Hilbert transform with incomplete data and applications in Tomography and Optics

The University Of Central Florida Board Of Trustees, Orlando FL

Investigators

Abstract

In computed tomography (CT), detectors usually cover the entire cross-section of the patient. Even when a small organ inside the patient needs to be visualized, the entire cross-section is irradiated. Development of robust algorithms for image reconstruction from truncated CT data (i.e., the data obtained by irradiating only a region of interest (ROI) inside the patient) will have numerous benefits, such as reducing the radiation dose to patients in many CT scans, opening the way to novel multimodality imaging platforms, etc. In this project, we will investigate stability of algorithms for reconstruction from truncated CT data. Similar mathematical approaches are useful for the study of image reconstruction from incomplete data in optical imaging, such as in microscopy and optical metrology applications. We will apply the methods developed for CT to optical imaging with the goal of increasing the field-of-view or reducing the amount of measurements, while maintaining spatial resolution in the ROI. This research may pave the way to the development of inexpensive, large-field-of-view direct phase imaging systems, which, in turn, would benefit cell biology research, and applications such as digital pathology and cell-tracking. In CT with truncated data, the key analytical tool is the Gelfand-Graev formula, which transforms the tomographic reconstruction problem to the problem of inverting the finite Hilbert transform (FHT) of the attenuation coefficient from incomplete data. When CT data are truncated, reconstruction of the attenuation coefficient is frequently non-unique. On the other hand, the contribution of the missing data to the ROI is analytic, and adding prior knowledge about the attenuation coefficient inside the ROI restores uniqueness. Another application where inversion of the FHT with incomplete data is useful is optical imaging. Inspired by the recent development of compressive optical imaging in spectroscopy and holography, optical systems based on the FHT have the potential to achieve high resolution, high speed phase contrast imaging. In a number of microscopy imaging applications, prior knowledge about the sample being investigated is possible to obtain. Thus, similar approaches can be used both for CT and for optical imaging. The objective of the research is to develop theory and algorithms for inverting the FHT with incomplete data. We will estimate stability of inverting the FHT with incomplete data by finding the singular value decomposition of the relevant operators. The method of the Riemann-Hilbert problem, perturbation theory, and the Titchmarsh-Weyl theory, are some of the mathematical tools that will be used in this project. We also plan to develop and test the corresponding reconstruction algorithms and test them on simulated and experimental data.