1. The wavelet transform can be used to detect singularities or discontinuities in signals by identifying large wavelet coefficients around points of abrupt change across multiple scales. 2. The wavelet transform modulus maxima (WTMM) method uses successive derivative wavelets to identify singularities by removing lower order polynomial terms from the signal at each scale. 3. Local maxima of the continuous wavelet transform are related to singularities in the signal, and their behavior across scales can be used to characterize the point-wise regularity of the signal, detect noise, and reconstruct the signal from its singularities.

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Farhad Gholami (EE-8811, Fall2013)
IEEE Transaction on Info. Theory , 1992
Singularity Detection and Processing
with Wavelets
S. Mallat , W. Hwang

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Introduction:
Abrupt changes in a signal, produce relatively large wavelet
coefficients centered around the discontinuity at all scales.
The wavelet transform makes it possible to localize
regularities(time, location) of signal.

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Set of CWT coefficients affected by the singularity increases with
increasing scale ( cone of influence).
The most precise localization of the discontinuity based
on the CWT coefficients is obtained at the smallest
scales.

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Concept of Lipschitz Exponent (LE):
There is a double cone (shown in white) whose vertex can be
translated along the graph, so that the graph always remains entirely
outside the cone.
A signal is regular if it can be locally approximated by a polynomial.

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Lipschitz regularity :

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Point-wise regularity of a signal related to the decay of its
wavelet transform:
Detecting singularities this way, numerically is complex !!!!
Reqularity measurments:

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The Wavelet Transform Modulus Maxima
(WTMM):
A method for detecting the fractal dimension of a signal.
By calculating CWT for subsequent wavelets (derivatives of the
mother wavelet) singularities can be identified.
Signal represented by polynomial:
.
Successive derivative wavelets remove the contribution of lower order terms
in the signal, allowing the maximum “hi” to be detected.
When taking derivatives, lower order =0.

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Wavelet transform local maxima are related to the singularities
of the signal.
Detection of singularities:

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Function is not singular if CWT has no local maxima.
This theorem indicates the presence of a maximum
at the finer scales where a singularity occurs.

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This theorem indicates the presence of a maximum
at the finer scales where a singularity occurs.
Modulus maxima of a 1-D signal

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Simulation results:

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Most of signals can be reconstructed using only local maximas
value and location of of dyadic (power of two) sequence
of scales very very good precision.
This is a basis for de-noising and edge detection algorithms
Reconstruction of signal using local Maxima:

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Lipschitz continuity is
related to the wavelet transform:

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Evolution of local maxima amplitude across scales we can determine
which ones are created by white noise.
De-noising Algorithm:
Remove all maxima whose amplitude increases on average when
scale decreases .
Noise LE of White noise is negative.

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Multiscale edge detection:
 A method of edge detection on across scales to obtain
satisfactory results on image+white noise noise.
 keep the scales small for locations with positive, increase scales for
locations with negative Lipschitz regularity

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Conclusion:
Wavelet local maxima detect all the singularities of a function.
and characterize their Lipschitz regularity.
This provides algorithms for characterizing the singularities.
Signal can separated from noise using evolution of local maxima
evolution as scale changes .
We can reconstruct the signal from remaining local maxima
(denoise).
Image edges like other singularities can be detected and
processed using local maxima.