Scanning paraxial optical tomography Vadim A. Markel and John C. Schotland

There has been considerable recent interest in the development of tomographic methods for imaging with diffuse light. Such methods have the potential to provide novel diagnostic tools while complementing existing medical imaging modalities. Clinical applications of current importance include breast imaging and functional brain mapping. A typical experimental conf iguration, often used in optical mammography, is the slab geometry in which N harmonically modulated point sources are located on one face of the slab and N point detectors are located on the opposite face [Fig. 1(c)]. The physical problem to be considered consists of reconstructing the optical properties of the interior of the slab from a complete set of N2 measurements taken on its surface. It is often assumed that many measurements (at a fixed modulation frequency) are needed for obtaining images with high spatial resolution, a requirement that is diff icult to realize in practice. Such large data sets also lead to image reconstruction algorithms with high computational complexity. Thus the development of reconstruction algorithms that are both computationally efficient and reduce the required number of spatial measurements below O N2 would be of considerable importance. To mitigate the principal diff iculties associated with the complete-data problem, we consider the problem in the paraxial geometry. In this geometry a single source is used to illuminate the medium, and the scattered light is collected by an on-axis detector along with a small number of off-axis detectors [Fig. 1(b)]. The entire source–detector array is then scanned over N points on the surface of the slab while the frequency of the source is varied over a specified range, resulting in O N spatial measurements. We refer to this method as scanning paraxial optical tomography. In this Letter we show that the linearized form of the corresponding inverse scattering problem may be solved analytically by means of an explicit inversion formula. This result has three important consequences. First, by trading spatial information for frequency information, we obtain an image reconstruction algorithm that reduces the required number of spatial measurements of the scattered f ield from O N2 to O N . Evidently, this reduction leads to a considerable simplif ication of the imaging experiment. Second, this algorithm has computational complexity that scales as N log N and is stable in the presence of added noise. This result should be compared with the N2 log N scaling of the computational complexity of the complete-data prob-