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
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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
7
CT Scanner
8
X-Ra
y
XRa
y
X-Ray
along
line
y
X-Ra
 f (ρ)dρ
X
R
ay
X-Ra
y
X-Ray
" Projection ":
ay
R
Xy
Ra
X
CT-Basics
ay
R
Xay
R
X
y
X-Ra
X-Ray
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
Rf ( , s )   f ( x, y )du 
L
 
  f ( x, y) ( x cos  y sin   s)dxdy
  
g(s,)
s
y
0
s
f(x,y)

x
2D Continuous Fourier Slice
Theorem
2D Fourier slice theorem
F ( Rf )( ,  )  fˆ ( cos ,  sin  )
where
fˆ (1 ,  2 ) 
 

f ( x, y )e  2i ( x1  y 2 ) dxdy
  
is the 2D continuous Fourier transform of f.
The1D Fourier transform with respect to s of Rf ( , s ) 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
y
L

x
• Summation along
straight lines with
θ<45°
• Trigonometric
interpolation at nongrid points
2D Discrete Radon Transform –
Formal Definition
For a line y  sx  t ( s  1) we define
Radon ({ y  sx  t}, I ) 
n / 2 1
~1
I
 (u, su  t )
u  n / 2
where
~
I 1 (u , y ) 
n / 2 1
 I (u, v) D
v n / 2
sin( t )
and Dm (t ) 
m sin( t / m)
m  2n 1.
m
( y  v)
is the Dirichlet kernel with
2D Definition - Illustration
y
Radon ({ y  sx  t}, I ) 
n / 2 1
~1
I
 (u, su  t )
L
u  n / 2
~
I 1 (u , y ) 
n / 2 1
 I (u, v) D
v n / 2
Dm (t ) 
sin( t )
m sin( t / m)
m
( y  v)
x
m  2n  1
2D Discrete Radon Definition –
Cont.
For a line x  sy  t ( s  1) we define
Radon ({ x  sy  t}, I ) 
n / 2 1
~2
I
 (sv  t , v)
v n / 2
~2
where I ( x, v) 
n / 2 1
 I (u, v) D
u  n / 2
and
Dm (t ) 
sin( t )
m sin( t / m)
m
( x  u)
m  2n  1
2D Definition - Illustration
y
Radon ({ x  sy  t}, I ) 
n / 2 1
~2
I
 (sv  t , v)
L
v n / 2
~
I 2 ( x, v ) 
n / 2 1
 I (u, v) D
u  n / 2
Dm (t ) 
sin( t )
m sin( t / m)
m
( x  u)
x
m  2n  1
Selection of the Parameter t
• Radon({y=sx+t},I)
• Sum over all lines
with non trivial
projections.
y
t=n-1
x
t=-n
•
nt  n
• Same arguments for
basically vertical lines.
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.
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: m  2n  1
The Translation Operator
The translation operator
(T  ) u 
n
 D
i n
i
m
(u  i   )
translates the vector  using trigonometric interpolation.
Example: translation of a vector with   1
 


 

  

 



T
The Shearing Operator
For the slope θ of a basically horizontal line:
( S1 I )(u , v)  (Tu tan I (u,)) v
For the slope θ of a basically vertical line:
( S2 I )(u, v)  (Tv cot I (, v)) u
Motivation: The shearing operator translates the samples along an
inclined line into samples along horizontal/vertical line.
The Shearing Operator Illustration
y
I
y
x
I
x
The Shearing Operator Illustration
y
I
y
x
I
x
Alternative Definition of the
Discrete Radon Transform
Radon ({ y  sx  t}, I ) 
n / 2 1
1 1
(
S
  I )(u, t )
u  n / 2
Radon ({ x  sy  t}, I ) 
n / 2 1
2 2
(
S
  I )(t , v)
v n / 2
I1 and I 2 are padded versions of I along the y-axis and
the x-axis respectively
2D Discrete Fourier Slice
Theorem
Using the alternative discrete Radon definition we prove:
 Iˆ( s1k , k ) s1  tan    [ / 4,  / 4]
F ( R I )( k )  
ˆ
 I (k , s2 k ) s2  cot    [ / 4,3 / 4]
where
Iˆ(1 ,  2 ) 
n / 2 1

n / 2 1
 2i (1u  2 v ) / m
I
(
u
,
v
)
e

u  n / 2 v n / 2
m  2n  1
Discretization of θ
• The discrete Radon transform was defined for a continuous
set of angles.
• For the discrete set Θ
1   / 2  arctan 2l / n | l    n / 2  l  n / 2
2  arctan 2l / n | l    n / 2  l  n / 2
  1  2
the discrete Radon transform is discrete in both Θ and t.
• For the set Θ, the Radon transform is rapidly computable
and invertible.
Illustration of Θ
y
y
x
x
Θ2
Θ1
Fourier Slice Theorem Revisited
For   arctan( 2l / n),   2
F ( R I )(k )  Iˆ(2lk / n, k )  PPI1 (k , l )
For    / 2  arctan( 2l / n),   1
F ( R I )(k )  Iˆ(k ,2lk / n)  PPI2 (k , l )
where
n / 2 1 n / 2 1
Iˆ( ,  )    I (u, v)e  2i (1u  2v ) / m .
1
2
u  n / 2 v n / 2
We define the pseudo-polar Fourier transform:
PPI1 (k , l )  Iˆ(2lk / n, k )
PPI (k , l )  Iˆ(k ,2lk / n)
2
The Pseudo-Polar Grid
PPI1 (k , l )  Iˆ(2lk / n, k )
PPI (k , l )  Iˆ(k ,2lk / n)
2
The pseudo polar Fourier transform is the sampling of Iˆ on a
special pointset called the pseudo-polar grid.
The pseudo-polar grid is defined by
P1    2lk / n, k  |  n / 2  l  n / 2,  n  k  n
P2   k ,2lk / n |  n / 2  l  n / 2,  n  k  n
P  P1  P2
The Pseudo-Polar Grid Illustration
y
y
x
x
P2
P1
The Pseudo-Polar Grid Illustration
y
x
The Fractional Fourier Transform
The fractional Fourier transform is defined as

( Fn 1 X )( ) 
n/2
 2iu /( n 1)
X
(
u
)
e

u  n / 2
  ,  n / 2    n / 2.
Can be computed for any  using O(n log n) operations.
We can use the fractional Fourier transform to compute
samples of the Fourier transform at any spacing.
Resampling in the Frequency
Domain
Given n samples in the frequency domain, we define the
resampling operator
Gk ,n  Fm  Fn1
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
Gk,n
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1
-1
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
2D Discrete Radon Algorithm
• Given PPI (i  1,2) we can compute RI using O(n 2 log n)
operations by using 1D Fourier transform.
• We show an O(n 2 log n) for computing PPI (i  1,2) .


y

y

x
Gk,n
x
2D Discrete Radon Algorithm –
Cont.
Description: (PP1I)
1. 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’.
2. Resample each row k in I’ using the operator Gk,n
with α = 2k/n.
3. Flip each row around its center.
Papers
http://www.math.tau.ac.il/~amir
http://pantheon.yale.edu/~yk253/
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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.