1. These slides based almost entirely on a set provided by Prof. Eric Miller Imaging from Projections Eric Miller With minor modifications by Dana Brooks

2. These slides based almost entirely on a set provided by Prof. Eric Miller Outline Problem formulation –What’s a projection? –Application examples –Why is this interesting? The forward problem –The Radon transform –The Fourier Slice Theorem The Inverse Problem –Undoing the Radon transform with the help of Fourier –Filtered Backprojection Algorithm Complications and Extensions

3. These slides based almost entirely on a set provided by Prof. Eric Miller A Projection The total amount of f(x,y) along the line defined by t and  x y t

4. These slides based almost entirely on a set provided by Prof. Eric Miller Application Examples CAT scans: –X ray source moves around the body –f(x,y) is the density of the tissue MRI –Not as clear cut what the “projection” is, but in a peculiar way, the math is the same (remind me to talk about this when we get to the MRI Imaging equation …) –f(x,y) is the spin density of molecules in the tissue Synthetic Aperture Radar –Satellite moves down a linear track collecting radar echoes of the ground –Used for remote sensing, surveillance, … –Again: math is the same (after much pain and anguish) –f(x,y) is the reflectivity of the earth surface

5. These slides based almost entirely on a set provided by Prof. Eric Miller Motivation In all cases, one observes a bunch of sum or integrals of a quantity over a region of space: these are “projections” The goal is to use a collection of these projections to recover f(x,y). Here we will talk about the full data case –Assume we see for all  and t Limited view tomography a topic for advanced course

6. These slides based almost entirely on a set provided by Prof. Eric Miller The Radon Transform x y t Polar equation for line: Function of t and  So the line exists only where this equation is true

7. These slides based almost entirely on a set provided by Prof. Eric Miller What does it do? Simplest case: f(x,y) a  function: only exists at a single point Proof only by limiting argument as products of  ’s not well defined Interpretation: A “function” in (t,  ) space which “is” 1 along a sinusoidal curve and zero elsewhere: note that a point in 2D  a curve Say y 0 = 0 and x 0 = 1 then this is an “image” which “is” 1 when t = cos 

8. These slides based almost entirely on a set provided by Prof. Eric Miller In Pictures x y  t The Image Called the Radon Transform (a.k.a.the sinogram) Kind of 2D impulse response (PSF) Note that we draw as a rectangular “image” in t and 

9. These slides based almost entirely on a set provided by Prof. Eric Miller More Examples t t  

10. These slides based almost entirely on a set provided by Prof. Eric Miller Fourier Slice Theorem Key idea here and for a large number of other problems Analytically relate the 1D Fourier transform of P to the 2D Fourier transform of f. Why? –If we can do this, then a simple inverse 2D Fourier gives us back f from the “data” P.

11. These slides based almost entirely on a set provided by Prof. Eric Miller Recall 2D Fourier Transform Analysis Synthesis “Space” variable x goes with “frequency” variable u “Space” variable y goes with “frequency” variable v (u,v) called “spatial frequency domain”

12. These slides based almost entirely on a set provided by Prof. Eric Miller Fourier – Slice Theorem (FST) Let F(u,v) be defined as on last slide Define S  (w) as the 1D Fourier transform of P along t for some frequency variable w FST says that S  is equal to F(u,v) along a line tilted at an angle  with respect to the (u,v) coordinate system To make this more precise …

13. These slides based almost entirely on a set provided by Prof. Eric Miller Fourier-Slice x y t 1D Fourier Transform u v F(u,v) along line w Variables w and  are the polar form of u and v So FST is:

14. These slides based almost entirely on a set provided by Prof. Eric Miller Reconstruction Implications Collect data from lots and lots of projections. Take 1D FT of each to get one line in 2D frequency space Fill up 2D spatial frequency space on a polar grid Interpolate onto rectangular grid Inverse 2D FT and we are done!! u v

15. These slides based almost entirely on a set provided by Prof. Eric Miller An Alternate Approach Filtered Backprojection The derivation of this algorithm is perhaps one of the most illustrative examples of how we can obtain a radically different computer implementation by simply re-writing the fundamental expressions for the underlying theory - Kak and Slaley, CTI This requires lots of Fourier Transforms This means we can’t begin processing until we have all slices Turns out there’s a more efficient way to organize things This requires “ugly” interpolation, worse at high frequencies

16. These slides based almost entirely on a set provided by Prof. Eric Miller FBP Motivation in Pictures u v By linearity, could in theory break up reconstruction into contribution from independent “wedges” in 2D Fourier space u v In practice, we measure over lines. Idea: build a 2D filter which covers the line, but has the same “weight” as the wedge at that frequency, w In other words “mush” triangle to a rectangle Then “sum up” filtered projections w For K projections, the width of the wedge at w is just w

17. These slides based almost entirely on a set provided by Prof. Eric Miller FBP Theory Now, change right side from polar to rectangular To get rectangular coordinates in space, polar in frequency:

18. These slides based almost entirely on a set provided by Prof. Eric Miller FBP Theory II Make use of two facts: To arrive at Filter (in space) Backproject

19. These slides based almost entirely on a set provided by Prof. Eric Miller FBP Interpretation Recall from linear systems So |w| filter is more or less a differentiator. Accentuated high frequency information leads to problems with noise amplification In practice, roll off response. w w

20. These slides based almost entirely on a set provided by Prof. Eric Miller FBP Interpretation Backprojection: Note that Q  (t) needs only one (filtered) projection Sum up over all angles Think of this as Q  (t) evaluated at the point t = xcos  + y sin  x y t Region we are reconstructing Along this line in “image space” set the value to Q  (t 0 ) All points get a value Do for all angles Add up

21. These slides based almost entirely on a set provided by Prof. Eric Miller FBP Example Orig. Recon Zoom

22. These slides based almost entirely on a set provided by Prof. Eric Miller Limited data I: Angle decimation

23. These slides based almost entirely on a set provided by Prof. Eric Miller Limited data II: Limited Angle

24. These slides based almost entirely on a set provided by Prof. Eric Miller Artifact Mitigation Take a more matrix-based “inverse problems” perspective Discretized Radon transform, data, and object to arrive at a forward model Where C has many fewer rows than columns Use SVD, TSVD, Tikhonov, or other favorite regularization scheme to improve reconstruction results Note: significant move from analytical to numerical inversion means a basic shift in how we are approaching the problem. No more FBP (at least not easily)

25. These slides based almost entirely on a set provided by Prof. Eric Miller Other Fourier Imaging Applications u v u v Standard SAR Collects data on wedge shaped regions of Fourier space Very limited view Similar math to X-ray Diffraction tomography Collects data on petal shaped regions of Fourier space Very limited view More sophisticated math than X ray Arises in geophysical and medical imaging problems

26. These slides based almost entirely on a set provided by Prof. Eric Miller Generalized Radon Transforms Radon transform = integral of object over straight lines Many extensions –Integration over planes in 3D –Over circles in 2D (different type of SAR) –Over much more arbitrary mathematical structures (asymptotic case of some acoustics problems with space varying background). –Of weighted object function (attenuated Radon transform)