Novel Image Reconstruction Methods in the Frequency Domain

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The PI will develop faster and more accurate mathematical algorithms for use in biomedical imaging, remote sensing, and novel optical device design. These image construction problems generally involve large and complex systems. The PI will build on recent theoretical breakthroughs to construct reduced systems which allow for more straightforward and efficient mapping of the data to the unknown physical quantities. The novel usage of the reduced models will for the first time allow for the generation of internal physical fields from exterior data only, providing a bridge to a broader range of modalities and experimental settings. Once the reduced model is generated from a small data set, the interior fields can be found for an arbitrarily large data set or for data given in other formats. The knowledge of interior fields greatly simplifies the imaging problems, and for certain applications such as medical ablation, it is the interior fields themselves that are of interest. Several students will be trained in the course of this research, including two full time Phd students. Undergraduate students will work for six months full time on this research problem as their co-op training. The PI continues to have a clear commitment to diversity in mathematics and will make every effort to involve women and/or underrepresented minorities in the co-op experience.The PI will generate new reconstruction methods and develop further theories which are crucial in medical imaging, remote sensing and nondestructive testing. The main goals are to (i) use reduced order models to generate interior solutions from boundary data, (ii) use these boundary data generated interior solutions to solve inverse problems for larger classes of data sets, (iii) derive, analyze and apply a new inverse Born series adapted to nonlinearity, and (iv) use boundary corrections and transmission eigenvalues to image periodic and nearly periodic microstructures. A new way of using reduced order models (ROMs) for inverse problems is introduced, that is, by embedding the ROM back into the continuous problem and generating interior fields from boundary data only. Highly accurate interior fields will be used to apply the ROM to large data sets to yield a completely new, fully nonlinear inversion method with low computational cost. Interior fields are of interest in their own right for applications such as medical ablation, and they provide a bridge between classical inverse problems and multi-physics hybrid methods. A crucial orthogonalization step in the procedure will be justified rigorously. New inverse scattering series will allow us to reconstruct nonlinear scatterers without the use of optimization or forward solvers except for that of the reference medium. We will analyze the series, show convergence estimates, and fully understand the series behavior. For microstructured media, asymptotics of the forward solution will enable us to capture fine scale features. Boundary corrections will be used to image the media along with transmission eigenvalues.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.