polar fft focm
TRANSCRIPT
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Fast Polar Fourier Transform
Michael Elad*Scientific Computing and Computational Mathematics
Stanford University
FoCM Conference, August 2002
Image and Signal Processing Workshop
IMA - Minneapolis
* Joint work with Dave Donoho (Stanford-Statistics), Amir Averbuch (TAU-CS-Israel), and Ronald Coifman (Yale-Math)
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Collaborators
Dave DonohoStatistics Department
Stanford
Amir AverbuchCS Department
Tel-Aviv University
Ronald CoifmanMath. Department
Yale
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Fast Polar Fourier Transform
FFT is one of top 10 algorithms of 20th century.
We'll extend utility of FFT algorithms to new
class of settings in image processing.
Create a tool which is:
Free of emotional involvement, &
Freely available over the internet.
Current Stage:
We have the tool, and its analysis,
Have not demonstrated applications yet.
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Agenda
1. Thinking Polar – Continuum
2. Thinking Polar – Discrete
3. Current State-Of-The-Art
4. Our Approach - General
5. The Pseudo-Polar Fast Transform
6. From Pseudo-Polar to Polar
7. Algorithm Analysis
8. Conclusions
Thinking Polar – Continuum Background&
Motivation
New Approach
and its
Results
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1. Thinking Polar - Continuum
For today f(x,y) function of (x,y)∈ℜ2
Continuous Fourier Transform
( ) ( )( ) ( ) { }∫ ∫ −−=ℑ= dxdyiyvixuexpy,xf y,xf v,uf ̂
( ) ( )
( ) { }∫ ∫ θ⋅−θ⋅−=
=θ⋅θ⋅=θ
dxdy)sin(iy)cos(ixrexpy,xf
)sin(r),cos(rf ̂,rf ~
Polar coordinates: u=r·cos(θ) , v=r·sin(θ)
Important Processes easy to continuum polar domain.
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-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
v
u
0 1 2 3 4 5 6 7
0
1
2
3
4
5
6
r
θ
1. Thinking Polar - Continuum
rθ
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Natural Operations: 1. Rotation
Using the polar coordinates, rotation is simply a shift in
the angular variable.
1. Thinking Polar - Continuum
Qθ0 – planar rotation by θ0 degrees
Rotation
In polar coordinates – shift in angular variable
( ) { }y,xQf y,xf 00 θθ =
( ) ( )0,rf ~,rf ~0
θ−θ=θθ
h k l C
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Natural Operations: 2. Scaling
Using the polar coordinates, 2D scaling is simply a 1D
scaling in the radial variable.
1. Thinking Polar - Continuum
Sα – planar scaling by a factor α
Scaling
In polar coordinates – 1D scale in radial variable
Log-Polar – shift in the radial variable.
( ) { }( )y,xSf y,xf αα =
( ) ( )θα⋅=θα ,rf
~
Const,rf
~
1 Thi ki P l C ti
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Natural Operations: 3. Registration
Using the polar coordinates, rotation+shift registration
simply amounts to correlations.
1. Thinking Polar - Continuum
f(x,y) and g(x,y):
Goal: recover .
Angular cross-correlation between
– Estimate θ0.
Rotation & cross-correlation on regular Fourier transform
gives the shift.
( ) { } { }00 y,xy,xQgy,xf 0
+= θ
{ }000 ,y,x θ
( ) ( )θθ ,rg~and,rf ~
1 Thi ki P l C ti
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Natural Operations: 4. Tomography
Using the polar coordinates, we obtain a method to
obtain the Inverse Radon Transform.
1. Thinking Polar - Continuum
Radon Transform:
Goal: Given Rf(t,θ), recover f.
Projection-Slice-Theorem: .
Reconstruction: .
( ) ( )∫∫ −θ+θδ=θ dxdyt)sin(y)cos(x)y,x(f ,tRf
( )( ) ( )θ=θℑ ,rf ~
,tRf 1
f f ̂f ~
Rf aaa
1 Thinking Polar Continuum
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More Natural Operations
1. Thinking Polar - Continuum
New orthonormal bases:
Ridgelets,
Curvelets,
Fourier Integral operations,
Ridgelet packets.
Analysis of textures.
Analysis of singularities.
More …
r
θ
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Agenda
1. Thinking Polar – Continuum
2. Thinking Polar – Discrete
3. Current State-Of-The-Art
4. Our Approach - General
5. The Pseudo-Polar Fast Transform
6. From Pseudo-Polar to Polar
7. Algorithm Analysis
8. Conclusions
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2. Thinking Polar - Discrete
Certain procedures very important to digitize
Rotation,
Scaling,
Registration,
Tomography, and
…
These look so easy in continuous theory – Can’t we
use it in the discrete domain?
Why not Polar-FFT?
2 Thinking Polar - Discrete
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The FFT Miracles
1D Discrete Fourier Transform
Uniformly sampled in time and frequency – FFT.
Complexity – O(5Nlog2N) instead of O(N2).
2. Thinking Polar - Discrete
2D Discrete Fourier Transform
Cartesian grid in space and frequency – Separability
Only 1D-FFT operations.
Smart memory management.
2 Thinking Polar - Discrete
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2D DFT – Cartesian Grid
2. Thinking Polar Discrete
N
2π
N
2π1
2
N
2
Nn,n2y
1x
21Nn2Nn2 −
−=
π=ωπ=ω
π-π
-π
π
xω
yω
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1D FFT to rows5N2logN
Cartesian2D-FFT 10N2logN
CartesianData
N-by-N
1D FFT to columns
5N2logN
2D FFT Complexity
Complexity: O(10N2log2N)
instead of O(N4
). Important Feature: All
operations are 1D
– leading to
efficient cache usage
2. Thinking Polar - Discrete
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Discrete Polar Coordinates?
2. Thinking Polar Discrete
Choice of grid?
π-π
-π
π
,NS
nr
1NS
0nr
1r
1
−
=
π=
rNS
π
Resulting with NSθ
rays with NSr
elements on each:
For Sθ=Sr=1, we
have N2 grid points. θ
πNS2
1NS
0n
2
2NS
n2
−
=θ
θ
π=θ
ωx
ωy
2. Thinking Polar - Discrete
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Grid Problematics
g
Grid spacing?
Fate of corners?
No X-Y separability !!π-π
-π
π
ωx
ωy
2. Thinking Polar - Discrete
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Polar FFT - Current Status
g
Current widespread belief - There cannot be a
fast method for DFT on the polar grid. See e.g.
The DFT: an owner’s manual, Briggs and
Henson, SIAM, 1995, p. 284.
Consequence of Non-existence: Continuous Fourier – vague inspiration only.
Fourier in implementations widely deprecated.
Fragmentation: each field special algorithm.
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Agenda
1. Thinking Polar – Continuum
2. Thinking Polar – Discrete
3. Current State-Of-The-Art
4. Our Approach - General
5. The Pseudo-Polar Fast Transform
6. From Pseudo-Polar to Polar
7. Algorithm Analysis
8. Conclusions
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3. Current State-Of-The-Art
Assessing T: Unequally-spaced FFT (USFFT)
Data in Cartesian set.
Approximate transform in non-Cartesian set.
Oriented to 1D – not 2D and definitely not Polar.
Assessing T †: For Tomography
Data in Polar coordinates in frequency.
Approximate inverse transform to Cartesian grid.
Inspired by the projection-slice-theorem.
3. Current State-of-the-Art
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USFFT - Rational
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
+ Destination Polargrid
O Critically sampledCartesian grid
o Over-sampledCartesian grid
ωx
ωy
3. Current State-of-the-Art
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USFFT - Detailed
Over-sample Cartesian grid.
Rapidly evaluate FT:
Values F.
Possibly - partial derivatives.
Associate Cartesian neighbors to each
polar grid point.
Approximate interpolation.
3. Current State-of-the-Art
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Our Reading of Literature
Boyd (1992) – Over-sampling and interpolation
by Euler sum or Langrangian interpolation.
Dutt-Rokhlin (1993,1995) - Over-sampling and
interpolation by the Fast-Multipole method.
Anderson-Dahleh (1996) – Over-sampling andobtaining the partial derivatives, and then
interpolating by Taylor series.
Ware (1998) – Survey on USFFT methods.
3. Current State-of-the-Art
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USFFT Problematics
Several involved parameters:
Over-sampling factor,
Method of interpolation, and
Order of interpolation.
Good accuracy calls for extensive over-sampling. Correspondence overhead: spoils vectorizability of
algorithm - causes high cache misses.
Emotionally involved.
d
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Agenda
1. Thinking Polar – Continuum
2. Thinking Polar – Discrete
3. Current State-Of-The-Art
4. Our Approach - General
5. The Pseudo-Polar Fast Transform
6. From Pseudo-Polar to Polar
7. Algorithm Analysis8. Conclusions
4 O A h G l
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4. Our Approach - General
Low complexity – O(Const·N2log2N)
Vectorizability – 1D operations only
Non-Expansiveness – Factor 2 (or 4) only
Stability – via Regularization Accuracy – 2 orders of magnitude over USFFT methods
We propose a
Fast Polar Fourier Transform
with the following features:
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Our Strategy
Fast and ExactFourier Trans.on a polar-like
grid
1Dinterpolationsto the polar
grid
Pseudo
PolarGrid
A d
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Agenda
1. Thinking Polar – Continuum
2. Thinking Polar – Discrete
3. Current State-Of-The-Art
4. Our Approach - General
5. The Pseudo-Polar Fast Transform
6. From Pseudo-Polar to Polar
7. Algorithm Analysis8. Conclusions
5 Th P d P l FFT
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5. The Pseudo-Polar FFT
Developed by Averbuch, Coifman, Donoho,
Israeli, and Waldén (1998).
Basic idea: A “Polar-Like” grid that enables
EXACT Fourier transform,
FAST computation,
1D operations only.
Applications: Tomography, image processing,Ridgelets, and more.
5. The Pseudo-Polar FFT
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The Pseudo-Polar Skeleton
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
ωx
ωy
NSr equi-spacedconcentric squares,
NSt ‘equi-spaced’(not in angle)
We separate our
treatment tobasically verticaland basically
horizontal lines.
5. The Pseudo-Polar FFT
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Fast Fourier Transform
ωx
ωy
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3 The destination samples
are uniformly sampled
vertically, Per each row, destination
samples are uniformly
sampled horizontally,
Fractional Fourier
Transform is the answer
(Chirp-Z), with complexity:O(20Nlog2N).
[Why?]
5. The Pseudo-Polar FFT
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PP-FFT versus 2D-FFT
1D FFT to columns
5N2logN
1D FFT to rows5N2logN
Cartesian2D-FFT 10N2logN
1D FFT to columns
5N2logN
1D FRFFT to rows20N2logN
PP-FFTvertical25N2logN
CartesianData
N-by-N
2D-FFTPP-FFT
5. The Pseudo-Polar FFT
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The PP-FFT - Properties
Exact in exact arithmetic.
No parameters involved !!
Complexity - O(50·N2log2N) versus O(N4).
1D operations only.
For the chosen grid (Sr=St=2) - κ≈5.
Agenda
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Agenda
1. Thinking Polar – Continuum
2. Thinking Polar – Discrete
3. Current State-Of-The-Art
4. Our Approach - General
5. The Pseudo-Polar Fast Transform
6. From Pseudo-Polar to Polar
7. Algorithm Analysis8. Conclusions
6 From Pseudo-Polar to Polar
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6. From Pseudo-Polar to Polar
Fast and ExactFourier Trans.
on a polar-likegrid
2 stages of 1Dinterpolations
to get to thepolar grid
6. From Pseudo-Polar to Polar
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The Interpolation Stages
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
The original Pseudo-Polar Grid
Warping to equi-spacedangles
Warping each ray to
have the same step
ωx
ωy
6. From Pseudo-Polar to Polar
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First Interpolation Stage
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
Every row is a
trigonometric polynomial of
order N,
FRFT on over-sampled
array and 1D interpolation,
Very accurate.
Rotation of the rays tohave equi-spaced angles
(S-Pseudo-Polar grid):
ωx
ωy
6. From Pseudo-Polar to Polar
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The Required Warping
Basically vertical lines:
12
NS
2NSm,
yt
xr
y
t
tNSm2,
NS2
−
−=
ω=ωπ=ω
l
l
12
NS
2NSmt
yx
t
tNS2
mtan
−
−=
π⋅ω=ω
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
π=ωy
Original ωx
New ωx
[Why?]
6. From Pseudo-Polar to Polar
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The Actual Interpolation
1D FFT to columns
5N2logN
1D FRFFT to rows20N
2
logN
PP-FFT Vertical25N2logN
CartesianData
N-by-N
S-PP-FFTPP-FFT
1D FFT to columns
5N2logN
1D Over-sampled (S) FRFFTto rows
20N2S·log(NS)
1D Interpolation
O{N2}
S-PP-FFT Vertical(20S+5)N2logN
6. From Pseudo-Polar to Polar
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Second Interpolation Stage
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
ωx
ωy
As opposed to the previous
step, the rays are not
trigonometric polynomialsof order N,
We proved that the rays
are band-limited (smooth)
functions,
Over-sampling and
interpolation is expected to
perform well.
6. From Pseudo-Polar to Polar
O S li l
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Over-Sampling Along Rays
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3 Over-sampling alongrays cannot be doneby taking the 1D rayand over-sampling it.
Sr>1:
More concentric squares.
Sr longer 1D-FFT’s at thebeginning of the algorithm.
Sr times FRFFT operations.
ωx
ωy
Th A t l I t l ti
6. From Pseudo-Polar to Polar
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The Actual Interpolation
CartesianData
N-by-N
1DFFT to over-sampled columns
N·5(NSr)·log(NSr)
NSr·20(NSt)·log(NSt)
1D Over-sampled (S) FRFFTto rows
1D Interpolation
O{(NSr)·N} O{N·N}
1D Interpolation
Polar-FFT Vertical
Full Polar FFT
O{40SrStN2
logN}
S-PP-FFT Vertical
T S i
6. From Pseudo-Polar to Polar
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Low complexity – O(Const·N2log2N)
Vectorizability – 1D operations only
Non-Expansiveness – Factor 2 (or 4) only
Stability – via Regularization Accuracy – 2 orders of magnitude over USFFT methods
We propose a
Fast Polar Fourier Transform
with the following features:
To Summarize
Agenda
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Agenda
1. Thinking Polar – Continuum
2. Thinking Polar – Discrete
3. Current State-Of-The-Art
4. Our Approach - General
5. The Pseudo-Polar Fast Transform
6. From Pseudo-Polar to Polar
7. Algorithm Analysis8. Conclusions
7. Algorithm Analysis
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7. Algorithm Analysis
We have a code performing the Polar-FFT in Matlab:
Where: X – Input array of N-by-N samples
St,Sr – Over-sampling factors in the approximations
Y – Polar-FFT result as an 2N-by-2N array with rows being
the rays and columns being the concentric circles.
Y=Polar_FFT(X);
OR
Y=Polar_FFT(X,St,Sr);
The Implementation
7. Algorithm Analysis
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The Implementation
The current Polar-FFT code implements Taylor
method for the first interpolation stage and spline
ONLY (no-derivatives) for the second stage.
For comparison, we demonstrate the performanceof the USFFT method with over-sampling S and
interpolation based on the Taylor interpolation
(found to be the best).
Error for Specific Signal
7. Algorithm Analysis
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10 20 30 40 50 60 70 80 90 100
10-3
10-2
10
-1
100
101
102
St·Sr
or S2
||Approximation error||1
Error for Specific Signal
Taylor USFFT
St=4
• Input is random32-by-32 array,
• USFFT methoduses oneparameterwhereas thereare two for theup-sampling inthe Polar-FFT.
• Thumb rule:Sr·St= S2.
St=2
St=3
St=1
Thumb rule: Sr=4St
Error For Specific Signals
7. Algorithm Analysis
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Error For Specific Signals
Similar curves obtained of ||*||∞ and ||*||2 norms.
Similar behavior is found for other signals.
Conclusion: For the proper choice of St and Sr, we
get 2-orders-of-magnitude improvement in the
accuracy comparing to the best USFFT method. Further improvement should be achieved for Taylor
interpolation in the second stage.
US-FFT takes longer due the 2D operations.
The Transform as a Matrix
7. Algorithm Analysis
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The Transform as a Matrix
All the involved
transformations (accurate
and approximate) are
linear - they can berepresented as a matrix of
size 4N2-by-N2.
Y a= AxOr
Y t=Tx
Approximate
True
Regularization of T/A
7. Algorithm Analysis
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Regularization of T/A
An input signal of N-by-N is transformed to anarray or 2N-by-2N.
For N=16, T size is 1024-by-256, and κ≈60,000(bad for inversion).
=
=
0
yxxy
Corner
PolarPolar
T
TT
Adding the assumption that the Frequencycorners should be zeroed, we obtain
and the condition number becomes κ≈5 !!!
Discarding the Corners?
7. Algorithm Analysis
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Discarding the Corners?
π-π
-π
π
2π
2π
2π−
2π−
If the given signal wassampled at 1.4142 the
Nyquist Rate, thecorners can beremoved.
If it is not, over-sampling it can be doneby FFT.
7. Algorithm Analysis
Error Analysis – Worst Signal
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Error Analysis – Worst Signal
( ) FFTPolarFFTPolar ex −− =− T A Approximation error is :
{ } ( )
22
22
x x
xMax / Arge,x
PolarFFTPolar2
worstworst
T A −=
−Worst error :
{ } ( )22
2
2
x xxMax / Arge,x
Polar
PolarFFTPolar2
rworstrworst
TT A −= −Worst relative error :
7. Algorithm Analysis
Worst Signal - Results
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Worst Signal Results
N=16 → T 1024×256, S=Sr=St=4
USFFTworst signal (abs. Value) λ=3.469
The worst casesignal in the freq.Domain (abs. and
shifted)
Polar-FFTworst signal (abs. Value) λ=0.0319
The worst casesignal in the freq.Domain (abs. andshifted)
7. Algorithm Analysis
Relative Worst Signal - Results
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Relative Worst Signal Results
Same parameters: N=16 → T 1024×256, S=Sr=St=4
USFFTworst signal (abs. Value) λ=0.0613
The worst casesignal in the freq.Domain (abs. and
shifted)
Polar-FFTworst signal (abs. Value) λ=0.0023
The worst casesignal in the freq.Domain (abs. andshifted)
7. Algorithm Analysis
Comparing Approximations
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Comparing Approximations
( ) ( )PolarFFTPolarPolarFFTPolar
H
T AT A −−
−−
Solve for the eigenvalue/vectors of the matrix
and obtained ( in ascending order).
Compare to by computing
using the above eigenvectors and compare to .
USFFT A
{ }2N
1k k k x,=
λ k λ
( ) 22k k xPolarUSFFT T A −=α
k λ
7. Algorithm Analysis
Comparing Approximations - Results
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Comparing Approximations Results
0 200 400 600 800 1000 120010
-10
10
-8
10-6
10-4
10-2
100
102
USFFT
Polar-FFT
Eigen-spaceof the Polar-FFT
Mean Squared Error
[N=32]
Agenda
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1. Thinking Polar – Continuum
2. Thinking Polar – Discrete
3. Current State-Of-The-Art
4. Our Approach - General
5. The Pseudo-Polar Fast Transform
6. From Pseudo-Polar to Polar
7. Algorithm Analysis8. Conclusions
8. Conclusions
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We have proposed a fast, accurate, stable, and
reliable Polar Discrete-Fourier-Transform.
By this we extend utility of FFT algorithms to
new class of settings in image processing.
Future plans: Extend the analysis and improve further,
Demonstrate applications,
Publish the code for the procedure and some
applications over the internet.
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Beyond this slides arethe appendix orredundant slides
USFFT for T†
3. Current State-of-the-Art
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Over-sample Polar grid (and possibly
partial derivatives).
Associate polar neighbors to each
Cartesian grid point.
Approximate interpolation to get the
Cartesian grid values.
Perform the Cartesian 2D Inverse-FFT.
Our Reading of Literature
3. Current State-of-the-Art
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g
Natterer (1985).
Jackson, Meyer, Nishimura and Macovski (1991).
Schomberg and Timmer (1995).
Choi and Munson (1998).
Direct Fourier method with over-sampling and
interpolation (also called gridding) – see
The Pseudo-Polar Sampling
A. The Fractional Fourier Transform
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p g
Basically vertical lines:
12
NS
2NSr
y
r
rNS2
−
−=
π=ω
l
l
12NS
2NS
my
tx
t
tNS
m2 −
−=
ω=ω
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
For St=Sr=1, we haveN2 grid points
ωx
ωy
The Pseudo-Polar FT – Stage 1
A. The Fractional Fourier Transform
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g
{ }1 2
1
N 1 N 1
1 y 1 2 2 yk 0 k 0t
f̂ k ,
2mexp ik f k ,k exp ik NS
− −
= =
=
= − ω − ω ∑ ∑
l
1444442444443
This part is obtained by 1D-FFT along the rows !!
( ) { }1 2
N 1 N 1
x y 1 2 1 x 2 yk 0 k 0
F , f k ,k exp ik ik − −
= =
ω ω = − ω − ω = ∑ ∑
1 2
N 1 N 11 2 1 y 2 y
k 0 k 0 t
2mf k ,k exp ik ik NS
− −
= =
= − ω − ω =
∑ ∑
The Pseudo-Polar FT – Stage 2
A. The Fractional Fourier Transform
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( ) [ ]∑−
=
ω
−==ωω1N
0k t
y11yx
1NS
2mik exp,k f ̂],m[F,F ll
The destination grid points are also 1D equi-spaced in
the frequency domain, BUT THEY ARE NOT IN THE
RANGE [-π,π], but rather [-ωy,ωy].
[ ]l,k f ̂ 1 This summation takes columns of (being equi-
spaced 1D signals) and computes Fourier transform of it.
This task is called Fractional Fourier/Chirp-Z Transform.
Fractional Fourier Transform
A. The Fractional Fourier Transform
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[ ]∑−
=
α⋅π
−=1N
0k N
km2iexpk f ]m[F
For α=1 we get the ordinary 1D-FFT,
For α=-1 we get the ordinary 1D-IFFT,
There exists a Fast Fractional Fourier Transform with the
complexity of O(20·Nlog2N), based on 1D-FFT operations.
See: Fast fractional Fourier transforms and applications, by D. H. Bailey and P. N.
Swarztrauber, SIAM Review , 1991, and also Bluestein, Rabiner, and Rader (1960’s).
FR-FFT Detailed
A. The Fractional Fourier Transform
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PostMultiplication
Convolution
Pre-Multiplication
[ ]
[ ] ( )[ ]
[ ] ( )∑
∑
∑
−
=
απαπ
−
=
−
=
α−π−⋅⋅⋅=
=
α⋅−−−π−=
=
α⋅π−=
1N
0k
2Nk iNmi
1N
0k
222
1N
0k
N
mk iexpek f e
Nmk mk iexpk f
Nkm2iexpk f ]m[F
22
[Back]
Interpolation As 1D Operation
B. From Pseudo-Polar to Polar
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{ }1 2
N 1 N 1
1 y 1 2 2 yk 0 k 0t
mexp ik tan f k ,k exp ik 2NS
− −
= =
π = − ω − ω =
∑ ∑
It is a 1D operation – But it is not the Fractional-FFT.Can be computed by over-sampled FRFFT and interpolation.
( ) { }1 2
N 1 N 1
x y 1 2 1 x 2 yk 0 k 0
F , f k ,k exp ik ik − −
= =
ω ω = − ω − ω = ∑ ∑
1
N 1
1 y 1k 0 t
m ˆexp ik tan f k ,2NS
−
=
π = − ω
∑ l