We present a review of methods for the forward and inverse problems in optical tomography. We limit ourselves to the highly scattering case found in applications in medical imaging, and to the problem of absorption and scattering reconstruction. We discuss the derivation of the diffusion approximation and other simplifications of the full transport problem. We develop sensitivity relations in both the continuous and discrete case with special concentration on the use of the finite element method. A classification of algorithms is presented, and some suggestions for open problems to be addressed in future research are made.M This article features multimedia enhancements available from the abstract page in the online journal; see http://www.iop.org.Figure 2. Spaces and operators used in optical tomography (based on figure 3.1 in [87]).(X (a) , X (b) ) are the solution spaces, Q is the space of sources, G is the space of solutions to the governing equation, Y M d are the data spaces. G is the Green function operating on a source, and M d are the measurement operators, operating on the solutions to the governing equation to give the data. P is the forward operator that maps the solution directly to the data. For completeness, the operator H M d is defined as the Dirichlet-to-Neumann map for data type M d .which define the measurables (see section 4). The P N approximationsThe P N approximations are obtained by spherical harmonic expansion of the quantities in (3.1). Relevant properties of spherical harmonics are summarized in the appendix. We express the 1 2 (l + m − 1) 1 2 ψ l−1,m−1 (r, t)− (l − m + 2) 1 2 (l − m + 1) 1 2 ψ l+1,m−1 (r, t)]

Quantitation of near infrared spectroscopic data in a scattering medium such as tissue requires knowledge of the optical pathlength in the medium. This can now be estimated directly from the time of flight of picosecond length light pulses. Monte Carlo modelling of light pulses in tissue has shown that the mean value of the time dispersed light pulse correlates with the pathlength used in quantitative spectroscopic calculations. This result has been verified in a phantom material. Time of flight measurements of pathlength across the rat head give a pathlength of 5.3 +/- 0.3 times the head diameter.

We review the current state-of-the-art of diffuse optical imaging, which is an emerging technique for functional imaging of biological tissue. It involves generating images using measurements of visible or near-infrared light scattered across large (greater than several centimetres) thicknesses of tissue. We discuss recent advances in experimental methods and instrumentation, and examine new theoretical techniques applied to modelling and image reconstruction. We review recent work on in vivo applications including imaging the breast and brain, and examine future challenges.

This paper is a review of recent mathematical and computational advances in optical tomography. We discuss the physical foundations of forward models for light propagation on microscopic, mesoscopic and macroscopic scales. We also consider direct and numerical approaches to the inverse problems which arise at each of these scales. Finally, we outline future directions and open problems in the field.

Abstract-Compressed Sensing Magnetic Resonance Imaging (CS-MRI) enables fast acquisition, which is highly desirable for numerous clinical applications. This can not only reduce the scanning cost and ease patient burden, but also potentially reduce motion artefacts and the effect of contrast washout, thus yielding better image quality. Different from parallel imaging based fast MRI, which utilises multiple coils to simultaneously receive MR signals, CS-MRI breaks the Nyquist-Shannon sampling barrier to reconstruct MRI images with much less required raw data. This paper provides a deep learning based strategy for reconstruction of CS-MRI, and bridges a substantial gap between conventional non-learning methods working only on data from a single image, and prior knowledge from large training datasets. In particular, a novel conditional Generative Adversarial Networks-based model (DAGAN) is proposed to reconstruct CS-MRI. In our DAGAN architecture, we have designed a refinement learning method to stabilise our U-Net based generator, which provides an endto-end network to reduce aliasing artefacts. To better preserve texture and edges in the reconstruction, we have coupled the adversarial loss with an innovative content loss. In addition, we incorporate frequency domain information to enforce similarity in both the image and frequency domains. We have performed comprehensive comparison studies with both conventional CS-MRI reconstruction methods and newly investigated deep learning approaches. Compared to these methods, our DAGAN method provides superior reconstruction with preserved perceptual image details. Furthermore, each image is reconstructed in about 5 ms, which is suitable for real-time processing.