BMME 560 & BME 590I Medical Imaging: X-ray, CT, and Nuclear Methods Tomography Part 2

Today Tomography –Radon Transform –Fourier Slice Theorem –Direct Fourier Reconstruction –Simple Backprojection

Tomographic Reconstruction Define terms f(x,y) p(t,  ) t t y  x s The object The projection “Object space” or “image space” “Projection space” The “rotated frame”

Tomographic Reconstruction The problem f(x,y) p(t,  ) t t y  x s Given p(t,  ) for 0<  <  Find f(x,y)

The Radon Transform Specifies the 1D projection of the 2D function f(x,y) at any angle  Nonzero only along the line of projection at t

Radon Transform Example Both delta functions have to be nonzero for the integral to be one. This occurs when the following conditions are met: Therefore, the transform is nonzero only when

Radon Transform Example Note that you can make use of the circular symmetry here.

The delta function selects x = t since it is nonzero and integrates to one only if this is true. Now set the limits of integration because the function is one only if t 2 + y 2 < r 2 We could have also reached this by reasoning that the projection for any t would be the length of that chord in the circle. Because of the circular symmetry, all projections will be the same. We can arbitrarily choose  = 0.

Sinograms imagesinogram t 0 22 

Sinograms imagesinogram t 0 22 

Sinograms imagesinogram t 0 22 

Sinogram Here is a sinogram. What does the object look like?

Tomographic Reconstruction f(x,y) p(t,  ) t t y  x s Recall that

Tomographic Reconstruction f(x,y) p(t,  ) t t y  x s It turns out that there is a relationship between the Fourier Transforms of f(x,y) and p(t,  )

Tomographic Reconstruction f(x,y) p(t,  ) t t y  x s Take the FT of p with respect to a frequency variable  t Reorder the integrals

Tomographic Reconstruction f(x,y) p(t,  ) t t y  x s Look at the 2DFT of f(x,y) These are the same if the following is true: The 1DFT of the projection equals the 2DFT of the image at certain spatial frequencies.

Tomographic Reconstruction f(x,y) p(y) y y x Consider if we project along x: The 2D FT of our object: The projection of our object: The 1D FT of the projection of our object:

Tomographic Reconstruction f(x,y) p(t,  ) t t y  x s Therefore This is called the Fourier Slice Theorem or the Projection Slice Theorem

Tomographic Reconstruction Look at this in the Fourier Space u v  This line is described by the parametric equations on  t tt So, each projection angle gives us information from one line in the 2D FT space.

Tomographic Reconstruction Does this suggest to you a method for reconstruction from projections? u v  tt

Tomographic Reconstruction Direct Fourier Reconstruction works, but is generally not done. What is its drawback? u v  tt

Tomographic Reconstruction This does tell us which angles we need for a full reconstruction. What is the rule? u v  tt

Tomographic Reconstruction Also, this proves that the Radon transform is an invertible transform (in the limit). u v  tt

Simple Backprojection Let’s try something else. What if we just cast the projections back across the image field?

Simple Backprojection A mathematical expression for simple backprojection Sum over the set of angles The projection that intersects the given location of the image space The estimate of the true image

Simple Backprojection Replace p by its 1D Fourier transform expression Which gives With t related to x, y, and  as before

Simple Backprojection Rewrite in terms of x and y since t = ? This tells something about the property of our simple backprojection estimate in the frequency space. But what?

Simple Backprojection Any 2D function can be written in terms of its Fourier transform in polar coordinates This looks similar to our expression for the simple backprojection estimate.

Simple Backprojection Compare these two –2D FT of any function –Simple backprojection estimate

Simple Backprojection According to the Fourier slice theorem, BUT,

Simple Backprojection So, what is the simple backprojection estimate? Put it in the form of a 2D FT in polar coordinates

Simple Backprojection An LSI system model for projection followed by simple backprojection: Fourier Transform filter Inverse Fourier Transform Projection Simple backprojection

Simple Backprojection The transfer function of that filter response

Key Point Simple backprojection causes a loss of high- frequency details. –Drops off as the inverse of the spatial frequency

Simple Backprojection Example True image Simple backprojection

Simple Backprojection Example True image Simple backprojection

Simple Backprojection Example True image Simple backprojection