Tomographic Reconstruction Using Ridge Functions

Transcription

1 st World Cogress o Idustral Process Tomography, Buxto, Greater Machester, Aprl 4-7, 999. Tomographc Recostructo Usg Rdge Fuctos Iva Kazatsev Uversty of Ghet St Peterseuwstraat Ghet, Belgum va.azatsev@els.rug.ac.be Abstract I ths paper tomographc recostructo based o the cocept of rdge fuctos (Loga ad Shepp) s cosdered. A recostructo approach for the rdge fuctos from a fte umber of arbtrary proectos s suggested wth the framewor of parallel beam geometry. The method deals wth mages that ca be preseted as a sum of rdge fuctos. We derve a formula to calculate the rdge fuctos from the set of arbtrary proectos. I the case of few proectos a approach for detecto of rdge fuctos wth uow drectos s suggested. Results of umercal smulato are preseted. Keywords : rdge fuctos, recostructo, tomography. INTRODUCTION Let f be a square tegrable fucto whch vashes outsde the ut ds D the plae (x,y). The Rado trasform R θ maps a fucto oto ts le tegrals alog some drecto determed by a agle θ ad a dstace s from the org []: Rθ f f ( scosθ ts θ, ssθ + tcos θ) dt We wll deote a sgle proecto by p(θ,s)r θ f(s). Theoretcally the verse Rado trasform s gve by p s s f ( x, y ) ( θ, )/ xcos ys s dsd θ.( ) + 0 Ths formula supposes the total owledge of p [0,)x[-,], thus a fte umber of proectos. I practcal tomographc recostructo [] oly a lmted umber of proectos s avalable. We represet the drectos the tuple ω(ω,...,ω ) ad the assocated proectos are deoted by Recostructg f from R ω f the case of equally spaced proectos s mostly doe by fltered bacproecto [3]. I ths techque the followg approxmato of () s used: f θ θ R f ( R f,..., R f ). ω ω ω ( x, y) r ( xcosω + ys ω ). ( ) FBP The fucto r s defed as r() s p( ω,) t ( s t) dt. () 3 The covoluto erel s a regularzed verso of the verse Fourer trasform of ρ.. RIDGE FUNCTIONS Loga ad Shepp suggested a alteratve to the aforemetoed covoluto-based approach. They troduced rdge fuctos h assocated wth a drecto θ whch are of the form hxy (, ) hx ( cosθ + ys θ). The followg theorem s due to Loga ad Shepp [4]: Theorem. Let ω(ω,..,ω ) be a tuple of R f R H,,...,. ω ω dstct agles. Let H be the uque fucto L (D) of the smallest orm whch satsfes The there exst fuctos h, h such that Hxy (, ) h( xcosω + ys ω). 433

2 st World Cogress o Idustral Process Tomography, Buxto, Greater Machester, Aprl 4-7, 999. Theorem states the exstece of the rdge fuctos. Ths wor reports o costructo algorthms for these fuctos, gve a arbtrary umber of proectos R ω f. The bass of the approach cossts the observato of Davso [5] ad Lous [6] that the Chebyshev polyomals a R R # U U U θ θ (cos( θ θ)) U are egefuctos of the operator a - RR # : # Here R θ s the bacproecto operator, a(t)(-t ) /. Usg these results, the followg theorem establshes a aalytcal formula whch eables to calculate the rdge fuctos drectly from the proectos. Theorem. The rdge fuctos h (s) from Theorem have the followg form: h() s η U() s p( ω,) t U() t dt,() 4 0 Fgure. Shepp ad Loga phatom We geerate 8 equspaced proectos each wth 8 samples. For comparso wth suggested approach the fltered bacproecto (FBP) techque ()-(3) s appled (Fgure ). Where U are the Chebyshev polyomlas of () the secod d, η are etres of the matrx - Λ (or Λ + - geeralzed verse), Λ (λ () ), λ s( ( ω ω )) s( ω ω ). ( 5 ) I the case of equally spaced agles the calculato ca be reduced to a computatoal more effcet formula. If the agles ω are equally spaced (ω (-)/,,,) the aalytcal verso of the matrx Λ s possble [7]: Theorem 3. For <, matrx Λ s sgular ad ts geeralzed verse Λ + s Λ + Λ. Fgure. Recostructo by fltered bacproecto The sum of 8 rdge fuctos s computed accordace wth formulae (4)-(5), summato over s cutted after 8 terms. Ths parameter of cuttg s chose emprcally, optmal choce of trucatg the seres (4) stays ope. The result s preseted Fgure 3. Theorem 4. For > ad m+l wth l0,,- the matrx Λ s osgular ad ts verse s m l Λ + (( m+ ) I ( m+ ) Λl). mm ( + ) Relevat cosderatos ca be foud also the paper [8].. Numercal expermets For our umercal modelg we use the Shepp-Loga phatom [3] wth teger grey level values of ellpses (Fgure ). The sze of dscretzed mage s 8x8. Fgure 3. Recostructo wth rdge fuctos We ca see that compared wth stadard FBP method, the recostucto wth rdge fuctos sufferes from oscllatos. Profles of the 64-th colum of test mage, fltered bacproecto ad recostructo wth rdge fuctos (from left to rght) are vsualzed Fgure

3 st World Cogress o Idustral Process Tomography, Buxto, Greater Machester, Aprl 4-7, Numercal expermets The phatom cossts of fve arrow ellpses wth ceters at the org; fucto f s costructed as a sum of fve approxmately rdge fuctos wth parameters α (8 o, 54 o, 90 o, 6 o, 6 o ). Axes of ellpses are a0.05 ad b0.99 wth ellpse s equato x /a +y /b (Fgure 5). Fgure 5. The test mage Fgure 4. Profles of the Shepp-Loga phatom ad the two recostructed mages alog the 64-th colum. 3. RECONSTRUCTION OF FEW RIDGE FUNCTIONS WITH KNOWN DIRECTIONS Let us assume that data acqusto system provdes us wth proectos of fucto f drectos ω(ω,,ω ). Suppose a pror that our fucto f s a sum of rdge fuctos h whch drectos α (α,,α ) are ow. We try to recostruct f from proectos R ω f (p(ω,s),,p(ω,s)) (p,,p ). Geeralzg the approach of prevous secto we ca derve the followg form of rdge fuctos h (s) terms of proecto data R ω f, smlar to (4): hα η U s p t U t dt () ω () (),() 6 0 Where h () are etres of the matrx Λ +, wth elemets of matrx Λ : ω α λ s( ( )) s( ω α ). ( 7 ) If proectos are uformly spaced [0,) ad umber s suffcetly large, recostructo wth rdge fuctos early cocdes wth FBP recostructo, as we have see secto. I ths secto the case of small umber of proectos s vestgated umercally. Destes for ellpses are chose the same ad equal 50 grey levels so that desty of cetral lght spot s 50. Agles uder whch fve equspaced proectos are geerated costtutes 5-tuple ω (0 o, 36 o, 7 o, 08 o, 44 o ). Results of fltered bacproecto are show Fgure 6. Fgure 6. Recostructed mage usg FBP Show Fgure 7 s recostructo obtaed from proectos R ω f usg formulae (6)-(7). Fgure 7. Recostructo usg rdge fuctos 435

4 st World Cogress o Idustral Process Tomography, Buxto, Greater Machester, Aprl 4-7, 999. We preset also more complcated example whch clearly shows lmts of applcablty of the approach cosdered. The test mage s a sum of fve ellpses some of whch are ot elogated structures (Fgure 8). Agles of maor axes clatos are α (0 o, 0 o, 45 o, 70 o, 90 o ). Fgure 8. The test mage Proectos are geerated ad bacproected (after stadard fltrato) uder the agles ω (0 o, 36 o, 7 o, 08 o, 44 o ) (Fgure 9). Fgure. Profles of the phatom ad the recostructed mages alog the 54-th colum. 4. DETECTION OF DIRECTIONS OF RIDGE FUNCTIONS Fgure 9. Recostructed mage usg FBP I Fgure 0 s show the alteratve recostructo based o a pror owledge about the mage structures drectos α. I practce, drectos of mage rdge fuctos ad other structures are uow. Gve mage f ad arbtrary proectos R ω f (p,,p ) uder drectos ω (ω,,ω ), we try to fd -tuple α (α,,α ) of agles such that correspodg rdge fuctos h,,h compose a mmal orm soluto H closest to fucto f. It was show [7] that as a measure of closeess we ca use the orm of mmal orm soluto. Let us deote H[α ω](x,y) a mmal soluto obtaed from proectos R ω f by formulae (6) (7) wth the assumpto that drectos of rdge fuctos are from -tuple α. The H[ αω ] h( α, t) p( ω, t) dt. Fgure 0. Recostructo usg rdge fuctos Show Fgure are profles of the phatom, FBP recostructo ad rdge fuctos recostructo alog the 54-th colum. Hece the orm of mmal soluto ca be computed wthout bacproecto procedure. Optmal set of agles α opt (α,,α ) ca be foud by cosumg search of global maxmum -dmesoal volume: α opt arg max H[ α ω]. α [ 0, ) 436

5 st World Cogress o Idustral Process Tomography, Buxto, Greater Machester, Aprl 4-7, 999. Related results wth detecto of elogated bdrectoal structures o dgtal mages ca be foud [9]. 5. CONCLUSIONS I these very frst ad prelmary expermets we try to chec out applcablty of the rdge fuctos recostructo of mages cotag elogated structures. Although, geeral, due to the bacproecto procedure, the fltered bacproecto method ca be cosdered also as a sum of some rdge (fltered ad bacproected) fuctos, we should coclude that the stadard techque of ρ-fltrato s more relable comparso wth expaso of proectos to the Chebyshev seres. The suggested approach eeds precse descrpto of the rage of ts applcablty. The expermetal results show that some cases of few-vews tomography the use of rdge fuctos ca extract formato about structures eve from osutably drected proectos. The approach ca be of essetal help dustral tomography of obects whch ca be examed oly wth lmted rage of observato agles. Much wor s eeded for refemet of the algorthm ad to automatcally detect the optmal set of rdge fuctos ad ther drectos. ACKNOWLEDGEMENTS [4] B.F. Loga ad L.A. Shepp, Optmal recostructo of a fucto from ts proectos, Due Math. Joural, 975, 4, pp [5] M.E.Davso, A sgular value decomposto for the Rado trasform - dmesoal Eucldea space, Numer. Fuct. Aal. ad Optmz., 98, 3, pp [6] A.K. Lous, Thoov-Phllps regularzato of the Rado trasform, Costructve Methods for the Practcal Treatmet of Itegral Equatos (Brhauser), 985, ISNM 73, pp. -3. [7] I.G. Kazatsev, Tomographc recostructo from arbtrary drectos usg rdge fuctos, Iverse Problems, 998, 4, pp [8] A.Capoetto ad M.Bertero, Tomography wth a fte set of proectos: sgular value decomposto ad resoluto, Iverse Problems, 997, 3, pp [9] I.G. Kazatsev, The Rado Trasform- Based Aalyss of Bdrectoal Structural Textures, Proc. 7 th It. Cof. CAIP 97, Lecture Notes Computer Scece, 96, 997, pp The author would le to tha the aoymous revewers for costructve commets. Ths paper was prepared durg the author s stay wth the ELIS Departmet of the Uversty of Get, Belgum. The author s grateful to Professor I. Lemaheu, D. Boude Msc, F. Jacobs MSc ad all members of the MEDISIP group for ther help ad support. REFERENCES [] F.Natterer, The Mathematcs of Computerzed Tomography (New Yor: Teuber ad Wley), 986. [] G.T. Herma, "Image Recostructo from Proectos: The Fudametals of Computerzed Tomography ( New Yor: Academc Press), 980. [3] L.A. Shepp ad B.F. Loga, The Fourer recostructo of a head secto, IEEE Tras. Nucl. Sc., 974,, pp