FFTs of Arbitrary Dimensions on GPUs Xiaobai Sun and Nikos - - PowerPoint PPT Presentation

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FFTs of Arbitrary Dimensions

• n GPUs

Xiaobai Sun and Nikos Pitsianis Duke University September 19, 2007 At High Performance Embedded Computing 2007 MIT-LL

• Sept. 19, 2007

FFTs of Arbitrary Dimensions on GPUs

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Overview

• Motivation

– FFTs of arbitrary dimensions and their applications – Graphics processing units (GPUs)

• Basic facts on dimensionality
• FFTs on GPUs
• 1. 2D FFT is chosen as the primitive one at API level
• 2. 2D FFT performance is conveyed to FFTs of other dimensions
• Experimental results
• Discussion of related issues and works

• Sept. 19, 2007

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Remove Keystone Remove Keystone Spatially Variant Refocus Spatially Variant Refocus Motion Comp Motion Comp Auto- Focus Auto- Focus

Motivation : FFT Applications

From S. Bellofiore and H. Schmitt at

Polar Format NUFFT Polar Format NUFFT

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Synthetic Aperture Ground Plane Slant Plane

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Δθ Range Samples Δθ PRFs

2-D FFT

1800 1800 2048 2048

Polar Format

1:1 Range 1.25:1 Azimuth

From S. Bellofiore and H. Schmitt at

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Motivation : GPU Architecture

GPU : Graphics Processing Unit

• Highly parallel multi-processors
• Affordable commodity product
• Initially dedicated to graphics processing and rendering
• Presently capable of co-processing on Desktop, Laptop
• Increasing programmability and API support
• Image processing & rendering
• GP-GPU

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Basic Facts on Dimensionality

• 1. In mathematics, FFTs are considered dimensionless in the sense that

the factorizations can be described in a unified, recursive representation with provided scaling factors some of which are dimension dependent. In computation, trivial scaling may be skipped.

• 2. In application, it is often required that phase-frequency information, or

spatial and geometric relation, be provided explicitly at input and output. FFT data are not shapeless.

• 3. In architecture, extra dimensions are induced by the data access

patterns most efficiently supported . FFT data are transfigured at different memory level. GPUs support fine-granularity, 2D access to memory frames at the API level

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2D FFT as the Primitive

• 2D FFT
• 2D data placement

– Two complex numbers per pixel vector (4 floating point numbers) :

• ne at the front, one at the back

– Even columns at the front layers, odd columns at the back layers

• 2D array operations through

– utilizing best the architectural support

• f 2D data access at API level

– Radix-2, radix 3 and mixed radices

• Direct 2D bit-reversal

– Up to certain sub-array size – 2D data partitioning in large data array Re( X ) Im( X )

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Radix 2, Radix 3 and Mixed Radices

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Direct Two Dimensional Bit Reversal

• Not one dimension after another
• Not recursion up to certain frame

block size (low bits)

• For large data size, block swaps

(bit reversal in high bits)

X( i, j ) X( Rm (i), Rn (j) )

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2D Bit Reversal

2K 2K 30.4 ms 2K 2K 19.6 ms

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2D Bit Reversal

1K 2K 17.5 ms 9.8 ms 14.8 ms

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18 19 20 21 22 23 50 10 15 20 25 30 35 40 2D FF T T im es log2 of data v

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e Time in msec Arithm etic Bit R ev ersa l

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18 19 20 21 22 23 10 20 30 40 50 60 T

• tal 2D FF

T T im es log2 of data v

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e Time in msec Write to GP U Arithm etic Bit R ev ersa l Re ad from G PU

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FFTs of Other Dimensions

• 1D FFT of size n
• 3D FFT of dimensions

Add a scaling stage in 2D FFT Skip a scaling stage in 2D FFT ( In a simple case )

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18 19 20 21 22 23 50 10 15 20 25 30 35 40 45 1D, 2D and 3D FFT s log2 of data v

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e Compute time in msec 1-D FF T 2-D FF T 3-D FF T

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Other Issues and Works

• Twiddle factors :

– Pre-calculated, partially calculated, calculate on the fly – Numerical behavior

• Data loading and unloading

– Data placement in main memory – A sequence of successive FFTs

• Automated tuning
• Other commodity products

– IBM Cell

• FPGAs