Topics. Projections. Review Filtered Backprojection Fan Beam Spiral CT Applications. Bioengineering 280A Principles of Biomedical Imaging

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1 Bioengineering 28A Principles of Biomedical Imaging Fall Quarter 24 X-Rays/CT Lecture 2 Topics Review Filtered Backprojection Fan Beam Spiral CT Applications s I θ (r) = I exp µ(x, y)ds Lr,θ = I exp µ(rcosθ ssinθ,rsinθ + scosθ)ds Lr,θ

2 s I θ (r) = I exp Lr,θ µ(rcosθ ssinθ,rsinθ + scosθ)ds p θ (r) = ln I θ (r) I = µ(rcosθ ssinθ,rsinθ + scosθ)ds Lr,θ Sinogram Sinogram µ 3 µ 4 p 2 µ µ 2 p p 3 p 4 Direct Inverse Approach p = µ + µ 2 p 2 = µ 3 + µ 4 p 3 = µ + µ 3 p 4 = µ 2 + µ 4 4 equations, 4 unknowns. Are these the correct equations to use? p µ p 2 = µ 2 p 3 µ 3 p 4 µ 4 No, equations are not linearly independent. p 4 = p + p 2 - p 3 Matrix is not full rank. 2

3 µ 3 µ 4 p 2 µ µ 2 p p 3 p 4 Direct Inverse Approach p 5 p = µ + µ 2 p 2 = µ 3 + µ 4 p 3 = µ + µ 3 p 5 = µ + µ 4 p µ p 2 = µ 2 p 3 µ 3 p 4 µ 4 4 equations, 4 unknowns. These are linearly independent now. In general for a NxN image, N 2 unknowns, N 2 equations. This requires the inversion of a N 2 xn 2 matrix For a high-resolution 52x52 image, N 2 =26244 equations. Requires inversion of a 26244x26244 matrix! Inversion process sensitive to measurement errors. Iterative Inverse Approach Algebraic Reconstruction Technique (ART) Backprojection

4 y Backprojection b(x, y) = B{ p( r,θ) } π = p(x cosθ + y sinθ,θ)dθ x x b(x, y) = p( r,θ = )Δθ = p(x )Δθ r Backprojection b(x, y) = B{ p( r,θ) } π = p(x cosθ + y sinθ,θ)dθ Backprojection π b(x, y) = B{ p( r,θ) } = p(x cosθ + y sinθ,θ)dθ 4

5 Theorem [ ] U(k x,k y ) = µ(x, y)e j 2π (k x x +k y y) dxdy = F 2D µ(x, y) U(k x,k y ) = P(k,θ) k x = k cosθ k y = k sinθ k = k 2 2 x + k y F P(k,θ) = p θ (r)e j 2πkr dr Fourier Reconstruction F Interpolate onto Cartesian grid then take inverse transform Fourier Interpretation N Density circumference N 2π k Low frequencies are oversampled. So to compensate for this, multiply the k-space data by k before inverse transforming. Kak and Slaney; 5

6 Polar Version of Inverse FT µ(x, y) = 2π π U(k x,k y )e j 2π (k xx +k y y ) dk x dk y = U(k,θ)e j 2π (k cosθx +k sinθy) kdkdθ = U(k,θ)e j 2π (xk cosθ +yk sinθ ) k dkdθ µ(x, y) = Filtered Backprojection π π π U(k,θ)e j 2π (xk cosθ +yk sinθ ) k dkdθ = k U(k,θ)e j 2πkr dkdθ = u (r,θ)dθ where r = x cosθ + y sinθ u (r,θ) = k U(k,θ)e j 2πkr dk = u(r,θ) F k = u(r,θ) q(r) [ ] Backproject a filtered projection Reconstruction Path F x F - Filtered Back- Project 6

7 Ram-Lak Filter q(r) = F k Ram-Lak Filter k q(r) = F k rect 2k max = [ ] = k e j 2πkr dk Not a realistic convolution kernel. k max k max k e j 2πkr dk k max =/Δs Reconstruction Path F x F - Filtered Back- Project Reconstruction Path * Filtered Back- Project 7

8 Example Kak and Slaney Additional Filtering k max =/Δs Sampling Requirements How many detectors do we need? How many angular views do we need? 8

9 Aliasing Kak and Slaney Artifacts Object Effect of Noise Aliasing due to insufficient number of detectors Aliasing due to insufficient number of views Aliasing Kak and Slaney 9

10 Aliasing Aliased Image Alias Component (aliased image)- (alias component) Kak and Slaney Alias components Kak and Slaney Sampling Requirements Beam Width 2/(Δs) W= 2/(Δs) δ=/w= Δs/2 Smoothed

11 Sampling Requirements Smoothed Detectors Δr Δs/2 Sampled Smooth Detector Sampling Requirements Beamwidth of detector Δs Sampling interval Δr Requirement is Δr Δs/2 View Aliasing Kak and Slaney

12 View Sampling Requirements View Sampling -- how many views? Basic idea is that to make the maximum angular sampling the same as the projection sampling. πfov N views = Δr N views,36 = πfov Δr = πn proj (for 36 degrees) N views,8 = πn proj 2 (for 8 degrees) CT System Generations 5 minutes/slice 2 seconds /slice.5 seconds /slice CT System 2

13 Fan Beam θ =α + β r = Rsinα r Fan Beam θ =α + β r = Rsinα β = r α max r π α max θ Spiral CT Nearest Neighbor Interpolation Linear Interpolation From 3

14 Longitudinal Aliasing in Spiral CT From Multislice CT CT Applications 4

15 Virtual Colonoscopy 5