ECE 468: Digital Image Processing
Lecture 8
Prof. Sinisa Todorovic
1
Image Reconstruction from Projections
X-ray computed tomography:
X-raying an object from different directions 3D object representation
2
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Image Reconstruction from Projections – Digital Image Processing - Slides | ECE 468, Study notes of Electrical and Electronics Engineering
Material Type: Notes; Professor: Todorovic; Class: DIGITAL IMAGE PROCESSING; Subject: Electrical & Computer Engineer; University: Oregon State University; Term: Unknown 1989;
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ECE 468: Digital Image Processing
Lecture 8
Prof. Sinisa Todorovic
1
Image Reconstruction from Projections
X-ray computed tomography:
X-raying an object from different directions 3D object representation
Example: Backprojecting a 1D signal
3
...As We Increase the Number of Backprojections
halo effect
Radon Transform
A point in the projection
is the ray-sum along
x cos θ
k
- y sin θ
k
= ρ
j
g(ρ
j
, θ
k
7
continuous space coordinates
key tool for reconstruction from projections
Radon Transform
discrete space coordinates
g(ρ, θ) =
∞
−∞
∞
−∞
f (x, y)δ(x cos θ + y sin θ − ρ)dxdy
g(ρ, θ) =
M − 1
x=
N − 1
y=
f (x, y)δ(x cos θ + y sin θ − ρ)
Example: Radon Transform
f (x, y) =
A , x
2
- y
2
≤ r
2
0 , o.w
9
Sinogram = Image of Radon Transform
Given point
Backprojection = copy the value of on the entire line
Backprojection from the Radon Transform
g(ρ
j
, θ
k
g(ρ
j
, θ
k
∀ρ ⇒ f
θ
k
(x, y) = g(x cos θ
k
- y sin θ
k
, θ
k
⇒ f (x, y) =
π
0
f
θ
(x, y)dθ
13
Given point
Backprojection = copy the value of on the entire line
Backprojection from the Radon Transform
g(ρ
j
, θ
k
g(ρ
j
, θ
k
∀ρ ⇒ f
θ
k
(x, y) = g(x cos θ
k
- y sin θ
k
, θ
k
⇒ f (x, y) =
π
0
f
θ
(x, y)dθ
Laminogram Obtained from Sinogram
Backprojection for a specific angle
Summation over all theta
f (x, y) =
π
θ=
f
θ
(x, y)
f
θ
k
(x, y) = g(x cos θ
k
- y sin θ
k
, θ
k
14
Laminogram Obtained from Sinogram
Backprojection for a specific angle
Summation over all theta
f (x, y) =
π
θ=
f
θ
(x, y)
f
θ
k
(x, y) = g(x cos θ
k
- y sin θ
k
, θ
k
1D Fourier Transform of the Projection
G(ω, θ) =
∫
∞
−∞
g(ρ, θ)e
−j 2 πωρ
dρ
17
1D Fourier Transform of the Projection
G(ω, θ) =
∫
∞
−∞
g(ρ, θ)e
−j 2 πωρ
dρ
by definition
G(ω, θ) =
∫
∞
−∞
∫
∞
−∞
∫
∞
−∞
f (x, y)δ(x cos θ + y sin θ − ρ)e
−j 2 πωρ
dx dy dρ
1D Fourier Transform of the Projection
G(ω, θ) =
∫
∞
−∞
g(ρ, θ)e
−j 2 πωρ
dρ
by definition
G(ω, θ) =
∫
∞
−∞
∫
∞
−∞
∫
∞
−∞
f (x, y)δ(x cos θ + y sin θ − ρ)e
−j 2 πωρ
dx dy dρ
=
∫
∞
−∞
∫
∞
−∞
f (x, y)e
−j 2 πω(x cos θ+y sin θ)
dx dy
17
1D Fourier Transform of the Projection
G(ω, θ) =
∫
∞
−∞
g(ρ, θ)e
−j 2 πωρ
dρ
by definition
G(ω, θ) =
∫
∞
−∞
∫
∞
−∞
∫
∞
−∞
f (x, y)δ(x cos θ + y sin θ − ρ)e
−j 2 πωρ
dx dy dρ
=
∫
∞
−∞
∫
∞
−∞
f (x, y)e
−j 2 πω(x cos θ+y sin θ)
dx dy
= F (ω cos θ, ω sin θ)
Reconstruction Using Filtered Backprojections
by definition
f (x, y) =
∫
∞
−∞
∫
∞
−∞
F (u, v)e
j 2 π(ux+vy)
du dv
19
Reconstruction Using Filtered Backprojections
by definition
f (x, y) =
∫
∞
−∞
∫
∞
−∞
F (u, v)e
j 2 π(ux+vy)
du dv
u = ω cos θ, v = ω sin θ, ⇒ dudv = ωdωdθ
f (x, y) =
∫
2 π
0
∫
∞
0
F (ω cos θ, ω sin θ)e
j 2 πω(x cos θ+y sin θ)
ω dω dθ
Reconstruction Using Filtered Backprojections
by definition
f (x, y) =
∫
∞
−∞
∫
∞
−∞
F (u, v)e
j 2 π(ux+vy)
du dv
u = ω cos θ, v = ω sin θ, ⇒ dudv = ωdωdθ
f (x, y) =
∫
2 π
0
∫
∞
0
F (ω cos θ, ω sin θ)e
j 2 πω(x cos θ+y sin θ)
ω dω dθ
f (x, y) =
∫
2 π
0
∫
∞
0
G(ω, θ)e
j 2 πω(x cos θ+y sin θ)
ω dω dθ
by Fourier Slice Theorem
19
Reconstruction Using Filtered Backprojections
G(ω, θ + 180
◦
) = G(−ω, θ)
Reconstruction Using Filtered Backprojections
f (x, y) =
π
0
∞
0
|ω|G(ω, θ)e
j 2 πω(x cos θ+y sin θ)
dω dθ
f (x, y) =
π
0
[
∞
0
|ω|G(ω, θ)e
j 2 πωρ
dω
]
ρ=x cos θ+y sin θ
dθ
G(ω, θ + 180
◦
) = G(−ω, θ)
1D filtering
20
Box + Ramp Filter
Algorithm for Filtered Backprojection
- Given projections g(!,") obtained at each fixed angle "
- Compute G(#,") = 1D Fourier Transform of each projection g(!,")
- Multiply G(#,") by the filter function |#| modified by Hamming window
- Compute the inverse of the results from 3.
- Integrate (sum) over " all results from 4.
22
Examples
ramp
filter
windowed
ramp filter
ramp
filter
windowed
ramp filter
zoom
naive backprojection