Abstract-Many practical applications require the reconstruction of images from irregularly sampled data. The spline formalism offers an attractive framework for solving this problem; the currently available methods, however, are hard to deploy for large-scale interpolation problems in dimensions greater than two (3-D, 3-D+time) because of an exponential increase of their computational cost (curse of dimensionality). Here, we revisit the standard regularized least-squares formulation of the interpolation problem, and propose to perform the reconstruction in a uniform tensor-product B-spline basis as an alternative to the classical solution involving radial basis functions. Our analysis reveals that the underlying multilinear system of equations admits a tensor decomposition with an extreme sparsity of its one dimensional components. We exploit this property for implementing a parallel, memory-efficient system solver. We show that the computational complexity of the proposed algorithm is essentially linear in the number of measurements and that its dependency on the number of dimensions is significantly less than that of the original sparse matrix-based implementation. The net benefit is a substantial reduction in memory requirement and operation count when compared to standard matrix-based algorithms, so that even 4-D problems with millions of samples become computationally feasible on desktop PCs in reasonable time. After validating the proposed algorithm in 3-D and 4-D, we apply it to a concrete imaging problem: the reconstruction of medical ultrasound images (3-D+time) from a large set of irregularly sampled measurements, acquired by a fast rotating ultrasound transducer.

Optical Diffusion Tomography (ODT) is a modern non-invasive medical imaging modality which requires mathematical modelling of near-infrared light propagation in tissue. Solving the ODT forward problem equation accurately and efficiently is crucial. Typically, the forward problem is represented by a Diffusion PDE and is solved using the Finite Element Method (FEM) on a mesh, which is often unstructured. Tensor B-spline signal processing has the attractive features of excellent interpolation and approximation properties, multiscale properties, fast algorithms and does not require meshing. This paper introduces Tensor B-spline methodology with arbitrary spline degree tailored to solve the ODT forward problem in an accurate and efficient manner. We show that our Tensor B-spline formulation induces efficient and highly parallelizable computational algorithms. Exploitation of B-spline properties for integration over irregular domains proved valuable. The Tensor B-spline solver was tested on standard problems and on synthetic medical data and compared to FEM, including state-of-the art ODT forward solvers. Results show that 1) a significantly higher accuracy can be achieved with the same number of nodes, 2) fewer nodes are required to achieve a prespecified accuracy, 3) the algorithm converges in significantly fewer iterations to a given error. These findings support the value of Tensor B-spline methodology for high-performance ODT implementations. This may translate into advances in ODT imaging for biomedical research and clinical application.

Solutions of 3-D elliptic PDEs form the basis of many mathematical models in medicine and engineering. Solving elliptic PDEs numerically in 3-D with fine discretization and high precision is challenging for several reasons, including the cost of 3-D meshing, the massive increase in operation count, and memory consumption when a high-order basis is used, and the need to overcome the "curse of dimensionality." This paper describes how these challenges can be either overcome or relaxed by a Tensor B-spline methodology with the following key properties: 1) the tensor structure of the variational formulation leads to regularity, separability, and sparsity, 2) a method for integration over the complex domain boundaries eliminates meshing, and 3) the formulation induces high-performance and memory-efficient computational algorithms. The methodology was evaluated by application to the forward problem of Optical Diffusion Tomography (ODT), comparing it with the solver from a state-of-the-art Finite-Element Method (FEM)-based ODT reconstruction framework. We found that the Tensor B-spline methodology allows one to solve the 3-D elliptic PDEs accurately and efficiently. It does not require 3-D meshing even on complex and non-convex boundary geometries. The Tensor B-spline approach outperforms and is more accurate than the FEM when the order of the basis function is > 1, requiring fewer operations and lower memory consumption. Thus, the Tensor B-spline methodology is feasible and attractive for solving large elliptic 3-D PDEs encountered in real-world problems.

Background Schistosomiasis, caused by Schistosoma mansoni, is of great significance to public health in sub–Saharan Africa. In the Democratic Republic of Congo (DRC), information on the burden of S. mansoni infection is scarce, which hinders the implementation of adequate control measures. We assessed the geographical distribution of S. mansoni infection across Ituri province in north-eastern DRC and determined the prevailing risk factors. Methods/Principal findings Two province–wide, community–based studies were conducted. In 2016, a geographical distribution study was carried out in 46 randomly selected villages across Ituri. In 2017, an in–depth study was conducted in 12 purposively–selected villages, across the province. Households were randomly selected, and members were enrolled. In 2016, one stool sample was collected per participant, while in 2017, several samples were collected per participant. S. mansoni eggs were detected using the Kato–Katz technique. In 2017, a point–of–care circulating cathodic S. mansoni antigen (POC–CCA) urine test was the second used diagnostic approach. Household and individual questionnaires were used to collect data on demographic, socioeconomic, environmental, behavioural and knowledge risk factors. Of the 2,131 participants in 2016, 40.0% were positive of S. mansoni infection. Infection prevalence in the villages ranged from 0 to 90.2%. Of the 707 participants in 2017, 73.1% were tested positive for S. mansoni. Prevalence ranged from 52.8 to 95.0% across the health districts visited. Infection prevalence increased from north to south and from west to east. Exposure to the waters of Lake Albert and the villages’ altitude above sea level were associated with the distribution. Infection prevalence and intensity peaked in the age groups between 10 and 29 years. Preschool children were highly infected (62.3%). Key risk factors were poor housing structure (odds ratio [OR] 2.1, 95% 95% confidence interval [CI] 1.02–4.35), close proximity to water bodies (OR 1.72, 95% CI 1.1–2.49), long-term residence in a community (OR 1.41, 95% CI 1.11–1.79), lack of latrine in the household (OR 2.00, 95% CI 1.11–3.60), and swimming (OR 2.53, 95% CI 1.20–5.32) and washing (OR 1.75, 95% CI 1.10–2.78) in local water bodies. Conclusions/Significance Our results show that S. mansoni is highly endemic and a major health concern in Ituri province, DRC. Infection prevalence and intensity, and the prevailing socioeconomic, environmental, and behavioural risk factors in Ituri reflect intense exposure and alarming transmission rates. A robust plan of action is urgently needed in the province.