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### 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.

Overview

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

Graphics Hardware 2003

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

The Discrete Fourier Transform

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

Graphics Hardware 2003

IDFT

The Discrete Fourier Transform- 2D transform can be computed by applying the transform in one direction, then the other.

Graphics Hardware 2003

The Fast Fourier Transform

- Divide and Conquer Algorithm
- Input sequence is divided into subsequences consisting of values from even and odd indices, respectively.

Graphics Hardware 2003

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?

Graphics Hardware 2003

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.

Graphics Hardware 2003

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

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

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.

Graphics Hardware 2003

Untangling Fourier Transform Pairs

Graphics Hardware 2003

Imaginary Values

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.

Graphics Hardware 2003

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

Graphics Hardware 2003

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

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.

Graphics Hardware 2003

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

Tangled

Imaginary

Tangled

Real

Tangled

Real

Tangled

Scale

Real

G

Scale

Imag.

G

Pass

Real

G

Pass

Imag.

G

Real

F

Imag.

F

Real

F

Imag.

F

Real, Tangled

Real

Untangled

Imag., Tangled

Imaginary

Untangled

Real, Tangled

Real

Untangled

Imag., Tangled

Imaginary

Untangled

I, F

Scale

I, G

Scale

I, F

Pass

I, G

Pass

R, F

R, G

R, F

R, G

ImplementationFFT

Untangle

FFT

Untangle

Frequency Spectra

Images

FFT

Untangle

FFT

Untangle

Graphics Hardware 2003

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

Get the Cg Code

- http://www.cgshaders.org ?
- http://www.cs.unm.edu/~kmorel/documents/fftgpu
- [email protected]

Graphics Hardware 2003

Questions?

Graphics Hardware 2003

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