Projection Slice TheoremBackproject a filtered projection! TT Liu, BE280A, UCSD Fall 2010! Fourier...
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1 TT Liu, BE280A, UCSD Fall 2010 Bioengineering 280A Principles of Biomedical Imaging Fall Quarter 2010 CT/Fourier Lecture 4 TT Liu, BE280A, UCSD Fall 2010 Projection Theorem Suetens 2002 U(k x ,0) = μ( x, y)e − j 2π ( k x x +k y y ) −∞ ∞ ∫ −∞ ∞ ∫ dxdy = μ( x, y)dy −∞ −∞ ∫ [ ] −∞ −∞ ∫ e − j 2πk x x dx = g( x,0) −∞ −∞ ∫ e − j 2πk x x dx = g(l ,0) −∞ −∞ ∫ e − j 2πkl dl g(l ,0) In-Class Example: μ( x, y) = cos2πx l TT Liu, BE280A, UCSD Fall 2010 Projection Theorem Suetens 2002 U(k x , k y ) = μ( x, y)e − j 2π ( k x x +k y y ) −∞ ∞ ∫ −∞ ∞ ∫ dxdy = F 2D μ( x, y) [ ] G(k,θ) = g(l ,θ)e − j 2πkl −∞ ∞ ∫ dl F U(k x , k y ) = G(k,θ) k x = k cosθ k y = k sinθ k = k x 2 + k y 2 g(l ,θ) l TT Liu, BE280A, UCSD Fall 2010 Projection Slice Theorem Prince&Links 2006 G( ρ,θ ) = g( l ,θ )e − j 2πρl −∞ ∞ ∫ dl = f ( x, y) −∞ ∞ ∫ −∞ ∞ ∫ δ( x cosθ + y sinθ − l)e − j 2πρl dx dy −∞ ∞ ∫ dl = f ( x, y) −∞ ∞ ∫ −∞ ∞ ∫ e − j 2πρ x cosθ +y sinθ ( ) dx dy = F 2D f ( x, y) [ ] u= ρ cosθ ,v= ρ sinθ
Transcript of Projection Slice TheoremBackproject a filtered projection! TT Liu, BE280A, UCSD Fall 2010! Fourier...
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TT Liu, BE280A, UCSD Fall 2010
Bioengineering 280A ���Principles of Biomedical Imaging���
Fall Quarter 2010���CT/Fourier Lecture 4���
TT Liu, BE280A, UCSD Fall 2010
Projection Theorem
Suetens 2002
€
U(kx,0) = µ(x,y)e− j2π (kxx+kyy )
−∞
∞
∫−∞
∞
∫ dxdy
= µ(x,y)dy−∞
−∞
∫[ ]−∞
−∞
∫ e− j2πkxxdx
= g(x,0)−∞
−∞
∫ e− j 2πkxxdx
= g(l,0)−∞
−∞
∫ e− j 2πkldl
€
g(l,0)
In-Class Example:
€
µ(x,y) = cos2πxl
TT Liu, BE280A, UCSD Fall 2010
Projection Theorem
Suetens 2002
€
U(kx,ky ) = µ(x,y)e− j2π (kxx+kyy )
−∞
∞
∫−∞
∞
∫ dxdy
= F2D µ(x,y)[ ]
€
G(k,θ) = g(l,θ)e− j2πkl−∞
∞
∫ dlF
€
U(kx,ky ) =G(k,θ)
€
kx = k cosθky = k sinθ
k = kx2 + ky
2
€
g(l,θ)
l
TT Liu, BE280A, UCSD Fall 2010
Projection Slice Theorem
Prince&Links 2006
€
G(ρ,θ) = g(l,θ)e− j2πρl−∞
∞
∫ dl
= f (x,y)−∞
∞
∫−∞
∞
∫ δ(x cosθ + y sinθ − l)e− j 2πρldx dy−∞
∞
∫ dl
= f (x,y)−∞
∞
∫−∞
∞
∫ e− j2πρ x cosθ +y sinθ( )dx dy
= F2D f (x,y)[ ] u= ρ cosθ ,v= ρ sinθ
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TT Liu, BE280A, UCSD Fall 2010
Fourier Reconstruction
Suetens 2002
F
Interpolate onto Cartesian grid then take inverse transform
TT Liu, BE280A, UCSD Fall 2010
Polar Version of Inverse FT
Suetens 2002
€
µ(x,y) = G(kx,ky−∞
∞
∫−∞
∞
∫ )e j 2π (kxx+kyy )dkxdky
= G(k,θ0
∞
∫0
2π∫ )e j2π (xk cosθ +yk sinθ )kdkdθ
= G(k,θ−∞
∞
∫0
π
∫ )e j 2πk(x cosθ +y sinθ ) k dkdθ
€
Note :g(l,θ + π ) = g(−l,θ)SoG(k,θ + π ) =G(−k,θ)
TT Liu, BE280A, UCSD Fall 2010
Filtered Backprojection
Suetens 2002
€
µ(x,y) = G(k,θ−∞
∞
∫0
π
∫ )e j 2π (xk cosθ +yk sinθ ) k dkdθ
= kG(k,θ−∞
∞
∫0
π
∫ )e j2πkldkdθ
= g∗(l,θ)dθ0
π
∫
€
g∗(l,θ) = kG(k,θ−∞
∞
∫ )e j2πkldk
= g(l,θ)∗F −1 k[ ]= g(l,θ)∗q(l)€
where l = x cosθ + y sinθBackproject a filtered projection
TT Liu, BE280A, UCSD Fall 2010
Fourier Interpretation
Kak and Slaney; Suetens 2002
€
Density ≈ Ncircumference
≈N
2π k
Low frequencies are oversampled. So to compensate for this, multiply the k-space data by |k| before inverse transforming.
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TT Liu, BE280A, UCSD Fall 2010
Ram-Lak Filter
Suetens 2002
kmax=1/Δs
TT Liu, BE280A, UCSD Fall 2010
Reconstruction Path
Suetens 2002
F
x F-1
Projection
Filtered Projection
Back- Project
TT Liu, BE280A, UCSD Fall 2010
Reconstruction Path
Suetens 2002
Projection
Filtered Projection
Back- Project
*
TT Liu, BE280A, UCSD Fall 2010
Example
Kak and Slaney
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TT Liu, BE280A, UCSD Fall 2010
Example
Prince and Links 2005 TT Liu, BE280A, UCSD Fall 2010
Example
Prince and Links 2005
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