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

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

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

Computerized tomography

CT-Basics

CT-Basics

Typical CT Images

CT Scanner

CT-Basics

CT-Basics

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

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.

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).

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

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).

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

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

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

2D Definition - Illustration

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.

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.

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.

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.

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.

The Shearing Operator -Illustration

The Shearing Operator -Illustration

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

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.

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

Illustration of Θ Θ2 Θ1

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

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

The Pseudo-Polar Grid - Illustration

The Pseudo-Polar Grid - Illustration

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.

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.

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

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.

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.