The FFT on a GPU

1
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
Energys National Nuclear Security
Administration under contract DE-AC04-94AL85000.
2
Overview

• Introduction
• Motivation, FFT review.
• FFT Techniques
• Exploitable FFT properties.
• Implementation
• Results
• Performance, applications, conclusions.

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?

4
The Discrete Fourier Transform

• Converts data in the spatial or temporal domain
  into frequencies the data comprise.

5
The Discrete Fourier Transform

• 2D transform can be computed by applying the
  transform in one direction, then the other.

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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.

7
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.

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).

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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.

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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

• 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
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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
FFT
FFT
Untangle
Untangle
Frequency Spectra
Images
FFT
FFT
Untangle
Untangle
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Fragment Programs

• Written in Cg, compiled for GeForce FX.

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Applications

• Digital image filtering.

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Applications

• Texture generation.
• Volume rendering.

22
Performance

• 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
• kmorel_at_sandia.gov

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Questions?