Reconstructing material properties by deconvolution of full-field measurement images: The conductivity case

This study concerns the reconstruction of material parameters from full-field measurements. In this context the typical available data is a set of digital images that is seldom handled as such when solving the inverse problem. Therefore, this work investigates a direct method to compute constitutive parameter maps from full-field measurement images. Within the prototypical framework of the periodic conductivity model, the starting point for the proposed approach is the Lippmann-Schwinger equation, which is satisfied by the fields measured internally. This integral equation is reinterpreted as a linear convolution model for the sought conductivity field. Considering that multiple experiments might be available and then combined, this problem is solved in the least-square sense. To do so, the Krylov subspace-based LSQR algorithm is employed. Full advantage is taken of the convenient expression of the featured Green's function in Fourier space and of the intensive use of the fast Fourier transform (FFT). Moreover, a spectral-based filtering regularization scheme is implemented to tackle noisy data. Overall, the proposed reconstruction algorithm only handles image-like quantities in an efficient mesh-free approach. The performance of the method is assessed on a set of synthetic 2D numerical examples both for isotropic and anisotropic material configurations.