On the Adjoint Operator in Photoacoustic Tomography

Photoacoustic Tomography (PAT) is an emerging biomedical "imaging from coupled physics" technique, in which the image contrast is due to optical absorption, but the information is carried to the surface of the tissue as ultrasound pulses. Many algorithms and formulae for PAT image reconstruction have been proposed for the case when a complete data set is available. In many practical imaging scenarios, however, it is not possible to obtain the full data, or the data may be sub-sampled for faster data acquisition. In such cases, image reconstruction algorithms that can incorporate prior knowledge to ameliorate the loss of data are required. Hence, recently there has been an increased interest in using variational image reconstruction. A crucial ingredient for the application of these techniques is the adjoint of the PAT forward operator, which is described in this article from physical, theoretical and numerical perspectives. First, a simple mathematical derivation of the adjoint of the PAT forward operator in the continuous framework is presented. Then, an efficient numerical implementation of the adjoint using a k-space time domain wave propagation model is described and illustrated in the context of variational PAT image reconstruction, on both 2D and 3D examples including inhomogeneous sound speed. The principal advantage of this analytical adjoint over an algebraic adjoint (obtained by taking the direct adjoint of the particular numerical forward scheme used) is that it can be implemented using currently available fast wave propagation solvers.Comment: submitted to "Inverse Problems

Hyperspectral Diffuse Optical Tomography Using the Reduced Basis Method and Sparsity Constraints

Diffuse Optical Tomography (DOT) has long been investigated as an effective imaging technique for soft tissue imaging, such as breast cancer detection. DOT has many benefits, including its use of non-ionizing light and its ability to produce high contrast images, but it also has low resolution. In recent years hyperspectral DOT (hyDOT) has been proposed, in an effort to improve that resolution by adding more information in the spectral domain. In this imaging modality, hundreds or even thousands of different wavelengths in the visible to near infrared range are used in the imaging process. Since tissue absorbs and scatters light differently at different wavelengths, it has been conjectured that this increase of information should provide images that give a better overall idea of the complete spatial reconstruction of the optical parameters. Although hyDOT has been investigated experimentally, a formal theoretical investigation into its mathematical foundations has not been thoroughly performed. This dissertation seeks to lay the groundwork for the mathematical formulation of this imaging modality. First, the forward problem for hyDOT is formulated and the spectral regularity of the solution investigated. We demonstrate that the solution to the governing PDE is very smooth with respect to wavelength. This spectral regularity allows for the application of a model reduction technique to the forward problem known as the Reduced Basis Method. Several proofs are given for the hyDOT forward solution and the spectral regularity term, including existence and uniqueness proofs and proofs showing the continuity of the solution with respect to the diffusion and absorption coefficients and the wavelength. The appropriate function spaces for the optical coefficients with respect to their dependence on the wavelength are explored and a new norm is proposed. Additionally, the hyDOT inverse problem is formulated. New cost functionals are proposed to solve the inverse problem that incorporate the spatial sparsity of the optical parameters and their spectral regularity. Finally, a gradient-based reconstruction algorithm that enforces the spatial sparsity with respect to wavelength, is shown to be very effective and robust in solving the hyDOT inverse problem when used on simulations with a simple geometry

Fast Radiative-Transfer-Equation-Based Image Reconstruction Algorithms for Non-Contact Diffuse Optical Tomography Systems

It is well known that the radiative transfer equation (RTE) is the most accurate deterministic light propagation model employed in diffuse optical tomography (DOT). RTE-based algorithms provide more accurate tomographic results than codes that rely on the diffusion equation (DE), which is an approximation to the RTE, in scattering dominant media. However, RTE based DOT (RTE-DOT) has limited utility in practice due to its high computational cost and lack of support for general non-contact imaging systems. In this dissertation, I developed fast reconstruction algorithms for RTE-based DOT (RTE-DOT), which consists of three independent components: an efficient linear solver for forward problems, an improved optimization solver for inverse problem, and the first light propagation model in free space that fully considers the angular dependency, which can provide a suitable measurement operator for RTE-DOT. This algorithm is validated and evaluated with numerical experiments and clinical data. According to these studies, the novel reconstruction algorithm is up to 30 times faster than traditional reconstruction techniques, while achieving comparable reconstruction accuracy

Solving the Monge-Amp\`ere Equations for the Inverse Reflector Problem

The inverse reflector problem arises in geometrical nonimaging optics: Given a light source and a target, the question is how to design a reflecting free-form surface such that a desired light density distribution is generated on the target, e.g., a projected image on a screen. This optical problem can mathematically be understood as a problem of optimal transport and equivalently be expressed by a secondary boundary value problem of the Monge-Amp\`ere equation, which consists of a highly nonlinear partial differential equation of second order and constraints. In our approach the Monge-Amp\`ere equation is numerically solved using a collocation method based on tensor-product B-splines, in which nested iteration techniques are applied to ensure the convergence of the nonlinear solver and to speed up the calculation. In the numerical method special care has to be taken for the constraint: It enters the discrete problem formulation via a Picard-type iteration. Numerical results are presented as well for benchmark problems for the standard Monge-Amp\`ere equation as for the inverse reflector problem for various images. The designed reflector surfaces are validated by a forward simulation using ray tracing.Comment: 28 pages, 8 figures, 2 tables; Keywords: Inverse reflector problem, elliptic Monge-Amp\`ere equation, B-spline collocation method, Picard-type iteration; Minor revision: reference [59] to a recent preprint has been added and a few typos have been correcte