1. 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 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. Graphics Hardware 20032 Overview Introduction –Motivation, FFT review. FFT Techniques –Exploitable FFT properties. Implementation Results –Performance, applications, conclusions.

3. Graphics Hardware 20033 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

4. Graphics Hardware 20034 The Discrete Fourier Transform Converts data in the spatial or temporal domain into frequencies the data comprise.

5. Graphics Hardware 20035 The Discrete Fourier Transform 2D transform can be computed by applying the transform in one direction, then the other. DFT IDFT

6. Graphics Hardware 20036 The Fast Fourier Transform Divide and Conquer Algorithm –Input sequence is divided into subsequences consisting of values from even and odd indices, respectively.

7. Graphics Hardware 20037 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?

8. Graphics Hardware 20038 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.

9. Graphics Hardware 20039 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).

10. Graphics Hardware 200310 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

11. Graphics Hardware 200311 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.

12. Graphics Hardware 200312 Untangling Fourier Transform Pairs

13. Graphics Hardware 200313 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 ValuesImaginary Values

14. Graphics Hardware 200314 Column-wise FFT We have two columns with real values. –Use same “tangled” approach. All other columns are complex numbers. –Use regular FFT. Real Paired for Complex

15. Graphics Hardware 200315 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 ValuesImaginary Values

16. Graphics Hardware 200316 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.

17. Graphics Hardware 200317 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.

18. Graphics Hardware 200318 Implementation Imaginary Tangled Real Tangled Real G Real F Imag. F Imag. G Scale RealUntangled Real, TangledImag., Tangled ImaginaryUntangled Scale R, F I, F R, G I, G Imaginary Tangled Real Tangled Real G Real F Imag. F Imag. G Pass RealUntangled Real, TangledImag., Tangled ImaginaryUntangled Pass R, F I, F R, G I, G FFT Untangle FFT Untangle Frequency Spectra Images

19. Graphics Hardware 200319 Fragment Programs Written in Cg, compiled for GeForce FX. ProgramInstructions ArithmeticTexture FFT273 Untangle42 Scale11 Tangle12 Pass01 Multiply664

20. Graphics Hardware 200320 Applications Digital image filtering.

21. Graphics Hardware 200321 Applications Texture generation. Volume rendering.

22. Graphics Hardware 200322 Performance Computation speed: 2.5 GigaFLOPS Texture read rate: 3.4 GB/sec Image SizeRendering Rate (Hz) Arithmetic (sec) Texture Lookup (sec) 1024 2 0.371.90.6 512 2 1.60.440.13 256 2 6.70.090.03 128 2 250.010.007

23. Graphics Hardware 200323 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.

24. Graphics Hardware 200324 Get the Cg Code http://www.cgshaders.org ?http://www.cgshaders.org http://www.cs.unm.edu/~kmorel/documents/fftgpu kmorel@sandia.gov

25. Graphics Hardware 200325 Questions?