Sparse Pseudo Inverse of the Discrete Plane Wave Transform

It has been observed that it is possible to represent the solution to certain interior scattering problems, such as simple cavities, in terms of solutions to the exterior problem and the pseudo inverse of the plane wave transform (PWT) operator. This may provide a pathway to sparse direct representations for interior problems provided that sparse representations of the exterior solution operator and the pseudo inverse of the PWT are available. This paper considers the latter possibility. The approach used consists of first compressing the PWT via spatially localized current modes that radiate far fields only to specific angular regions. It is shown that the resulting operator has a sparse QR factorization, which permits the efficient direct implementation of the desired pseudo inverse operation.