EE 4372 Tomography. Carlos E. Davila, Dept. of Electrical Engineering Southern Methodist University

Transcription

1 EE 4372 Tomography Carlos E. Davila, Dept. of Electrical Engineering Southern Methodist University EE 4372, SMU Department of Electrical Engineering 86

2 Tomography: Background 1-D Fourier Transform: F( ω) = f( x) e dx 2-D Fourier Transform: jωx j( ω x ω y) + F( ω1, ω 2) = f( x, y) e 1 2 dxdy EE 4372, SMU Department of Electrical Engineering 87

3 Inverse 2-D Fourier Transform j( ω x+ ω y) ω 1 ω 2 ω 1 ω 2 f( x, y) = F(, ) e 1 2 d d EE 4372, SMU Department of Electrical Engineering 88

4 Polar Coordinates 2-D Fourier Transform can be expressed as: ω ω F( Ω,) 1 = Ω cos 2 = Ω sin ω 2{ Ω ω 1 EE 4372, SMU Department of Electrical Engineering 89

5 Tomography: Geometry s y x-ray paths { t x tissue slice f(x,y) t P ( t ) EE 4372, SMU Department of Electrical Engineering 9

6 Geometry (cont.) P ( t) is the line integral of f(x,y) along direction of x-ray paths (projection): Axis transformations: P ( t) = f( t, s) ds t s = cos sin sin cos x y x y = cos sin sin cos t s EE 4372, SMU Department of Electrical Engineering 91

7 Fourier Slice Theorem 1-D Fourier Transform of P ( t) : j t Ω S ( Ω ) = P ( t) e dt ( ) Consider F ω, ω on the line ω 2 = : 1 2 F( ω 1, ) = f( x, y) e 1 dxdy = f( x, y) dy e ω 1 dx j ω x EE 4372, SMU Department of Electrical Engineering 92 j x

8 Fourier Slice Theorem (cont.) x=t, y=s, = [ ] F( ω 1, ) = P ( t) 1 e dx On the line ω 2 =, F( ω 1, ω 2) can be obtained by finding the 1-D Fourier transform of P ( t) j ω F( Ω,) ( ) = = S o x Ω Ω = ω 1 EE 4372, SMU Department of Electrical Engineering 93

9 Fourier Slice Theorem (cont.) In general: P ( t) = f( t, s) ds jωt S ( Ω ) = P ( t) e dt = f( t, s) ds e dt j Ω t EE 4372, SMU Department of Electrical Engineering 94

10 Fourier Slice Theorem (cont.) Changing variables to (x,y): jω ( xcos xsin ) + S ( Ω ) = f( x, y) e dxdy = F( Ω,) By taking F{ P ( t ) } for 1, 2,, N, we can obtain F( ω 1, ω 2) on radial lines at angles 1, 2,, N. If N is large enough, we get a good estimate of F( ω 1, ω 2), F$ ( ω, ω ) 1 2 EE 4372, SMU Department of Electrical Engineering 95

11 Fourier Slice Theorem (cont.) (, ω ) F ω 1 2 ω 2... { } S ( Ω ) = F P ( t) 2 2 S ( Ω ) = F P ( t) { } 1 1 ω 1 EE 4372, SMU Department of Electrical Engineering 96

12 Image Reconstruction f(x,y) recovered by inverse 2-D Fourier transform: j( ω x+ ω y) ω1 ω 2 ω 1 ω 2 f( x, y) F$ (, ) e d d 1 2 (1) This recovery step is accomplished via the Filtered Back Projection Algorithm EE 4372, SMU Department of Electrical Engineering 97

13 Filtered Back Projection Algorithm Assume F( ω 1, ω 2) is known exactly, in polar coordinates, (1) becomes: 2 jω ( xcos + ysin ) f( x, y) = F( Ω, ) e ΩdΩ d = jω x + y F( Ω, ) e ΩdΩ d jω ( xcos( + ) + ysin( + )) + + ( cos sin ) F( Ω, ) e ΩdΩ d EE 4372, SMU Department of Electrical Engineering 98

14 Filtered Back Projection (cont.) Note: F( Ω, + ) = F( Ω, ) cos( ) cos + = sin( ) sin + = + Ω F( Ω,) F( Ω,) EE 4372, SMU Department of Electrical Engineering 99

15 Filtered Back Projection (cont.) jω ( xcos + ysin ) f( x, y) = F( Ω, ) e ΩdΩ d + = jω ( xcos ysin) + F( Ω, ) e ΩdΩ d jω x + y F( Ω, ) e ΩdΩ d j xcos ysin + Ω ( + ) F( Ω, ) e ( Ω ) dω d ( cos sin ) EE 4372, SMU Department of Electrical Engineering 1

16 Filtered Back Projection Algorithm (cont.) jω ( xcos + ysin ) f( x, y) = F( ) e t Ω, Ω dω d = S ( Ω ) e Ω dω d Q j Ω ( t) t filtered projection: Q ( t) = P ( t) h( t) is not quite the same as P called Back Projection). ( t) (algorithm would then be EE 4372, SMU Department of Electrical Engineering 11

17 Filtered Back Projection Algorithm (cont.) j t h( t) e Ω dt = Ω f( x, y) Q ( xcos ysin ) d = + jω t Ω Ω Ω Q = S ( ) e d EE 4372, SMU Department of Electrical Engineering 12

18 Filtered Back Projection Algorithm (cont.) ex) Projection functions for a square: y t x P ( t) P /2 (t) t EE 4372, SMU Department of Electrical Engineering 13

19 Filtered Back Projection Algorithm (cont.) back projection partially reconstructs square: y t P ( t) d x P ( t) P /2 (t) t EE 4372, SMU Department of Electrical Engineering 14

20 Filtered Back Projection Algorithm (cont.) FBP algorithm back-projects Q (t), in practice, P (t) only known for finite number of samples (x-ray sensors): P [n]= P (nt), n=,..., N-1 estimate Q (t) as Q [n] = P[ n] h[ n] N must be large enough to minimize aliasing Ω is not band-limited. EE 4372, SMU Department of Electrical Engineering 15

21 Filtered Back Projection Algorithm (cont.) ω ω.25 h[n] n -1/9 2-1/25 2-1/ 2 EE 4372, SMU Department of Electrical Engineering 16

22 Scanning Configuration First generation scanners used parallel beams to measure P ( t), fourth generation scanners use: source patient stationary detector array beam is fan-shaped source rotates around patient EE 4372, SMU Department of Electrical Engineering 17