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Erik Strand
pit
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369a0115
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369a0115
authored
5 years ago
by
Erik Strand
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Start explaining the projection slice theorem
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@@ -7,3 +7,32 @@ title: Final Project
## Background
-
[
Radon Transform
](
http://www-math.mit.edu/~helgason/Radonbook.pdf
)
by Sigurdur Helgason
## 2d Reconstruction
In two dimensions, the theory of image reconstruction from projections is pretty simple. Assume some
density function $$f :
\m
athbb{R}^2
\r
ightarrow
\m
athbb{R}$$ (with compact support). The projection
of this density function to the x axis is
$$
p(x) =
\i
nt_
\m
athbb{R} f(x, y) dy
$$
Meanwhile, the Fourier transform of $$f$$ is
$$
\h
at{f}(
\h
at{x},
\h
at{y})
=
\i
nt_
\m
athbb{R}
\i
nt_
\m
athbb{R} f(x, y) e^{-2
\p
i i (
\h
at{x} x +
\h
at{y} y)} dx dy
$$
Note that the slice along the $$
\h
at{x}$$ axis in frequency space is described by
$$
\b
egin{align
*
}
\h
at{f}(
\h
at{x}, 0)
&=
\i
nt_
\m
athbb{R}
\i
nt_
\m
athbb{R} f(x, y) e^{-2
\p
i i
\h
at{x} x} dx dy
\\
&=
\i
nt_
\m
athbb{R}
\l
eft(
\i
nt_
\m
athbb{R} f(x, y) dy
\r
ight) e^{-2
\p
i i
\h
at{x} x} dx
\\
&=
\i
nt_
\m
athbb{R} p(x) e^{-2
\p
i i
\h
at{x} x} dx
\\
&=
\h
at{p}(
\h
at{x})
\e
nd{align
*
}
$$
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