This Ph.D. thesis dissertation focuses on the study of the undecimated wavelet 
transform UWT, as well as its application to signal denoising and detection. This 
transform has several advantages over the traditional discrete wavelet transform, DWT. 
It is translation invariant, avoids the loss of time resolution in the successive 
decomposition scales and provides some additional information. Although this 
information is redundant from the point of view of linear processing, it can provide 
great improvements when combined with non-linear processing methods, in 
applications such as signal analysis or denoising.
In this context, the main objectives of this thesis are two. The first one is the study of 
the undecimated wavelet transform in depth, trying to establish a general framework for 
the different names and implementations existing in the literature. The second objective 
is the application of this transform to the development of denoising algorithms. In this 
respect, the problem of grain noise reduction in ultrasonic non destructive evaluation is 
addressed. This kind of noise is due to reflections in the internal discontinuities of the 
materials, which have frequency content similar to that of the defect echoes to be 
detected. For this reason, they cannot be suppressed by the classical band-pass filtering 
techniques, suitable for white noise reduction. The present thesis deals with this kind of 
noise, and proposes specific algorithms to take advantage of the different time-space 
distribution of the signal and noise in the undecimated wavelet domain.
The methodology used to achieve the previous objectives, includes the development 
and implementation of new signal processing methods based on MATLABŪ computing 
language. Throughout the thesis, four different schemes for ultrasonic grain noise 
reduction are proposed, based on the different possibilities for implementation and 
reconstruction provided by the UWT. These four schemes are applied to a wide set of 
typical ultrasonic signals, obtained from synthetic as well as experimental procedures.
The results show that the new UWT-based methods improve the quality of the traces 
and the robustness of the processing, in terms of both signal to noise ratio and pulse 
shape recovery, in relation to the methods that use the traditional DWT. These 
improvements are obtained regardless of the wavelet family considered. In fact, the 
UWT processing shows clear preference for the lower order wavelets, irrespective of 
the wavelet family or the kind of threshold used. In this way, the thesis provides a set of 
denoising methods that, besides improving the quality of the traces significantly, prove 
to be very robust in relation to the choice of processing parameters.