Application of ICAMM to Impact-Echo Testing - On Statistical Pattern Recognition in Independent Component Analysis Mixture Modelling - page 114

In the experiments, the dimension of V ð PCA Þ is 140 (20 components of V ðð PCA ÞÞ
n
by N ¼ 7 sensors), and it is reduced to only 50 components in V ðð PCA ÞÞ , thus
retaining 92 % of the total variance.
Now, in order to account for the variability of the impact generation (different
strength, different locations,..), we assume that F ðð PCA ÞÞ

0 can be expressed as a mean
excitation m plus a random variation s. Substituting in equation ( 5.14 )

V ðð PCA ÞÞ ¼ X

F

ð m þ s Þ¼ As þ b ; A ¼ X

F

; b ¼ X

F

H ðð PCA ÞÞ
j

H ðð PCA ÞÞ
j

H ðð PCA ÞÞ
j

m

j ¼ 0 j ¼ 0 j ¼ 0

ð 5 : 15 Þ

Equation ( 5.15 ) demonstrates the suitability of an ICA model for the multi-
channel impact-echo scenario. In a given specimen, an ICA model can be applied
to estimate A and b from a set of training feature vectors

n obtained by
repeated impacts on the material. Note that linear transformation A depends on the
transfer functions between the focus and the sensors and between the excitation
point and the focus, whereas the bias term b additionally depends on the mean
impact excitation. This indicates that, in principle, a different ICA model should be
required for every specific defect (defective zone with particular geometry), every
specific deployment of the sensors, and every specific impact location. Thus, we
can formulate the problem of classification of materials with different quality
conditions, which are inspected by impact-echo in the ICAMM framework.
Equation ( 5.15 ) can be formulated to the case of a given material class
C k ð k ¼ 1...K Þ , considering a different set of parameters A ð k Þ , s ð k Þ for every class
of material. The following ICAMM expression can be written,

V ðð PCA ÞÞ

A ð k Þ ¼ X ; b ð k Þ ¼ X

F F

V ðð PCA ÞÞ
ð k Þ

H ðð PCA ÞÞ
j ð k Þ

H ðð PCA ÞÞ
j ð k Þ

¼ A ð k Þ s ð k Þ þ b ð k Þ ; m ð k Þ ;

j ¼ 0 j ¼ 0

ð 5 : 16 Þ

where,
V ðð PCA ÞÞ
ð k Þ

compressed spectra of the multichannel impact-echo setup for the
defective material class k

A ð k Þ mixture matrix for the defective material class k

s ð k Þ compressed spectra from the focuses f j ,j ¼ 0 ; ...F for the defective

material class k

Finally, let us express Eq. ( 5.11 ) considering separately the magnitude and the
phase term of the complex numbers, i.e.,