Spline- and tensor-based signal reconstruction : from structure analysis to high-performance algorithms to multiplatform implementations and medical applications

The problem of signal reconstruction is of fundamental practical value for many applications associated with the field of signal and image processing. This work considers a particular setting where the problem is formulated using a spline-based variational approach. We mainly concentrate on the following four problem-related aspects: 1). analysis of the problem structure, 2). structure-driven derivation of high-performance solving algorithms, 3). high-performance
algorithm implementations and 4). translation of these results to medical applications.
The second chapter of this work presents a tensor-based abstraction for formulation and efficient solving of problems arising in the field of multidimensional data processing. In contrast to traditional matrix abstraction the proposed approach allows to formulate tensor structured problems in an explicitly multidimensional way with preservation of the underlying structure of
computations that, in turn, facilitates the derivation of highly efficient solving algorithms. In addition to being a very helpful tool for our specific problem, the proposed tensor framework with its differentiating features is well suitable for implementing a tensor programming language that offers self-optimized computations of tensor expressions by semantic analysis of their terms.
The third chapter presents a practical example how the proposed tensor abstraction can be used for solving a problem of tensor B-spline-based variational reconstruction of large multidimensional images from irregularly sampled data. Based on our tensor framework we performed a detailed analysis of the problem formulation and derived highly efficient iterative solving algorithm, which offers high computational performance when implemented on computing platforms such as multi-core and GPGPU. We successfully applied the proposed approach to a real-life medical problem of ultrasound image reconstruction from a very large set of four-dimensional (3-D+time) non-uniform measurements.
The fourth chapter presents an alternative approach to the problem of variational signal reconstruction that is based on inverse recursive filtering. We revisited a B-spline-based formulation of this problem via a detailed analysis of the problem structure moving from the uniform towards non-uniform sampling settings. As a result we derived highly efficient algorithms for computing smoothing splines of any degree with an optimal choice of regularization parameter. We extended the presented approach to higher dimensions and showed how a rich variety of non-separable multidimensional smoothing spline operators and the corresponding solutions can be computed with high efficiency. We successfully applied the proposed inverse recursive filtering approach to the problem of medical Optical Coherence Tomography.
We conclude our work by presenting a high-level approach to software/hardware co-design of high-performance streaming data processing systems on FPGA. This approach allows to develop hybrid application specific system designs by combining the flexibility of multi-processor-based systems and high-performance of dedicated hardware components. A high-level programming model that is at the center of the approach along with an integrated development environment, implemented based on its principles, allow software and signal processing engineers who are not FPGA experts to design high-performance hardware architectures in a short time. We show some examples of how the developed framework can be efficiently used for implementation of our tensor- and spline-based algorithms.