PSI - Issue 80

Guangxiao Zou et al. / Procedia Structural Integrity 80 (2026) 93–104 Author name / Structural Integrity Procedia 00 (2023) 000–000

98

6

2.3. MVDR imaging

MVDR imaging represents an advanced form of DAS imaging, employing adaptive artifact suppression techniques originally developed in the field of beamforming.

2.3.1. Spatiotemporal Correlation matrix The first step in the MVDR imaging method is to calculate the spatiotemporal correlation matrix, R xy , for each pixel ( x , y ). This matrix is computed from the back-propagated residual signals within the time window corresponding to the excitation: R xy =  t ∈ excitation window⃗ u xy ( t )⃗ u H xy ( t ) dt = sub U xy sub U H xy (13) where the superscript ’H’ denotes the conjugate transpose,and sub U xy is the sub-matrix of U xy containing only the columns that correspond to the excitation window. The element in the i -th row and j -th column of the correlation matrix R xy quantifies the degree of correlation between the signals of the i -th and j -th channels at pixel ( x , y ). 2.3.2. Steering vector The steering vector is an important parameter in both the MVDR and MUSIC imaging methods. If damage is truly present at a pixel location ( x , y ), all the back-propagated signals corresponding to that pixel should align perfectly in time. Any variation between the signals from di ff erent channels should only be in their amplitudes. As shown in Eqs. (4) and (5), these magnitude values are predictable, and they have proportional relationship to the excitation signal, caused by the scatterer characteristics and the geometric spreading of the wave (the inverse square-root of the propagation distance). The steering vector for a pixel ( x , y ) is a vector that models this expected relationship, describing how the magnitude of the scattered signal in each channel should ideally appear:⃗ e xy =    ψ 1 xy  d ∗ 1 xy , · · · ψ mxy  d ∗ mxy , · · · ψ Mxy  d ∗ Mxy    T (14) Here, d ∗ mxy represents the product of the transmitter-to-pixel and pixel-to-receiver distances for the m -th channel. The scattering coe ffi cient ψ mxy defines the energy scattered by a defect at that location. By assuming uniform (omnidirec tional) scattering, ψ mxy can be simplified to one. 2.3.3. Pixel value calculation In MVDR imaging method, a weighting vector⃗ w xy for each pixel location ( x , y ) is found from a constrained optimization problem:⃗

w H R

H⃗ e

w xy = arg min⃗ w⃗

xy⃗ w subject to⃗ w

xy = 1

(15)

It can be solved by using a Lagrange multiplier:⃗

R − 1

xy⃗ e xy⃗

w xy =

(16)

e H

xy R −

1 xy⃗ e xy