9. Computed Tomography (CT) 9.1 Absorption of X-ray

Transcription

1 Absorption of X-ray Absorption of X-ray X-ray radiography Object I κ Incident Transmitted d I(y) = I e κ(y)d κ : attenuation coefficient In the case of X-ray, it depends on the atomic number. (Heavy atom large κ.) I If κ is depth dependent, κd d κ(x,y)dx κ(x,y)dx (κ(x,y) = Not Object) Only the integral of κ along optical path can be obtained from X-ray radiography. 149/197

2 Projection from several directions 9.2 Projection from several directions I I(ξ,θ = ) I ξ I(ξ,θ = π 2 ) The distribution of κ including depth distribution, which is called tomography, can be obtained from several projected data with different directions. Computed Tomography ξ 15/197

3 Projection from several directions Number of projections and Num. of internal nodes 3x3 unknowns 3 projs. (observed) 3 projs. (observed) Num. of unknowns > Num. of Obs. Cannot solve. Num. of unknowns = Num. of Obs. Cannot distinguish. To obtain more projection, other projections with different directions are needed. θ [,π] 151/197

4 Schematic of forward and backward-projection 9.3 Schematic of forward and backward-projection Forward projection (measuring process) Forward projection Integral along beam path p(ξ,θ) = η κ(x,y)dη θ (x x(ξ,θ), y y(ξ,θ)) Accumulate along path deg 45 deg = 9 deg 135 deg /197

5 Schematic of forward and backward-projection Backward projection (1) Simple backprojection 1 Map averaged value of the projection data along path for each angle. = 2 Take an average of mapped data for each pixel Blurrier than original. 8 Original = deg 45 deg deg 135 deg /197

6 Schematic of forward and backward-projection Backward projection (2) Filtered backward projection In the simple backward projection, the reconstracted result is blurred. To reduce the blurring, edge enhancement filter is applied to the projection data. Edge enhancement filter g n = f n kf n = f n k(f n 1 2f n +f n+1 ) = kf n 1 +(2k +1)f n kf n+1 (k = 1) g n = +1 i= 1 w i f n i (w 1,w,w +1 ) = ( 1,+3, 1) 154/197

7 Schematic of forward and backward-projection Filtered backward projection 1 Edge enhancement of projection data: p (e.g.) p j = +1 w i p j i i= 1 (w 1,w,w +1 ) = ( 1,3, 1) p 1 p Apply simple backward projection using p. 1 = In this example, the reconstructed field is identical to original. deg 45 deg deg 135 deg /197

8 Radon Transform 9.4 Radon Transform Coordinate system η I (ξ,θ) y r θ ξ x I(ξ, θ) ξ Inner product of e x : xe x e x +ye x e y = ξe x e ξ +ηe x e η η y x = ξcosθ ηsinθ θ ξ x Expression of point r r = xe x +ye y { x = +ξcosθ ηsinθ y = +ξsinθ +ηcosθ { ξ = +xcosθ +ysinθ η = xsinθ+ycosθ = ξe ξ +ηe η 156/197

9 Radon Transform Radon Transform Projected data (known) I(ξ,θ) = I e L κ(r)dl L {r(ξ,η;θ) ξ = ξ (const.)} Sinogram (known) p(ξ,θ) log I(ξ,θ) I Radon Transform Integral over straight line in 2-D space. p(ξ,θ) = κ(r(ξ, θ)) dl L Extend from line integral to 2-D area integral ξ = xcosθ+ysinθ L [ ]dl = [ ]dη ξ=ξ = [ ]δ(ξ ξ )dξ dη η y dl(ξ = ξ ) = [ ]δ(ξ ξ (x,y))dxdy = [ ]δ(ξ (xcosθ+ysinθ))dxdy x ξ ξ 157/197

10 Projection slice theorem 9.5 Projection slice theorem Projection (Sinogram) p(ξ,θ) = κ(x,y)δ(ξ (xcosθ+ysinθ))dxdy (1) FT with respect to ξ P(k ξ,θ) = κ(x,y)δ(ξ (xcosθ+ysinθ))e jkξξ dxdydξ = κ(x,y)e jkξ(xcosθ+ysinθ) dxdy (2) 158/197

11 Projection slice theorem 2-D Fourier Transform in polar coordinate system Forward transform F(k x,k y ) = f(x,y)e j(kxx+kyy) dxdy (k x = kcosθ, k y = ksinθ) = f(x,y)e jk(xcosθ+ysinθ) dxdy F (k,θ) (3) Inverse transform f(x,y) = 1 4π 2 F(k x,k y )e +j(kxx+kyy) dk x dk y = 1 4π 2 2π ( dk x dk y = 2π kdθdk) F (k,θ)e +jk(xcosθ+ysinθ) k dθdk (4) 159/197

12 Projection slice theorem Eq. (2) and Eq. (3) are same. Projection Slice Theorem P(k ξ,θ)isexpressed byfourier transformofκ(x,y) inpolar coordinate system. Since P(k ξ,θ) is known, κ(x,y) is obtained by the inverse FT using (4). κ(x,y) = 1 4π 2 2π P(k ξ,θ)e +jk ξ(xcosθ+ysinθ) k ξ dθdk ξ (5) 16/197

13 Projection slice theorem Projection from opposite direction κ(x,y) = 1 4π 2 = 1 4π 2 2π π P(k ξ,θ)e +jk ξ(xcosθ+ysinθ) k ξ dθdk ξ P(k ξ,θ)e +jk ξ(xcosθ+ysinθ) k ξ dθdk ξ 2π π 2π k ξ dθdk ξ = k ξ dθdk ξ + k ξ dθdk ξ π Projection from opposite direction ( ) p(ξ,θ±π) = p( ξ,θ) θ θ±π P(k ξ ξ ξ,θ ±π) = P( k ξ,θ) e +jkξ(xcos(θ±π)+ysin(θ±π)) = e jk ξ(xcosθ+ysinθ) 2π 2nd Term = P(k ξ,θ)e +jkξ(xcosθ+ysinθ) ( k ξ dθdk ξ θ = θ π ) π π = P( k ξ,θ )e jkξ(xcosθ +ysinθ ) k ξ dθ ( ) dk ξ k ξ = k ξ π = P(k ξ,θ )e +jk ξ (xcosθ +ysinθ ) ( k ξ )dθ ( dk ξ ) π = P(k ξ,θ )e +jk ξ (xcosθ +ysinθ ) k ξ dθ dk ξ 161/197

14 Reconstruction by using Fourier transform 9.6 Reconstruction by using Fourier transform κ(x,y) = 1 4π 2 = 1 2π π π q(ξ(x,y),θ) = ξ {}}{ P(k ξ,θ)e +jk ξ(xcosθ+ysinθ) kξ dθdk ξ [P(k ξ,θ) k ξ ]e +jkξξ dk ξ dθ = π 1 2π } {{ } =F 1 k {P((k ξ,θ) k ξ } q(ξ,θ) ξ [P(k ξ,θ) k ξ ]e +jk ξ(xcosθ+ysinθ) dk ξ π q(ξ(x,y),θ)dθ }{{} Average with θ. κ(x,y) = 1 2 } F 1 k ξ {F ξ {p(ξ,θ)} ξ H(k ξ ) k ξ θ (H(k ξ ) = k ξ in the case of Ramp function) 162/197

15 Reconstruction by using Fourier transform 1 Sinogram : p(ξ, θ) 2 FWD FT with ξ : P(k ξ,θ) = 3 Filtering (Weight with k ξ ) : k ξ P(k ξ,θ) p(ξ,θ)e jk ξξ dξ 4 INV FT with k ξ : q(ξ,θ) = 1 P(k ξ,θ) k ξ e +jkξξ dk ξ 2π 5 Backward projection (Coordinate transform and integrate with θ) : κ(x,y) = 1 π q(xcosθ+ysinθ,θ)dθ 2π Two FT (FWD and INV) are needed for a certain θ. 163/197

16 Reconstruction by using Fourier transform Filtered Back-Projection Before IFT, k ξ is multiplied. Since this factor k ξ is considered as a filter in the spectral domain, the method based on FT is called Filtered Back-projection (FBP). In the actual computation, k ξ [, ] [ k max,+k max ]. k max is Nyquist frequency determined by sampling interval. To avoid ringing artifact caused by high frequency component, another filter can be applied. (e.g. Shepp-Logan filter) H(k ξ ) = 2k max sin πk ξ π k max Ramp filter Shepp-Logan filter 2k max k max +k max 164/197

17 Reconstruction by using Fourier transform Handling of discrete data p(ξ n,θ m ) = p(n ξ,m θ) (n,m) : Integer e.g. (Interpolation for ξ) q(xcosθ+ysinθ,θ) q(ξ n+1,θ) q(ξ n,θ) κ(x i,y j ) = κ(i x,j y) (i,j) : Integer ξ ξ n ξ n+1 xcosθ+ysinθ In order to evaluate q(xcosθ+ysinθ,θ) from q(ξ,θ) interpolation are needed. 165/197

18 Reconstruction by using Fourier transform Reconstruction using convolution { c(x) = a(x )b(x x )dx C(k) = A(k)B(k) q(ξ,θ) = 1 2π kmax } c(x) = 1 A(k)B(k)e +jkx dk 2π k max P(k ξ,θ)h(k ξ )e +jk ξξ dk ξ = ξmax ξ max p(ξ,θ)h(ξ ξ )dξ h(ξ) If H(k ξ ) = k ξ, h(ξ) = 1 2π = kmax k ξ e +jkξξ dk ξ k max 1 2π k2 max (ξ = ) 1 k max sin(k max ξ)+ 1 π ξ ξ 2 (cos(k maxξ) 1) (ξ ) 5 5 ξ/ ξ Subtract by neighbors Edge Enhancement 166/197

19 Reconstruction by using Fourier transform 1 Sinogram : p(ξ, θ) 2 Convolution : q(ξ,θ) = p(ξ,θ)h(ξ ξ )dξ 3 Back-projection : κ(x,y) = 1 π q(xcosθ +ysinθ,θ)dθ 2π Only one convolution for each θ. No FT. κ(x, y) p (x,y;) q (x,y;) ( p,q : contrast is enhanced to display (erf(i/σ)) ) Fourier transform Convolutinal integral Number of multiplifications for each θ DFT 2 2N 2 FFT 2 2N logn All points N 2 Neighboring M pts. (M N) MN Faster computation than DFT, if the convolution is applied to neighboring points only. 167/197

20 Reconstruction by using Fourier transform Filtered Back-projection and Simple BP Filtered Back-Projection κ(x,y) = 1 } F 1 2 k ξ {F ξ {p(ξ,θ)} ξ H(k ξ ) Simple Back-projection H(k) = 1 κ(x,y) = 1 2 = 1 2π F 1 { k ξ π F ξ {p(ξ,θ)} ξ } p(ξ,θ)dθ No need to FT Fast The reconstructed distribution is blurred. k ξ θ k ξ θ = 1 2 p(ξ,θ) θ Iterate two procedures of projection and back-projection. 168/197

21 Iterative Reconstruction 9.7 Iterative Reconstruction Applying the forward projection (FP) to the reconstructed field obtained by the backward projection (BP), we can evaluate the error. The BP of the error is added to the field obtained in the previous step. The simple BP is used for the BP algorithm, since the simple BP is fast. κ(x, y) p(ξ, θ) κ True p Obs κ 2 F B (1) (3) p 1 (5) κ 1 (4) p 1 κ 1 (2) (1) BP for the proj. : κ 1 = B { p Obs} (2) FP for the field : p 1 = F {κ 1 } (3) Under-estimation : p 1 = p Obs p 1 (4) BP for the under-est. : κ 1 = B{ p 1 } (5) Update the field : κ 2 = κ 1 +α κ 1 ( α : Relaxation factor for stable reconstruction < α 1 ) 169/197

22 Iterative Reconstruction Simple Back-projection Sinogram p(ξ n,θ m ) = κ(x,y)dl κ(x,y L nm ) l nm L nm κ(x,y L nm ) = p(ξ ( ) n,θ m ) l nm = dl l nm L nm Fraction of projection is mapped onto the internal distribution. κ(x i,y j ) = 1 p(ξ n,θ m ) a i,j,n,m N n l nm m n a i,j,n,m : Overlap area fraction between the pixel (x i,y j ) and the beam L nm with width ξ. 17/197

23 Example of reconstruction by simulation 9.8 Example of reconstruction by simulation True κ(x,y) Filtered Back-Projection (Filter : Ramp function) κ(x, y) [,2] κ(x, y) [-.2,+.2] Sinogram p(ξ, θ) κ(x,y) [,2], N x = N y = 1, N ξ = 1, N θ = 45( θ = 4deg) κ 2 κ 2 =.15 Line artifact 171/197

24 Example of reconstruction by simulation True κ(x,y) Iterative reconstruction N = 1 N = 2 N = 1 N = 2 Sinogram p(ξ, θ) κ(x,y) [,2], N x = N y = 1, N ξ = 1, N θ = 45( θ = 4deg) κ 2 =.42 κ 2 =.25 κ 2 =.17 Reduction of edge blurring κ 2 = /197