PSI - Issue 80

Guangxiao Zou et al. / Procedia Structural Integrity 80 (2026) 93–104

99

Author name / Structural Integrity Procedia 00 (2023) 000–000

7

Then, the MVDR imaging value for each pixel location (x,y) is :

H xy R xy⃗ w xy

P x , y =⃗ w

(17)

The intuition behind MVDR imaging is that if damage exists at (x, y),the back-propagated residual signals from all channels align in time. Consequently, these signals exhibit no phase change, only a change in amplitude. The amplitudes of these back-propagated residual signals will be proportional to the amplitude of the excitation signal x ( t ) with the steering vector⃗ e xy , but corrupted by an additive white noise:⃗ n xy ( t ):⃗

u xy ( t ) =⃗ e xy | ˆ x ( t ) | +⃗ n xy ( t )

(18)

And therefore:⃗

xy H xy

w H

xy R xy⃗ w xy =⃗ w H

H xy ( t ) dt⃗ w xy

u xy ( t )⃗ u

t ∈ excitation window⃗

2⃗ e H

H xy ( t ) +⃗ n xy ( t )⃗ n

H xy ( t )) dt⃗ w xy

xy + 2⃗ e xy | ˆ x ( t ) |⃗ n

=⃗ w

(⃗ e xy | ˆ x ( t ) |

t ∈ excitation window

H xy⃗ w xy

H xy

H xy ( t ) dt⃗ w xy

H xy⃗ e xy⃗ e

| ˆ x ( t ) | 2 dt

=⃗ w =

+⃗ w

n xy ( t )⃗ n

t ∈ excitation window

t ∈ excitation window⃗

H xy

H xy ( t ) dt⃗ w xy

| ˆ x ( t ) | 2 dt

+⃗ w

n xy ( t )⃗ n

(19)

t ∈ excitation window

t ∈ excitation window⃗

under the constraint⃗ w H⃗ e xy = 1. This implies that although the MVDR algorithm finds an optimal weight vector⃗ w xy to minimize the output power⃗ w H R xy⃗ w at every pixel, for a pixel at the true damage location ( x , y ), the resulting MVDR pixel value is preserved, corresponding to the energy of the excitation signal. This is because the weight vector⃗ w xy only a ff ects the noise term, leaving the signal from the actual source intact, as shown in Eq.(19). For all other pixels, especially for those far from the true damage location, the back-propagated residual signals will not align within the excitation time window and they may only contain some noise or interferences. Then, the MVDR algorithm will suppress their contribution, resulting in very small, near-zero pixel values.

2.4. MUSIC imaging

The first step in the MUSIC imaging algorithm is to perform an eigen decomposition of the spatiotemporal corre lation matrix R xy :

H

R xy = Q Σ Q

(20)

Here, Q is a matrix whose columns, [⃗ q 1 ,⃗ q 2 , ...,⃗ q M ], are eigen vectors. The matrix Σ is a diagonal matrix, and its diagonal entries are eigen values arranged in descending order. If the true damage exists at the pixel location ( x , y ), by combining Eqs. (13) and (18) the spatiotemporal correlation matrix R xy can also be expressed as :