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We consider the problem of imaging extended reflectors in three-dimensional acoustic waveguides using a planar array that is parallel to the waveguide’s cross-section. Our data consist of the multiple-frequency array response matrix. To form an image, we back-propagate a projection of the data on the propagating modes in the waveguide. The projection operator is adequately defined for any array-aperture size and shape that cover fully or partially the waveguide cross-section. The properties of the imaging method are analyzed theoretically and its effectiveness is assessed by numerical experiments for scatterers and arrays of varying shape and size.

We consider the problem of imaging sparse scenes from a few noisy data usingan $l_1$-minimization approach. This problem can be cast as a linear system ofthe form $A \, \rho =b$, where $A$ is an $N\times K$ measurement matrix. Weassume that the dimension of the unknown sparse vector $\rho \in{\mathbb{C}}^K$ is much larger than the dimension of the data vector $b \in{\mathbb{C}}^N$, i.e, $K \gg N$. We provide a theoretical framework that allowsus to examine under what conditions the $\ell_1$-minimization problem admits asolution that is close to the exact one in the presence of noise. Our analysisshows that $l_1$-minimization is not robust for imaging with noisy data whenhigh resolution is required. To improve the performance of $l_1$-minimizationwe propose to solve instead the augmented linear system $ [A \, | \, C] \rho=b$, where the $N \times \Sigma$ matrix $C$ is a noise collector. It isconstructed so as its column vectors provide a frame on which the noise of thedata, a vector of dimension $N$, can be well approximated. Theoretically, thedimension $\Sigma$ of the noise collector should be $e^N$ which would make itsuse not practical. However, our numerical results illustrate that robustresults in the presence of noise can be obtained with a large enough number ofcolumns $\Sigma \approx 10 K$.

We consider the problem of imaging sparse scenes from a few noisy data usingan $l_1$-minimization approach. This problem can be cast as a linear system ofthe form $A \, \rho =b$, where $A$ is an $N\times K$ measurement matrix. Weassume that the dimension of the unknown sparse vector $\rho \in{\mathbb{C}}^K$ is much larger than the dimension of the data vector $b \in{\mathbb{C}}^N$, i.e, $K \gg N$. We provide a theoretical framework that allowsus to examine under what conditions the $\ell_1$-minimization problem admits asolution that is close to the exact one in the presence of noise. Our analysisshows that $l_1$-minimization is not robust for imaging with noisy data whenhigh resolution is required. To improve the performance of $l_1$-minimizationwe propose to solve instead the augmented linear system $ [A \, | \, C] \rho=b$, where the $N \times \Sigma$ matrix $C$ is a noise collector. It isconstructed so as its column vectors provide a frame on which the noise of thedata, a vector of dimension $N$, can be well approximated. Theoretically, thedimension $\Sigma$ of the noise collector should be $e^N$ which would make itsuse not practical. However, our numerical results illustrate that robustresults in the presence of noise can be obtained with a large enough number ofcolumns $\Sigma \approx 10 K$.