Imola K' Fodor, Chandrika Kamath Center for Applied Scientific Computing Lawrence Livermore National

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Imola K. Fodor, Chandrika KamathCenter for
Applied Scientific ComputingLawrence Livermore
National LaboratoryIPAM WorkshopJanuary,
2002
Exploring the use of wavelet thresholding for
denoising images
UCRL-JC-145671. This work was performed under the
auspices of the U.S. Department of Energy
by University of California Lawrence Livermore
National Laboratory under contract W-7405-Eng-48.
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Many data mining applications require removing
noise from the data

• Noise contaminates many scientific data sets
• instrumental noise
• data acquisition process
• interfering natural phenomena
• Denoising is a first pre-processing step in such
  cases
• Many different denoising techniques
• spatial filters, wavelets, simple thresholding,
  level sets, total variational methods,
  essentially non-oscillatory schemes, hidden
  Markov models, ridgelets, curvelets,
• No comprehensive evaluation of all methods exists

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We explore wavelet-based and spatial filter-based
2D denoising methods

• Denoising
• estimate signal from noisy observation
• we assume i.i.d. Gaussian noise, independent from
  the signal
• find estimate with desired properties, e.g.
  minimum mean square error (MSE)

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Wavelets decompose the data into different
multiresolution levels

• Decimated wavelet transform with J2 levels

Horizontal
Vertical
Diagonal
Original
Decomposed level 1, level2
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Denoising by thresholding of wavelet coefficients
entails several steps
Image
Wavelet transform
Threshold detail wavelet coefficients
Inverse wavelet transform
De-noised image
Wavelet Haar Daublets Symmlets Coiflets ...
Boundary treatment Zero-pad Periodic Symmetric R
eflective Constant
Number of levels J
Shrinkage rule calculate threshold
Shrinkage function apply threshold
Noise scale
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The shrinkage function determines how the
threshold is applied
Hard
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The calculation of some thresholds requires an
estimate of the noise scale

• Choice of coefficients D global or
  level-dependent?
• or for
• Choice of estimator
• median absolute deviation (MAD)
• sample standard deviation
• norm

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The shrinkage rule determines how the threshold
is calculated

• Universal
• global
• minFDR minimize false discovery rate
• global
• Top based on quantiles of the coefficients
• global
• HypTest test hypothesis of zero coefficients
• level dependent
• SURE minimize Steins Unbiased Risk Estimate
• adapt combines SURE and Universal
• level and shrinkage function dependent
• Bayes use a Bayesian framework
• level dependent

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Our implementation is more general than
conventional methods in literature

• Conventional either
• 1 (global) or
• J (level-dependent) or
• 3J (subband-dependent) threshold(s)
• Sapphire choice of
• 1 (global) or
• J (level-dependent) or
• 3J (subband-dependent) threshold(s)
• Preliminary observation
• subband-dependent de-noising outperforms
  level-dependent de-noising on some test images

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Spatial filtering and simple thresholding are
alternative ways to denoise images

• Spatial filters are commonly used in the signal
  processing community. Our software includes
• mean filters and alpha-trimmed mean filters
• Gaussian filters
• (scaled) unsharp masking filters
• median filters
• mid-point filters
• minimum mean squared error filters
• combinations of the above filters
• Simple thresholding
• drop the pixels below 5RMS in the FIRST images

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Results with Lena using soft shrinkage, symm12
wavelet, periodic boundary, J3
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Results with Lena using hard shrinkage, symm12
wavelet, periodic boundary, J3
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Results with Lena using garrote and semisoft
shrinkage, symm12, periodic boundary, J3
S_ single global threshold
P_ pyramid of thresholds (J or 3J)
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The Lena test image with simulated Gaussian noise
added

Original
Noise added, sigma20 MSE399.50
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Wavelet denoising on the Lena test image

Global universal hard thresholding MSE103.95 Noti
ce the artifacts and ringing near the edges
SureShrink MSE61.59
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Results with the Lena image using various spatial
filters
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Spatial filtering on the Lena test image

Min-MSE (5x5) followed by Gaussian (3x3) MSE56.80
Min-MSE (5x5) followed by mean (3x3) MSE64.80
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Summary

• SureShrink and BayesShrink are best wavelet-based
  denoisers across a range of images and noise
  levels
• Soft is the best shrinkage function
• MAD is the best noise estimator
• Choice of wavelets, number of multiresolution
  levels, and boundary treatment rule have little
  influence
• Combination of spatial filters (5x5 minimum MSE
  filter followed by 3x3 Gaussian filter) often
  yields smaller error rates than SureShrink and
  BayesShrink