Symmetries, inversion formulas, and image reconstruction for optical tomography

Abstract

We consider the image reconstruction problem for optical tomography with diffuse light. The associated inverse scattering problem is analyzed by making use of particular symmetries of the scattering data. The effects of sampling and limited data are analyzed for several different experimental modalities, and computationally efficient reconstruction algorithms are obtained. These algorithms are suitable for the reconstruction of images from very large data sets. We consider the image recostruction problem for optical tomography with diffuse light. The associated inverse scattering problem is analyzed by making use of particular symmetries of the scattering data. The effects of sampling and limited data are analyzed for several different experimental modalities, and computationally efficient reconstruction algorithms are obtained. These algorithms are suitable for the reconstruction of images from very large data sets.

Inverse Scattering and Acousto-Optic Imaging

“…(i) We observe that the inverse problem of AOI is expected to be well posed since (19) yields a band limited approximation to the kernel K by an inverse Fourier transform. In contrast, the linear inverse problem of DOT is severely ill posed, requiring the inversion of a Laplace transform, a problem which has logarithmic stability [29,30]. This illposedness is responsible for the relatively low resolution of images in DOT.…”

Methods in diffuse optical imaging

“…In the particular case of an infinite slab and the use of Fourier patterns, the structured light technique corresponds to the analytical block-decomposition method of Markel & Schotland [21], which has been successfully applied to the reconstruction of complex structures in measured data [22,23]. An illustration is shown in figure 11, where an FEM model of a slab is used as the object.…”

Superresolution and corrections to the diffusion approximation in optical tomography

“…To proceed, we note that, following Ref. [6], the Green's function G(r,ŝ; r ′ ,ŝ ′ ) may be expanded in angular harmonics ofŝ and…”

“…These methods have been shown to be computationally efficient and may be applied to data sets consisting of a very large number of measurements [5,6]. The approach taken is to construct the singular value decomposition of the linear operator K in the proper Hilbert space setting and then use this result to obtain the pseudoinverse solution to (6). In this manner, it is possible to account for the effects of sampling and thereby obtain the best (in the sense of minimizing the appropriate L 2 norm) bandlimited approximation to δα.…”

“…Within the accuracy of the DA, reconstruction algorithms based on both numerical [3] and analytic solutions [4,5,6] to the ISP have been described. Regardless of the method of inversion, the spatial resolution of reconstructed images is expected to be limited to ℓ * .…”