Glitches detection by a local regularity analysis method using wavelets

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Glitches detection by a local regularity analysis
method using wavelets

• Christophe Ordenovic
• Laboratoire d'astrophysique de Marseille

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Holderian analysis

• f is locally a-Holderian at t0 if
• f (t0 h) f (t0) lt c.h a
• a measures the local regularity of f
• a gt 1 f continuous and differentiable
• a in 0 , 1 f continuous, non differentiable
• a in -1, 0 f non continuous
• a lt -1 f no longer continuous gt tempered
  distribution

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Link wavelet Holderian analysis

• Hypothesis
• f is a - hoderian at to
• Y wavelet well localised in time and Fourier
  space
• Y has its first m vanishing moments ( m gt a )
• Result Holschneider, Jaffard, Mallat
• Holderian exponent can be linked with wavelet
  coefficient modulus by the relation wf(b,a)
  lt c. a a
• Holder exponent can be asympotically estimated by
  a linear fitting of log w(b,a) log c a log
  a when scale a -gt0
• Estimation along maxima lines Mallat
  continuous curves a(bi) belonging the local
  maxima modulus

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Algorithm

• Compute wavelet coefficients wf(b,a)
• Use a Mexican hat (2 vanishing moments)
• Scale a in scale_min, scale_max
• Compute w(b,a) maxima lines on time scale space.
• For each maxima lines Holderian exp. estimation
• Log-log plane
• Linear fit estimation
• Linear Slope calculation -gt Holder exponent
• Non linear derivate value for scale_min -gt
  Holder
• Implemented with 'LastWave' software

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Example of Holderian analysis

• Dirac distribution Holderian exponent -1

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Glitches detection

• Ideal case Glitch Dirac distribution, but ...
• Electronic response -gt signature (shape)
• Glitch signal 'less' regular than interferogram
  signal
• Bounding a glitche signature
• Begining of signature
• Top of signature
• End of signature
• Discretization parameter algorithm limiting
  factor
• Too few samples discontinuous everywhere
• Too much samples continuous everywhere
• ideal sampling of glitche signature (??)

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Examples

• Low resolution
• Signal from FTS Simulator
• Position -0.1,0.1 cm
• Stage speed 0.01mm/s
• Glitches amplitudes 70 of central peak
  amplitude
• Glitche event probability 0.001
• Holderian exponent thres 0
• Determination coefficient thres 0.9

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Complete interferogram and flagged samples with
its wavelet analysis and its extrema
representation (modulus maxima lines). The major
'begin', 'top', and 'end' glitches samples have
been detected by the algorithm
Zoom on a perfectly detected glitch begin, top
and end samples have been detected.
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Example

• High resolution

Position -3.2,3.2 cm Stage speed
0.03 Glitches amplitudes 70 of central peak
amplitude Glitche event probability
0.001 Holderian exponent thres
0.9 Determination ceofficient thres 0.95
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