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

2. Overview Introduction FFT Techniques Implementation Results
Motivation, FFT review.
FFT Techniques
Exploitable FFT properties.
Implementation
Results
Performance, applications, conclusions.
Graphics Hardware 2003

3. 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?
Graphics Hardware 2003

4. The Discrete Fourier Transform
Converts data in the spatial or temporal domain into frequencies the data comprise.
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5. The Discrete Fourier Transform
2D transform can be computed by applying the transform in one direction, then the other.
DFT
IDFT
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6. The Fast Fourier Transform
Divide and Conquer Algorithm
Input sequence is divided into subsequences consisting of values from even and odd indices, respectively.
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7. Index Magic Do not use recursion. Indexing is non-obvious.
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?
Graphics Hardware 2003

8. 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.
Note that the paper has a typo.
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9. 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).
Graphics Hardware 2003

10. 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
Graphics Hardware 2003

11. 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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12. Untangling Fourier Transform Pairs
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13. 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.
Real Values
Imaginary Values
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14. 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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15. 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.
Real Values
Imaginary Values
Graphics Hardware 2003

16. Available Resources nVidia GeForce FX 5800 Ultra. Cg
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.
Graphics Hardware 2003

17. 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.
Graphics Hardware 2003

18. Implementation Frequency Spectra Images FFT Untangle FFT Untangle FFT
Imaginary
Tangled
Real
Real G F
Imag.
Scale
Untangled
Real
Real, Tangled
Imag., Tangled
Imaginary
Scale
R, F
I, F
R, G
I, G
Images
Frequency Spectra
FFT
Untangle
FFT
Untangle
Imaginary
Tangled
Real
Real G F
Imag.
Pass
Untangled
Real
Real, Tangled
Imag., Tangled
Imaginary
Pass
R, F
I, F
R, G
I, G
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19. Fragment Programs Written in Cg, compiled for GeForce FX. Program
Instructions
Arithmetic
Texture
FFT 27 3
Untangle 4 2
Scale 1
Tangle
Pass
Multiply 66
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20. Applications
Digital image filtering.
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21. Applications Texture generation. Volume rendering.
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22. 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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23. 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.
Graphics Hardware 2003

24. Get the Cg Code http://www.cgshaders.org ?
Graphics Hardware 2003

25. Questions?
Graphics Hardware 2003