Recovering inhomogeneities in a waveguide using eigensystem decomposition

We present an eigensystem decomposition method to recover weak inhomogeneities in a waveguide from knowledge of the far-field scattered acoustic fields. Due to the particular geometry of the waveguide, which supports only a finite number of propagating modes, the problem of recovering inhomogeneities in a waveguide has a different set of challenges than the corresponding problem in free space. Our method takes advantage of the spectral properties of the far-field matrix, and by using its eigenvalues and its eigenvectors we obtain a representation of the linearized solution to the inverse problem in terms of products of fields which are linear combinations of the propagating modes of the waveguide with weights given by the eigenvectors of the far-field matrix. The problem of finding the unknown inhomogeneity reduces to a problem of determining some coefficients in a finite system of linear equations, whose coefficients depend on the background medium and the eigenvalues and the eigenvectors of the far-field matrix. By numerically implementing our inverse algorithm we reconstructed the inhomogeneities present in the waveguide. We show that (1) even with as few as seven propagating modes we obtain a good recovery of the size and shape of the inhomogeneity; (2) multiple inhomogeneities can be well recovered and, as expected, (3) the recovery improves when the number of propagating modes increases.