Abstract
This paper concerns the reconstruction of a scalar diffusion coefficient $\sigma(x)$ from redundant functionals of the form $H_i(x)=\sigma^{2\alpha}(x)|\nabla u_i|^2(x)$ where $\alpha\in\mathbb{R}$ and $u_i$ is a solution of the elliptic problem $\nabla\cdot \sigma \nabla u_i=0$ for $1\leq i\leq I$. The case $\alpha=\frac12$ is used to model measurements obtained from modulating a domain of interest by ultrasound and finds applications in ultrasound modulated electrical impedance tomography (UMEIT), ultrasound modulated optical tomography (UMOT) as well as impedance acoustic computerized tomography (ImpACT). The case $\alpha=1$ finds applications in Magnetic Resonance Electrical Impedance Tomography (MREIT). We present two explicit reconstruction procedures of $\sigma$ for appropriate choices of $I$ and of traces of $u_i$ at the boundary of a domain of interest. The first procedure involves the solution of an over-determined system of ordinary differential equations and generalizes to the multi-dimensional case and to (almost) arbitrary values of $\alpha$ the results obtained in two and three dimensions in [10] and [5], respectively, in the case $\alpha=\frac12$. The second procedure consists of solving a system of linear elliptic equations, which we can prove admits a unique solution in specific situations.
Highlights
Medical imaging modalities aim to combine high resolution with high contrast between healthy and unhealthy tissues
Ultrasound Tomography and Magnetic Resonance Imaging are high resolution modalities that sometimes suffer from low contrast
We describe and investigate the compatibility conditions associated with such a redundant system
Summary
Medical imaging modalities aim to combine high resolution with high contrast between healthy and unhealthy tissues. Optical Tomography and Electrical Impedance Tomography display such high contrasts but often suffer from poor resolution. Ultrasound Tomography and Magnetic Resonance Imaging are high resolution modalities that sometimes suffer from low contrast. The ultrasound modulation of electrical or optical properties of tissues and the combination of simultaneous electrical and magnetic resonance measurements both offer the possibility to combine high resolution with high contrast. For the acquisition of ultrasoundmodulated measurements, we refer the reader to, e.g., [1, 5, 10, 12, 14] for works in the mathematical literature. For the acquisition of internal information on electrical conductivities by magnetic resonance imaging, we refer the reader to, e.g., [13, 17, 18, 19]
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