X-Ray Notes, Part II

Transcription

1 oll 6 X-ra oes : Page X-Ra oes Par oise in X-ra Sses par n an -ra sse iages picall are creae fro inensi values ha are relae o he nuber of phoons ha srike a eecor eleen in a finie perio of ie. The phoons are generae b elecrons ranol sriking a source an hus he phoons a he eecor are also rano in naure. We picall escribe his kin of rano process as one having a rae paraeer λ unis: evens/ie an an observaion ie T. Le X be he rano variable R.V. ha escribes he nuber of evens phoons sriking he eecor eleen in ie T. X will be a Poisson isribue rano variable wih paraeer λt. E.g. X ~ PoissonλT Derivaion of Poisson Disribuion Below we will erive he Poisson isribuion fro a se of inepenen Bernoulli R.V. s. Le Δ be soe sall ie inerval an T/Δ be he nuber of inepenen rials. The probabili of an even phoon in inerval Δ will be λδ. Each Bernoulli rail will hen be an R.V.: Y i ~ BernoulliλΔ Y even A wih probabili p λδ i even B wih probabili q p We also assue ha Δ is chosen o be sall enough so ha he probabili ha here are wo evens is ver sall laer we will le Δ go o ero so his is a non-issue. ow we consier he su of he evens which iels a binoial R.V. X Y i i X ~ Binoial λt

2 oll 6 X-ra oes : Page The probabili ensi funcion is f Probabili{X } he probabili ha here were evens in ie T. For a binoial R.V. his is erive fro he following: A A B B of even pe A of even pe B which will occur wih probabili p q an here are o ge of even pe A. This iels he following p..f.: Please also observe ha The ean of X is: X E[ X ] f!!!! p!! f q p q! ifferen was!! p! p!! q an leing an - p! p!! q p f p n a siilar fashion we can show ha E [ X ] p + p p an σ X p p pq Finall we will le Δ T/Δ p λδ an q. n he following keep in in ha q p p λt λt/p. The Poisson probabili isribuion is herefore:

3 oll 6 X-ra oes : Page 3 [ ] [] [][ ]!! li li! li! li!!! li li / T e e T p q T q q p q p f T T T p p q λ λ λ λ λ λ Δ Δ Δ + + L L [The eponenial lii coes fro ε ε ε ε / e e.] The ean an variance are: T T pq T T p X X λ λ λ σ λ λ Δ Δ Δ Δ Δ Δ Δ Δ Δ li li li li Here X is a Poisson R.V. wih paraeer λt: X ~ PoissonλT. SR of a Poisson easureen n general he piel values in an -ra iage are isribue accoring o a Poisson R.V. f he ean value of he phoon couns for a piel is μ hen he signal o noise raio of for ha piel will be: μ μ μ σ X X SR The SR increases as he square roo of he nuber of phoons. Thus he SR increases as he square roo of he ose o he paien. Finall b averaging ogeher wo neighboring piels we can roughl ouble he phoon couns an iprove he SR b.

4 oll 6 X-ra oes : Page 4 The above figure shows Poisson isribuions as he ean increases fro 3 o 5. We can see ha he isribuion becoes ore seric an Gaussian. The above figure akes Poisson isribuions an noralies he b heir ean ha is we subrac he ean an ivie he -ais b he ean. This plo show eonsraes ha he wih of he isribuion as a fracion of he ean. As he ean ges larger he isribuion ges proporionael narrower he s. ev. vs. ean raio is saller SR is higher.

5 oll 6 X-ra oes : Page 5 Source ssues The Parallel X-ra aging Sse Earlier we consiere a parallel ra sse wih an incien inensi ha passes hrough a 3D objec having a isribuion of aenuaion coefficiens μ an projecs o an iage : ep μ There are esseniall no pracical eical projec -ra sses where he source has parallel ras. There are soe scanning sses ha igh be appropriae for inusrial inspecion operaions for eaple: bu hese kins of sses are oo slow for eical applicaions. Pracical X-ra Sources There are wo ain issues associae wih pracical -ra sources:. Geoeric isorions ue o poin geoer eph epenen agnificaion.

6 oll 6 X-ra oes : Page 6. Resoluion loss blurring ue o finie large source sies Poin Source Geoer Firs we will fin epressions for he iage inensi for a poin source geoer: i ep μ r Coens:. is he coorinae sse in he oupu eecor plane.. is he coorinae sse of he objec.

7 oll 6 X-ra oes : Page 7 3. oice ha i a spaiall varian incien inensi replaces. 4. oice ha he inegraion is along soe pah r wih variable of inegraion r. nensi Variaions The incien inensi is aial a he cener of he coorinae sse an falls off owars he eges. This has wo coponens an increases in isance fro he source an he ras obliquel sriking he eecor. nensi has reall power/uni area. We can wrie an epression for he inensi i as: i phoons ean phoon E uni areaeposure ie where k is a scaling coefficien is he nuber of phoon ha are eie uring he observaion ie we assue here ha phoons are eie isoropicall over a sphere an Ω/4π is fracion of he surface of a sphere ha is subene b piel area a. [Ω is known as he soli angle an has unis of seraians of which here are 4π over he surface of a sphere. This is siilar o here being π raians over circuference of a circle.] For a piel of area a a soe posiion angle θ awa fro he origin he par of a sphere covere will be acos θ. Thus: Ω a cosθ a cosθ or Ω 4π 4πr r We now efine he inensi a he origin o be. A he origin θ an he isance fro he source o he eecor is r hus Ω a/ an: k i 4π k a Ω 4π

8 oll 6 X-ra oes : Page 8 oe ha he inensi falls off wih / as he eecor oves awa fro he source. The consan k can now be foun in ers of : Subsiuing: Observing ha i cos θ we ge: r k k a 4π Ω cosθ 4π r 3 3 cos i θ r we can pu his epression in he coorinae sse of he eecor using r + an r + : r i + r 3 r + The cos 3 θ er or is oher represenaions is calle he incien inensi obliqui er an his has wo coponens: he cos θ er for an increase in isance fro he source o he eecor an he cosθ er for ras obliquel sriking he eecor. The cos θ er is reall a /r er he inverse square law for fallou of inensi. The cosθ er can be easil visualie if ou hink of a flashligh bea hiing a wall obliquel he oblique bea spreas he phoons over a larger area of he wall. 3/ Oblique Pah negraion f we look a soe poin in he objec a eph we see ha i will srike he : eecor a a posiion

9 oll 6 X-ra oes : Page 9 where is he agnificaion facor for an objec a eph. We can now wrie he aenuaion coefficien a locaion in ers of he oupu coorinae sse: μ μ Also insea of inegraing along he pah r we can rewrie he epression o inegrae in : r + + r This epression sas ha if wih inegrae in insea of r he inegral will nee o be r increase b + in orer o accoun of he longer pah lengh in r han. This er is soeies known as he pahlengh obliqui er.

10 oll 6 X-ra oes : Page Finall we pu i all ogeher an we ge an epression for he oupu inensi fro a poin source: + + r r ep 3/ μ Eaple For he eaple we will reuce he iensions of he proble o an an hus r. ow le s look a a recangular objec a eph : W L rec rec μ μ The epression for he iage inensi will be: + + W L 3/ rec rec ep μ The use of he agnificaion facor allowe he funcion of o be convere o a funcion of for each locaion in he eecor plane. The firs rec in he above epression has wih L/ an is cenere a. The secon rec has wih W an is cenere a. The inegral is he area uner he overlap of hese wo rec funcions.

11 oll 6 X-ra oes : Page The inegral is: for l W < or > L W / W for l W > + or L < + W / L W + oherwise f we ignore all obliqui ers we ge he following: ncluing he pahlengh an incien inensi obliqui ers we ge:

12 oll 6 X-ra oes : Page Uner a parallel ra geoer we ge he following: As we can see he eph epenen agnificaion has significanl isore he appearance of he objec in he iage. We can efine a fracional ransiion wih be: L L W / + W / W L L + 4 W / 4 + W / Thus we can iniie he geoeric isorions b placing he objec as far fro he source as possible ake large. Finie Large Sources To gain an unersaning of his issue we will firs consier a hin objec. Specificall we will le he aenuaion coefficien be: an hen: μ τ δ

13 oll 6 X-ra oes : Page ep ep r r i i τ δ τ We le / he objec agnificaion facor an we will ignore he pahlengh obliqui er o ge: i i ep τ where ep-τ is he ransission funcion. gnoring all obliqui ers we ge: ow we consier a finie source funcion s an a ver sall pinhole ransission funcion: The iage will now be an iage of he source wih he source agnificaion facor : ks

14 oll 6 X-ra oes : Page 4 where k is a scaling facor ha is proporional o he area of he pinhole / ec. f we wan he above o represen he ipulse response of he sse we nee o ake he pinhole equal o δ an accoun for all of he scaling ers [ δ is no a realiable ransission funcion since can never ecee neverheless we will allow i for aheaical convenience.] The area of he pinhole is δ. The capure efficienc of he pinhole is he fracion of all phoons eie fro he source ha pass hrough he pinhole. This will be equal o: pinhole area η 4 Leing he oal nuber of phoon eie be: π 4π s an he oal nuber of phoons o ge hrough he pinhole will be: η. 4π This us be he sae nuber a he eecor: ks The scaling coefficien will herefore be: k 4π k 4π so: s 4π ow we le he pinhole be a posiion ha is δ- - :

15 oll 6 X-ra oes : Page 5 The iage of he source is no locae a where is he objec agnificaion facor. Thus he ipulse response funcion is: s h 4 ; π ow we can calculae he iage for an arbirar ransission funcion using he superposiion inegral: s s s h ** 4 4 an sub 4 ; π π π Thus he final iage is equal o he convoluion of he agnifie source an he agnifie objec. The objec is blurre b he source funcion. The frequenc oain equivalen is: { } 4 v u T v u S F D π Consier / which iels an. The objec is agnifie b a facor of an is blurre b he unagnifie source.

16 oll 6 X-ra oes : Page 6 Coens:. The leas blurring coe when is ae sall. Thus i is esirable o ake he eph plane as far fro he source as possible:. Then -/ an. As we was above aking also reuces geoeric isorions. The coon pracice for -ra iaging hen is o posiion he subjec ieiael ne o or on op of he eecor.. f he hickness of he bo is a liiing facor hen le. This will ake he sse close o a parallel ra geoer wih an. The ain proble wih his approach is. / an SR 3. We woul also like he ake s as sall as possible o reuce blurring bu s an aking i sall igh reuce he nuber of phoons creae an hus reuce SR. 4. For a cople objec we can ake μ τ δ an each plane i i will have is own agnificaion facors. This is no paricularl useful bu i can give ou soe iea of how blurring an agnificaion igh affec ifferen pars of a real objec ifferenl. Overall Sse Response ow we can a he eecor response o he oher sse eleens: 4 π The ipulse response funcion will hen be: h 4 π or for a circularl seric source funcion: s ** ** h r h 4 π s ** h r r s ** h r

17 oll 6 X-ra oes : Page 7 Objec Blurring One issue is how uch oes he eecor response blur he objec. is iporan o realie ha he eecor blurs he agnifie objec. Our inuiion woul be o ake he objec as large as possible b aking / ver large. This woul icae oving he objec as close o he source as possible which is eacl opposie as wha we woul like o o o iniie source blurring. Consier also ha he agnifie source also blurs he agnifie objec source an objec have ifferen agnificaion facors. One wa o look a his is o eaine he response in he coorinae sse of he objec raher han he eecor : ks ** ** h r he effecive agnificaion of he source is: an he effecive agnificaion of he eecor response is: These are in copeiion: o ake he source blurring sall ake o ake he eecor response sall ake Coens:. For os fil sses he eecor response is ver sall an he source is alos alwas bigger. Therefore we woul like o ake.. For oher kins of sses e.g. igial fluoroscop sses he eecor resoluion is uch larger e.g..5 an for hese sses an inereiae a be appropriae.

18 oll 6 X-ra oes : Page 8 Deecor ssues Earlier we iscusse he effec of source sie an locaion on spaial resoluion an agnificaion isorions in -ra iaging. ow we will iscuss eecor issues. n selecing eecor characerisics we will have a resoluion/sr rae-off his coe priaril fro he fac he hicker eecors have beer SR bu a larger ipulse response. Conversion of -Ras o Fil Phoographic fils are generall no ver sensiive o -ras so -ras us firs be convere o visible ligh b a scinillaing screen: We will now evelop epressions o represen he ipulse response of he eecor. Suppose we have a -ra phoon ener he scinillaing screen an i ineracs a soe eph which we ll call an generaes a shower of ligh phoons isoropicall fro a poin of which soe evenuall srike he eecor. The geoer is esseniall he sae as a poin -ra source sriking he eecor. oabl: 3 h r h cos θ h + r 3 bu h b he inverse square law hus: h r k + r 3/ The corresponing frequenc oain equivalen is:

19 oll 6 X-ra oes : Page 9 H ρ πk ep πρ Wihou loss of generali we will selec k o noralie his epression o have a peak frequenc response of. H ρ ep πρ oice ha righ ne o he fil : H ρ h r δ Finall we can calculae an average frequenc response b aking: H ρ H ρ p where p is he probabili ensi funcion for an ineracion occurring a eph. To eerine his we firs recognie ha he scinillaing screen has is own linear aenuaion coefficien μ. The nuber of phoons ha pass hrough a an eph is: ep μ an he nuber absorbe will be: abs ep μ The oal fracion absorbe in he eecor is: η ep μ where η is eecor efficienc which increases wih. We can efine he cuulaive isribuion funcion as: ep μ P η an hus he probabili ensi funcion is: P μ p ep μ η The average frequenc response is hen: μ H ρ ep π ρ ep μ η μ ηπρ + μ ep πρ + μ

20 oll 6 X-ra oes : Page For large ρ his epression looks like: μ H ρ πηρ The high spaial frequencies pla a large role in icaing he shape of he ipulse response close o he peak e.g. h r near r an he low spaial frequencies will icae he appearance of he ails of h r. Thus near r he average ipulse response will ake on he shape: h r μ πηr recall he inverse Fourier-Bessel ransfor of /ρ is /r. The average ipulse response hen is ver peake infinie in apliue. One consequence of his is he coon easures of resoluion or blurring e.g. like FWH Full Wih a Half aiu have no eaning. One wa o evaluae he perforance of he eecor sse is o efine a cuoff frequenc ρ k as he frequenc a which he response falls o k H. For saller values of k his is: μ ρk πηk This in essence give he aiu spaial frequenc ha can be eece where k represens he level of eecabili. For eaple k. is a coon value an having a higher cuoff frequenc ρ k is esire o iprove spaial resoluion. We can now begin o see he SR resoluion rae-off. As increase he eecor efficienc η increases which leas o ore -ra phoons being eece an hus he SR iprove. This however causes ρ k o be saller resuling in lower spaial resoluion.

21 oll 6 X-ra oes : Page Recall ha he SR is proporion o he square roo of he nuber -ra phoons an in orer o see he he us be eece so he SR is proporional o he roo of he nuber -ra phoons ha are eece. SR is herefore proporional o η. Eaple Le s look a a eecor wih he μ.5 - an.5 an we will use k.. η.3 ρ k 8 an he liiing spaial resoluion is approiael: 5μ ρ ow if we ouble he hickness o.5 : η.53 ρ k k 4.5 an he liiing spaial resoluion is approiael: μ ρ k Coens: n general increasing iproves boh η an ρ k. Wha happens if we pu he fil on he back of he scinillaor? s he response beer or worse? Two Screen Deecors wih Double Eulsion Fils To ease he raeoff beween resoluion an SR we can use a ouble eulsion fil wih a wo screen scinillaor:

22 oll 6 X-ra oes : Page We assign a coorinae sse here o ease our analsis: Since no ineracion occur in he fil we can neglec is hickness: For ineracions occurring in he firs screen : h r k + r 3/

23 oll 6 X-ra oes : Page 3 which iels a frequenc response of: H ρ ep πρ ep πρ For ineracions occurring in he secon screen : Finall: H ρ ep πρ ep πρ ep πρ for < H ρ ep πρ for < where +. The eecor efficienc is again: η ep μ. We can now fin he average frequenc response in a siilar anner as before: μ ep μ ep π ρ ep μ H ρ + η πρ μ For large ρ his epression looks like: ep π ρ + πρ μ μ H f we ake / hen: an he cuoff frequenc will be: where [ η ] / Eaple μ ρ πηρ [ ep μ + ep μ ] μ H ρ η πηρ πηρ [ ] / μ ep μ [ ] / μ ρk πηk [ η ] / is he iproveen facor over he single eulsion fil sse. Le s look a he previous eaple wih a eecor wih he μ / hen an we will use k.. η.3 [ η ] / ρ k 3.7

24 oll 6 X-ra oes : Page 4 an he liiing spaial resoluion is approiael: 76μ ρ Alernaivel we can hol ρ k consan b seing. 4 : k η.45 ρ k ρ k 8 5μ