# Background to optical modelling and reconstruction

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The 1997 review of modelling and image reconstruction techniques (Arridge and Hebden 1997) is still a useful introduction to the basic theoretical concepts. Progress since 1997 has largely focussed on developing more realistic and efficient models of light transport in tissue, and on solving the ill-posed inverse problem in an increasingly rigorous way. In particular, it is becoming increasingly common to include prior information of the anatomy and optical properties of tissue in both the modelling and the image reconstruction.

Three processes are required to reconstruct an optical image. First, the transport of light in tissue must be modelled. Second, this model must be used to predict the distribution of light in the object under examination. This stage, the forward problem, allows the measurements to be simulated from the model, and generates (explicitly or implicitly) a sensitivity matrix (the Jacobian of the forward mapping) which relates the measurements to the internal optical properties. Finally, the image is reconstructed by inverting the Jacobian and solving the inverse problem. In x-ray CT, scatter is minimal, so the forward problem becomes a series of integrals along lines connecting the sources and detectors (a Radon transform), and the inverse problem is linear and well-posed. In optical imaging, however, scatter is dominant, meaning that the forward problem becomes a series of integrals over the entire volume. Each measurement is therefore sensitive to the whole volume, meaning that the inverse problem is ill-posed and underdetermined and that complex, computer-intensive reconstruction techniques are required. For recent reviews of image reconstruction in optical imaging, see Arridge (1999), Kohlemainen (2001) and Schweiger et al. (2003).

It should be noted that similar issues are being addressed in electrical impedance tomography (EIT), which is a related medical imaging technique in which surface measurements of electrical impedance are used to reconstruct volume images of tissue conductivity (Metherall et al. 1996, Borcea 2002, Lionheart 2004).

## ReferencesEdit

- Arridge S R (1999), "Optical tomography in medical imaging" Inverse Problems 15 R41-R93.
- Arridge S R and J C Hebden (1997), "Optical imaging in medicine: II. Modelling and reconstruction" Phys. Med. Biol. 42(5) 841-853.
- Borcea L (2002), "Electrical impedance tomography" Inverse Problems 18 R99-136.
- Kohlemainen V (2001), "Novel approaches to image reconstruction in diffusion tomography" University of Kuopio PhD Thesis.
- Lionheart W R B (2004), "EIT reconstruction algorithms: pitfalls, challenges and recent developments" Physiol. Meas. 25 125-142.
- Metherall P, D C Barber, R H Smallwood, and B H Brown (1996), "Three-dimensional electrical impedance tomography" Nature 380 509-512.
- Schweiger M, A P Gibson, and S R Arridge (2003), "Computational aspects of diffuse optical tomography" IEEE Computing in Science and Engineering Nov/Dec 2003 33-41.