Radial Basis Functions: Biomedical Applications and Parallelization

Radial basis function (RBF) is a real-valued function whose values depend only on the distances between an interpolation point and a set of user-specified points called centers. RBF interpolation is one of the primary methods to reconstruct functions from multi-dimensional scattered data. Its abilities to generalize arbitrary space dimensions and to provide spectral accuracy have made it particularly popular in different application areas, including but not limited to: finding numerical solutions of partial differential equations (PDEs), image processing, computer vision and graphics, deep learning and neural networks, etc. The present thesis discusses three applications of RBF interpolation in biomedical engineering areas: (1) Calcium dynamics modeling, in which we numerically solve a set of PDEs by using meshless numerical methods and RBF-based interpolation techniques; (2) Image restoration and transformation, where an image is restored from its triangular mesh representation or transformed under translation, rotation, and scaling, etc. from its original form; (3) Porous structure design, in which the RBF interpolation used to reconstruct a 3D volume containing porous structures from a set of regularly or randomly placed points inside a user-provided surface shape. All these three applications have been investigated and their effectiveness has been supported with numerous experimental results. In particular, we innovatively utilize anisotropic distance metrics to define the distance in RBF interpolation and apply them to the aforementioned second and third applications, which show significant improvement in preserving image features or capturing connected porous structures over the isotropic distance-based RBF method. Beside the algorithm designs and their applications in biomedical areas, we also explore several common parallelization techniques (including OpenMP and CUDA-based GPU programming) to accelerate the performance of the present algorithms. In particular, we analyze how parallel programming can help RBF interpolation to speed up the meshless PDE solver as well as image processing. While RBF has been widely used in various science and engineering fields, the current thesis is expected to trigger some more interest from computational scientists or students into this fast-growing area and specifically apply these techniques to biomedical problems such as the ones investigated in the present work

An Efficient Algorithm for Video Super-Resolution Based On a Sequential Model

In this work, we propose a novel procedure for video super-resolution, that is the recovery of a sequence of high-resolution images from its low-resolution counterpart. Our approach is based on a "sequential" model (i.e., each high-resolution frame is supposed to be a displaced version of the preceding one) and considers the use of sparsity-enforcing priors. Both the recovery of the high-resolution images and the motion fields relating them is tackled. This leads to a large-dimensional, non-convex and non-smooth problem. We propose an algorithmic framework to address the latter. Our approach relies on fast gradient evaluation methods and modern optimization techniques for non-differentiable/non-convex problems. Unlike some other previous works, we show that there exists a provably-convergent method with a complexity linear in the problem dimensions. We assess the proposed optimization method on {several video benchmarks and emphasize its good performance with respect to the state of the art.}Comment: 37 pages, SIAM Journal on Imaging Sciences, 201

Retrospective Illumination Correction of Retinal Images

A method for correction of nonhomogenous illumination based on optimization of parameters of B-spline shading model with respect to Shannon's entropy is presented. The evaluation of Shannon's entropy is based on Parzen windowing method (Mangin, 2000) with the spline-based shading model. This allows us to express the derivatives of the entropy criterion analytically, which enables efficient use of gradient-based optimization algorithms. Seven different gradient- and nongradient-based optimization algorithms were initially tested on a set of 40 simulated retinal images, generated by a model of the respective image acquisition system. Among the tested optimizers, the gradient-based optimizer with varying step has shown to have the fastest convergence while providing the best precision. The final algorithm proved to be able of suppressing approximately 70% of the artificially introduced non-homogenous illumination. To assess the practical utility of the method, it was qualitatively tested on a set of 336 real retinal images; it proved the ability of eliminating the illumination inhomogeneity substantially in most of cases. The application field of this method is especially in preprocessing of retinal images, as preparation for reliable segmentation or registration

Quadtree Structured Approximation Algorithms

The success of many image restoration algorithms is often due to their ability to sparsely describe the original signal. Many sparse promoting transforms exist, including wavelets, the so called ‘lets’ family of transforms and more recent non-local learned transforms. The first part of this thesis reviews sparse approximation theory, particularly in relation to 2-D piecewise polynomial signals. We also show the connection between this theory and current state of the art algorithms that cover the following image restoration and enhancement applications: denoising, deconvolution, interpolation and multi-view super resolution. In [63], Shukla et al. proposed a compression algorithm, based on a sparse quadtree decomposition model, which could optimally represent piecewise polynomial images. In the second part of this thesis we adapt this model to image restoration by changing the rate-distortion penalty to a description-length penalty. Moreover, one of the major drawbacks of this type of approximation is the computational complexity required to find a suitable subspace for each node of the quadtree. We address this issue by searching for a suitable subspace much more efficiently using the mathematics of updating matrix factorisations. Novel algorithms are developed to tackle the four problems previously mentioned. Simulation results indicate that we beat state of the art results when the original signal is in the model (e.g. depth images) and are competitive for natural images when the degradation is high.Open Acces

An investigation of methods of surface estimation with application to the interpolation of antenna patterns

The problem of estimating a surface from a set of discrete measurements lying along straight lines is considered. This situation arises when one attempts to determine the S-Band antenna gain pattern for the Space Shuttle, from measurements taken at several ground stations. The results of previous investigators concerned with the performance of surface approximation techniques for the present application, are extended in this study by examining the case where the data samples are corrupted by measurement noise. Results have been obtained using least-squares approximation with bicubic B-spline basis functions, and for an interpolation algorithm in conjunction with a spatial smoothing filter. Because of the nature of the data acquisition and the impracticality of the least-squares algorithm when many sample points are used, the application of the Kalman filter to the surface estimation problem is discussed, although no numerical results were obtained using this approach. It is shown that a direct application of Kalman filter theory yields a filter algorithm which would be extremely difficult to implement. Based on the applications of reduced-order, suboptimal filters to image processing, a suboptimal approximation to the Kalman filter, applied to the surface estimation problem, is considered. The use of a decentralized estimation approach to this problem is briefly examined --Abstract, page ii