This project assembles a team of applied and computational mathematicians and physicists to develop, analyze and implement reconstruction algorithms for the broken-ray Radon transform (BRT) and its generalizations. The BRT describes the propagation of single-scattered particles or waves. This situation is typical of x-ray imaging at clinical energies or optical imaging of nearly transparent tissues and model organisms. The intent of the proposed research is to provide theoretically, numerically justified and practically applicable reconstruction algorithms for the BRT with applications to both x-ray computed tomography and optical tomography in the weak-scattering regime. In particular, the investigators propose to derive and analyze BRT-based scanning protocols and corresponding inversion techniques to reconstruct the absorption and scattering coefficients. Associated scanning protocols, which provide the optimum balance between spatial resolution and stability to noise, are to be developed. In addition, questions of uniqueness and stability (in the scale of Sobolev spaces) are a concern. Techniques of microlocal analysis may be used to characterize the propagation of singularities. Efficient numerical algorithms for inverting the BRT are to be implemented and tested using data derived from radiative transport forward solvers that account for both single- and multiple-scattering, hence connecting the research to the experimental regime.
One of the grand challenges in imaging is to address the problem of scattering. It is generally believed that only unscattered particles or waves carry useful information about the medium through which they have traveled. The Investigators aim to show that this is not the case. By making use of mathematical methods and computational approaches that exploit the presence of scattering, they seek to transform a variety of biomedical and security-related x-ray and optical imaging technologies. This research is a collaboration between applied and computational mathematicians and physicists and their work with three graduate students. Broad dissemination of the results of the research is anticipated through publications and generation of publicly available software.
The main goal of this Project was to develop the necessary mathematical tools for new tomographic imaging that utilize scattered X-rays. All existing X-ray tomographic technologies are based on the assumption that the X-rays travel through the imaged medium along straight lines. But this is not actually so. A lot of X-ray particles (called photons) change direction due to the process known as scattering. In existing approaches to imaging, one tries not to detect these scattered photons, to block them by various means from reaching the detectors. This is done because the scattered photons are viewed as noise and, as such, as being detrimental to image quality. We however were able to show theoretically and in simulations that detection of these scattered photons can be very useful. In fact, we have found that detection of single-scattered photons opens new possibilities and has a potential to make tomographic imaging more flexible and precise. But to achieve this end, we had to modify the underlying assumptions that are used in image reconstruction. One interesting consequence of using the scattered photons is that this enables one to image different contrasts of the medium separately. These contrasts are related to the scattering and absorbing properties of the medium. This is somewhat similar to the advantages of viewing a color rather than a black-and white image of an object, although in our case only two "colors" are visualized. Another advantage of detecting scattered photons is the possibility to utilize incoplete data sets for image reconstruction. To obtain a tomographic image of an object, the conventional instruments require the measurements of all rays that traverse the object in all possible directions (within a two-dimensional slice) and, for each given direction, with all possible displacements (relative to a fixed point in the center of the medium). This is so even if one needs to reconstruct only a small sub-region of interest. In other words, measurement of complete data is always required in traditional CT tomography, even if the region of interest is relatively small. This is one of the reasons why the radiation dose in the conventional CT tomography can not be further reduced. However, when scattered photons are registered, the collection of complete data is no longer necessary. One can focus instead on a small region of interest and record a data set, which is just sufficient to perform reconstruction in that region, even if the medium outside of it is inhomogeneous. This finding is potentially very important and such "limited-data" reconstructions are in principle impossible in traditional approaches. This possibility to perform tomographic reconstructions with limited data was first discovered by our collaborator on this Collaborative Project, A.Katsevich. We have then developed the limited-data methods from a somewhat different point of view. This approach was convincingly proved in simulations by both Prof.Katsevich and in our group.