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The FFT on a GPU
Graphics Hardware 2003
July 27, 2003
Kenneth Moreland Edward AngelSandia National Labs 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.
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Overview
• Introduction– Motivation, FFT review.
• FFT Techniques– Exploitable FFT properties.
• Implementation• Results
– Performance, applications, conclusions.
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• 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?
Motivation
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The Discrete Fourier Transform
• Converts data in the spatial or temporal domain into frequencies the data comprise.
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The Discrete Fourier Transform
• 2D transform can be computed by applying the transform in one direction, then the other.
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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.
uFWuFuF uN
oe
xfxf 2e 12o xfxf
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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.
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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.–
– Values at and are real (because they are conjugates with themselves).
uNFuFu *, 0F 2
NF
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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.
f
g
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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.
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Untangling Fourier Transform Pairs
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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.
0 1N 2N 12 N12 N 1N
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 forComplex
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Packing 2D Transforms of Real Functions
• Rows transformed from complex values are already packed appropriately.
• The two rows transformed from real values are untangled and packed to follow suite. 0
0
1M 2M 12 M12 M 1M
1
12 N
2N
12 N
1N
Real Values Imaginary Values
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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
ImaginaryTangled
RealTangled
RealG
RealF
Imag.F
Imag.G
Sca
le
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le
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ntan
gled
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angl
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Imag
., T
angl
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inar
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Scale Scale
R, F
I, F
R, G
I, G
ImaginaryTangled
RealTangled
RealG
RealF
Imag.F
Imag.G
Pas
s
Pas
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gled
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., T
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Pass Pass
R, F
I, F
R, G
I, G
FFT FFTUntangle Untangle
FFT FFTUntangle Untangle
Frequency S
pectraIm
ages
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Fragment Programs
• Written in Cg, compiled for GeForce FX.
Program Instructions
Arithmetic Texture
FFT 27 3
Untangle 4 2
Scale 1 1
Tangle 1 2
Pass 0 1
Multiply 66 4
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Applications
• Digital image filtering.
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Applications
• Texture generation.
• Volume rendering.
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Performance
• Computation speed: 2.5 GigaFLOPS• Texture read rate: 3.4 GB/sec
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
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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
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Questions?
