  Localization and reconstruction of small defects in acoustic or
electromagnetic waveguides is of crucial interest in nondestructive evaluation
of structures. The aim of this work is to present a new multi-frequency
inversion method to reconstruct small defects in a 2D waveguide. Given one-side
multi-frequency wave field measurements of propagating modes, we use a Born
approximation to provide a L2-stable reconstruction of three types of defects:
a local perturbation inside the waveguide, a bending of the waveguide, and a
localized defect in the geometry of the waveguide. This method is based on a
mode-by-mode spacial Fourier inversion from the available partial data in the
Fourier domain. Indeed, in the available data, some high and low spatial
frequency information on the defect are missing. We overcome this issue using
both a compact support hypothesis and a minimal smoothness hypothesis on the
defects. We also provide a suitable numerical method for efficient
reconstruction of such defects and we discuss its applications and limits.
