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The degrees of freedom afforded by mixtures of ICA suggest that it is a good
candidate for a broad range of problems. As was commented in Chap. 1 , there are
different perspectives to undertake the ICA mixture modelling of the physical
phenomenon under analysis. From the perspective of ''most physical'' interpre-
tation, this section includes a modelling of the impact-echo data as an ICA mix-
ture. It is clearly important to have as much knowledge as possible about the
underlying physical phenomenon, in order to better interpret the results for the
general performance of the method.
We proposed an ICA model for the impact-echo problem in Sect. 5.3 .This
model considered the transfer functions between the impact location and the point
defects that are spread in a material bulk as ''sources'' for blind source separation.
In this work, we formulate a model based on ICAMM that takes into account the
resonance phenomenon involved in the impact-echo method. The proposed model
extends to defects with different shapes, such as cracks or holes, and defines the
quality condition determination of homogeneous and defective materials as an ICA
mixture problem.
In ICA mixture modelling, it is assumed that feature (observation) vectors x k
corresponding to a given class C k ð k ¼ 1...K Þ are the result of applying a linear
transformation defined by matrixA k to a (source) vector s k , whose elements are
independent random variables, plus a bias vector b k , i.e.,
x k ¼ A k s k þ b k
k ¼ 1 ; ... ; K
ð 5 : 8 Þ
Let us find the ICA mixture model for the impact-echo problem. The impact-
echo signals can be considered as a convolutive mixture of the input signal and the
defect signals inside the material, as shown in Fig. 5.4 .
In Fig. 5.4 , there is one attack point that generates the wave r 0 ð n Þ¼ p ð n Þ ;
Finternal focuses (point flaws) that generate the waves f j ð n Þ j ¼ 1 ; ... ; F; and N
sensors that measure the waves v i ð n Þ i ¼ 1 ; ... ; N. To simplify, we consider the
impact as another focus; thus,f j ð n Þ j ¼ 0 ; ... ; F, withf 0 ð n Þ¼ r 0 ð n Þ .
We assume that the impact-echo overall scenario can be modelled as a multi-
ple-input-multiple-output linear time invariant system (MIMO-LTI) [ 22 ]. This
implies that the complex spectrum at sensor i (Fourier transform of v i ð n Þ ) will be
the sum of all the contributions due to focusj ¼ 0 ; ... ; F. Moreover, the contri-
bution of every individual focus j will be the product of the complex spectrum at
focus j by the frequency response of the path between focus j and sensor i.In
practice, the discrete Fourier transform (DFT) is to be used to allow numerical
computation of the complex spectrum. Considering that the length of the recorded
signal is long enough to capture the transients, and so to overcome the time-
overlapping effect of the DFT, the MIMO-LTI model may be expressed in an
algebraic form by means of properly defined vectors and matrices.
Let us call V i a vector formed by the samples obtained from the computation of
the DFT of v i ð n Þ . Hence, V i is a vector representation of the complex spectrum of
v i ð n Þ . Considering the MIMO-LTI model, the total complex spectrum V i will be
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