Image Processing Reference
In-Depth Information
The final global SNR, after the combination of all the 2D displays belonging to the different
frequency bands, SNR 2DTFlinear , can be obtained supposing that the 2D representations for
each band are independent and perfectly synchronized (Rodríguez et al 2004b):
SNR
()2
dB
 
L SNR
()
dB
(5)
2
DTFlinear
ini
being, L, the number of the selected frequency bands.
Consequently, in this case, the resulting SNR 2DTFlinear presents an important factor of
improvement over the SNR ini . This factor is the double of the number of frequency bands
used in the combination. It must be noted that comparing expressions (5) and (3), the SNR
improvements is multiplied by L, but the computational complexity of the algorithm is also
multiplied by the same factor L . In the experimental results section of this chapter, the
accuracy of this expression will be confirmed comparing (5) with simulations using as linear
time-frequency tool the undecimated wavelet packet transform (Shensa 1992, Coifman and
Wickerhauser 1992). In any case, it must be noted that this expression is also valid for any
linear time-frequency transform.
3.3 Wigner-Ville Transform (WVT) combination technique
The non-linear time-frequency distributions present some advantages over linear
transforms, but some non-linear terms (“cross-terms”) appear degrading the quality of the
distributions and usually the computational cost is incremented. One of the most popular
non-linear time-frequency representations is the Wigner-Ville transform (WVT) (Claasen
and Mecklenbrauker 1980), which has been previously utilized in ultrasonic applications
with good results (Chen and Guey 1992, Malik and Saniie 1996, Rodríguez et al 2004a).
The WVT presents an useful property for dealing with ultrasonic traces: its positivity for
some kind of signals (Cohen 1995). In order to illustrate the suitability of this transform for
the processing of the ultrasonic pulses typical in NDE applications, we will show that they
fulfil that property. For it, an ultrasonic pulse-echo like to those acquired in such NDE
equipment can be approximately modelled by the following expression:
2
(
at
)
p t
( )

Ae
cos(
)
(6)
0
where A is the pulse amplitude, a is a constant related to the duration and bandwidth of the
pulse ( a >0), and ω 0 is the central frequency of its spectrum.
The WVT of the ultrasonic pulse modelled by (6) is (Rodríguez 2003):
A
2
2
2
(
-
at
/
2
)
(
)
/
a
WVT
(
t
,
)
=
e
0
p
(7)
1
(
a
)
2
The expression (7) shows that the WVT of an ultrasonic pulse with Gaussian envelope has
only positive values. The chirp with Gaussian envelope is the most general signal for which
the WVT is positive through-out the time-frequency plane (Cohen 1995). The ultrasonic
grain noise does not carry out this property, so resulting that the sign of the WVT values
represents a useful option to discriminate this type of difficult-to-eliminate noise of the echo
pulses coming from real flaws.
The combination method begins in this case by calculating the WVT in all the ultrasonic
traces. After the band selection is performed, the negative values (that correspond mainly
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