Abstract
This paper concerns the reconstruction of a diffusion coefficient in an elliptic equation from knowledge of several power densities. The power density is the product of the diffusion coefficient with the square of the modulus of the gradient of the elliptic solution. The derivation of such internal functionals comes from perturbing the medium of interest by acoustic (plane) waves, which results in small changes in the diffusion coefficient. After appropriate asymptotic expansions and (Fourier) transformation, this allow us to construct the power density of the equation point-wise inside the domain. Such a setting finds applications in ultrasound modulated electrical impedance tomography and ultrasound modulated optical tomography. We show that the diffusion coefficient can be uniquely and stably reconstructed from knowledge of a sufficient large number of power densities. Explicit expressions for the reconstruction of the diffusion coefficient are also provided. Such results hold for a large class of boundary conditions for the elliptic equation in the two-dimensional setting. In three dimensions, the results are proved for a more restrictive class of boundary conditions constructed by means of complex geometrical optics solutions.
Highlights
Optical tomography (OT) and electrical impedance tomography (EIT) are medical imaging techniques that utilize the large contrast between the optical and electrical response of certain unhealthy tissues and that of healthy tissues
Several recent imaging techniques aim to combine the high contrast in OT and EIT with a high resolution modality, for instance based on magnetic resonances (MREIT) [19, 21, 22] or, for what is of interest for us here, ultrasound
This paper aims to generalize these results to the case n = 3 and to show that ultrasound modulated electrical impedance tomography (UMEIT) is a stable inverse problem, in the sense that in an appropriate norm, errors on the measurement of the power densities Hij result in errors on the reconstruction of σ that are of the same order
Summary
Optical tomography (OT) and electrical impedance tomography (EIT) are medical imaging techniques that utilize the large contrast between the optical and electrical response of certain unhealthy tissues and that of healthy tissues. This paper aims to generalize these results to the case n = 3 (which generalize to the case n ≥ 4, see [20], we shall not present the details here) and to show that UMEIT is a stable inverse problem, in the sense that in an appropriate norm, errors on the measurement of the power densities Hij result in errors on the reconstruction of σ that are of the same order This should be contrasted with the case of EIT and OT, where the error on σ is roughly proportional to the logarithm of (and much larger than) the error on the available (Cauchy) data.
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