The FFT on a GPU Graphics Hardware 2003 July 27, 2003 Kenneth MorelandEdward Angel Sandia National LabsU. of New Mexico Sandia is a multiprogram laboratory.

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.

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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).

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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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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

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Scale Scale

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Pass Pass

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I, F

R, G

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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

• [email protected]

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Questions?