Three-dimensional optical tomography of objects in turbid media using the round-trip matrix

Abstract

A new algorithm based on multi-static data and vector subspace classification to eigenvectors of a round-trip matrix is introduced for optical imaging and localization of objects embedded in a turbid medium. The transport of light from multiple sources through excitation of the embedded objects to the array of detectors is represented by a response matrix that can be constructed from experimental data. The 'round-trip (RT) matrix’ is constructed by multiplying the response matrix by its transpose for continuous-wave (adjoint matrix for frequency domain) illumination. Mathematically, the RT matrix is equivalent to transfer of light from the sources via the embedded objects to the array of detectors and back, and is similar to the time-reversal matrix used in the general area of array processing for acoustic and radar time-reversal imaging. The eigenvectors with leading non-zero eigenvalues of the RT matrix correspond to embedded objects, which are orthogonal to the vectors in the noise subspace. The vector subspace method along with Green’s functions calculated from an appropriate model for light propagation through turbid media is then used to determine the locations of the embedded objects. We tested this algorithm in simulation for light transmitting through a 50 l _tr thick (l _tr ~ 1 mm is transport mean free path) parallel slab turbid medium with up to six embedded absorptive objects. The method was able to globally locate all six objects with surprising accuracy. This “round-trip tomographic imaging” approach is fast, applicable to different geometries and to different forward models.