledge must be avalable about the tssue dstrbuton n the orgnal mage. The resultng error wth the statstal method based on EM depends upon the orretness of the pror dstrbuton. t s easer to model soft tssues but an rregular anatom suh as edges and/or fne detal does not ft well n statstal models. To elude suh areas n the MR mage Gullemaud et al. [] ntrodue an addtonal tssue lass alled other and all rregular anatom s assumed to belong to other. Dependng on the desgn of E and E the loal mnmum usng EM ma not be a good appromaton of the bas feld and tssue dstrbuton see our results wth the method

18 desrbed n [6] later n ths hapter. To aheve smoothness for the estmated bas feld f blurrng n E or E s used between EM teratons; the bas feld determned b EM-based statstal methods often looks lke a fuzz orgnal mage see Fgure.6. Ths result suggests a omputaton error and ma remove some anatomal nformaton after the orreton has been appled. Wells et al. [3] and other authors have reported good orreton results; however t s hard to estmate the performane of man statstal methods sne the publshed results are often based upon unque tranng data and sometmes aheved after man algorthm parameter adustments e.g. Wells et al. [3] menton two ears of tranng data analss. The statstal methods requred tssue dstrbuton models usuall developed for one spef bod part. The statstal methods are usuall appled to bran mages. Fgure.6. Correton of a phantom bran mage usng Wells et al. [3]. Unform mage top left based mage top rght ts bas feld f found b Wells algorthm bottom left and orreton bottom rght..4.4 Hstogram Analss Methods Large areas of unform tssue n MR mages orrespond to hstogram peaks see Fgure.7. Ths propert s wdel used n ntenst-based segmentaton of MR mages to 3

19 determne the number of tssue lasses ther means and other dstrbuton haratersts. Hstograms of unform MR mages usuall have well-defned peaks whh are eas to separate. The non-unformt ntrodues addtonal rregulartes nto the hstogram: the peaks n suh hstograms are lower and tssue dstrbutons ma sgnfantl overlap posng an addtonal dffult for subsequent separaton see Fgure.8 an eample of a MR bran mage wth an rregular hstogram. Fgure.7. Effet of non-unformt on mage hstogram. Top row: non-unform phantom mage of the bran wth 4% non-unformt left ts multplatve bas rght; bottom row: hstogram of based mage left and orgnal unform mage rght. The MR non-unformt orreton method should mprove the mage hstogram. Ths s wdel used b the methods that emplo hstogram analss. These methods form a separate lass of MR artfat orreton tehnques and ther analss an nlude a ombnaton of loal and global hstograms. For nstane Brnkmann et al. [7] ompare loal and global hstogram mean and medan ratos; Chrstensen [8] uses hstogram 4

20 dervatve analss; DeCarl et al. [9] utlze an ntenst-based segmentaton usng a global hstogram and then estmate the bas feld f usng loal hstogram analss. Fgure.8. A ase where hstogram peak dentfaton s dffult. MR sagttal bran mage left and ts hstogram rght. Hstogram based algorthms normall need onl a few mage related omputatons to omplete and therefore are fast; the do not requre an tranng data. However the followng reasonng ma lead to questons about ther dret applablt to bas orreton. An mage hstogram represents the dstrbuton of dfferent ntenst levels n an mage. t s a mappng of two-dmensonal or three-dmensonal wth approprate hange n notaton mages to a one-dmensonal hstogram graph H where H 4 and ranges between and ma[ ]. Equaton 4 llustrates that the transformaton H s degenerate and an nverse transformaton does not est. Ths means that the hstogram does not unquel dentf the mage. A purel hstogram based approah reonstruts a ertan global mage haraterst bas feld from the loal haraterst hstogram. Sne the hstogram transformaton s degenerate some nformaton for the reonstruton of ts global haraterst ma be mssng. For eample smoothness or even peewse ontnut of a bas feld s not guaranteed; therefore addtonal assumptons are almost alwas used. Hstogram-based methods should requre more empral adustments than the other methods 5

21 beng dsussed. To summarze hstogram based methods are fast smple to mplement but ther aura an be nsuffent..4.5 Rentegraton Methods The mage gradent s another mportant haraterst from whh the bas feld estmaton an be obtaned. mage gradent analss methods assume that n the areas where the tssue s unform the gradent vetor to the gradent vetor of a bas feld: D at eah pont s roughl equal D D f. 5 Formall t follows from 3 that the bas feld an be obtaned b an applaton based on the re-ntegraton of the resultng gradent feld. Assumpton 5 and 3 alone are not suffent for the bas feld estmaton e.g. the addtve nose and small anatomal detal would ntrodue an unaeptable error. The models derved wth rentegraton methods are desgned to suppress the nose and elude the bad areas of an mage from onsderaton. Then the applaton of a seleted bas feld reonstruton s performed. Vokurka et al. [5] for nstane desgned a speal flter to be appled durng the gradent feld omputaton. The report that the resultng gradent feld s smooth and regular enough to allow the bas feld re-ntegraton. However the orgnal paper does not ontan suffent epermental data onl three datasets were analzed to determne the effen of ths approah and sne re-ntegraton s assoated wth hgh omputaton error etended eperments are needed to show the stablt of a partular re-ntegraton. n another re-ntegraton method La and Fang [8] mnmze the error b performng re-ntegraton on a fnte element grd so that the error an be mnmzed on eah fnte element separatel. 6

22 .4.6 Surfae Fttng Methods All the prevousl overed orreton methods are based on one partular foundaton for the spef desgn of the orreton. There are numerous methods that ma use dfferent models but are ver smlar n one formal haraterst; the model the bas feld as a smooth and slow varng funton that an be appromated wth a fnte bass. Ths approah s the defnng haraterst of surfae fttng; man of the prevousl mentoned tehnques belong n ths ategor. n general surfae fttng methods appromate the bas feld f wth a fnte bass: f B N B 6 where f B s the bas appromaton B are bass funtons and are the resultant bass oeffents. Usng 6 the problem of fndng the bas feld s to determne the oeffents. Ths has the obvous advantages.e. nstead of searhng for the bas feld value at ever pont e.g. done b statstal methods Wells et al. [3] onl a few oeffents need to be found. The resultant smoothness s obtaned automatall from the propertes of bass funtons. Surfae fttng an emplo an of the tehnques mentoned and ontan features of ther dfferent algorthms. For eample some methods []-[] onstrut an error funtonal usng the bas model 6 and an teratve mnmzaton method to determne the bass oeffents. The newer fuzz lusterng methods [6] [3] and [4] use a fuzz-set approah wth statstal methods plus the EM algorthm to determne the bass oeffents. Surfae fttng methods ommonl use an teratve mnmzaton algorthm where the number of teratons s not alwas known n advane. For eample the spm method [5] Seton.9.7 requred from 5 to 9 teratons on dfferent datasets sometmes runnng for as long as twent mnutes. 7

23 .5 MR Non-unformt Correton: Wh Another Method? n ths researh a new method for orreton of MR non-unformt s developed. n Seton.4 several methods are presented to solve ths problem. What s our ratonale for reatng another method? To answer ths queston onsder the omple struture of the varous MR artfats desrbed n Setons. and.3 whh suggest that an effent orreton method must provde smplfaton where ts bas feld model onl appromates the real bas feld. The applablt of an orreton method depends upon ts model propertes and the sope of ths model. n a orreton method desgn a ompromse between the generalt and aura of the method s performane on atual mages of mmedate mportane must be made. The urrent trend s to appl MR orreton methods to bran mages n a pre-proessng step to do segmentaton volume alulaton and 3D renderng. A maort of the publatons referened addressed the orreton of or verfed the performane of ther proposed method wth bran mages. n addton reent studes show that even the most elaborate methods annot entrel remove MR non-unformt: none of the algorthms that we evaluated performed deall under all rumstanes Arnold et al. [6] omparng s bas orreton algorthms. Thus t s lear that a general method that performs equall well on a broad range of MR mages would be of nterest. Establshed orreton methods ommonl perform several teratons on ther nput. As the sze of data nreases ther resultant eeuton tme nreases several tmes on the order of the ube of the lnear mage dmenson. Clnans demand fast eeuton tmes for almost all mage proessng and an undefned tme s undesrable. For these reasons a method wth a fed short eeuton tme would be preferable for use n lnal envronments. 8

24 To summarze we onsder a orreton method generalt robustness and short proessng tme to be an optmal ombnaton for lnal and researh envronments wth the usual demand from MR mage proessng. n the setons that follow we wll present our new orreton method named dervatve surfae fttng dsf wth these propertes and evaluate t on volumes of MR mage data..6 New Method Dervaton Ths seton ontans the mathematal foundaton of our proposed orreton algorthm. To ustf our desons we wll provde a mnmal mathematal bakground and prove several useful fats and relatons..6. Non-unformt Model Used n Dsf Although the multplatve MR bas model desrbed 3 s the foundaton for the maort of estng orreton methods t has never been used n ths form for omputaton. The reason s obvous: an method that determnes f from the produt f from 3 s ver lkel to perform numerous dvsons whh are not omputatonall effent. The log-transformaton s routnel performed to onvert the multplatve form nto an addtve form: [ f n ] log[ ] log. 7 Varous authors [8] [3] often delare that the nose term s small enough to be negleted. Ths s not obvous from 7 and a more prese estmaton of resultng error s neessar. Let s denote g n and g f '. Lemma.. Let g g >. For natural logarthm the nequalt 9

25 log g log g g log g log g 8 holds when g g. 9 < Proof: Suppose g s fed. Consder the funton We have G g log g log g log g. g g G g g g g g g g so t nreases b g. G g onl when g g and therefore G g when g g g. g The funton g g dereases b g when g > g. Therefore G g when g g see Fgure.9 and from nreasng of the log funton follows 8. _ < Fgure.9. Area of safe log-transform. The graph of Lemma.. g g llustrates the proof of g

26 Denotng: log f log f log and log n log n from 7 and 8 t follows that log log f n log log log log when the nequalt 9 s met. We wll use ths mportant relaton to mprove the robustness of our model. To redue n and therefore the error defned b an edge preservng smoothng flter smlar to those desrbed n [7] [8] wll be used. Wth n 7 beomes f. log log log After the multplatve bas model has been onverted nto an addtve one we an better desgn a numeral method to estmate f log. log log.6. MR mage and Bas Feld Modelng As shown n Setons. and.3 the bas feld f log ma be onsdered smooth and slow varng based on the nature of atual MR artfat. f log an be appromated b a fnte set of bass funtons B 6: B f B. log N Sne we are appromatng a smooth funton smooth bass elements B suh as polnomals should be used. Sne homogeneous tssues are represented b pels of smlar ntenst we model an unbased MR mage as a peewse onstant funton where areas of mnmal ntenst varaton orrespond to a sngle tssue. A regular MR mage ontans some of the followng: representatons of large organs vessels bones and smaller organs. Tpall large organs and bakground oup most of the mage spae and sne the represent unform obets

27 from MR magng pont of vew we an assume that the areas of low ntenst varaton preval n a regular MR mage. The areas of an mage wth fne strutures or edges need to be eluded from the model to mnmze the omputaton error. Ths an be done b ntrodung a pel weght funton w whh determnes the nfluene of loal mage haratersts of eah pel on the fnal result. The detaled substantaton of our hoe of w for use n dsf s provded n Seton Computaton of Bass Coeffents To determne f log from t s neessar that we etrat an deal mage log frst. For eample Brehbühler et al. [] evaluates the dfferene between log and pre-defned tssue ntenstes for ths purpose. n our vew an orgnal mage ma be too rregular and suh an operaton would nevtabl ntrodue addtonal error and smoothng the orgnal mage suggested n [3] would lead to a loss n edge defnton and an nreased error n the areas of rregular anatom. Our approah s based on the propert of log llustrated n prevous setons to be peewse onstant for the most of the mage. The onl haraterst of nterest for deteton of a slow varng bas feld s the low frequen varaton throughout the mage. The partal dervatve operator appled to an mage produes a gradent feld n the dervatve dreton and an be used as a natural measure of slow varaton. We an appl a med partal dervatve operator D { ; ; > ; L K} α α to both the mage and the bas feld modeled b a polnomal bas. A partal dervatve of a onstant s zero and an be onsdered zero everwhere eept the tssue D α log boundares and fne anatomal strutures removed from alulatons b the use of the weght

28 funton w α. Applng Dα to both sdes of and omttng arguments for smplt we obtan usng : α B α B α α D log log log α f Dα log f log B 3 α α where f B D f and B D B log α log α N α. The set of funtons { α B N } L ma ontan lnearl dependent elements and no longer be sutable for appromaton. however Lemma.. Let real dfferentable funtons f L f be lnearl ndependent wth at N least one f onst. f L k so that then f k onst. N N k f 4 Proof: Frst t should be noted that onl one onstant an be ontaned n a lnearl ndependent set of funtons. Takng the ndefnte ntegral from both parts of 4 we obtan N N f d d f d C. From the defnton of ndefnte ntegral t follows that N N f C C f C C C. 5 Suppose f onst f k C. Let ~ f. Usng 5 we have ~ f f f f C. 6 Sne and b our supposton k the left hand sde of 6 s a non-trval lnear k ombnaton of f L f and we have a ontradton. _ N t follows from Lemma. that removng onstants from the set of bass funtons guarantees the lnear ndependene of ther dervatves. After neessar nde hanges we 3

29 α an assume wthout lmtng the generalt the lnear ndependene of{ B L M} M N orm N. The problem of fndng the bas feld an now be reformulated as fndng the best M appromaton B α of a known funton α f usng the bass{ B L M}. α log Suppose f belongs to the normal lnear spae. Fndng the best appromaton means that we need to fnd an element B α suh that M f M B α nf f M L M B α. 7 f suh an element ests t s alled the element of the best appromaton. Bakhvalov et al. [9] shows that the element of the best appromaton ests. To fnd t n our ase t s onvenent to onsder the norm and salar produt f Ω f g Ω f q dd f g q dd 8 where q and [ L w] [ Lh] Ω w and h are the lnear dmensons of an mage. The norm defnes a Hlbert spae for whh the element of best appromaton s unque. The proof of ths an also be found n [9]. From 7 oeffents { } mnmum for the epresson L of the element of best appromaton provde a M M M M α α α Φ L M f B f B f B. Φ Ths epresson reahes ts mnmum when ondtons are satsfed. We have k 4

30 5. Φ α α α α α α k M k M M k k B B f B B f B f B From ths we obtan the sstem of lnear equatons wth unknowns { } M L that orrespond to loal mnmum of Φ: M k k M k B f B B. L α α α 9 One of the solutons of ths sstem orresponds to the element of best appromaton so we need to know how man solutons the sstem has. Sne the elements { } M B L α are lnearl ndependent due to ther seleton based on Lemma. the matr [ ] α α M B B B s postvel defned [9].e. from g g B M t follows that g. Sne M B s postvel defned ts determnant annot be zero and the sstem 9 has a unque soluton whh defnes due to ts unqueness the element of best appromaton. For dsrete dgtal mages ntegrals n 8 are replaed b summatons: d d q g f g f q f f. Usng w q α and ombnng 3 9 and we obtan: M k B w B B w M k k log L α α α α α α. For ever α has a unque soluton α α α M C L whh an be obtaned dretl usng Gaussan elmnaton. Lemma.3. Let L where L s a normal lnear spae wth norm L. Let L M L be

31 the elements that appromate on L: δ L M δ L beng the M appromaton errors. Then f ~ the nequalt M ~ ma δ L L holds. M Proof. We an represent as δ so that M ~ L M M δ M M M M δ L L M M δ L M M ma δ k k L ma δ k k L whh proves. t should be observed that the estmate ~ annot be mproved: n the ase when δ δ L δ M s an equalt. _ The soluton α C of sstem provdes an appromaton to the true bas oeffent vetor C L. Let Α be the set of all partal dervatves for whh the solutons are M obtaned. Combnng solutons for allα Α we an develop an appromaton of C ~ C Α α A C α 3 From Lemma.3 the error of ths appromaton δ ~ C C ~ L s guaranteed to be smaller than or equal to the error of everc α α δ α C C. f the dstrbuton of L α δ s smmetr about zero the proof of Lemma.3 suggests that the error reduton ma be sgnfant whh s mportant for pratal omputatons. 6

32 .7 The Algorthm.7. A General Sheme of Dsf Algorthm Based on Setons we an now develop a general form of the dsf algorthm for the MR bas orreton. n ths seton the maor steps nvolved are presented. The general algorthm steps and data flow are shown on Fgure... ntalzaton. The nput nludes an orgnal m n mage matr set Α ontanng the orders of partal dervatves and the bass parameter set Ω. The latter depends on tpe of the bass dsussed n the net seton.. Edge preservng smoothng of. Ths step redues the omputatonal error on further steps. 3. Log-transform of. We an assume for all. To mnmze the error assoated wth the addtve nose omponent n n 7 the log-transform s performed to hold true n aordane wth 9: [ ] log log. 4. Calulaton of mage partal dervatve matres α log α Α. Ths s done usng the partal dervatve sheme desrbed n Seton Generaton of bass. Based onω a set of m n bass matres { B } L N s generated. For ever α Α matres α { B } L M orrespondng are onstruted usng an analtal epresson for eah α D. When the analtal form of B s not avalable we an use the fnte dfferene sheme used to alulate α log. α 6. Calulaton of the weght matres w α Α. The weghts are used to remove from onsderaton the nter-tssue areas and fne detal whh annot be desrbed b the peewse onstant model. Justfaton of our approah for alulaton of 7 α w s

33 Bass ntalzaton parameters SMOOTHNG LOG Calulaton of mage partal dervatves log α { } log Calulaton of weght matres Calulaton of bass { B } { B } α { B } Calulaton of bass dervatves { α log } { α w } Construton of lnear sstems S S Α Gaussan elmnaton Gaussan elmnaton Construton of averaged soluton ~ f log Reverse transformaton and salng ~ T S Fgure.. The general steps of the dsf algorthm. Dashed arrows ndate optonal data flow. 8

34 presented n Seton Construton of lnear sstem S α for everα Α and solvng Sα wth Gaussan elmnaton. n ths step the S α s are defned usng summatons n and sne the are postvel defned the an be solved wth Gaussan elmnaton. 8. Construton of a fnal soluton. n ths step poolng of the solutons obtaned on Step 7 usng 3 s done to redue the resultng error. 9. nverse log-transform and salng. After the fnal soluton for the bas feld ~ f log M ~ B s obtaned the nverse log-transform log s appled to the orreted mage: ~ log f ~ log log 4 Fnall the salng transformaton T S ma mn ~ ~ ma mn 5 ~ ~ ~ mn mn ~ s appled to preserve the ntenst range of orgnal mage. The transform T S produes the fnal form of the orreted mage. There ests a problem of orgnal and orreted mage hstogram msmath. We dsuss the wa to mprove Seton.7.6. T S ~ n.7. Edge Preservng Smoothng The random addtve nose reduton s a fast pre-proessng step whh mproves the robustness of the results of further proessng. Aordng to the appromaton error b the fnte bass n the log-doman depends dretl on the magntude of addtve nose omponent n. At the same tme a denosng flter should not remove or blur the edges n 9

35 an mage sne that would affet the values of the partal dervatves and thus nterfere wth our dsf algorthm. We have hosen to mplement the edge preservng flter wth the followng onvoluton formula: ˆ k ep α k ep α [ ] σ β β σ 6 where Î s the resultng ntenst the summatons are done over the flter ore dmensons and k α β σ are oeffents that determne the strength of the flter and ontrbutons of ts varous omponents. Edge preservaton s aheved b eponentaton of the ontrbuton weghts whh are based on the ntenst dfferene and dstane from the entral pont. Fgure. llustrates the applaton of ths flter to a bran mage..7.3 Partal Dervatve Estmaton Sne our solutons of resultant lnear sstems depend on the error n alulaton of partal dervatves ther estmates should be robust and the error mnmal. n ths seton we wll eamne varous partal dervatve shemes and ther ablt to produe satsfator estmates. Let the funton be defned on Cartesan grd { } h h. Wthout lmtng generalt we wll onsder onl the frst-order partal dervatves n detal and provde neessar remarks about hgher orders. For two-pont shemes smmetr fnte dfferene shemes produe better appromatons. For{ } and are appromated: 3

36 h h h h ~ L ~ L h h 7 wth resdual errors R R h 6 h 6 ξ ζ ξ [ h h] ζ [ h h]. 8 Fgure.. Effet of edge preservng smoothng on a bran mage. Contrast optmzed magnfed area n the soure left and n the denosed MR mage rght. The ntra-tssue ntenst varaton beomes smoother whle edges are preserved. The nferene s based on Talor seres epansons and Roll theorem [9] p. 79. Although the appromatons 7 have resdual errors O h the eat values of the funton on the grd knots are not known beause mages ontan random nose. So n the alulaton of partal dervatves 7 the appromate value ~ n s used nstead of the eat value. The mamum resdual s R ma ξ ζ ma and the random nose 6 magntude s E ma n. From 7 and 8 we obtan for smlarl for 3

37 ~ ~ n n r r Rma h h h h h h E h. The term M O nreases as h dereases so dereasng h leads to nreases n the resultng h error. The mnmum of the epresson E R ma h s when h 3 h E R ma. From eperene R ma ~ ma for the tpal MR mage and the random nose vares wthn.ma E.9 ma from whh.4 h.8 9 s the optmal nterval grd step. The mnmum dgtzaton step n a dgtal mage s and our estmate shows that h s aeptable for use n a fnte dfferene sheme for the alulaton of partal dervatves. Formall t s possble to use h < b nterpolatng the values between the pels. However an suh nterpolaton would be based on the dsrete values wth the same samplng ponts whh leads to a fnte dfferene sheme based on more ponts and the resultng error wll not be redued. To mantan the same error wth hgher order partal dervatves the number of grd ponts n the approprate sheme should be nreased. What wll happen f we add more grd ponts to 7? Suppose N knots are used n a fnte dfferene appromatng a frst order partal dervatve: D α h h K N t ~ a t N t t a t t t K t K N t a n t t t 3 wth K N a t t. 3

38 The nose n at pont an be modeled as a random varable wth Gaussan dstrbuton wth µ andσ M. The sum N a n t t t K t also has a Gaussan dstrbuton wth µ and Sne M σ. K N N a t t a N t t at N 3 K N t at t σ as N. Beauseσ determnes the varaton of error n ths fnte dfferene sheme the error that results from naurate measurements at mage samplng ponts wll be redued b the fator defned n 3. t s desrable to use a small neghborhood wndow for a partal dervatve sheme. The desgn of our algorthm requres aurate deteton of tssue nterfae areas to elude them from onsderaton and some borders n an mage ma have ver sharp defnton. For ths reason the ntroduton of the ponts loated far from the pont of nterest n the alulaton of a loal dervatve would redue the aura of suh alulaton. n a nnepel neghborhood enterng n a pont of nterest onsder the patterns depted on Fgure.. n the ntal frst order partal dervatve appromaton 7 the ponts shown as on Fgure. are used. The ponts shown as and 3 an also produe a smmetr dervatve estmate. The dervaton for usng the ponts labeled on Fgure. wll be presented and an be derved smlarl. 33

39 34 Fgure.. Ponts partpatng n mproved partal dervatve shemes. S-pont patterns used for left and rght are shown. Consder the Talor seres epansons usng : L ± ± ± ± ± ± ± ± ± h h h h from whh we obtan L ± ± ± ± ± ± ± h h h. 3 Now we an also onsder an epanson for ± : L ± ± ± h h h h from whh h O h h. Combnng ths wth 3 the fnal form of the four-pont fnte dfferene appromaton L 4 for follows:

40 h O L h O h h O h h h h h h L. Thus we obtan the dagonal dfferene sheme appromaton for the partal dervatve. To obtan a s-pont lnear appromaton ombne ths wth 7: h O h h O L h O L L. 33 The analogous epresson for s h L Fgure.3. Evaluaton of the s-pont and two-pont approahes. The hstogram of orreted mage usng L urve and L 6 urve. The varane of random error for 33 and 34 s 8 8 of the orgnal random nose varane E 3. Sne L 6 and L 6 are two-pont appromatons 9

41 s also vald for these and usng h allows ths naurate measurement error to be lose to a theoretal mnmum. However ths redues the appromaton aura for the dervatve from Oh n L to Oh n L 6. Even so L 6 produed more sutable dervatve estmates for dsf. We ompared dsf performane usng L and L 6. L 6 resulted n hgher peaks n the hstogram of orreted mages see Fgure.3 and thus provded better dstnton between dfferent soft tssues. Calulaton of hgher order partal dervatves s based on smlar onsderatons..7.4 Seleton of Appromaton Funtons The estng methods for a one-dmensonal appromaton are well developed and standardzed. The performane of urrent omputers allows most one-dmensonal problems to be solved wth standard methods developed from theoretal researh. The omplet of these problems rses sharpl as ther dmensonalt nreases and mult-dmensonal methods usuall do not provde the same level of aura as onedmensonal methods. For ths reason appromaton funtons are usuall seleted for eah partular problem. Even f a set of appromaton funtons seems sutable ther use requres theoretal substantaton. n MR mages the appromaton funtons B should be both smooth and slow varng whh means that hgher order dervatves of B should be appromatel zero. n the one-dmensonal problem usng ether a polnomal or trgonometr bass for the smooth funton appromaton one ould epet to use ths same approah for two- and three- dmensonal medal mage problems. Mult-dmensonal appromaton funtons an be obtaned from the sngle-dmensonal funton set usng a Cartesan produt: from the appromaton funtons of one varable B we an generate funtons of two varables.e. B B B L N. f the funton set B L B s lnearl N 36

42 ndependent then the D funton set B L B wthout the onstant funton N N see Seton.6.3 s also lnearl ndependent. ndeed suppose the opposte: n suh there should est a non-trval lnear ombnaton Ths means N N N B B. B N B for all values of. Let be some fed value for whh some ~ N B s not zero. Suh value ests sne B L B are lnearl ndependent and we have a ontradton: N N ~ B. Therefore the Cartesan produt generates a set of lnearl ndependent bass funtons. Ths onstruted two-dmensonal bass s N. To redue ts omplet we an lmt the number of ts hgher-order members: N for all. n ths ase the B bass sze s redued n two dmensons to N N We onsdered the two tpes of funtons that are most frequentl used n one-dmensonal appromaton problems: sets of polnomals P n of degree n and trgonometr funtons T n epπn.. 37

43 Fgure.4. Comparson of bass funtons. Polnomal bass funtons left vs. trgonometr rght. Fgure.4 shows the eamples of polnomal and trgonometr two-dmensonal bass funtons where N 3. Both P n and T n ma not appromate the bas feld deall. The problem wth polnomal funtons s ther unlmted growth on borders whh an potentall lead to a loss n aura. t ma also be useful to onsder other sstems of polnomals wth speal propertes. n partular orthogonal polnomals are ommonl used as nterpolaton funtons beause the have man useful propertes. For eample zeros of orthogonal polnomals annot be multple and are dstrbuted asmptotall unforml on an gven lne segment [9]. Wth dsf we used two sstems of orthogonal polnomals: Legendre polnomals and Hermte polnomals. Legendre polnomals have a norm n d L n n n n! d n L n n and ther oeffents are omputed usng the reurrene relaton Hermte polnomals n Ln n Ln nln. 38

44 n H e n d d n n e have a norm H n n n! π and ther oeffents are omputed from the relaton H n H n nh n. Trgonometr funtons have lmted growth but are more omple omputatonall than polnomals. Addtonall T n πnepπn would not appromate a onstant unforml. Ths would redue aura when the bas feld s small. To selet the set of funtons for appromaton t s useful to estmate the error of soluton n eah ase. Suppose we are solvng a lnear sstem A b 35 and ts oeffents are known onl appromatel and deall the sstem A should be solved. Let X be the soluton of 35 followng estmate of r s orret 9: b A A b b η 36 * X the soluton of 36 and X X * r. The r A η X A. n the left hand sde s known presel n ths ase and we an wrte The quantt r A η. 39

45 τ r η b r b sup sup A η X b X η η X epresses the onneton between relatve errors of the rght hand sde and the soluton: r X η τ 37 b and s alled the ondton measure of a sstem. We an also onsder the haraterst of a sstem ν A supτ based on the left-hand sde of a sstem onl. t s alled the ondton number of a matr A so 37 an be rewrtten as b r X η ν A. b t s lear that ν A and ts magntude s proportonal to the relatve error. To ompute ν A sne we have b A sup sup A b X A ν A A. Table.. Effet of bass seleton on ondton number of resultng sstem. Condton numbers of the sstem for dfferent bass funtons are shown. P n L n H n T n ν A To selet bass funtons wth the lowest error we randoml hose ten MR mages and evaluated the ondton number of for eah MR mage usng dfferent bass funtons. 4

46 These results are summarzed n Table. Based on these results P n was hosen as the bass resultng n a lnear sstem wth the smallest ondton number..7.5 Weght Funtons w a Some ponts n an mage ma have values that produe errors n the alulaton of the oeffents for. Weght funtons w α are used to prevent an negatve effet suh ponts ma have upon resultng error. There are two possble soures of sngulart: a ver low sgnal and a hgh magntude of the gradent. Wthout lmtng generalt we an defne the ponts wth a ver low sgnal Λ as those where the mage has an ntenst wth magntude smaller than a smallε Λ : Λ ε. Suppose n E. f E ~ ε the ondton 9 of Lemma. wll not hold for Λ and we annot onsder the nose omponent n 7 small enough to perform a logtransformaton. Sne random nose usuall vares between % and 7% of the ntenst range t s suffent to elude ponts belongng to Λ from where Λ ε.. Λ ma Aordngl we an defne the weght funton for a ver low sgnal as. ma wα Λ. 38 otherwse Smlarl we an defne a set Γ wth a hgh gradent magntude b spefng the upper bound: Γ ε. Γ 4

47 Note that the weght funton s zero when > ε Γ. n prate the dstrbuton of gradent values s not known n advane and alulatng ε Γ for ever mage s more aurate than usng a fed value for all mages. For that we an selet ε Γ ε β based on a fed β: ε β β β where the pont s suh that f S β β { : [ L m] [L ]} β β n then S mn β. From statsts β represents a β-perentle of S. To appromatel omputeε β t s β suffent to determne a sequene { } of all mage ponts ordered b gradent magntude. Sne the number of ponts s mn ε β β β β a. [ mn.5] Ths has the same omplet as the quk sort algorthm for the one-dmensonal arra onsstng of mage elements. f m > n ts omplet s O mnlog m. The orrespondng weght funton s defned as β β wα Γ 39 otherwse We have onl dsussed weght funtons that an be ether or. t s also possble to develop a ontnuous for eample w~ w α Γ ep e ma α Γ. Clearl w ~ α Γ dereases eponentall as 4

48 r ma nreases. n ponts where the gradent s low r ~ the weght funton s lose to one w ~ α Γ ~ and when the gradent s hgh r ~ then w ~ α Γ ~. Sne ths ontnuous weght funton ma provde addtonal soures of error a omparson of dsf usng a dsrete and a ontnuous weght funton was neessar. We tested dsf usng α and w ~ α Γ wth ever eamned mage n our random set. The result w Γ showed that the orreton wth w ~ α Γ depended sgnfantl on the of edges n an mage and ther spatal dstrbuton; ths effet was not observed usng. Ths ma be due w α Γ to the ontnuous weghts ntrodung more random fators nto the sstem whh result n a less predtable outome. For ths reason we used w α Γ n the fnal verson of dsf to obtan the more robust soluton. The weght funton that allows eludng or sgnfantl redung both nstablt fators s obtaned from the ombnaton of 38 and 39: w wα Γ wα Λ..7.6 Resultant mage Salng The nverse log-transform 4 produes the mage ~. Ths resultant mage often has an ntenst dstrbuton dfferent than the orgnal mage. The post-aquston orreton methods tend to shrnk the hstogram of the orgnal mage [6] whh an omplate a proper ntenst based tssue dentfaton see Fgure.5. For ths reason a more detaled analss of the salng transformaton s needed. n ths work onl lnear mage transformatons are onsdered although onl the eat ntenst regstraton of orgnal and orreted mages would provde an deal math. 43

49 However regstraton s a omple problem beond the sope of ths stud. n ths seton an mprovement to the mathes between the orgnal and orreted mage hstograms usng onl lnear transformatons are dsussed. Let ~ be the mage obtaned b nverse log transformaton at Step 8 of the dsf algorthm. We seek salars a b suh that the fnal output mage where E s ˆ a ~ be 4 m n matr and E would provde the best ntenst range math wth the orgnal mage. Transformaton 4 preserves peewse ontnuous funtons and therefore the result of non-unformt orreton s also preserved. However the ntenst range of the result ma not math the ntenst range of orgnal mage mn mn ma ma. Requrng ths leads to the followng ondtons on a and b: a ~ ma ma ~ mn mn b mn ~ ~ ma ma ~ ~ mn mn ma and 4 beomes 5. Clnans often requre preservng the orgnal ponts of the mage so transformaton 5 s onl avalable as an opton n the dsf algorthm. Sne addtve random nose s alwas present the mamum ntenst observed n an mage an be onsdered a random varable and we an determne ts dstrbuton. That s f n E we an onsder the brghtest tssue T hgh n an mage to have ntenstes n the range [ E E] where hgh s the tssue average. All the ponts belongng to T hgh hgh hgh P LP k represent a sample of sze k from the normal dstrbuton wth unknown parameters. The order statsts P L P k an be obtaned from ths sample. Ther dstrbuton denst funton s desrbed b the epresson [3] 44

50 k! P ~ p k [ F ] [ F ] f! k! Fgure.5. Spm orreton. Hstogram of orgnal mage shown on top and orreton produed b spm algorthm [5] shown on bottom. The average spm ntenst 4. 3 whereas spm where f s normal probablt denst funton pdf: µ f ep σ π σ and F s normal umulatve dstrbuton funton df: F f t dt. The mamum of ths sample P k s dstrbuted as k [ F ] f P k ~ p k k. 4 Aordngl the probablt the observed ntenst mamum s wthn [ E E] s hgh hgh 45

51 hgh E Pr p t dt ma k. E hgh Usng the sample mean and varane for the brghtest tssue we an evaluate ths ntegral numerall. We seleted E. Pr ma and omputed the onfdene nterval lmts ma tα P k ma tα for ten bt grasale MR mages wth α. 95. t α vared between.6 ma and.5 ma throughout ths stud. Ths s the error n the ntenst range that ma result from 5 when the orreton transformaton was assumed to be lnear. Sne t s not lnear the atual error ma be hgher. Beause of mage ntenst randomness n an gven pont averagng estmators provde a more robust landmark for the aurate estmaton of transformaton parameters n 4. t s onvenent to onsder three estmators:. The average ntenst of the set Λfrom Seton.7.5 α low Λ Λ. the global mage average ntenst avg 3. hgh. Usng low s not desrable sne Λ s an area of low sgnal-to-nose rato. We dsussed ths n a prevous seton that the nferene based on the ponts from ths area s error-prone. Sne we onl need two parameters to defne a lnear transformaton we hoose avg and hgh. Hene: from whh ~ ~ avg a avg be hgh a hgh be 46

52 ~ ~ avg hgh hgh avg hgh avg a ~ ~ b ~ ~. 4 avg hgh avg hgh The transformaton defned b 4 permts an adequate math of the hstograms. When the non-unformt n the orgnal mage s not sgnfant t s possble to fnd a b more auratel usng a least squares mnmzaton: ~ [ ] [ ~ ˆ a b ] mn. Settng the dervatves b a b to zero we obtan the followng lnear sstem: Notng that ~ [ a ~ b ] [ a ~ b ] mn we an solve ths sstem as follows:. S S ~ S ~ S ~ ~ ~ ~ lead to the followng epressons for a b: mns ~ S a b S as ˆ. 43 mns S S mn ~ ~ These oeffents an be used for an aurate least squares soluton. n summar our orgnal goal was to mprove the math between hstograms of the orgnal and the orreted mages. We dsussed mage haratersts that were not dretl related to ts hstogram to obtan the lnear transformaton. To defne ths relaton reall that the hstogram s obtaned b ntenst summatons 4 and therefore the lnear transformaton of an mage produes the hstogram ~ ~ H a be ah b. Therefore the desrable propertes of fnal mage are also refleted n ts hstogram. 47

53 .7.7 mplementaton of Dsf Orgnall dsf was mplemented n MATLAB 6 the envronment sutable for quk although sometmes neffent development and testng of numeral algorthms. Sne MATLAB s avalable for maor operatng sstems verfaton of dsf performane s possble both n Wndows and UNX based operatng sstems. n development of dsf pseudoode we defned the followng proedures: - Man: the man ontrol funton whh takes the mage and algorthm parameters as an nput and outputs the orreted mage; - Generate_bass_matres takes bass parameters as an nput and outputs the matres of bass funtons and analtall alulated partal dervatves of bass funtons; - Generate_dervatve_matres - takes log-transformed mage as an nput produes a set of partal dervatve matres of ths mage as an output; - Partal_dervatve takes a matr and partal dervatve order as an nput returns partal dervatve matr obtaned b onvoluton wth a fnte dfferene sheme appromatng ths partal dervatve. See Append A for omplete dsf pseudoode lstng. For etensve testng and use an mage proessng algorthm should be mplemented effentl. The effent mplementaton s usuall based on seletng a spef operatng sstem and usng platform dependent development tools. n man researh envronments platforms from the UNX faml are preferred for development and testng of new software. Ths hoe n man ases s defned hstorall b estng nfrastruture and avalablt of low ost software for researh purposes. For eample the maort of free medal mage proessng tools are developed under Un-lke platforms. Of several avalable mplementatons of MR nhomogenet orreton algorthms referred n ths work onl one 48

54 [5] ould run under the Wndows platform and onl beause the ode was wrtten n MATLAB. Fgure.6. UnVewer man wndow. Despte the seemng attratveness of the UNX platform Wndows was hosen for mplementaton of ths algorthm. The reason for ths s that sne the etensve testng was requred to valdate dsf performane t was desrable to run t n man dfferent loatons on a dverse nput. Therefore our purpose was to develop the software for work n most lnal envronments and the Wndows platform s more sutable for ths purpose. Prevousl our group had developed the DCOM Dgtal Communatons n Medne PACS Pture Arhvng and Communaton Sstem. Part of t was a Wndows-based DCOM vewer alled UnVewer apable of dsplang and manpulatng mages n all medal magng modaltes as the ome from the sanner see Fgure.6 for the man vew of the UnVewer. We norporated dsf n UnVewer whh made t wdel avalable for revew testng and use. C mplementaton of dsf s urrentl a part of UnVewer whh s avalable from: 49

55 .8 Bas Evaluaton of Dsf Numeral algorthm evaluaton should aheve four goals:. Valdaton of mplementaton;. Testng performane on datasets wth known deal output; 3. Etensve testng of performane on large amounts of real data; 4. Comparson to other methods. n ths seton we desrbe the bas tests performed to verf that dsf dereases nonunformt n MR mages..8. Snthet mages Aordng to theoretal results n omputaton theor proven frst b Alan Turng n 936 the haltng problem s n general unsolvable so t s mpossble to desgn a proedure that determnes whether a gven algorthm halts on some arbtrar gven nput or not. For ths reason the valdt of a partular algorthm mplementaton s n general mpossble to prove mathematall. The onl possblt here s to verf that the mplementaton output s onsstent wth theoretal algorthm output through a seres of eperments. The maor proedure s:. One or several eperments that epose the ke algorthm features;. Theoretal output of the algorthm s alulated and the eperments on ts mplementaton are arred out; 3. The output s ompared wth theoretal estmates and based on ther math the onluson about mplementaton orretness s drawn. n the ase for dsf t s neessar to fnd a non-trval mage whh an be orreted wth predtable result. To do that a peewse onstant mage wth non-unformt desrbed 5

56 b 3 an be used. We emulated a peewse onstant funton as the monohrome hessboard mage Θ C where square brakets denote an ntegral part of a postve number C > s a fed onstant onstants representng ell szes. To emulate the multplatve bas feld we used the parabol funton. The based mage s defned as: w h Θ * [ w] [ h] w h. The result of orreton b dsf s shown on Fgure.7 rght. Comparson wth Θ showed that the varaton of ntenst wthn an sngle lass does not eeed.% so we an onlude that our mplementaton s onsstent wth the dsf algorthm. Fgure.7. Model mage orreton. Artfall dstorted mage left found bas feld enter orreted mage rght.8. Phantom mages To evaluate dsf on data wth a known bas feld we used smulated MR bran mage volumes avalable from MGll Unverst [3]. MR mage volumes were hosen wth mm sle thkness and 4% non-unformt. S emulated volumes onsstng of 8 sles eah 5

57 were tested three of normal brans wth T T and proton denst weghtng and three wth leson brans. Correton produed results smlar to those shown on Fgure.8. Fgure.8. Phantom mage orreton. n the top row: orgnal dstorted mage left true bas feld mddle based mage hstogram rght; n the bottom row: orreted mage left bas feld found b algorthm mddle orreted mage hstogram rght. Fgure.9. Aal mage orreton. Orgnal mage top left ts hstogram top rght bas feld found bottom left and the hstogram of orreted mage bottom rght. 5

58 .8.3 Hstogram Evaluaton t follows from Seton.4.4 that hstogram vsual analss an show whether an mage was mproved after orreton. Heght and wdth of hstogram peaks provde nformaton about the varane observed n dfferent soft tssues of MR mage and effent orreton method should redue the varane and nrease the peak heghts. The atual eperments wth phantom mages onfrm that for dsf Fgure.8. We also ompared the hstograms of real MR mages wth orretons produed b dsf. The omparson was performed on two omplete mage volumes and a number of separate bran mages from dfferent soures and of dfferent qualt. n all ases the hstogram peaks nreased n heght after orreton. The eample of hstogram omparson s shown on Fgure.9..9 Comparson wth Seleted Publshed Methods n ths seton we desrbe the etended analss of dsf performane on a large volume of phantom and real mages n omparson wth seleted prevousl publshed MR non-unformt orreton methods..9. MR Artfat Correton Methods Chosen for Comparson wth Dsf Arnold et al. [6] dvded the estng MR artfat orreton methods nto two groups: non-loall adaptve where the parameters of the bas feld at a partular pont are determned usng global mage nformaton and loall adaptve where the bas feld at a gven pont s determned from loal neghborng ponts. For omparson wth dsf we seleted two prevousl publshed methods representng eah of these groups: spm and bfm. Spm s a non-loall adaptve method developed b Ashburner and Frston [5]. t uses pre-segmentaton of the bran mage to etrat the whte matter as the frst appromaton. After that the bas feld s teratvel appromated b EM usng mamum log-lkelhood. MATLAB mplementaton of spm s norporated nto freel avalable SPM 53

59 software pakage [33] developed n the Department of magng Neurosene Unverst College London UCL. Bfm Bas-orreted Fuzz C-Means algorthm s a reentl publshed b Ahmed Yaman et al. [6] loall adaptve method. The dea of fuzz C-means s to determne the tssue prototpe luster medan tssue ntenst for ever soft tssue n an mage and defne an obetve funton for parttonng nto lusters: J u v 44 where v are prototpe lusters and [ ] u determne fuzz membershp of the pont n the th luster. Bfm etends 44 b ntrodung the bas feld adustment parameters; the resultng optmzaton problem s solved teratvel b sequental appromatons. We mplemented bfm n MATLAB. To valdate our mplementaton we hose the same BranWeb phantom mages [3] used n orgnal bfm paper and ompared our output wth results reported b Ahmed and Yaman et al [6]..9. Testng Crtera n seton. the two man reasons for developng MR non-unformt orreton methods were dsussed:. mprovng of vsual qualt. mprovng the ntenst unformt wthn a sngle soft tssue for subsequent automated proessng. Therefore non-unformt method testng should answer the queston whether these two goals are aheved. A vsual omparson s nevtabl subetve; therefore onlusons about the orreton method s vsual performane wll be made after omparng orretons on large volumes of data from dfferent subets. 54

60 55 The unformt wthn a sngle tssue s a more subtle haraterst and ts aurate vsual deteton s dffult so a numeral evaluaton s used. f eah tssue s s modeled as a random varable the natural measure of ts non-unformt an be derved from sample varane: [ ] s s s s µ σ where s s the number of ponts n s and s s s µ s the sample mean. The atual magntude of s σ depends on the amount of varaton and on the mage ntenst range whh does not allow omparng s σ for dfferent mages. To avod ths the normalzed verson of s σ alled the oeffent of varaton wll be used as the tssue ntenst measure of unformt: s s s v µ σ. The oeffent of varaton s nvarant to a unform salng ntenst transformaton: sne [ ] s s s s s s λµ λ λ λ µ and [ ] [ ] s s s s s s s s σ λ µ λ λ µ λ λ σ s v s s s s s v λµ σ λ λ µ λ σ λ. 45 However vs s not nvarant to a unform addtve ntenst transformaton: s v s s s s s v λ µ σ λ µ λ σ λ 46

61 and therefore oeffents of varaton annot be ompared for dfferent tssues. Sne the maort of MR orreton algorthms were evaluated n the past wth bran mages we hose to evaluate the oeffent of varaton on the prnpal soft tssues of the bran whte matter WM and gre matter GM. As dsussed n Seton.7.6 MR orreton methods ma sgnfantl modf the orgnal soft tssue means. n some ases maged tssues beome harder to separate.e. for two soft tssues s s the quantt µ s s µ s µ s ma derease after orreton. Ths would mean degradng the mage qualt nstead of mprovng t and Arnold et al. [6] desrbe ths as a ommon problem n MR orreton algorthms. Lkar et al. [] suggested a measure to estmate the overlap between two tssues s s alled oeffent of ont varaton as v s s σ s σ s. µ s µ s The oeffent of ont varaton reflets the relaton between µ s s and the varane of 56 s s. Clearl v s small for well-separated tssues and nreases as µ s s dereases. Cv s s an be shown usng epressons smlar to to be nvarant to both salar multplatve and addtve ntenst transformaton and as suh an effentl haraterze the soft tssue overlap. To evaluate v and v prelmnar lassfatons of soft tssues are desrable. Sne ths s a ver tme-onsumng proess we used a ombnaton of soures to obtan these lassfatons. These soures are provded n the net seton. Apart from mage qualt enhanement rtera desrbed above several other haratersts of MR orreton algorthms eamned n ths stud an be ompared. As dsussed n Seton.5 our goal s to develop a fast and robust algorthm so the omparson

62 rtera must also reflet ths. To ompare robustness we tested dsf spm and bfm on phantom mages wth dfferent nose levels Seton.9.5. To determne omparatve speeds we measured ther MATLAB ode eeuton tme Seton Testng Datasets n ths seton the MR datasets used n ths stud for the numeral evaluaton of dsf spm and bfm are desrbed. These do not nlude over datasets orreted b dsf that were evaluated vsuall.. S phantom mage volumes from the BranWeb smulator [3] were used. These nlude T T and proton denst PD weghted varatons of normal and multple sleross leson 3D bran mages. For all these mages the resoluton s wth a mm sle thkness. ntenst of eah pel n these mages s represented usng bts provdng 496 shades of gra. A smulated multplatve nonunformt bas feld f was hosen wth a 4% varaton whh means.8 f.. The random nose level defned for tssue s s [ n ] s σ s % was fed at 3 %. n further referenes these datasets are named T N T L T N T L PD N PD L where aptal letters represent the pulse sequene and ndees N and L normal and leson brans aordngl see Table.. Addtonall fve varatons of the T normal brans wth nose levels % % 3 % 5 % and 7 % were used to ompare the nose senstvt of the orreton algorthms. The soft tssue segmentatons were etrated from orgnal peewse onstant mages usng an ntenst range math. 57

63 . S real T MR normal bran mage sets Sets -6 Table. and ther manual segmentatons provded b the Center for Morphometr Analss at Massahusetts General Hosptal avalable through The spatal resoluton for these sets ranges between and bts per pel. 3. Four real T MR bran mage sets from dfferent soures Sets 7- Table. for whh we performed the soft tssue segmentaton manuall. 4. S MR and one RF mage set of varous bod parts from dfferent soures Sets - 7 Table. were also nluded..9.4 Coeffent of Varaton Evaluaton The oeffents of varaton obtaned n ths stud are shown n Table.. To obtan a graphal nterpretaton of these results we defned the normalzed gradent for the oeffent of varaton of tssue s: v sor v sorg dv s % 47 v s org where s org represents tssue s n the orgnal mage and s or n the output mage of orreton algorthm. Usng 47 we obtaned the satter plot of dvwm versus dvgm for spm dsf and bfm Fgure.. We an also defne the normalzed gradent for the oeffent of the ont varaton of two tssues s s : v s or s or v s org s org dv s s %. 48 v s s org org The relatve hange n oeffents of ont varaton resultng from the non-unformt orreton for sets - s plotted on Fgure.. Dsf and spm redued the WM and GM oeffents of varaton for all smulated sets. 58

64 Table.. Datasets used for numeral omparson of dsf spm and bfm. Set Modalt Bod part Bts per pel Resoluton T n MR T Normal bran T l MR T Leson bran T n MR T Normal bran T l MR T Leson bran PD n MR PD Normal bran PD l MR PD Normal bran Set MR T Bran Set MR T Bran Set 3 MR T Bran Set 4 MR T Bran Set 5 MR T Bran Set 6 MR T Bran Set 7 MR T Bran Set 8 MR T Bran Set 9 MR T Bran Set MR T Bran Set MR Chest Set MR Abdomen Set 3 MR Chest Set 4 MR Heart Set 5 RF Knee Set 6 MR Lumbar spne Set 7 MR Shoulder Fgure.. Performane omparson. Satter plot of dvgm usng 47 versus dvwm for sets -. One pont for bfm n the left bottom orner s not shown to preserve the sale. 59

65 Table.3. Coeffents of varaton of orretons performed b algorthms beng ompared. Eah haraterst GM oeffent of varaton vgm WM oeffent varaton vwm WM and GM oeffent of ont varaton vwm GM measured for unorreted volumes wth sr dsf spm and bfm. vgm vwm vwm GM Set sr dsf spm bfm sr dsf spm bfm sr dsf spm bfm T N T L T N T L PD N PD L Set Set Set Set Set Set Set Set Set Set Fgure.. Performane omparson. DvWMGM plot usng 48 for sets -. Ponts orrespondng to bfm orretons of sets 7 and 8 are not shown to preserve the plot sale. On authent datasets dsf redued or produed the same oeffent of varaton for the GM n all ases and for the WM n 8% of the ases. The other two algorthms spm and bfm 6

66 redued or produed the same GM oeffent of varaton n 4% and 9% respetvel for authent datasets and redued the WM oeffent of varaton n 7% and 9%. Sne spm does a pre-segmentaton of the bran mage volume for whte matter as a frst appromaton the results of the bas feld etrapolaton from whte matter to the entre mage must be ver aurate to produe a onsstent orreton. Spm an be epeted to aheve good results n orretng whte matter nhomogenet but the mprovement for the entre mage depends on error of WM bas feld estmaton etrapolaton method used and the orretness of an assumpton that the bas an be etrapolated from the whte matter to the entre mage volume. Spm s GM orreton n 7% of authent ases dereased modestl and sometmes even nreased the GM varaton oeffent. Wth the same authent data the WM oeffent of varaton was redued n most ases whh ponts to an noheren n estmatng the total bas feld. The bfm algorthm produed orretons wth a sgnfantl dereased ontrast between whte and gre matter dstrbutons n 94% of the ases whh rases a queston of whether t reall mproved those datasets. Dsf mproved T phantoms as well as spm although ts resultng bas orretons for T and PD were somewhat smaller. However t unforml mproved both WM and GM oeffents of varaton b not ntrodung an addtonal non-unformt. Spm was partularl unstable on both the hgh and ver low nonunformt mage datasets and requred a large number of teratons up to 6 on Sets 3 6 and 9 requrng up to mnutes whh s not aeptable n atual use..9.5 Senstvt to Nose Wth authent MR mages the sgnal-to-nose rato an var from sanner to sanner and t depends on the aquston pulse sequene and an fators that ma be present. t s therefore mportant to evaluate the performane of an orreton algorthm on mages wth randoml hangng nose parameters. For suh an evaluaton we appled spm dsf and bfm 6

67 to the smulated normal T mage volume wth % non-unformt and fve dfferent levels of random nose Seton.9.3. We ompared the dfferene between the GM oeffent of varaton for the orreted volume and orgnal volume; these results are shown on Fgure.. Fgure.. Senstvt to nose. Sold urve: v b GM-v GM; dash-and-dot urve: v spm - v GM; dashed urve: v dsf GM-v GM. v GM represents GM oeffent of varaton of an unbased nos soure v b GM of a based nos soure v dsf GM and v spm GM of the orreton produed b dsf and b spm respetvel. Bfm results are not shown on Fgure. beause the were too errat and would nterfere wth the graph sale used. As shown dsf orreton was stable even wth hgh levels of random nose. Ths s due to the smooth bas feld model that s not senstve to a sgnal of hgher frequen and to the use of nose-anelng s-pont partal dervatve appromatons Other Bod Parts As dsussed n Seton.5 man MR orreton algorthms were desgned spefall for bran mages and the maort of evaluatons n the lterature has been performed on bran mages. Sne our goal was to develop a more general orreton 6

68 Fgure.3. Vsual evaluaton of orretons. Comparatve orreton results for sets -6 are shown: a orgnal mage b orreted wth our method orreted wth spm d orreted wth bfm. 63

69 algorthm dsf was desgned for an MR mage regardless of the bod part. Dsf even produes good results on non-mr mages f the have the same slow varng multplatve pattern of nhomogenet and areas of well defned homogeneous tssue. These nlude for eample man CT hest mages. Sne UnVewer software was nstalled n a number of loatons we were able to appl dsf to over mages of MR and other modaltes from dfferent soures and observed a vsual mprovement on most of these mages. For analss we randoml hose several mage datasets of varous bod parts orrespondng to sets -7 n Table.. A omparson of vsual results for the three methods evaluated s shown on Fgure.3. On these mage datasets spm removed some non-unformt n sets 3 and 6 but ntrodued addtonal non-unformt n sets 4 and 5. Spm also tended to redue the hgher ntenstes n the mage whh n ombnaton wth hgh output ontrast result n a loss of anatomal nformaton. Bfm produed a vsble derease n the tssue ontrast and removed some anatomal detal whh was tpal for all datasets orreted b ths algorthm n ths stud. Dsf redued non-unformt n all the mage datasets although an etra algorthm pass mght rarel be requred; t dd not result n an vsual loss of anatomal detal..9.7 Eeuton Tmes Sne dsf has been norporated nto UnVewer t performs the orreton of MR mages n a volume n 7 seonds wth a fed orreton tme. Dsf learl s onsdered to eeute fast enough to proess hgh volumes of data. However sne fast mplementatons of bfm and spm were not avalable we annot make an onlusons about ther performane. To ompare the eeuton tmes of these three algorthms we ran them n MATLAB on Pentum V 3.6 MHz PC. The relaton of the eeuton tmes usng a more effent mplementaton ma var. 64

70 We randoml hose four mage datasets and measured the eeuton tmes of eah of the three orreton algorthms Table.4. Dsf was desgned to use a sngle pass spm and bfm had to do several passes teratons. Spm s teratons vared n dfferent mage datasets between 5 and 8. Table.4. Runnng tmes for evaluated algorthms on four datasets. Dataset Eeuton tme n MATLAB mn:se dsf spm bfm Set :3 :5 5: Set :4 4:4 : Set 4 :55 :5 : Set 5 :56 : 9:. Dsusson and Conlusons The method presented n ths hapter s based on the assumpton that the MR bas feld s multplatve smooth and slow varng and the partal dervatves of an MR mage an be appromated b ts orrespondng partal dervatves of the modeled bas feld. Surfae fttng of the bas feld usng a polnomal bass guarantees smoothness and stablt of a modeled bas sgnal; the seleton of bass funtons was ustfed b ther omputatonal propertes. A smlar bass model desrbed n [] uses Legendre polnomals; however our method was more stable usng a smpler polnomal bass. Wdel used statstal methods based on the EM algorthm suh as n [3] and [] use ntermedate low-pass flterng after eah teraton to redue omputaton error and produe a smoother bas feld. Due to the nature of the bas model dsf produes a bas feld estmate that s alwas smooth nose nsenstve and not affeted b loal mage dstortons. Our model also does not requre pror knowledge of the tssue ntenst dstrbuton. The onl nput parameter for our algorthm s the perentle valueβ that haraterzes the hgh frequenes n the nput MR sgnal. Ths parameter was found to be dfferent for the mages 65

71 of dfferent bod parts and modaltes but dsf performed onsstentl wth fed β on the mages of the same bod part obtaned from dfferent soures. For ths stud we appled dsf and two prevousl publshed methods spm and bfm to snthet mages s smulated mage volumes from BranWeb [3] ten authent bran mage volumes from Massahusetts General Hosptal Center for Morphometr Analss [34] and other soures and s datasets of other bod parts. Several parameters were estmated: varaton oeffents for WM and GM vsual qualt of orreton senstvt to random nose n data and runnng tmes. Dsf dereased the WM and GM oeffents of varaton for most of the bran datasets and was robust. Spm n several ases dereased the WM oeffent of varaton more than dsf but was also less stable and ntrodued addtonal non-unformt n several ases espeall for gre matter. Both dsf and spm performane was not affeted b nreasng the random nose n smulated datasets. Our algorthm vsuall mproved % of s mage datasets 5 mages n total of other bod parts one of them was not a MR modalt; both spm and bfm were less stable on these and appeared to remove anatomal strutures from orgnal data. Due to ts non-teratve desgn dsf s runnng tme depends onl on the mage data sze whereas spm s number of teratons ranged between 5 and 8 and was hard to predt. The bfm algorthm appeared to sgnfantl redue soft tssue ontrast and remove anatomal detal n pratall all eamned mage datasets. The speed of our algorthm makes t partularl useful for real lnal applatons. Man aurate MR orreton algorthms usuall use teratons to appromate a soluton and eah step often requres advaned omputaton. Long omputaton tmes make t dffult to use man urrent MR orreton algorthms n lnal software. Our method uses a sngle teraton although t an be appled repeatedl and was ntall desgned to be omputatonall effent. Usng the mplementaton desrbed n Seton.7.7 the orreton of 5656 MR bran mage volume takes about 7 seonds on a urrent PC. Due to 66

72 short runnng tmes our algorthm an also be used as a quk mage f and/or a preproessng for volume segmentaton or renderng proedures. 67

73 Chapter Automated Medal mage Volume Renderng. ntroduton The vsualzaton of medal mage volumes has beome a large growng researh area. t has attrated man researhers throughout the world. Vsualzaton has ts roots n the 96 s and 97 s when the theoretal foundaton for D mage proessng was establshed. However ntensve nvestgaton of volumetr mage generaton from medal data dd not begn n earnest untl the seond half of the 98 s. One reason for ths nrease was a demand to fnd effent methods for 3D mage proessng and vsualzaton. The rapd development of medal magng equpment produed large three-dmensonal volumes of dgtal medal mages. Tradtonal methods of the manual san-b-san eamnaton b radologsts are tedous on the sets ontanng hundreds of hgh resoluton mages. Automaton s requred to rapdl etrat useful features from ths oean of data. On the other hand the automat proessng of volumetr data requres omputatonal power that s proportonal to the ube of the volume s lnear dmensons. n an ase there s a mnmum tme threshold for volume mage proessng whh when not aheved automat volumetr vsualzaton remans manl wthn the bounds of pure theor.e. t wll not be used n a produton envronment. Fortunatel ths threshold has been aheved n reent ears and ths allowed real world lnal applatons usng 3D-vsualzaton. The rapd growth of medal mage volume vsualzaton has gven rse to numerous methods for volume renderng whh an be broadl lassfed nto surfae and dret volume renderng. Surfae renderng generates and dsplas the obet s boundar surfae and need to be generated frst. Dret volume renderng uses ever volume element n the generaton of the 3D sene. 68

74 . Medal mage Volumes There are two dfferent approahes to the representaton of the smallest data element n a dsretzed mage volume. Ths s due to the dualt of a dsrete nput sgnal: t an be onsdered a set of samples along a ertan grd from the ontnuous obet; eah sample an be onsdered an average value over a ertan area. Ths dualt s analogous to one n the phss of quantum theor: aordng to Hesenberg s unertant prnple ether the eat poston of a partle or the eat tme n ths poston an be measured presel but not both smultaneousl. The unertant prnple for a dsrete sgnal ma be formulated: sgnal magntude nterpreted as ntenst for mages and ts eat spatal poston the orrespondng pont n a volume annot both be determned presel at the same tme. A sample from a medal mage volume s frequentl nterpreted n terms of as an averaged sgnal over a small subvolume. Ths s ustfed b the nature of the medal mage aquston proess desrbed for MR mages n setons..3. n aordane wth medal mage volume V s a set of three-dmensonal elements voels v that an be nterpreted as materal ponts eah havng a olor v. The doman of a voel s olor funton v depends on the nput sgnal nterpretaton used to represent the volumetr data. Sometmes n volume renderng t s neessar to treat voels aordng to to alulate loal mage haratersts suh as ts gradent. Wth v s are understood to be samples of a volumetr funton F z on a regular grd knots on a regular grd n the Cartesan oordnate sstem are spaed evenl along the oordnate aes. The maort of medal mage data s grasale wth rare eeptons suh as ertan ultrasound mages and n ths researh we onl onsder grasale mages. For suh mages v has an ntenst value n v for n bts per pel. 69

75 The sosurfae V for a gven volume V ntenst and fed small ε t s understood to be a two dmensonal surfae havng non-empt ntersetons wth all voels wth a gven ntenst range: v V v V and v < ε. 49 Ths sosurfae represents the laer n the volumetr obet that ontans the elements wth ntenstes lose to a fed value. t follows from ths defnton that there are an nfnte number of was to onstrut the sosurfae for a fed volume V and olor so t s better to onsder a lass of sosurfaes defned b. n terms of a funtonal F ths sosurfae s a representaton of ts surfae value F z and an unertant n ts defnton follows from the mssng nformaton related to the values of F between the grd knots..3 Surfae Renderng One of the two approahes used n surfae renderng of 3D volumes s sosurfang. The lassal method of sosurfae etraton the Marhng Cubes Algorthm was proposed b Lorensen and Clne [43] and Wvl et al. [44]. Eah voel s onsdered the topologal equvalent of a ube and the planar appromaton of the sosurfae F z wthn ths ube s sought. To alulate ntersetons of V wth the ube edges voel trlnear nterpolaton s used. Eah verte of a ube an be ether greater than or less than the threshold value gvng 56 dfferent senaros [4]. Eah of these 56 onfguratons represents one or more trangles onsttutng the sosurfae wthn the voel of nterest and ever onfguraton s stored n a look-up table for subsequent fast aess. Ths analss of eah voel n the volume produes a trangulated sosurfae. To reate the resultant 3D mage ever trangle for eah voel s rendered aordng to the seleted lghtng model. After ths trangulaton s onstruted renderng an be fast enough espeall usng 7

76 graphs hardware to render trangles to provde aeptable nteratve rates of rotaton and salng operatons for the resultant 3D mage..4 Surfae Renderng and sosurfae Representaton How does the sosurfae orrelate wth the medal mage volume vsualzaton? n other words what ondtons should the orgnal data satsf for the etrated sosurfae to represent meanngful strutures to the lnan? To answer ths queston let s onsder MR data whh s a good eample for the tpal volume renderng soure data. Under ertan pulse sequenes the ntenstes of soft tssues n MR mages an be statstall separated to model the MR mage as a peewse onstant funton multpled b a slowl varng bas feld see Chapter. The applaton of a threshold whh s lose to one of the soft tssue ntenstes would lead to a nos and sometmes meanngless set of partall dsont voels wth some of the soft tssue voels belongng to the sosurfae and some not Fgure.. Fgure.. ntenst thresholdng. D MR mage left and the part that would be part of the sosurfae f ntenst thresholdng s appled rght. 7

77 From ths eample t s ntutvel lear that to represent a boundar for a subvolume V V all voels v V should also have ntenstes ether hgher or lower than the spefed threshold. Therefore addtonal voel ntenstes that do not belong to V should be on the hgher/lower sde of.e. for all v V v V \ V one of nequaltes v v < > v v 5 should alwas hold. Ths parttonng s often possble for the task of separatng the obet from ts bakground and the sosurfae determnaton should allow reaton of the entre obet outlne suh as the MR head sosurfae presented on Fgure.. Other eamples of suh separatons are provded n Seton.8.. Fgure.. MR head soure data left and the sosurfae onstruted b applng a bakground threshold rght Sne lnans are often nterested n nternal organs and strutures the dret applablt of ths sosurfang s not possble n all ases and other onsderatons nvolvng addtonal mplt knowledge about the nput must also be used to onstrut boundar surfaes. Ths requres addtonal preproessng to onvert the orgnal volume to a form where the onstruton of an sosurfae usng 5 s applable. Detaled lassfaton of suh ases s beond the sope of ths researh. 7

78 .5 Volume Renderng Volume renderng methods an be essentall lassfed nto two groups:. mage order and. Obet order. n the mage order approah also alled bakward renderng proessng s done from the mage plane to the volume. n the obet order approah also alled forward renderng proessng s done from volume to mage [4]. The lassal mage order approah desrbed n Levo et al. [38] onsders the volume as a loud of partles eah of whh absorbs a ertan amount of the lght that goes through t. The denst µ of partles vares throughout the volume and the amount of lght reeved b the mage plane s obtaned b the followng ntegral aordng to notaton used n [4]: L s µ t dt C s µ s e ds 5 where L s the length of the lght ra and Cs s the amount of lght refleted at loaton s along the ra. n man applatons 5 s appromated b a dsrete ntegral sum and ever method that nvolves alulaton of has a ompromse between hgher aura of the 3D representaton of an obet and the greater speed of renderng aheved b usng less samples along the ra. The tpal obet order approah usng the splattng tehnque was desrbed b Westover [45 46]. Voels n the volume are essentall proeted onto a vewng plane formng splats for further omposton n the mage plane. The splattng algorthm orders the voels n the target volume n suh a wa that for a gven sene the voels nearest the observer are alwas proessed frst. Then eah voel s proeted nto a vewng plane usng a smoothng flter to determne ts mage spae oupaton and ths s blended wth prevousl proeted voels usng transparent olor blendng. The obet order methods onvert voels 73

79 dretl to geometr obets on vewng plane and for that reason are also alled dret methods. n dret renderng methods an sosurfae s determned b renderng opaquel all voels wth values greater than some threshold [38]. The voels are onverted dretl to geometr obets on a vewng plane wthout ntermedate steps. When a surfae s reated abrupt thresholdng sometmes reates the surfae wth gaps or holes whh obvousl affets the vsual qualt and ntrodues artfats. To make the omputer generated 3D mage look more natural transferrng between the mageable voels v V and the remanng voels r R s aomplshed b lmtng the value of the ntenst gradent wthn the loalt N V of the boundar sosurfae V. Ths tehnque s sometmes alled fuzz or shell thresholdng [35]. Ths should not be onfused wth the volume renderng tehnques whh emplo n-depth volume analss to obtan sem-transparent mages based on transfer funtons [39]. Another problem wth dret methods s ther nreased omputatonal omplet sne ever element n the volume s beng proessed for ever new 3D sene generaton. For nstane volume renderng [35] requres 6m n 4 alulatons for m samples n eah dmenson of V and n pels n the resultant mage. Reent advanes n dret methods as well as nreased proessor speeds have redued the generaton tme for one 3D mage frame from several mnutes to fratons of a seond. Software renderng algorthms have not et aheved a ombnaton of hgh qualt 3D mages wth reall usable nteratve rates the hol gral of volume renderng researh. On the other hand hardware graphs ards supportng dfferent renderng approahes suh as VolumePro ra astng hardware desrbed n a landmark paper b Pfster et al. [4] have advaned nteratve frame rates. Hardware renderng however nherentl has the drawbak of ts low fleblt n ts hoe of renderng algorthms and the advantages the ma offer. 74

80 .6 Renderng Based on Boundar Voels of a Segmented Volume The problem of fast 3D obet renderng has been addressed n [36]. f the 3D mage s segmented nto two voel sets: a volume of nterest V and the remanng volume R t s possble to ompute the set of boundar voels B suh that for ever voel v belongng to B ts neghborhood Nv ontans voels from both V and R. The elements from V wthout ts boundar.e. nternal voels denoted V\B wll be obsured b voels from B. Thus the problem of renderng the entre volume V s redued to the renderng of ts boundar voels B. n order to onstrut the 3D mage voels from R are proeted onto a vew plane and rendered usng one of three solutons [36]:. omputng a voel proeton pv onto a vew plane and renderng the polgon obtaned; ths an result n alasng gaps between dfferent polgons blak holes et.;. generous area fll suh as rular [37]; 3. usng a sale fator small enough to ft ever voel proeton nto one pel. Ths method though annot be used on small datasets whh poses the maor hallenge to render. Bulltt et al.[37] suggest representng eah voel purposes. f sb s suh that b s b b B as a sphere s b for drawng b ts proeton on the vewng plane s a rle and an be easl rendered. Two neghborng voels are proeted nto partall overlapped rles. n ths wa the omputatonall epensve alulaton of a ub voel proeton s avoded. The fnal sene s produed usng a Z-buffer: eah pel nsde the rle s assgned a depth whh s ompared to the depth of the prevousl drawn at that poston and the resultng ntenst s hosen from the pel wth mnmal depth. 75

81 .7 The Problem of Effent Renderng Volume and surfae renderng methods are numerous and the sope of ths researh does not allow an ehaustve revew. For those nterested n elaborate and detaled lassfaton the artle b Ken Brodle and Jason Wood [4] an be of help. Our purpose here was to llustrate two maor approahes to volume vsualzaton; based on ths llustraton ustfaton of our new renderng method an be presented. Our fnal goal s to automate the proess of medal mage volume vsualzaton and develop an effent tool for ts atual real-tme use. We have seen that volume vsualzaton an be performed n man was and thus a more spef defnton of effen s needed. Effent medal mage volume vsualzaton wll be heneforth understood to possess the ombnaton of the followng propertes:. Adequate sosurfae renderng qualt: Ths nludes smoothness mnmal artfats and lear lghtng.. Real tme renderng speed: The renderng of the omplete 3D sene should not take more and preferabl muh less than seond on a urrent off-the-shelf PC. The effent vsualzaton provdes both aeptable renderng speed and renderng qualt even though the tradeoff between them must alwas be made. 3. Fleblt: Sne man nternal strutures wth dfferent ntenst haratersts are of nterest to the lnan ether all volume parts should be easl dentfable or eposng eah part should not requre lnall unaeptable proessng tmes. Therefore vsualzng an sosurfae defned wthn the soure 3D volume should not requre too muh overhead. Under ths effen requrement the hoe of bas algorthms to produe vsualzaton s smplfed. Renderng the entre volume s n general slower than surfae renderng due to the number of omputatons 5 and therefore an sosurfae method s 76

82 preferable. However obtanng the sosurfae nvolves substantal pre-proessng e.g. marhng ubes [43] analzes from 6 to 56 onfguratons for eah voel and hangng an ntenst threshold ma also requre substantal tme thus ompromsng our fleblt requrement. The hbrd approah based on volume segmentaton [35-37] an produe a new frame for renderng qukl O n 3 where n s the mamum lnear dmenson of soure volume. Ths dret renderng approah nvolves onl a small fraton of voels that onsttute the ntegral 5 and s therefore muh faster. Usng these advantages we wll develop an effent automat medal mage volume vsualzaton algorthm..8 Pre-segmentaton of a Medal mage Volume As dsussed n Seton.6 before a fast renderng an be appled to a volume the area of nterest should be segmented. For eah partular bod part e.g. organ numerous algorthms have been developed to detet a partular organ and/or tssue. For nstane Ra et al. [47] mplement atve ontours for MR lung segmentaton Lew et al. [3] Zhang [4] et al. and Wells et al. [3] use fuzz lusterng and statstal methods to segment bran mages nto whte matter gre matter and erebrospnal flud. The man prnple for ths varet of methods s to obtan an ntal ntenst range and spatal poston of the organ of nterest. Most effent urrent methods usuall requre some manual proessng and ma not be used dretl n an automated vsualzaton sstem. However the sosurfae 49 that orresponds to a seleted ntenst range ma serve as a good appromaton for ertan strutures. n the followng setons we wll demonstrate that suh vsualzaton an be ompletel automated..8. sosurfae Range Seleton n Chapter an MR mage representaton was developed as a funton whh s peewse onstant on most of ts doman and behaves otherwse on remanng fraton of 77

83 doman. For bran mages large peewse onstant areas orrespond to whte matter gre matter and erebrospnal flud tssues; some other strutures lke bone n omputed tomograph CT hest mages an be also dentfed usng ntenst thresholdng. n ths seton we present a method for an automated deteton of homogeneous tssue ranges from the mage hstogram. As dsussed n Seton.8.3 homogeneous tssue produes a peak n the mage hstogram. t follows dretl from the hstogram defnton 4 from Seton.4.4 that the peak heght for the mage ntenst H depends lnearl on the area of homogeneous tssue. Let N be the number of tssue lasses S L n the mage and the mean ntenstes S N of these lasses S are known to form an nreasng sequene: S < S < L <. 5 S N Then the problem of detetng the mean ntenstes S for N lasses n mage takes the form of fndng the N hghest loal mamums of the mage hstogram. Normall the peaks orrespondng to whte matter and gre matter are vsuall dentfable. However the aurate automat deteton of these peaks s dffult due to ther lak of defnton and a flterng sheme needs to be developed. As a frst step we suggest removng a resonant frequen palsade effet. Ths effet s produed b the aquston pulse sequene: ertan ntenstes are more frequent than neghborng ones whh results n a palsade learl vsble on Fgure.3 left. To overome ths we an use a smoothng edge preservng flter defned b 6. Ths ntrodues smoother ntenst transtons for neghborng ponts thus redung the resonant frequenes sgnfantl Fgure.3 rght. After Gaussan hstogram flterng s fnshed peak frequen etraton an be done. Eamnng numerous MR mage hstograms has led us to an observaton that n man real applatons the hstogram peaks have a shape that resembles a reversed parabola. f a 78

84 Fgure.3. Effet of Gaussan blur on MR mage hstogram. Hstogram of orgnal MR bran mage left and blurred mage rght. suffentl large loal subsegment s seleted a parabola appromaton of H an be produed usng least squares. The reverse parabola ndates a loal mamum; the parabola s urvature n the pont of the mamum depends solel on ts loseness to the mamum pont and therefore an serve as a haraterst of the mamum. llustraton of relaton between parabola urvature denoted as a and mamum ponts on the appromaton segment are presented on Fgure.4. After ths nformal desrpton we wll now develop ths dea mathematall. Consder the loal subsegments s [ ] [ mn ma ] mn L ma δ 53 Wthout lmtng generalt we an assume mn. To appromate H on s wth a parabola P P a a a the least squares sum funton F s omposed: δ [ H a a a ] F. The equatons for loal mnmum are onstruted b settng dervatves wth respet to a a and a to zero: 79

85 8 [ ] [ ] [ ]. δ δ δ a a a H a F a a a H a F a a a H a F Fgure.4. Appromaton of a hstogram segment b parabola. Appromaton segments top row and orrespondng parabola fts bottom row. The negatve urvature of parabola s nsgnfant at the absene of loal mamum left a.7 and nreases sharpl n the vnt of the pont of mamum rght a -. Ths an be rewrtten n a matr form X H X H X H a a a X X X X X X X X X 54 where δ p p X 55

86 and δ p p H X H. 56 The unknowns a a a a a a from 54 defne the best n a least squares sense parabol appromaton Pˆ of H on the segment s. Sweepng through the dsrete doman of H we obtan a faml of parabolas P Pˆ Pˆ L Pˆ. ma n the areas where > ma δ or < etrapolaton of H s used to obtan the values beond ts doman. For nreased aura quadrat or ub splnes an be used wth boundar ondtons on H that nlude frst and seond order dervatves. n our eperene the etrapolaton aura s not rtal and lnear or even onstant etrapolaton produes satsfator results. Fgure.5. Result of parabola hstogram flterng. Smoothed MR mage hstogram top and resultng A bottom. To omplete the onstruton of a robust mamum flter t s suffent that a equals the urvature of the parabola and the loal mamum of ts negatve ours when the 8

87 best ft of the parabola shape and loal H s aheved. At the same tme the areas where a do not ontan a loal hstogram peak. Thus the loal mamums of the funton > a a < A otherwse 57 provde an appromaton to the hstogram peaks. Sne A s obtaned from averagng segment s loal hstogram rregulartes do not affet t and ts loal mamums an be dentfed automatall detals are provded n the algorthm desrpton below. See Fgure.5 for an eample of hstogram peak etraton. Wth ths mathematal foundaton t s possble to desrbe an algorthm for the etraton of hstogram peaks. nput: Number of lasses n the mage N. ntalzaton: Calulate mage hstogram funton H usng 4. Calulate the length δ of the dsrete segment s. The length of s vares dependng on the number of peaks n the hstogram. On one hand small δ leads to deteton of mnor peaks due to flutuatons n ntenst wthn homogeneous tssue and therefore does not provde the aurate ntenst mean value; on the other hand larger δ leads to a lak n resoluton of loal mamums of A. We an assume that the length of the samplng segment should be proportonal to the ntenst range and nversel proportonal to the number of lasses N. We aheved optmal qualt n most ases wth δ. 58 ma N Step : for all from to ma alulate A defned b 57. Parabol oeffent a s found wth 54 usng 55 and 56. Step : for all from to ma detet onneted segments support of A : s L m onn s onn that onsttute the 8

88 sup A A >. A one-dmensonal pont belongs to a onneted dsrete segment s onn where s onn ontans more than one pont f one of ts mmedate neghbors or - belongs to s onn. Therefore a smple nearest neghbor eamnaton trval for one dmenson would allow generatng all onneted segments. Eah onneted segment s onn sup A ontans one loal mamum orrespondng to a large hstogram peak. ndeed there should be at least one mamum sne A on the boundares of s and A > nsde s onn. On the other hand f the peak s sgnfant and onn orresponds to a large area n the mage t s separated from the other peak b the area where A due to seleton 58 of the appromaton segment length. Fgure.5 llustrates ths proposton. Step 3: Calulate the mamums on eah onneted segment: ma H ma H L m s onn and sort them to form the non-dereasng sequene: Hˆ ma Hˆ L ˆ. ma ma H m The arra obtaned forms the sequene of hstogram peaks that orrespond to the ntenst unformt areas tssue lasses S L S ordered b oupan area n the mage. That s m S L orrespond to tssue lasses that are present n the mage from the most to least S m defned. Therefore the frst N members of ths sequene math the N best defned tssue lasses as desrbed b 5. Therefore the fnal output of the algorthm s S L. S N 83

89 .8. Automated Seleton of ntenst Ranges To produe a onsstent segmentaton the automat seleton of ntenst ranges ma [ s ] s for eah tssue S 5 s needed. Regardless of seleton method a porton of the mn ntenstes ρ between the two tssues [ S ] used for lassfaton should be S defned. Wthout lmtng generalt we assume ths porton to be equal ρ for all S. n the smplest ase the lmts of S are defned as: s s mn ma S S ρ [ S S ] ρ S. S ρ S 59 Varng ρ allows the etraton of tssues wth dfferent mean ntenst varatons. We found.6 ρ to produe best results for bran mages. From Fgure.4 and Fgure.5 t s lear that dfferent tssues ma have dfferent mean varatons the wdths of hstogram peaks and therefore borders defned b 59 ma not alwas be optmal. A smple observaton that the length of onneted support for A s onn orrespondng to S s proportonal to the wdth of the hstogram peak allows the adustment of 59 to be s s mn ma S S s s onn onn s s onn onn ρ S ρ S 6 whh allows an adequate orrespondene between wdths of neghborng hstogram peaks. The resultng algorthm named Automat Tssue Deteton atd was eeuted on a number of mages to valdate the proposed method. n all ases fltered ntenst peaks 84

90 Fgure.6. Automat bran mage segmentaton. Fgure.7. Automat hest mage segmentaton. 85

91 showed a lose math wth hstogram peaks. Fgures Fgure.6 and Fgure.7 llustrate the automated segmentaton of a bran mage wth N 3 and a hest mage wth N 4..9 Gravtatonal Shadng Algorthm The automat algorthm developed n the seton.8 etrats homogeneous tssues S L from an mage. These nput data ma now serve as a foundaton for an effent S N n terms of Seton.7 3D renderng algorthm. Sne dret renderng methods provde more fleblt n dsplang dfferent ntenst ranges a dret renderng wth eah voel proeted onto a vewng plane s preferable. We dsussed n Seton.7 that dret alulaton of the ntegral 5 s tmeonsumng but the knowledge of pre-segmented strutures n the volume permts the reduton of the number of alulatons along eah ra; desrbed n Seton.6. f eah voel s represented b a sphere a fast 3D renderng an be performed usng ths sheme. However the qualt of suh a renderng obtaned b overlappng rles over the vewng plane serousl degrades at lower resolutons. To mprove the dspla qualt at low resoluton we developed a renderng tehnque based on the gravtatonal voel nvarants. n the followng setons we provde detals on a segmentaton based 3D renderng..9. Etraton of the Boundar Voels The renderng proedure developed n ths researh an nlude vsualzaton of one or more of the tssues S b the use of an sosurfae S. The voels that onsttute S are potentall vsble and an be etrated for eah of S on the frst step of the algorthm. For eah voel n three dmensons v z V z a set of voels termed neghborhood Nv z s defned for ths etraton. The mnmal neghborhood nludes v z and ts s losest voels along the dretons parallel to the oordnate aes. For smplt we assume the 86

92 voels are evenl spaed along all oordnate aes wth a unt nterval. Hene the mnmal neghborhood s defned as { v v v } N. 6 z N 6 v z ± z ± z z± Neghborhoods wth more ponts N and N 6 are smlarl defned. N and N 6 are more omputatonall epensve but the produe more aurate results. The output of the atd algorthm wrtten n terms of ndator funtons δ s: v z S δ v z δ z. otherwse Defnng the neghborhood haraterst funton: for whh t s onvenent to defne p z z δ v v N p z S as: δ v z v z S p z mod p. 6 ndeed p z when ether the voel does not belong to S or none of the neghborng voels belong to S. On the other hand when p z p all voels from N p are nternal to S and thereforev z S. Thus the set S an be alulated for ever 3 usng 6 wth O pn omparsons and addtons. Ths operaton elmnates 9 99 % of the voels n the 3D mage sets used n ths researh for 3D sene generaton..9. Vewng Plane Proeton Vsualzng the tssue S s now redued to vsualzng the 87 S alulated at the prevous step. The 3D sene s assumed to be generated usng an orthograph proeton. Ever voel s proeted onto vewng plane P vew aordng to a vewng transformaton VT. The vewng transformaton nludes the obet s proper salng rotaton and translaton ombned wth the automat salng and translaton needed to ft the 3D obet to the

93 vewng area. A detaled overvew of a 3D ppelne desgn an be found n Waggenspak [48] or F. S. Hll [49]. We prevousl dsussed the term voel n the meanng of materal pont approah Seton.. Ths materal pont however has non-zero dmensons whh should be refleted n proetons onto the vewng plane. Bultt and Award [37] used rles as appromatons of suh proetons assumng the orgnal voel was sphereshaped. To form an obet s vsble surfae from ts voel proetons a Z buffer approah [48] [49] s used. A Z buffer ontans the entr for eah pel n the vewng plane wth two tpes of nformaton: ntenst and depth. A Z buffer algorthm for renderng S follows:. ntalze the Z buffer to ontan an empt entr wth three felds for eah pel n the vewng area: { z } Z Z Z where Z orresponds to ntenst at the gven pont for monohrome lghtng z Z s the depth ndator and Z s the tssue nde.. For ever voel v S do steps 3 and 4; 3. Calulate the proeton P v Z of v onto the vewng plane. Ever pel P v s obtaned from the orgnal voel usng a vewng transformaton; ts thrd oordnate z s nterpreted as the dstane of maged voel from the observer. The shadng tehnque named gravtatonal shadng gs developed n ths researh s desrbed n Seton f Z s empt or z z Z proeed to the net pel. Otherwse assgn the felds of Z to the values orrespondng to P v. 5. Ever pel n the vewng plane s assgned an ntenst Z. 88

94 .9.3 Lghtng Model and Trlnear nterpolaton To produe an mage that atuall looks three-dmensonal the ntenst of eah pel needs to appromate the ondtons of a naturall lghted 3D obet. Routnel the lghtng ntenst wth one lght soure s defned b a ombnaton of the three omponents: dffuse speular and ambent:. dff spe amb Sne the dffuse omponent haraterzes the dretonal refletons an aurate determnaton of ths omponent s essental to adequate 3D mage generaton. We wll desrbe the alulaton of onl ths omponent for our 3D renderng algorthm. For further detaled desrpton of lghtng models used n omputer graphs refer to [48] [49]. The dffuse omponent s determned from dff r r r os 6 l d r n where l s the ntenst of lght soure r d - the dffuse refletvt of the surfae r - the vetor from a pont on surfae to the lght soure n r - the surfae normal. Sne l and r d are onstants and r s known for eah pont the problem of alulatng dff s equvalent to the problem of fndng the surfae normal at a gven pont. To fnd an aurate appromaton of n r for raw voel data we wll use a trlnear nterpolaton sheme ommonl used to fnd mssng values nsde a ube. Consder the unt ube Fgure.8. ts orrespondng 8 vertes are denoted V ube { V V L V } and wth trlnear nterpolaton the value nsde the ube at poston z an be determned from k V k V ube k k z V V. 63 z 89

95 Fgure.8. Trlnear nterpolaton ube. Usng 63 we an derve formulas for the ntenst gradent at ever pont nsde the ube: z V V V z z z z V k k k z V V V V V V V ube ube ube z k V V V V k k k z k k z k k z. 64 To determne the resultng ntenst of the rendered voel an averaged gradent s used. Sne the trlnear nterpolaton formula s entrall smmetr the gradent value at the enter pont V tr V provdes an aurate estmaton of the normal. Usng 64 we obtan : tr tr V tr V tr V tr Vk Vube Vk Vube Vk V ube V k V k V k k. 65 9

96 Formula 65 allows alulatng the normal of a unt voel. f the volume s ansotrop a orrespondng salng transformaton should also be performed D mage Zoomng Qualt A normal alulated usng 6 and 65 provdes adequate 3D mage qualt for hgh resoluton volumes. At lower resolutons or wth hgher magnfaton however the renderng qualt of ths voel-b-voel sheme degrades allowng separate voel proetons to beome noteable. Fgure.9 presents a head vsualzaton usng unform shadng of rle voel proetons wth normals alulated usng 6 and 65. The soure volumetr data was obtaned from the publ doman [5]. Fgure.9. Head vsualzaton usng unform voel proeton shadng. Entre volume left magnfed part of the volume rght. There are two possbltes for mprovng the vsualzaton qualt of low-resoluton data:. Usng a hgher order appromaton n the normal alulaton formula whh nludes more neghborng ponts. Ths approah allows smoother gradent transton and therefore ould produe a more naturall lookng mage. However ths tehnque has lmts wthn the urrent model: hgher order appromatons substantall nrease 9

97 pre-proessng tme and make the vsualzaton less fleble. n addton as voel proetons beome more dstnt at hgher zooms the voel borders an no longer be ompensated for wth a smoother gradent See Fgure. for llustraton of the nterfae between two voels at a hgh zoom.. Usng a non-unform voel shadng to redue gran effets permts voel proetons to be rendered usng smooth transtons. deall ever pel n the proeton should be the result of a mro ra astng through the voel. Pratal mplementaton of ths prnple s not straghtforward sne the ra astng s overhead s sgnfant at hgh magnfaton. Fgure.. Two-voel proeton. The nterfae between two voels at hgh zoom s hard to elmnate usng unform voel shadng. Hene t s desrable to ombne the hgher order normal appromaton wth more ntellgent voel proeton shadng. n the followng setons a sheme to appromate a non-unform voel proeton that s omputatonall effent s derved..9.5 Gravtatonal Conept An natural phenomenon an be onsdered from the pont of vew of several dfferent theores. Estng phsal theores often omplement eah other n provdng a workng model. For nstane orpusular theor eplans quantum effets observed n eletrt whereas wave representaton of radomagnet pulses based on Mawell equatons allows aurate desrpton of man other eletromagnet nteratons. n 3D mage 9

98 vsualzaton fundamental phsal laws suh as those governng lght refleton and absorpton form a foundaton for renderng algorthms. However n ase of renderng wth rle prmtves these laws have lmted applaton sne the nteraton of mrostrutures.e. voels produe a prevalng nfluene on the observed depton of a 3D obet. These effets are generall referred to as low-resoluton artfats and ma be onsdered the omputer graphs analog of nanopartle nteratons that nether follow the patterns of the behavor of matter n a maro world nor the patterns at a submoleular level. For nanopartles and nanomaterals speal laws govern nteratng fores and movement. Analogall we an onsder speal voel nvarants for more aurate renderng results. Suppose we render the part of a dgtal volume V bounded b two ntenst values <. Eah rendered pel s drawn wth two values:. ts ntenst usng a lghtng model desrbed b 6 and 65;. ts opat defnng the result of sem-transparent blendng. The blendng rule for drawng a pel wth ntenst over the pel wth ntenst and opato s defned b [5]: o. 3 o n the ontnuous ase opat s obtaned b ntegraton along the ra. We aept that the bakground has zero opat and the opat of nternal obet s voel s. However f the voel on a boundar s rendered we an assume that opat n the pont nsde the voel depends on ts dstane from the border. We onsder opat as an analog of the phsal mass wth a denst funton: ρ where r r s the radus vetor to the urrent pont and ρ rr mn ρ 66 r ρ r ρ. 93

99 The denst funton defned b 66 for pont nsde a boundar voel haraterzes ts loseness to the border of the segmented volume and s related to the resultng opat of the voel proeton at pont : o r R r r ρ d 67 where R s the ra ast through the voel that ends on poston n the vew-plane. However ths formula annot be used for alulatons sne the dret applaton of 66 s requred at ever 3D pont along the ra. We ma onsder another approah alulatng an nvarant that haraterzes the dstrbuton of the denst nsde the voelv.e. ts enter of gravt C v z. The enter of gravt for a three-dmensonal volume v s determned b [5]: m ρdv ρdv z v m v m v zρdv 68 where m ρ dv. v To develop a formula for the enter of gravt a trlnear nterpolant 63 s used. An analtal epresson for the enter of gravt annot use the denst 66 sne ths denst funton s not defned b an analtal epresson. To obtan the derved analtal epresson we ma assume that f the voel s on the boundar the denst equals ether ρ or ρ throughout the entre voel. That s all ponts nsde the voel are loser to one border of segmentaton ntenst band [ ] than to the other: for all z v ether ρ z ρ ρ. 69 or ρ z f the ntenst band s wde ths assumpton produes a workng model. Wth ths assumpton the enter of gravt for ρ s: 94

100 m m where V m V k Vk V ρdv k [ k z ] V k Vk V Smlarl the epressons for dddz 8 k [ k z ] dddz Vk 4 and Vk V Vk V z are: Vk k 4 4 Wth ths the epressons for enter of gravt are: z V k V k Vk V. 7 Vk V. 7 z m m m z z z m m m z Other Forms of Gravtatonal nvarant 7 an be used onl f 69 s vald; hene t would be benefal to have a more general epresson for the voel enter of gravt. For ths purpose nstead of trlnear nterpolaton of ntenst values a dret nterpolaton of the denst funton based on values alulated at the vertes of the voel an be used: k ρk ρube k [ k z ] ρ ρ z where aordng to 66 ρ k mn Vk Vk. Usng ths nterpolaton 95

101 m 8 ρk V and the oordnates of a gravtatonal nvarant for the voel are smlar to 7 7 wth V k replaed b ρ : k ρ k z m m m ρk V ρk V ρk V ρ ρ ρ k k k k allows dret omputaton of the gravtatonal nvarant for the subsequent shadng. nvarants based on trlnear nterpolaton are omputatonall effent frst-order appromatons. f hgher aura s needed a hgher order nterpolaton nvolvng a rapdl growng number of knots would result n a substantal nrease n omputatonal omplet. For ths reason we onsdered another approah to alulate the gravtatonal nvarant. Suppose that the average ntenst values on the ube faes are known and denoted as for the faes { } { } V V remanng faes. of unt ube Fgure. and z z V V V V for Fgure.. Unt ube wth entral aes. For a three-dmensonal voelv uvw these orrespond to 96

102 97 ~ ~ ~ v V v V v V w v u z w v u w v u. 74 The denstes alulated wth 66 z ρ ρ ρ and orrespond to z V V V and. Suppose that the mass n the one-dmensonal ase s onentrated along the aes O O and Oz onnetng the enters of opposte faes Fgure.. Denotng denst along these aes as z ρ ρ ρ and we an determne the one-dmensonal enters of gravt along eah of these aes as 68: dz dz z z d d d d z z ρ ρ ρ ρ ρ ρ. 75 f the voel ntenst aordng to approah from Seton.4 represents an average over the unt volume then the seleton of adaent voel values 74 s more approprate than the model n the prevous seton where the values of the ube vertes were dsrete samples of the sgnal. Addtonall approah agrees wth the phsal proess of medal mage aquston and ts realzaton s preferable and produes more aurate geometr alulatons. Sne z ρ ρ ρ and are one-dmensonal funtons of one varable ther nterpolaton requres fewer omputatons than the general three-dmensonal problem. For a lnear appromaton a two-pont formula for b a ρ s obtaned from the sstem of equatons b a b a b a ρ ρ ρ ρ ρ and smlarl for z ρ ρ and. Substtutng these for ρ n 75: [ ] [ ] d d 3 6 ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ. 76

103 98 The epressons for z and are obtaned b replang the nde b and z. The threepont formula an be smlarl obtaned wth the quadrat appromaton b a ρ. ts oeffents are obtaned from the followng sstem of equatons: b a b a b a b a ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ and for m [ ] a d b a m 3 3 ρ ρ ρ. Combnng these [ ] b m d b a m 3 3 ρ ρ ρ ρ ρ ρ. 77 As before epressons for z and are obtaned b replang the nde. A pont wthn the voel bounds obtaned usng 77 s termed a quas enter of gravt..9.7 Effent Shadng of Sem-transparent Voel Proeton When the voel gravtatonal nvarant enter of gravt 7 or 73 quas enter of gravt 77 s alulated t s possble to alulate voel proeton shadng. To determne ths shadng onsder an arbtrar pont nsde the sphere z P z where the dstane between P and an nvarant pont Cv s denoted z P v C d. ntall we onsdered the relaton between opat and z P v C d as: ~ A v C d P v C d P O z z. 78

104 Fgure. presents the planar seton β of sphere Sv to represent voel v; the plane β s defned b three ponts: sphere enter O P z and C v. The pont A les on the nterseton of the ra CPz and Sv. The sphere s ompletel opaque n ts enter of gravt 78 and transparent on the borders to allow smooth transton nto neghborng spheres. Fgure.. Sphere seton llustratng the relaton of opat and dstane from ts gravtatonal nvarant. Eah pont n the two-dmensonal spheral voel proeton an be assgned a value that orresponds to the resultant opat along the vewng ra smlar to 67: O u v w R O P dudvdw. 79 uvw Unfortunatel 78 uses a rato of Euldean dstanes and the ntegral 79 beomes too omple to be omputatonall-feasble. For ths reason a more pratal approah that results n a smlar vsual effet was derved. n order to smplf the results t s onvenent to perform the alulatons n a oordnate sstem that has ts orgn at the enter of Sv Fgure.3. To derve a workng appromaton for 78 the opat an be modeled as beng lnearl dependent on the dstane from the γ plane on Fgure.3.e.: O P z ~ d γ P z. 8 99

105 Fgure.3. Spheral voel representaton and ts oordnate sstem. Pont C s the enter of gravt A les on nterseton of S and OC ra. γ s the plane ontanng A and perpendular to OC r s the radus of S OA r. Usng 8 the opat nreases lnearl n the OC dreton. The magntude of the opat gradent throughout the sphere depends upon the rato r r where OC r o. Wth ths model the opat gradent perpendular to the γ plane s lose to zero when enter of gravt s near the enter of the sphere and nreases as the rato r r nreases. The oordnates of pont A on Fgure.3 are: r r t r z t z z t t A A A A A A. The equaton for the γ plane s:. ] [ ] [ ] [ rr z z z r r z z z z z A A A Aordngl the dstane from the arbtrar pont P z nsde the sphere to γ s:

106 z z d P γ rr. To obtan the result of a proeton onto the pont of vewng plane ntegraton P P smlar to 79 along the lne parallel to z as wthn the sphere beomes: O P P Z S Z S d P γ dz Z S rr P P z z rr Z S dz rr P rr P P z z P Z S Z S Z S ± are the ponts of nterseton of the proeton lne { } where Z P p S the sphere S z r. From these results we obtan: wth P P O P P rr P P r P P. 8 The desred spheral voel proeton shadng was developed wth 8 whh we all gravtatonal shadng gs. Gs s performed wthn step 3 of the Z buffer algorthm Seton.9. and t beomes part of our 3D graphs renderng ppelne. Gs requres that the enter of gravt for ever voel must be determned; ths alulaton s performed onl one and an be arred out wth ether 73 or 8 n the ntalzaton step. The remanng nput for gs nludes the enter of eah voel v n the proper oordnate sstem vewng transformaton matr VT the rotaton/salng matr R of the vewng transformaton and the lght soure poston L z L L L. Then the shadng algorthm for ever voelv proeeds as follows:. Calulate the normal n v 65. Determne the voel base ntenst v wth the lghtng model 6.. Determne the sphere radus r: dff r R. 83

107 3. Obtan the vewng transformaton result VT v and hange the voel oordnate sstem to the one on Fgure.3 b the translaton of the enter of the sphere to the orgn. 4. Calulate v and the opat of ever pel nsde the spheral proeton 8 dff and pass these results to the Z buffer algorthm.. Evaluaton of Gs The fast renderng algorthm desrbed n Setons was mplemented n C wth a Wndows XP platform wthout the use of an thrd-part graphal lbrares. To valdate the orretness of gs an mplementaton of the voel shadng funton was developed n MATLAB. Ths mplementaton orretl renders the three-dmensonal sphere proetons. Fgure.4 shows the proeton of a sphere wth dfferent loatons for ts enter of gravt. To assess the performane of gs we used the propertes of effent 3D vsualzaton from Seton.7: adequate renderng qualt speed and fleblt. Fgure.4. Test shadng of sphere proeton. Sphere of radus s entered n orgn. Centers of gravt from left to rght: Surfae renderng qualt We ompared gs wth a dret renderng method that uses unform rular proeton shadng. We hose the fast segmented volume renderng fsvr based on ts desrpton from

108 Bulltt and Alward [37] omplemented wth the normal alulaton 65 and the Z buffer algorthm Seton.9.. The frst omparson used a 3D phantom dataset onsstng of a ube. Ths sze of rendered voels permts an estmaton of the subvoel shadng qualt for both methods Fgure.5. To ompare the renderng qualt wth authent medal mage volumes the followng datasets were used:. A bt MR sagttal mages of the head wth manuall removed erebral orte [5]. ntal segmentaton was obtaned usng bakground thresholdng. Renderng results are shown on Fgure.6;. A bt MR aal mages of the head. ntal segmentaton was obtaned wth a method smlar to Set. Renderng results are shown on Fgure A % based -bt sagttal phantom MR normal bran volumes from the BranWeb smulator [3]. Ths set was pre-proessed to demonstrate the apabltes of our automated segmentaton and vsualzaton ppelne usng the algorthms developed n ths researh. ntall t was orreted usng dsf Chapter to remove non-unformt that ould nterfere wth atd Setons our segmentaton algorthm. n the seond stage the erebral orte was removed usng BET Bran Etraton Tool based on an etraton algorthm from S. M. Smth [53] whh s norporated nto the MRro medal mage analss software [54]. Conseutvel atd was appled to a strpped volume to detet the gre and whte matter tssue ntenst ranges. Based on ths segmentaton vsualzaton usng fsvr and gs was performed separatel for gre matter Fgure.8 and whte matter Fgure.9. 3

109 Fgure.5. Renderng of a ubal volume. Result produed b fsvr left and gs rght. Fgure.6. Renderng of Set. Results of fsvr left olumn and gs rght olumn. 4

110 Fgure.7. Renderng of Set. Results of fsvr left olumn and gs rght olumn. Fgure.8. Renderng of Set 3 GM. Results of fsvr left olumn and gs rght olumn. 5

111 Fgure.9. Renderng of Set 3 WM. Results of fsvr left olumn and gs rght olumn. Fgure.. Renderng of Set 4. Result of fsvr left and gs rght. 6