ICASSP 2004 Poster Slides

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ICASSP 2004 Poster Slides

• Adam Zelinski

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Abstract

• This poster presents an automatic approach for
  minimizing the number of additions required for a
  multiplierless implementation of a signal
  transform, under an arbitrary quality constraint.
• Multiplierless implementation means
  multiplications by constants are realized as
  networks of shifts and additions (e.g., y 5x ?
  y xltlt2 x).
• Cost of implementation number of additions
• Higher precision higher output quality but also
  higher cost there is a tradeoff.

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Design Flow Challenges
Choosing a robust algorithm requires
understanding DSP concepts and literature.
An exponentially large number of different
precision configurations exist.
Reducing the precision of a constant
unpredictably impacts output.

• Given a transform
• We automatically select a numerically robust
  algorithm
• We automatically tune the constant precisions

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DSP Transform Algorithms

• We consider the following transforms

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SPIRAL

• Automatically implements, optimizes DSP
  algorithms.
• Searches across many formulas, finding one with
  minimal runtime.
• We use it to generate robust algorithms.

transform
controls
algorithm generation
algorithm
controls
search engine
algorithm compilation
C code
runtime
runtime measurement
platform-adapted implementation
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Robust Algorithm Example DCT-II 8

• Formula generated by SPIRAL

• Data flow diagram

• Rotation-based algorithms are selected for
  robustness.

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Increasing Algorithm Robustness

• Automatically convert Rotations to Lifting Steps
  (LS)

Targets for approximation

• Rounding error in 1st LS (3rd LS analogous)

not magnified

• Rounding error in 2nd LS

e is magnified unless ? in 0, ?/2 or 3?/2,
2?
Solution angle manipulation
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Multiplierless Implementation
Constant multiplies are converted to shifts and
additions.
Arobust
Amultiplierless

• An algorithm constant c is approximated as
  . (n denotes of fractional bits)
• Example

Direct
c 0.10011100100101
6 adds (6 shifts)
Canonical Signed Digit (CSD)
c 0.10100100100101
5 adds (5 shifts)
Addition Chains(our method)
4 adds (5 shifts)
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Converting Constant Multiplies to Shifts and Adds
Addition Chains outperform CSD
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Quality Measures

• Transform quality must be maintained when
  minimizing cost.
• Measures we consider
• Coding Gain
• MP3 decoder compliance rating(Non-Compliant,
  Limited Accuracy, or Fully Compliant)
• Peak Signal to Noise Ratio of JPEG decompressed
  image, D, to original image, O

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Search Space

• A precision list, , is
  associated with an algorithm having n constants.
  
• If max. bitwidth, B, is 19, at most 5 adds are
  needed per constant
• Goal Find a precision list s.t. of additions
  is minimized and quality threshold is met.
• Size of search space 6n32-point DCT-II has 80
  constants, size 680, exhaustive search
  infeasible

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Global Greedy Search

• Global
• Same bitwidth assumed for all constants
• Exhaustive search over all B1 possibilities
• Greedy
• Set each constant to max. precision
• Each constant is reduced in turn to require one
  fewer addition quality is evaluated (n
  evaluations)
• Choose the config. whose quality is highest
• Continue until Qthresh is not satisfied by any
  config.

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Evolutionary Search

• Mimics natural process of evolution
• Random configs. are chosen
• For a set of generations, random members are
  introduced, mutated, and crossbred. Only the
  fittest proceed to the next generation.

Mutation
Randomly change precision.
Crossbreeding
Swap precisions.
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Experiments
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Experimental Results
Number of additions is significantly reduced
transform quality is maintained.
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Experimental Results
Progress of the evolutionary algorithm during E1.
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Applying Global Search to the DCT-II within MP3.
Experimental Results
Quality Measure MP3 Compliance Rating Limited
Accuracy(RMS lt 1.4e-4, MaxDiff lt 8) Achieved
whenglobal bitwidth 9 Fully Compliant(RMS lt
8.8e-6, MaxDiff lt 6.1e-5) Achieved when global
bitwidth 13
Reference input is decoded and compared to
reference output. RMS (Root Mean Squared) error
and MaxDiff error are computed.
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Searching to find a low-cost DCT-II for JPEG.
Experimental Results
Quality Measure PSNR (dB) Thresholds 32, 32.5,
, 34.5 (dB) Global best for lowest
thresholds Greedy, Evol best for larger
thresholds. Greedy, Evol often produce same
result for a given threshold.
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Acknowledgments

• This work was supported by NSF through awards
  0234293, 0310941, and 0325687.

Primary References

• J. Liang and T.D. Tran, Fast Multiplierless
  Approximations of the DCT with the Lifting
  Scheme, IEEE Trans. on Sig. Proc., vol. 49, no.
  12, pp. 30323044, 2001.
• M. Pueschel, et. al., SPIRAL A Generator for
  Platform-Adapted Libraries of Signal Processing
  Algorithms, Journal of High Performance
  Computing and Applications, special issue on
  Automatic Performance Tuning, 18(1), pp. 21-45,
  2004. http//www.spiral.net