Sub-image Searching Using Genetically-Evolved Wavelet Transforms

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Sub-image Searching Using Genetically-Evolved
Wavelet Transforms

• By Chris Wedge

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Evolved Transform Coefficients

• Wavelets capable of lossless compression in a
  perfect conditions
• Dr. Frank Moore showed superior coefficients
  could be evolved in imperfect conditions
• Very time-intensive

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Evolution on Sub-images

• Don Tinsley and Jason Kettell explored evolution
  with sub-images
• Similar performance gains but with computation
  time drastically reduced
• Noted performance disparity between sub-image and
  super-image
• Can disparity be exploited?

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Wheres Waldo?
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Search Algorithms

• Four algorithms used, though basically all the
  same iteratively apply transform
• Strict repeated transform application
• Interesting side-effects with quantization
• Repeated transform application, but quantize only
  once
• Repeated transform application, only on Y
• Again, side-effects present
• Repeated transform application, only on Y, but
  quantize only once
• Search focus placed on developing
  high-performance transforms

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Performance Evaluation

• Two methods of evaluation
• Quantitative
• Compare mean-squared error (MSE) values between
  sub-image and super-image
• Anecdotal
• MSE comparisons may indicate improvement, but
  searching meant for human consumption

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Meet The Crew
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Simple Evolution

• Evolve versus non-representative sub-image
• Repeatedly apply resulting transforms using
  search algorithms

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Simple Evolution - Parameters

• 24 total runs
• Fixed parameters
• Daubechies-4 (D4) wavelet used as basis wavelet
• Population (M) 5000
• Generations (G) 2500
• Multi-resolution (MR) 1
• Sub-image size 32x32 pixels (regular is 512x512)
• Variable parameters
• 4 images
• 3 quantization (Q) levels 0, 32, 64
• 2 threshold (T) levels 0, 16

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Simple Evolution - Results

• 24 runs
• 1 run each per Q, T, image combination
• Mean runtime 63min 57sec, St Dev 0.00144
• MSE reductions over D4 on sub-image
• Q64 3.5, 15.0, 8.8, 16.2
• Q32 4.8, 11.0, 5.2, 22.3
• Q0
• T16 7.9, 6.7, 1.8, 12.4
• T0 25.6, 30.3, 19.3, 49.4
• MSE reductions over D4 on super-image
• Q64 7.9, -1.3, 6.8, 83
• Q32 3.9, 6.7, 4.6, 84.1
• Q0
• T16 7.6, 1.9, 1.4, 95.6
• T0 24.9, 44.8, 18.9, 99.6
• (lenna, goldhill, monet, and dissimilar,
  respectively)

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Simple Evolution - Results

• ?

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Simple Evolution - Conclusions

• MSE reduction of sub-image over parent image
  seems somewhat arbitrary
• More runs needed to get a better picture, but
  want a general method which always works, so
  little point exploring that
• T has no effect if it is set to a value less than
  Q
• Not surprising. Should have been obvious before
  running the tests
• MSE reduction at Q 0 is by far the highest
• Very surprising! Wavelets theoretically capable
  of lossless compression. Attributed to
  imprecision of floating-point arithmetic
• Dissimilar, the toy image, had some impressive
  results! Unfortunately, they are far from the
  desired results, and not exactly realistic

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Detour!

• Non-representative sub-image did not perform as
  well as expected
• Representative sub-image performance
• Revisiting Tinsleys, Kettells work to
  concretize
• More runs
• More fixed parameters (size, etc)
• But what is representative?
• No determination algorithm, criteria mentioned,
  largely subjective
• Ended up using miniature versions of the image

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Detour! - Parameters

• 63 total runs
• Fixed parameters
• D4 wavelet used as basis wavelet
• M 5000
• G 2500
• MR 1
• T 0
• Mini-image size 32x32 pixels (regular is 512x512)
• Variable parameters
• 4 images
• 3 Q levels 0, 32, 64

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Detour! - Results

• Evolution versus single image
• 60 runs
• 5 per combination of Q level, image
• Mean runtime 63min 51sec, St Dev 0.00065
• Mean MSE reductions over D4
• Q64 5.5 - 8.3
• Q32 2.4 - 2.7
• Q0 16.7 - 25.1

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Detour! - Results

• Evolution versus all images
• 3 runs
• 1 for each Q level
• Wanted 5 each, but bugs, time constraints got in
  the way
• Mean runtime 250min 57sec, St Dev 0.00212
• MSE reduction over D4
• Q64 6.4 - 9.5
• Q32 3.9 - 5.2
• Q0 17.3 - 27.2

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Detour! - Results

• 4.5 improvement

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Detour! - Conclusions

• MSE improved over D4 in every run
• Excluding Q0, higher Q values may increase
  improvement, but need more Q levels tested
• Improvement even noticed when transform applied
  to different images
• Single-image mean runtime approximately 98 lower
  than in Dr. Moores runs
• Four-image evolution outperformed single-image
  evolution, but more runs needed
• Four-image mean runtime approximately 91 lower
  than in Dr. Moores runs

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Back to searching

• Simple evolution failed to do the trick
• Want very good performance on desired sub-image
• Also want very poor performance on the whole
• But how to do both simultaneously?

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Co-evolution

• Alter the GA to use a weighted fitness function
• Total fitness sub-image goodness
  super-image badness.
• Can weight the importance of each aspect to
  drive evolution

SUBIMAGE_WEIGHT 50 SUPERIMAGE_WEIGHT
100 fitnessM (subMSE / minSubImageMSE)
SUBIMAGE_WEIGHT (maxParentMSE / parentMSE)
SUPERIMAGE_WEIGHT
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Co-evolution - Parameters

• 25 total runs so far
• Fixed parameters
• D4 basis wavelet
• M 500
• G 200
• MR 1
• T 0
• Sub-image size 32x32 pixels (super is 512x512)
• Variable parameters
• 3 images
• 3 Q levels 0, 32, 64
• 3 weightings used (sub vs super) 50 vs 100, 100
  vs 50, 100 vs 100

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Co-evolution - Results

• Mean runtime 242min 26sec, St Dev 0.00149
• 25 runs
• 1 for each image, Q level, weight combination
• Results vary wildly!
• Two Monet runs unfinished

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Co-evolution - Results
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Co-evolution - Conclusions

• Does seem to be a slight trend in favor of
  weights
• D4 MSEs of sub-image vs super-image is
  inconsistent
• Low M, G does not give much time to evolve

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Finishing Up

• Last two Monet co-evolves
• Co-evolution using mini-images as the super-image
• Runtime reduction
• Increased G, M

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That Which Could Not Be

• Finishing additional four-image mini-image runs
• In general, more runs
• More parameter testing explored
• Unless mini co-evolution proves fruitful, unable
  to get search working

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Extensions

• Forward transforms
• Variable-length transforms
• Evolve versus Y, U, V instead of just Y
• Initially seed coefficients randomly
• Adapt for distributed / parallel computing

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Fin

• Questions?