The Continuous wavelet transform is given
by
(1)
where the * represents the
complex conjugate of the function. The
CWT is simply a correlation of the signal, f(t),
with a scaled basis function y*(t). In general, the basis function is complex and
is termed the “mother wavelet”. The
wavelet coefficient, W(a,t)
is also complex and can be represented in the Fourier domain as
.
(2)
The CWT is a linear operation and
has an exact inverse transform given by the double integral
(3)
where C is found from a Parseval-like
integral
(4)
that requires the basis function to have zero mean for this
integral to be bounded at . We use Morlet’s wavelet (Goupillaud et al.
1985) in which
. (5)
As can be seen, Morlet’s wavelet is a heavily damped sinusoid around t=0
allowing the scale parameter to be approximately interpreted as Fourier period
(inverse frequency) in the CWT domain since we use .
Hard Thresholding
An estimate of the noise power is
made in a time window before the first arrival of the event. This is then used with the following
criterion to remove wavelet coefficients from the map. Coefficients are kept if they are greater than
the mean plus 3 standard deviations (99.8% confidence level; Starck et al. 2010), otherwise they are set to zero. Mathematically, this is represented by
(6)
where
(7)
(8)
and
. (9)
The limits t2 and t1 represent the limits of the noise time lag window.
Soft Thresholding
Alternatively, thresholding can
be done in a somewhat less severe manner by taking into account the amplitude
of the noise in a “keep or shrink” process where the coefficients that survive
are modified by the inferred noise level:
(10)
where
.
(11)
Donoho (1995) suggested a
thresholding factor for the standard deviation that is equal to ,
where n is the number of CWT coefficients.
This number is often close to the value of 3 as shown in equation (9) for
realistic time series.
References
Goupillaud, P., A. Grossmann, and J. Morlet,
1985, Cycle-octave and related transforms in seismic signal analysis, Geoexploration,
23, pp. 85-102.
Starck, J.-L., F. Murtagh, and J.
M. Fadili, 2010, Sparse
Image and Signal Processing, Cambridge University Press, New York.
Donoho, D. L., 1995, De-Noising by soft-thresholding, IEEE Trans. Inf. Theory, 41, 613-627.