Direct Reconstruction Methods in Optical Tomography

Optical tomography is a biomedical imaging modality that uses scattered light as a probe of structural variations in the optical properties of tissue [1]. In a typical experiment, a highly-scattering medium is illuminated by a narrow collimated beam and the light that propagates through the medium is collected by an array of detectors. In first generation systems, the sources and detectors are coupled to the medium by means of optical fibers. The number of measurements that can be obtained in this manner varies from 10 to 10 source-detector pairs. More recently, noncontact imaging systems have been introduced, wherein a scanned beam and a lens-coupled CCD camera is employed to replace the illumination and detection fiber-optics of firstgeneration systems, as shown in Fig. 1.1. Using such a noncontact method, extremely large data sets of order 10 measurements can be obtained. The inverse problem of optical tomography is to reconstruct the optical properties of a medium of interest from boundary measurements. The mathematical formulation of the corresponding forward problem is dictated primarily by spatial scale, ranging from the Maxwell equations at the microscale, to the radiative transport equation at the mesoscale, and to diffusion theory at the macroscale. The standard approach to the inverse problem is framed in terms of nonlinear optimization. It is important to note that although optimization methods are extremely flexible, they lead to iterative algorithms with very high computational cost. Direct reconstruction methods offer an alternative approach that can fill this gap. By direct reconstruction, we mean the use of inversion formulas and associated fast algorithms.