Nonreciprocal broken ray transforms with applications to fluorescence imaging

Lucia Florescu,John C Schotland,Vadim A Markel

doi:10.1088/1361-6420/aacec7

Lucia Florescu, John C Schotland + Show 1 more

Open Access

https://doi.org/10.1088/1361-6420/aacec7

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Abstract

Broken ray transforms (BRTs) are typically considered to be reciprocal, meaning that the transform is independent of the direction in which a photon travels along a given broken ray. However, if the photon can change its energy (or be absorbed and re-radiated at a different frequency) at the vertex of the ray, then reciprocity is lost. In optics, non-reciprocal BRTs are applicable to imaging problems with fluorescent contrast agents. In the case of x-ray imaging, problems with single Compton scattering also give rise to non-reciprocal BRTs. In this paper, we focus on tomographic optical fluorescence imaging and show that, by reversing the path of a photon and using the non-reciprocity of the data function, we can reconstruct simultaneously and independently all optical properties of the medium (the intrinsic attenuation coefficients at the excitation and the fluorescence frequency and the concentration of the contrast agent). Our results are also applicable to inverting BRTs that arise due to single Compton scattering.

Highlights

The broken-ray transform (BRT), referred to as the V-line transform, has attracted significant recent attention [1,2,3,4,5,6,7,8,9]
A related recent development is inversion of the conical Radon transform, which arises in tomographic applications of the Compton camera [10,11,12,13,14,15,16,17,18], in the latter case the vertex of the V-line is located at the detection surface rather than within the medium
The BRT is defined in terms of integrals of a function along two rays with a common vertex

Summary

Introduction

The broken-ray transform (BRT), referred to as the V-line transform, has attracted significant recent attention [1,2,3,4,5,6,7,8,9]. We consider a non-reciprocal BRT applicable to the problem of fluorescence imaging in a weakly-scattering or non-scattering medium In this case, the photon travels along a line until it is absorbed by a fluorophore molecule and is re-emitted in a different direction and at a different (generally lower) frequency. We show that the non-reciprocity of the BRT allows us to access additional information about the medium by interchanging the source and detector in each pair This approach allows us to reconstruct the attenuation coefficient of the medium at both the excitation and the fluorescence frequencies independently, as well as the concentration of the fluorophore.

Coupled transport equations

Data function

Inverse problem

Inversion with complete data

Inversion with incomplete data

Summary