PSI - Issue 5

Andreas Kyprianou et al. / Procedia Structural Integrity 5 (2017) 1192–1197 Andreas Kyprianou / Structural Integrity Procedia 00 (2017) 000 – 000

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Fig. 2. An example of contour computed using edge detection

3. Spatio-temporal continuous wavelet transform

This section starts with the exposition of continuous wavelet transform and its theory Mallat (1999) and Kaiser (1994} in order to facilitate the presentation of SPT-CWT as an extension of the conventional CWT. The CWT introduces scale into signal analysis in a conceptually similar manner by which the Fourier transform introduces frequency. The Fourier transform F ( ω ) of an one dimensional signal f ( t ) is given by,

e i t     

  

  f t dt

(1)

F



where ω is the angular frequency, t is the independent variable which is interpreted as time and i the imaginary unit. The frequency ω is introduced in the analysis through the harmonic function e - iωt which in its argument the frequency ω operates on the independent variable t through multiplication. The SPT-CWT is an extension of the CWT that can be used to analyse the sequence, f ( x , t ), of time series responses from the cantilever beam of section 2. The first step toward the SPT-CWT is the definition of a wavelet function called spatio-temporal wavelet, ψ ( x , t ) that depends on two independent variables, x and t , which are interpreted as position and time. The properties concerning its support and average should be maintained over the x - t plane. To enable multiscale analysis translation and dilation operators acting on both the independent variables x and t should be defined. The translation operators can be defined directly by simply translating ψ along x by c and along t by τ . However, the dilation operators must emulate the inverse relationship that the human vision physiology establishes between spatial and temporal resolution when humans, through vision, perceive the motion of moving objects; i.e. when humans observe high speed moving objects the perceived time resolution is high and the perceived spatial resolution is low; and vice versa. Therefore, by considering that dilation by σ operates simultaneously on both x and t by ψ c , τ , σ ( x , t )=(1\ σ ) ψ (( x - c )/ σ , ( t - τ )/ σ ) the inverse relationship between the temporal and spatial resolution is established by introducing a fourth parameter v , as a tuning parameter, operating on (( x - c )/ σ , ( t - τ )/ σ ) as follows

  

  

 v x c 

1





(2)

,

  c v ( , ) , , , x t

v t









