1. 2D and 3D Fourier Based Discrete Radon Transform
Amir Averbuch
With
Ronald Coifman – Yale University
Dave Donoho – Stanford University
Moshe Israeli – Technion, Israel
Yoel Shkolnisky – Yale University

2. Research Activities
Polar processing (Radon, MRI, diffraction tomography, polar processing, image processing)
Dimensionality reduction (hyperspectral processing, segmentation and sub-pixel segmentation, remote sensing, performance monitoring, data mining)
Wavelet and frames (error correction, compression)
Scientific computation (prolate spheroidal wave functions)
XML (fast Xpath, handheld devices, compression)
Nano technology (modeling nano batteries, controlled drug release, material science simulations) - interdisciplinary research with material science, medicine, biochemistry, life sciences

3. Participants: previous and current
Dr Yosi Keller – Gibbs Professor, Yale
Yoel Shkolnisky - Gibbs Professor, Yale
Tamir Cohen – submitted his Ph.D
Shachar Harrusi – Ph.D student
Ilya Sedelnikov - Ph.D student
Neta Rabin - Ph.D student
Alon Shekler - Ph.D student
Yossi Zlotnick - Ph.D student
Nezer Zaidenberg - Ph.D student
Zur Izhakian – Ph.D student

4. Computerized tomography

5. CT-Basics

6. CT-Basics

7. Typical CT Images

8. CT Scanner

9. CT-Basics

10. CT-Basics

11. Introduction – CT Scanning
X-Ray from source to detector
Each ray reflects the total absorption along the ray
Produce projection from many angles
Reconstruct original image

12. 2D Continuous Radon Transform
The 2D continuous Radon is defined as
The transform behind the model of the previous slide.
Maps a 2D function to its line integrals.

13. 2D Continuous Fourier Slice Theorem
2D Fourier slice theorem
where
is the 2D continuous Fourier transform of f.
Important property of the 2D continuous Radon transform.
Associates the Radon transform with the Fourier transform.
How to use the 2D Fourier transform to compute the Radon transform
The1D Fourier transform with respect to s of is equal to a central slice, at angle θ, of the 2D Fourier transform of the function f(x,y).

14. Discretization Guidelines
We will look for both 2D and 3D definitions of the discrete Radon transform with the following properties:
Algebraic exactness
Geometric fidelity
Rapid computation algorithm
Invertibility
Parallels with continuum theory

15. 2D Discrete Radon Transform - Definition
Summation along straight lines with θ<45°
Trigonometric interpolation at non-grid points
Traverse the x-direction.
“Resample” the discrete image (using trigonometric interpolation).

16. 2D Discrete Radon Transform – Formal Definition
For a line
we define
where
I1 – Continuous extension along the y-direction.
Illustration on the next page.
and
is the Dirichlet kernel with

17. 2D Definition - Illustration
Explain why we need s<1

18. 2D Discrete Radon Definition – Cont.
For a line
we define
where
Till now we handled only lines with slope less than 45°.
and

19. 2D Definition - Illustration

20. Selection of the Parameter t
Radon({y=sx+t},I)
Sum over all lines with non trivial projections.
Same arguments for basically vertical lines.
We first regard to basically horizontal lines (s<45°).
We can bound t since we bound the slope.
Convenient range that includes all projections.
t is discrete.

21. Selection of the Parameter m
Periodic interpolation kernel.
Points out of the grid are interpolated as points inside the grid.
Summation over broken line.
Wraparound effect.
Show slide 10 to recall the kernel.
Explain why there are points outside the image range
Assume m=n.

22. Selection of the Parameter m – Cont.
Pad the image prior to using trigonometric interpolation.
Equivalent to elongating the kernel.
No wraparound over true samples of I.
Summation over true geometric lines.
Required:
Explain the equivalence between padding the image and using longer kernel.
Explain why the summation is over true geometric lines.

23. The Translation Operator
translates the vector
using trigonometric interpolation.
Example: translation of a vector with
We next sketch the construction of the Fourier slice theorem.

24. The Shearing Operator For the slope θ of a basically horizontal line:
For the slope θ of a basically vertical line:
Motivation: The shearing operator translates the samples along an inclined line into samples along horizontal/vertical line.

25. The Shearing Operator -Illustration

26. The Shearing Operator -Illustration

27. Alternative Definition of the Discrete Radon Transform
After the definition, explain again using slide 20.
The motivation for the alternative definition – computing the adjoint Radon transform and the Fourier slice theorem.
and are padded versions of along the y-axis and the x-axis respectively

28. 2D Discrete Fourier Slice Theorem
Using the alternative discrete Radon definition we prove:
where
The trigonometric polynomial is the Fourier transform of the padded image.

29. Discretization of θ
The discrete Radon transform was defined for a continuous set of angles.
For the discrete set Θ the discrete Radon transform is discrete in both Θ and t.
For the set Θ, the Radon transform is rapidly computable and invertible.
The slopes in Theta are equally spaced in slope and not in angle

30. Illustration of Θ Θ2 Θ1

31. Fourier Slice Theorem Revisited
For
For
where
We define the pseudo-polar Fourier transform:

32. The Pseudo-Polar Grid
The pseudo polar Fourier transform is the sampling of on a special pointset called the pseudo-polar grid.
The pseudo-polar grid is defined by

33. The Pseudo-Polar Grid - Illustration

34. The Pseudo-Polar Grid - Illustration

35. The Fractional Fourier Transform
The fractional Fourier transform is defined as
Can be computed for any
using
operations.
We can use the fractional Fourier transform to compute samples of the Fourier transform at any spacing.

36. Resampling in the Frequency Domain
Given samples in the frequency domain, we define the
resampling operator
Fm – Pad to length m. Apply FRFT with α. Return n+1 central elements.
Gk,n is the key operator for the computation of the PP transform.

37. 2D Discrete Radon Algorithm
Given we can compute using operations by using 1D Fourier transform.
We show an for computing
Explain the first bullet. Reference to slide 25.
Gk,n

38. 2D Discrete Radon Algorithm – Cont.
Description: (PP1I)
Pad both ends of the y-direction of the image I and compute the 2D DFT of the padded image. The results are placed in I’.
Resample each row k in I’ using the operator Gk,n with α = 2k/n.
Flip each row around its center.

39. Papers http://www. math. tau. ac. il/~amir http://pantheon. yale
Optical Snow Analysis using the 3D-Xray Transform, submitted.
Fast and Accurate Polar Fourier Transform, submitted.
Discrete diffraction tomography, submitted.
2D Fourier Based Discrete Radon Transform, submitted.
Algebraically accurate 3-D rigid registration, IEEE Trans. on Signal Proessing.
Algebraically Accurate Volume Registration using Euler's Theorem and the 3-D Pseudo-Polar FFT, submitted.
Fast Slant Stack: A notion of Radon Transform for Data in a
Cartesian Grid which is Rapidly Computible, Algebraically Exact, Geometrically
Faithful and Invertible, SIAM Scientific Computing.
Pseudo-polar based estimation of large translations, rotations and scalings in images, IEEE Trans. on Image Processing.
The Angular Difference Function and its application to Image Registration, IEEE PAMI.
3D Discrete X-Ray Transform, Applied and Computational Harmonic Analysis
3D Fourier Based Discrete Radon Transform, Applied and Computational Harmonic Analysis
Digital Implementation of Ridgelet Packets, Beyond wavelets – chapter in book.
Multidimensional discrete Radon transform, chapter in book.
The pseudopolar FFT and its Applications, Research Report
A signal processing approach to symmetry detection, IEEE Trans. on Image Processing
Fast and accurate pseudo-polar protein docking, submitted.