Computerized tomography. CT Reconstruction. Physics of beam attenuation. Physics of beam attenuation

Transcription

1 Compterize tomograph Compterize tomograph (CT) is a metho for sing x-ra images to reconstrct a spatiall aring ensit fnction. CT Reconstrction First generation CT scanner 1 2 Phsics of beam attenation CT works b collecting x-ra images one slice at a time. Consier a parallel beam of x-ras passing throgh an object being image orthographicall: Phsics of beam attenation f we consier a single ra passing throgh, we ll fin that it s intensit rops off as: = µ x where µ is the linear attenation coefficient. o x We can re-write this as ifferentials an permit µ to ar along the ra: = µ ( x) x x=- An x-ra photon interacts with the material b: x= f the material is mae of a single sbstance of aring ensit, then µ(x) can be moele as proportional to that ensit. absorption scatter Absorbe photons are simpl lost. Re-arranging: = µ ( x) x We will assme that scattere photons are all reirecte awa from the sensor. 3 ntegrating: o = µ ( x) x 4

2 Phsics of beam attenation Performing the integration of the left sie: o = ln[ ] = ln[ ] ln[ ] ln o o = o Eqating to the right sie: ln = ( x) x µ o Raising to an exponent: = exp µ ( x) x o Soling for etector intensit: Phsics of beam attenation f we back p a little bit, we can remoe the negatie sign b inerting the argment of the log: Allowing to ar: o ln = µ ( x) x o ln = µ ( x, x ) = g ( ) ( ) Ths, we can take the etector ata, an, sing this log, we can interpret the reslt as an integral projection of the attenation fnction. = oexp µ ( x) x Consiering beams that pass throgh at arios - positions: ( ) = oexp µ ( x, ) x 5 o g() 6 ART ART Using projections from mltiple angles, o can tr to sole for the interior istribtion. One approach is essentiall to create a large linear sstem an sole iteratiel. For example: a b 12 c a b 9 = c n practice, ART has proen comptationall expensie an sensitie to noise. nstea, we can se some fancier math to erie an elegant soltion Sch a techniqe is calle an Algebraic Reconstrction Techniqe, or ART. 7 8

3 The Forier transform Recall (from CSE 557?) that the Forier transform of a fnction can be written as: {( f x)} = f( x)exp[ i2 π x] x = F( ) where is spatial freqenc. The inerse Forier transform is simpl: The 2D Forier transform We can generalize this to 2D: { f( x, )} = f( x, )exp[ i2 π ( x+ )] x = F(, ) 2D where is spatial freqenc in x, an is the spatial freqenc in. Likewise, the 2D inerse Forier transform is: { F ( )} = F ( )exp[ i2 π x ] = f( x) 1 { F (, )} = F (, )exp[ i2 π ( x+ )] = f( x, ) 1 2D Note that an f(x) implies a niqe F() an ice ersa, so if we know one, we can compte the other: Again, gien one fnction, we can niqel compte the other. f( x) F( ) f(x,) 1 F(,) 1 f( x) F( ) x 9 10 Linear transforms of Forier omains Linear transforms of Forier omains We can also write the Forier transform relation in terms of ector argments: f( x) F( ) f(x,) F(,) t s eas to show that scaling one omain correspons to inerse scaling the other: x ( ) 1 f ax F( ) a a n fact, if we replace a with a matrix A, it is not har to show that: ( x) T ( T ) f A A F A f(x',') ' x' F(',') ' ' For rotations, this implies: T = f( Rx) R F( R )? 11 12

4 Forier transforms an projections Forier transforms an projections So, what o Forier transforms hae to o with x-ra projections? Let s change terminolog slightl an sa f(x,)=µ(x,). We e alrea note that: f(x,) F(,) F (, ) = f( x, )exp[ i2 π ( x+ )] x g() F(0,) = G() What happens if we ealate this at F(0,)? F(0, ) = f( x, )exp[ i2 π ( 0 + )] x = f( x, )exp[ i2 π ] x = f( x, ) xexp[ i2 π ] = g( )exp[ i2 π ] = { g( )} Projection at an angle Projection at an angle r What happens if we project the olme at an angle? f(x,) g (r) F(,) r f(x,) g (r) f(x',') ' ' ' F(',') ' g(') F(0,') = G(') ' r ρ f(x,) F( ρ, ) g (r) = G( ρ) F(,) = F( ρ, ) 15 16

5 Forier projection slice theorem Secon generation scanner n other wors, if we express F(, ) in polar coorinates F(ρ, ): F( ρ, ) = { g ( r)} = G ( ρ) This reslt is calle the Forier projection slice theorem or the central slice theorem. Using this theorem, we can reconstrct an object from its projections b: 1. Poplating the Forier omain with oriente Forier lines 2. Taking the inerse Forier transform n practice, all of these operations can be performe in the spatial omain Thir generation scanner Forth generation scanner 19 20

6 A meical scanner Bibliograph Macoski, Meical maging Sstems, Prentice Hall, nc., Englewoo Cliffs, New Jerse,