Abstract
It is widely recognised that the most popular manner of image representation is obtained by using an energy-preserving transform, like the Fourier transform. However, since the advent of computerised tomography in the 70s, another manner of image representation has also entered the center of interest. This new type is the projection space representation, obtained via the Radon transform. Methods to invert the Radon transform have resulted in a wealth of tomographic applications in a wide variety of disciplines. Functions that are reconstructed by inverting the Radon transform are scalar functions. However, over the last few decades there has been an increasing need for similar techniques that would perform tomographic reconstruction of a vector field when having integral information. Prior work at solving the reconstruction problem of 2-D vector field tomography in the continuous domain showed that projection data alone are insufficient for determining a 2-D vector field entirely and uniquely. This thesis treats the problem in the discrete domain and proposes a direct algebraic reconstruction technique that allows one to recover both components of a 2-D vector field at specific points, finite in number and arranged in a grid, of the 2-D domain by relying only on a finite number of line-integral data. In order to solve the reconstruction problem, the method takes advantage of the redundancy in the projection data, as a form of employing regularisation. Such a regularisation helps to overcome the stability deficiencies of the examined inverse problem. The effects of noise are also examined. The potential of the introduced method is demonstrated by presenting examples of complete reconstruction of static electric fields. The most practical sensor configuration in tomographic reconstruction problems is the regular positioning along the domain boundary. However, such an arrangement does not result in uniform distribution in the Radon parameter space, which is a necessary requirement to achieve accurate reconstruction results. On the other hand, sampling the projection space uniformly imposes serious constraints of space or time. In this thesis, motivated by the Radon transform theory, we propose to employ either interpolated data obtained at virtual sensors (that correspond to uniform sampling of the projection space) or probabilistic weights with the purpose of approximating uniformity in the projection space parameters. Simulation results demonstrate that when these two solutions are employed, about 30% decrease in the reconstruction error may be achieved. The proposed methods also increase the resilience to noise. On top of these findings, the method that employs weights offers an attractive solution because it does not increase the reconstruction time, since the weight calculation can be performed off-line.This thesis also looks at the 2-D vector field reconstruction problem from the aspect of sampling. To address sampling issues, the standard parallel scanning is treated. By using sampling theory, the limits to the sampling steps of the Radon parameters, so that no integral information is lost, are derived. Experiments show that when the proposed sampling bounds are violated, the reconstruction accuracy of the 2-D vector field deteriorates over the case where the proposed sampling criteria are imposed. It is shown that the employment of a scanning geometry that satisfies the proposed sampling requirements also increases the resilience to noise.