Diffusion Regularisation Methods of the Non-linear Inverse Prob- Lem for Diffuse Optical Tomography

Diffusion Optical Tomography (DOT) is a non-invasive functional imaging modality that aims to image the optical properties of biological organs. The forward problem of the light propagation of DOT can be modelled as a diffusion process and is expressed as a differential diffusion equation with boundary conditions. This model can be considered as the form of a general large scale non-linear inverse problem. In this paper, we formulate a solution to the DOT inverse problem as a minimisation of some functional energy of the solution and the data. One component of this functional is a measure of the discrepancy between the measured data and the data produced by representation of the modelled object. Minimisation of this term alone would force the solution to be consistent with the data and the solver used for this purpose. But in practice data is always accompanied with noise and with some jump discontinuities, these will unavoidably yield unsatisfactory solutions, so some regularisation to restore the solution is required. We add a diffusion regularisation potential with a statistical threshold to the objective functional. This term aims to reduce the noise associated with the data and to preserve the edges in the solution by a diffusion flux analysis of the local structures in the object. To accelerate the iterative solver we introduce a particular method called the Lagged Diffusivity Newton-Krylov method. The whole proposed strategy, which makes use of an a prior diffusion information and an specific algorithm, has been developed and evaluated on simulated data. 1. DIFFUSE OPTICAL TOMOGRAPHY Diffusion Optical Tomography (DOT) is an emerging medical imaging modality that aims to image the optical properties of biological tissue, particularly the peripheral muscle, breast and the brain [12, 3, 18, 7]. It is non-invasive, portable and can produce images of clinically relevant parameters which cannot be obtained by other diagnostic methods, such as blood volume and oxygenation. The data acquisition systems for optical tomography consist of a light source as an infrared laser, which is used to illuminate the body surface at different source locations. The light which has propagated through the volume under investigation is measured at multiple detectors locations on the surface. The intensity and the path-length distribution of the detected light provides information about the optical properties of the transilluminated tissue. This boundary data measurement can be used to recover the spatial distribution of internal absorption and scattering coefficients. The forward problem of the light propagation of DOT can be modelled as a diffusion process [1]. Mathematically, this is expressed by the diffusion equation, which is given in the frequency domain by, r ∈ Ω ⊂ R (N = 2, 3): ( −∇ · κ (r)∇+ μa (r) + iω c ) Φ (r, ω) = q (r, ω) (1) with Robin-type boundary condition, m ∈ ∂Ω: Φ (m,ω) + 2κ(m)A ∂Φ(m,ω) ∂n = 0 (2) where ω is the frequency parameter, Φ is the photon density, c is the velocity of light, q is an isotropic source of light in the medium, A is a boundary term which incorporates the refractive index mismatch at the tissue-air boundary, n is the outward normal at ∂Ω, κ and μa are the diffusion and absorption coefficients, respectively. We define here: κ = 1 3 (μa + μs) (3) where μs is the reduced coefficient. The measurable quantity is given by the boundary exitance: Γ (m,ω) = −κ(m) (m,ω) ∂n (4) D04