Transcript The FFT on a GPU Graphics Hardware 2003 July 27, 2003 Kenneth Moreland Sandia National Labs Edward Angel U.

The FFT on a GPU
Graphics Hardware 2003
July 27, 2003
Kenneth Moreland
Sandia National Labs
Edward Angel
U. of New Mexico
Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,
for the United States Department of Energy’s National Nuclear Security Administration
under contract DE-AC04-94AL85000.
Overview
• Introduction
– Motivation, FFT review.
• FFT Techniques
– Exploitable FFT properties.
• Implementation
• Results
– Performance, applications, conclusions.
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Motivation
• The Fourier transform is a principal tool for digital
image processing.
– Filtering.
– Correction.
– Compression.
– Classification.
– Generation.
• As such, should not our graphics hardware
support such a tool?
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The Discrete Fourier Transform
• Converts data in the spatial or temporal domain
into frequencies the data comprise.
1 N 1
F  f x   F u    f x WNux
N x 0
N 1
F F u   f x    F u WNux
1
u 0
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WN  e j 2
N
The Discrete Fourier Transform
• 2D transform can be computed by applying the
transform in one direction, then the other.
1 N 1 M 1
ux
vy


F  f x, y   F u, v  
f
x
,
y
W
W

M
N
MN y 0 x 0
N 1 M 1
F F u, v   f x, y    F u, v WMuxWNvy
1
v 0 u 0
DFT
IDFT
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The Fast Fourier Transform
• Divide and Conquer Algorithm
– Input sequence is divided into subsequences
consisting of values from even and odd indices,
respectively.
f e x   f 2 x 
f o x  f 2 x  1
F u   F e u   WNu F o u 
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Index Magic
• Do not use recursion.
– Use dynamic programming: iterate over entire array
computing all values for each recursive depth
together, like mergesort.
• Indexing is non-obvious.
– Unlike mergesort, recursive step does not divide
array into contiguous chunks.
– At any iteration, what partition does a given index
belong to, and where can one find the applicable
values of the sub-partitions?
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Index Magic
• Common solution: rearrange data by reversing
the bits of indices.
– FFT can occur with contiguous partitions.
– Requires an extra data copy.
• Our solution, determine indexing in place.



Ai n  Ai 1 n  u N 2i  W2ui Ai 1 n  u N 2i  N 2i
u  n div N 2i
Note that the paper has a typo.
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
Fourier Symmetry of Real Sequences
• In general, the frequency spectra of even real
functions contain imaginary values.
– Captures magnitude and phase shift of sinusoids.
• Brute force FFT doubles computation and storage
costs.
• But, Fourier transforms of real functions have
symmetry.
*


N  u 

u
,
F
u

F
–
N
– Values at F 0 and F  2  are real (because they are
conjugates with themselves).
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Fourier Transform of Real Functions
• Pick two functions, let them be
f(x) and g(x).
• Let h(x) = f(x) + j g(x).
– Note that there is no loss of
information.
• Can perform FFT of h in half the
time as performing the brute
force FFT of f and g individually.
– Simply point to one row of
image as real components and
another as imaginary
components.
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f
g
Untangling Fourier Transform Pairs
• Fourier transform is linear.
– H(u) = F(u) + j G(u)
• We can “untangle” using symmetry of F and G.
– Add and subtract H(u) and H(N – u) to cancel out
conjugate terms of F and G.
H u   H N  u   F u   F N  u   jGu   GN  u 
 2F u R  2 jGu R
H u   H N  u   2 jF u I  2Gu I
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Untangling Fourier Transform Pairs
F u R 
H u R  H N  u R 
F u I  12 H u I  H  N  u I 
G u R  12 H u I  H  N  u I 
G u I   12 H u R  H  N  u R 
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1
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Packing Transforms of Real Functions
• We can store Fourier
transform in an array
the same size as the
input.
– Throw away
conjugate duplicates.
– Throw away
imaginary values
known to be zero.
13
0
N  1 N2  1 N2 N2  1 N  1
Real Values
Imaginary Values
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Column-wise FFT
• We have two columns
with real values.
– Use same “tangled”
approach.
• All other columns are
complex numbers.
– Use regular FFT.
Real
Real
Paired for
Complex
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Packing 2D Transforms of Real Functions
N 1
• Rows transformed
from complex values
are already packed
appropriately.
• The two rows
transformed from
real values are
untangled and
packed to follow
suite.
N
2

1
N
2
N
2
1

1
0
0 M  1
M
2
Real Values
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1
M
2
M
2
 1 M  1
Imaginary Values
Available Resources
• nVidia GeForce FX 5800 Ultra.
– Full 32-bit floating point pipeline and frame buffers.
– Fully programmable vertex and fragment units.
• Cg
– High level language for vertex and fragment
programs.
• Traditional CPU: 1.7 GHz Intel Zeon
– Freely available high performance FFT
implementations.
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Implementation
• Using a SIMD model for parallel computation.
– Draw quadrilateral parallel to screen.
– Rasterizer invokes the same fragment program “in
parallel” over all pixels covered by quadrilateral.
– Inputs/output dependent on location of pixel the
fragment program is running.
• We require many rendering passes.
– Use “render to texture” extension.
– Use two frame buffers: one for retrieving values of
last pass and one for storing results of current
computation.
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Implementation
Imag.
F
I, G
R, G
I, F
Imaginary
Untangled
R, F
I, G
Real
F
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R, G
Real
Tangled
Scale
Untangle
I, F
Imag.
G
Scale
R, F
Real
G
Pass
Imaginary
Tangled
FFT
Imaginary
Untangled
Untangle
Real
Untangled
Imag., Tangled
Imag.
F
Real
Untangled
Imag., Tangled
Scale
Real
F
Imag.
G
Real, Tangled
FFT
Pass
Images
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Real
Tangled
Real
G
Untangle
Pass
Pass
Frequency Spectra
Imaginary
Tangled
FFT
Real, Tangled
Untangle
Scale
FFT
Fragment Programs
• Written in Cg, compiled for GeForce FX.
Program
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Instructions
Arithmetic
Texture
FFT
27
3
Untangle
4
2
Scale
1
1
Tangle
1
2
Pass
0
1
Multiply
66
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Applications
• Digital image filtering.
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Applications
• Texture generation.
• Volume rendering.
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Performance
Image Size
Rendering
Rate (Hz)
Arithmetic
(sec)
Texture
Lookup (sec)
10242
0.37
1.9
0.6
5122
1.6
0.44
0.13
2562
6.7
0.09
0.03
1282
25
0.01
0.007
• Computation speed: 2.5 GigaFLOPS
• Texture read rate: 3.4 GB/sec
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Conclusions
• The Fourier transform on the GPU has many
potential applications.
• A well established FFT on the CPU (FFTW) still
has an edge over GPU implementation.
– Both software and hardware of GPU are first
generations.
– Room for improvement.
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Get the Cg Code
• http://www.cgshaders.org ?
• http://www.cs.unm.edu/~kmorel/documents/fftgpu
• [email protected]
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Questions?
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