ECE 468: Digital Image Processing
Lecture 15
Prof. Sinisa Todorovic
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Outline
• Image reconstruction from projections (Textbook 5.11)
• Radon Transform (Textbook 5.11.3)
• Fourier-Slice Theorem (Textbook 5.11.4)
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Computed Tomography
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Radon Transform
A point in the projection
is the ray-sum along
x cos �k + y sin �k = ⇥j
g(⇥j , �k)
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continuous space coordinates
Radon Transform
g(⇤, ⇥) =� ⇥
�⇥
� ⇥
�⇥f(x, y)�(x cos ⇥ + y sin ⇥ � ⇤)dxdy
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continuous space coordinates
Radon Transform
g(⇤, ⇥) =� ⇥
�⇥
� ⇥
�⇥f(x, y)�(x cos ⇥ + y sin ⇥ � ⇤)dxdy
discrete space coordinates
g(⇤, ⇥) =M�1�
x=0
N�1�
y=0
f(x, y)�(x cos ⇥ + y sin ⇥ � ⇤)
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Properties of the Radon Transform
g(⇢, ✓ + 180
�) =
Z 1
�1
Z 1
�1f(x, y)�(x cos(✓ + 180
�) + y sin(✓ + 180
�)� ⇢) dx dy
=
Z 1
�1
Z 1
�1f(x, y)�(�x cos ✓ � y sin ✓ � ⇢) dx dy
= g(�⇢, ✓)
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Properties of the Radon Transform
g(⇢, ✓ + 180
�) =
Z 1
�1
Z 1
�1f(x, y)�(x cos(✓ + 180
�) + y sin(✓ + 180
�)� ⇢) dx dy
=
Z 1
�1
Z 1
�1f(x, y)�(�x cos ✓ � y sin ✓ � ⇢) dx dy
= g(�⇢, ✓)
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Properties of the Radon Transform
g(⇢, ✓ + 180
�) =
Z 1
�1
Z 1
�1f(x, y)�(x cos(✓ + 180
�) + y sin(✓ + 180
�)� ⇢) dx dy
=
Z 1
�1
Z 1
�1f(x, y)�(�x cos ✓ � y sin ✓ � ⇢) dx dy
= g(�⇢, ✓)
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Properties of Objects from Sinogram
• Sinogram symmetric = Object symmetric
• Sinogram symmetric about image center = Object symmetric and parallel to x and y axes
• Sinogram smooth = Object has uniform intensity
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Fourier Slice Theorem
relates
1D Fourier Transform of the projection
with
2D Fourier Transform of the original image
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1D FT of the Projection -- Properties
G(!, ✓ + 180�) =?
G(!, ✓ + 180�) =Z 1
�1g(⇢, ✓ + 180�)e�j2⇡!⇢d⇢
=Z 1
�1g(�⇢, ✓)e�j2⇡!⇢d⇢
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1D FT of the Projection -- Properties
G(!, ✓ + 180�) =?
= �Z �1
1g(⇢, ✓)ej2⇡!⇢d⇢
=Z 1
�1g(⇢, ✓)e�j2⇡(�!)⇢d⇢
= G(�!, ✓)
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Fourier Slice Theorem
G(⇤, �) =� ⇥
�⇥g(⇥, �)e�j2�⇤⇥d⇥
by definition
G(⌅, ⇥) =� ⇥
�⇥
� ⇥
�⇥
� ⇥
�⇥f(x, y)�(x cos ⇥ + y sin ⇥ � ⇤)e�j2�⇤⇥dx dy d⇤
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Fourier Slice Theorem
G(⇤, �) =� ⇥
�⇥g(⇥, �)e�j2�⇤⇥d⇥
by definition
=� ⇥
�⇥
� ⇥
�⇥f(x, y)e�j2⇥⇤(x cos �+y sin �)dx dy
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Fourier Slice Theorem
G(⇤, �) =� ⇥
�⇥g(⇥, �)e�j2�⇤⇥d⇥
by definition
=� ⇥
�⇥
� ⇥
�⇥f(x, y)e�j2⇥⇤(x cos �+y sin �)dx dy
= F (⇥ cos �,⇥ sin �)24
Fourier Slice Theorem relates
1D Fourier Transform of the projection
with
2D Fourier Transform of the original image
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Reconstruction Using Backprojections
by definition
f(x, y) =� ⇥
�⇥
� ⇥
�⇥F (u, v)ej2�(ux+vy)du dv
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Reconstruction Using Backprojections
by definition
f(x, y) =� ⇥
�⇥
� ⇥
�⇥F (u, v)ej2�(ux+vy)du dv
u = ⇥ cos �, v = ⇥ sin �, � dudv = ⇥d⇥d�
f(x, y) =� 2⇥
0
� �
0F (⇥ cos �, ⇥ sin �)ej2⇥⇤(x cos �+y sin �)⇥ d⇥ d�
polar coordinates in the frequency domain
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Reconstruction Using Backprojections
f(x, y) =� 2⇥
0
� �
0F (⇥ cos �, ⇥ sin �)ej2⇥⇤(x cos �+y sin �)⇥ d⇥ d�
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Reconstruction Using Backprojections
f(x, y) =� 2⇥
0
� �
0F (⇥ cos �, ⇥ sin �)ej2⇥⇤(x cos �+y sin �)⇥ d⇥ d�
f(x, y) =� 2⇥
0
� �
0G(⇥, �)ej2⇥⇤(x cos �+y sin �)⇥ d⇥ d�
by Fourier Slice Theorem
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Reconstruction Using Backprojections
f(x, y) =� 2⇥
0
� �
0G(⇥, �)ej2⇥⇤(x cos �+y sin �)⇥ d⇥ d�
f(x, y) =� ⇥
0
� �
0|⇥|G(⇥, �)ej2⇥⇤(x cos �+y sin �) d⇥ d�
G(⇥, � + 180�) = G(�⇥, �)
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Reconstruction Using Backprojections
f(x, y) =⇤ ⇥
0
�⇤ �
0|⇥|G(⇥, �)ej2⇥⌅⇤ d⇥
⇥
⇤=x cos �+y sin �
d�
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Reconstruction Using Backprojections
f(x, y) =⇤ ⇥
0
�⇤ �
0|⇥|G(⇥, �)ej2⇥⌅⇤ d⇥
⇥
⇤=x cos �+y sin �
d�
1D filtering
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Algorithm for Filtered Backprojection
1. Given projections g(ρ,θ) obtained at each fixed angle θ
2. Compute G(ω,θ) = 1D Fourier Transform of each projection g(ρ,θ)
3. Multiply G(ω,θ) by the filter function |ω| modified by Hamming window
4. Compute the inverse of the results from 3.
5. Integrate (sum) over θ all results from 4.
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