2D and 3D Fourier Based Discrete Radon Transform

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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

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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

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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

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Computerized tomography
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CT-Basics
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CT-Basics
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Typical CT Images
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CT Scanner
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CT-Basics
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CT-Basics
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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

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2D Continuous Radon Transform
The 2D continuous Radon is defined as
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2D Continuous Fourier Slice Theorem
2D Fourier slice theorem
where
is the 2D continuous Fourier transform of f.
The1D Fourier transform with respect to s of
is equal to a central slice, at angle
?, of the 2D Fourier transform of the function
f(x,y).
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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

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2D Discrete Radon Transform - Definition

• Summation along straight lines with ?lt45
• Trigonometric interpolation at non-grid points

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2D Discrete Radon Transform Formal Definition
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2D Definition - Illustration
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2D Discrete Radon Definition Cont.
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2D Definition - Illustration
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Selection of the Parameter t

• Radon(ysxt,I)
• Sum over all lines with non trivial projections.
• Same arguments for basically vertical lines.

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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.

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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

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The Translation Operator
The translation operator
Example translation of a vector with
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The Shearing Operator
For the slope ? of a basically horizontal line
For the slope ? of a basically vertical line
Motivation The shearing operator translates the
samples along an inclined line into samples along
horizontal/vertical line.
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The Shearing Operator -Illustration
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The Shearing Operator -Illustration
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Alternative Definition of the Discrete Radon
Transform
and are padded versions of along the
y-axis and the x-axis respectively
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2D Discrete Fourier Slice Theorem
Using the alternative discrete Radon definition
we prove
where
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Discretization of ?

• The discrete Radon transform was defined for a
  continuous set of angles.
• For the discrete set Tthe discrete Radon
  transform is discrete in both T and t.
• For the set T, the Radon transform is rapidly
  computable and invertible.

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Illustration of T
T2
T1
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Fourier Slice Theorem Revisited
where
We define the pseudo-polar Fourier transform
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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
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The Pseudo-Polar Grid - Illustration
P2
P1
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The Pseudo-Polar Grid - Illustration
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The Fractional Fourier Transform
The fractional Fourier transform is defined as
We can use the fractional Fourier transform to
compute samples of the Fourier transform at any
spacing.
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Resampling in the Frequency Domain
Given samples in the frequency domain, we
define the resampling operator
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2D Discrete Radon Algorithm
Gk,n
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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 a 2k/n.
• Flip each row around its center.

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Papershttp//www.math.tau.ac.il/amirhttp//pant
heon.yale.edu/yk253/

• 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.