Microlocal analysis of non-linear operators arising in Compton CT

We present a novel microlocal analysis of a non-linear ray transform, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:mrow><mml:mi>R</mml:mi></mml:mrow> </mml:mrow> </mml:math> , arising in Compton scattering tomography (CST). Due to attenuation effects in CST, the integral weights depend on the reconstruction target, <i>f</i>, which has singularities. Thus, standard linear Fourier integral operator (FIO) theory does not apply as the weights are non-smooth. The V-line (or broken ray) transform, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:mrow><mml:mi>V</mml:mi></mml:mrow> </mml:mrow> </mml:math> , can be used to model the attenuation of incoming and outgoing rays. Through novel analysis of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:mrow><mml:mi>V</mml:mi></mml:mrow> </mml:mrow> </mml:math> , we characterize the location and strength of the singularities of the ray transform weights. In conjunction, we provide new results which quantify the strength of the singularities of distributional products based on the Sobolev order of the individual components. By combining this new theory, our analysis of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:mrow><mml:mi>V</mml:mi></mml:mrow> </mml:mrow> </mml:math> , and classical linear FIO theory, we determine the Sobolev order of the singularities of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:mrow><mml:mi>R</mml:mi></mml:mrow> <mml:mi>f</mml:mi></mml:mrow> </mml:math> . The strongest (lowest Sobolev order) singularities of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:mrow><mml:mi>R</mml:mi></mml:mrow> <mml:mi>f</mml:mi></mml:mrow> </mml:math> are shown to correspond to the wavefront set elements of the classical Radon transform applied to <i>f</i>, and we use this idea and known results on the Radon transform to prove injectivity results for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:mrow><mml:mi>R</mml:mi></mml:mrow> </mml:mrow> </mml:math> . In addition, we present novel reconstruction methods based on our theory, and we validate our results using simulated image reconstructions.