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  • ---
    title: Final Project
    ---
    
    # CT Imaging from Scratch
    
    
    ## 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 $$f : \mathbb{R}^2 \rightarrow \mathbb{R}$$ (with compact support). The projection
    of this density function to the x axis is
    
    $$
    p(x) = \int_\mathbb{R} f(x, y) dy
    $$
    
    Meanwhile, the Fourier transform of $$f$$ is
    
    $$
    
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    \hat{f}(u, v)
    = \int_\mathbb{R} \int_\mathbb{R} f(x, y) e^{-2 \pi i (u x + v y)} dx dy
    
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    Note that the slice along the $$u$$ axis in frequency space is described by
    
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    \hat{f}(u, 0)
    &= \int_\mathbb{R} \int_\mathbb{R} f(x, y) e^{-2 \pi i u x} dx dy \\
    &= \int_\mathbb{R} \left( \int_\mathbb{R} f(x, y) dy \right) e^{-2 \pi i u x} dx \\
    &= \int_\mathbb{R} p(x) e^{-2 \pi i u x} dx \\
    &= \hat{p}(u)