Fast inverse approach for the deconvolution of ultrasonic TFM images using a spatially varying PSF in NDT

Multi-element probes are widely used in Non-Destructive Testing (NDT) for their ability to produce images. Full Matrix Capture (FMC) is a standard acquisition process that consists in acquiring all inter-element responses. The common reconstruction procedure, namely the Total Focusing Method (TFM), is a linear reconstruction technique that does not account for the instrumental response of the transducers. Thus, its resolution remains limited. The inversion of the direct model considering the acoustic response of the transducers and formulated on the FMC data is a heavy procedure in terms of computing resources and storage. In order to reduce the size of the problem, the proposed approach considers the beamformed TFM image as a back-projection of the data in the space domain. The proposed model linearly links the TFM image and the reflectivity map of the media under inspection. The PSF (Point Spread Function) associated to a pixel is the TFM reconstruction from data acquired on a reflectivity map where only this pixel is reflecting the signal. This raises a 2D "deconvolution" problem where the PSF is varying spatially. To limit the number of computations, the proposed method assumes that the variation of the PSF is slow, so that it can be interpolated accurately. The interpolation model involves convolution products that can be quickly computed using Fast Fourier Transforms (FFTs). The reconstructed image is obtained within an iterative procedure enforcing sparsity and spatial smoothness. The resolution improvement is evidenced from experimental data acquired from a stainless steel block containing Side Drilled Holes (SDH) inspected with a 128-element probe.