Numerical methods for coupled reconstruction and registration in digital breast tomosynthesis.

Digital Breast Tomosynthesis (DBT) provides an insight into the fine details of normal fibroglandular tissues and abnormal lesions by reconstructing a pseudo-3D image of the breast. In this respect, DBT overcomes a major limitation of conventional X-ray mam- mography by reducing the confounding effects caused by the superposition of breast tissue. In a breast cancer screening or diagnostic context, a radiologist is interested in detecting change, which might be indicative of malignant disease. To help automate this task image registration is required to establish spatial correspondence between time points. Typically, images, such as MRI or CT, are first reconstructed and then registered. This approach can be effective if reconstructing using a complete set of data. However, for ill-posed, limited-angle problems such as DBT, estimating the deformation is com- plicated by the significant artefacts associated with the reconstruction, leading to severe inaccuracies in the registration. This paper presents a mathematical framework, which couples the two tasks and jointly estimates both image intensities and the parameters of a transformation. Under this framework, we compare an iterative method and a simultaneous method, both of which tackle the problem of comparing DBT data by combining reconstruction of a pair of temporal volumes with their registration. We evaluate our methods using various computational digital phantoms, uncom- pressed breast MR images, and in-vivo DBT simulations. Firstly, we compare both iter- ative and simultaneous methods to the conventional, sequential method using an affine transformation model. We show that jointly estimating image intensities and parametric transformations gives superior results with respect to reconstruction fidelity and regis- tration accuracy. Also, we incorporate a non-rigid B-spline transformation model into our simultaneous method. The results demonstrate a visually plausible recovery of the deformation with preservation of the reconstruction fidelity

3D shape based reconstruction of experimental data in Diffuse Optical Tomography

Diffuse optical tomography (DOT) aims at recovering three-dimensional images of absorption and scattering parameters inside diffusive body based on small number of transmission measurements at the boundary of the body. This image reconstruction problem is known to be an ill-posed inverse problem, which requires use of prior information for successful reconstruction. We present a shape based method for DOT, where we assume a priori that the unknown body consist of disjoint subdomains with different optical properties. We utilize spherical harmonics expansion to parameterize the reconstruction problem with respect to the subdomain boundaries, and introduce a finite element (FEM) based algorithm that uses a novel 3D mesh subdivision technique to describe the mapping from spherical harmonics coefficients to the 3D absorption and scattering distributions inside a unstructured volumetric FEM mesh. We evaluate the shape based method by reconstructing experimental DOT data, from a cylindrical phantom with one inclusion with high absorption and one with high scattering. The reconstruction was monitored, and we found a 87% reduction in the Hausdorff measure between targets and reconstructed inclusions, 96% success in recovering the location of the centers of the inclusions and 87% success in average in the recovery for the volumes

Basis mapping methods for forward and inverse problems

This paper describes a novel method for mapping between basis representation of a field variable over a domain in the context of numerical modelling and inverse problems. In the numerical solution of inverse problems, a continuous scalar or vector field over a domain may be represented in different finite-dimensional basis approximations, such as an unstructured mesh basis for the numerical solution of the forward problem, and a regular grid basis for the representation of the solution of the inverse problem. Mapping between the basis representations is generally lossy, and the objective of the mapping procedure is to minimise the errors incurred. We present in this paper a novel mapping mechanism that is based on a minimisation of the L2 or H1 norm of the difference between the two basis representations. We provide examples of mapping in 2D and 3D problems, between an unstructured mesh basis representative of an FEM approximation, and different types of structured basis including piecewise constant and linear pixel basis, and blob basis as a representation of the inverse basis. A comparison with results from a simple sampling-based mapping algorithm shows the superior performance of the method proposed here

Combined Reconstruction and Registration of Digital Breast Tomosynthesis

Digital breast tomosynthesis (DBT) has the potential to en- hance breast cancer detection by reducing the confounding e ect of su- perimposed tissue associated with conventional mammography. In addi- tion the increased volumetric information should enable temporal datasets to be more accurately compared, a task that radiologists routinely apply to conventional mammograms to detect the changes associated with ma- lignancy. In this paper we address the problem of comparing DBT data by combining reconstruction of a pair of temporal volumes with their reg- istration. Using a simple test object, and DBT simulations from in vivo breast compressions imaged using MRI, we demonstrate that this com- bined reconstruction and registration approach produces improvements in both the reconstructed volumes and the estimated transformation pa- rameters when compared to performing the tasks sequentially

Adaptive adjustment of the number of subsets during iterative image reconstruction

A common strategy to speed-up image reconstruction in tomography is to use subsets, i.e. only part of the data is used to compute the update, as for instance in the OSEM algorithm. However, most subset algorithms do not convergence or have a limit cycle. Different strategies to solve this problem exist, for instance using relaxation. The conceptually easiest mechanism is to reduce the number of subsets gradually during iterations. However, the optimal point to reduce the number of subsets is usually depends on many factors such as initialisation, the object itself, amount of noise etc. In this paper, we propose a simple scheme to automatically compute if the number of subsets is too large (or too small) and adjust the size of the data to consider in the next update automatically. The scheme is based on idea of computing two image updates corresponding to different parts of the data. A comparison of these updates then allows to see if the updates were sufficiently consistent or not. We illustrate this idea using 2 different subset algorithms: OSEM and OSSPS

Patch-based anisotropic diffusion scheme for fluorescence diffuse optical tomography-part 1: technical principles

Fluorescence diffuse optical tomography (fDOT) provides 3D images of fluorescence distributions in biological tissue, which represent molecular and cellular processes. The image reconstruction problem is highly ill-posed and requires regularisation techniques to stabilise and find meaningful solutions. Quadratic regularisation tends to either oversmooth or generate very noisy reconstructions, depending on the regularisation strength. Edge preserving methods, such as anisotropic diffusion regularisation (AD), can preserve important features in the fluorescence image and smooth out noise. However, AD has limited ability to distinguish an edge from noise. In this two-part paper, we propose a patch-based anisotropic diffusion regularisation (PAD), where regularisation strength is determined by a weighted average according to the similarity between patches around voxels within a search window, instead of a simple local neighbourhood strategy. However, this method has higher computational complexity and, hence, we wavelet compress the patches (PAD-WT) to speed it up, while simultaneously taking advantage of the denoising properties of wavelet thresholding. The proposed method combines the nonlocal means (NLM), AD and wavelet shrinkage methods, which are image processing methods. Therefore, in this first paper, we used a denoising test problem to analyse the performance of the new method. Our results show that the proposed PAD-WT method provides better results than the AD or NLM methods alone. The efficacy of the method for fDOT image reconstruction problem is evaluated in part 2

Radiance Monte-Carlo for application of the radiative transport equation in the inverse problem of diffuse optical tomography

We introduce a new Monte-Carlo technique to estimate the radiance distribution in a medium according to the radiative transport equation (RTE). We demonstrate how to form gradients of the forward model, and thus how to employ this technique as part of the inverse problem in Diffuse Optical Tomography (DOT). Use of the RTE over the more typical application of the diffusion approximation permits accurate modelling in the case of short source-detector separation and regions of low scattering, in addition to providing time-domain information without extra computational effort over continuous-wave solutions

Inversion formulas for the broken-ray Radon transform

We consider the inverse problem of the broken ray transform (sometimes also referred to as the V-line transform). Explicit image reconstruction formulas are derived and tested numerically. The obtained formulas are generalizations of the filtered backprojection formula of the conventional Radon transform. The advantages of the broken ray transform include the possibility to reconstruct the absorption and the scattering coefficients of the medium simultaneously and the possibility to utilize scattered radiation which, in the case of the conventional X-ray tomography, is typically discarded.Comment: To be submitted to Inverse Problem