  This study considers the Block-Toeplitz structural properties inherent in
traditional multichannel forward model matrices, using Full Matrix Capture
(FMC) in ultrasonic testing as a case study. We propose an analytical
convolutional forward model that transforms reflectivity maps into FMC data.
Our findings demonstrate that the convolutional model excels over its
matrix-based counterpart in terms of computational efficiency and storage
requirements. This accelerated forward modeling approach holds significant
potential for various inverse problems, notably enhancing Sparse Signal
Recovery (SSR) within the context LASSO regression, which facilitates efficient
Convolutional Sparse Coding (CSC) algorithms. Additionally, we explore the
integration of Convolutional Neural Networks (CNNs) for the forward model,
employing deep unfolding to implement the Learned Block Convolutional ISTA
(BC-LISTA).
