Mathematical Tools for Imaging Reconstruction

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The research program has a two-pronged approach aimed at identifying important areas in imaging that are ripe for enhanced mathematical treatment. The PI's main goal is to apply newly developed mathematical tools to such imaging problems as well as to consider new imaging situations that suggest the need to develop or adapt new mathematical techniques. The classical theory of scalar valued spherical functions unifies a number of classical approximation methods. These include Jacobi polynomials, Bessel functions, Laguerre polynomials, Hermite polynomials, and Legendre functions. The PI has extended this theory to matrix valued situations. This will be the main tool in solving the imaging problems. Natural candidates involve areas where polarization or anisotropy (as in optical tomography) play important roles. These imaging methods will be applied in biomedical areas, including X-ray, magnetic resonance, X-ray crystallography, and optical tomography. The National Research Council rported that there was a potential for great progress in the emerging field of optical or diffuse tomography, where one aims at obtaining images with the use of very low energy probes, like infrared lasers. From the beginning there was a need to deal both with attenuation and scattering. The techniques developed in this research should improve the ability to reconstruct images using these techniques. Other applications envisaged are repeated mammographies and certain brain blood flow studies (measuring oxygen content) in neonatal clinics. The PI will work closely with those who are developing the instrumation hardware.