A Tensor B-Spline Approach for Solving the Diffusion PDE With Application to Optical Diffusion Tomography

Abstract

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.

Tensor B-Spline Numerical Methods for PDEs: a High-Performance Alternative to FEM

“…These truncated B-splines result in non-separable kernels. We refer the reader to [14,15] where an efficient method for the integration of such nonseparable kernels was proposed. Usually, the number of such kernels is small in comparison to the number of domain kernels.…”

“…9, a) requires assembling a sparse matrix from B-spline kernels. It was shown that this approach appears to be the least efficient [14,15]. This follows from the overhead due to the sparse matrix format, from non-regular memory access, from a very low flop-to-byte ratio [21,22], and from problems concerning load imbalance [23].…”

“…At each iteration it is multiplied with c k . This algorithm is the fastest [14,15] but still is not feasible for large systems (Fig. 9, b).…”

“…Compared to matrices, such tensors allow us to represent multidimensional structures in a more compact and natural way. This computational tensor algebra approach [12] makes it possible to develop highly efficient numerical algorithms, and several benefits of combining tensors and Bsplines have been shown in the context of multidimensional signal reconstruction [13] and for solving diffusion PDEs in Optical Diffusion Tomography [14,15].…”

A New Robot Navigation Algorithm Based on a Double-Layer Ant Algorithm and Trajectory Optimization

“…Hence these discontinuous segments must be smoothed. B-spline curve is one of the most efficient curve interpolations and has been widely applied in many disciplines, such as medical imaging [27], geometric modeling [28], surface reconstruction [29] and position control [30]. With the properties of the B-spline curve, this interpolation scheme is practically useful for path smoothing [31], [32].…”