Fourier- and spline-based approaches for sparse-data problems in medical tomography - La Riviere, Patrick Jean (ProQuest Dissertations & Theses, 2000)
Sparse-data problems requiring the fitting of smoothing or interpolating curves to sampled data arise commonly in medical imaging. This work develops and evaluates applications of Fourier- and spline-based interpolation and smoothing approaches to some of these problems. The Fourier-based approaches harness the computational efficiency of the Fast Fourier transform (FFT) algorithm to estimate a set of uniform, interpolated or smoothed samples of a function from one or more sets of its samples. The spline-based approaches rely on the fitting of cubic splines. While many of the results derived in the work have general significance, we focus on two specific applications: few-view emission computed tomography (ECT) and helical computed tomography (CT). Few-view ECT entails reconstruction of ECT images from a smaller number of projections than is usually used. We pursue sinogram preprocessing techniques that allow, under appropriate conditions, for reconstruction by filtered backprojection without the star artifacts that usually plague such images. Specifically, we focus on a two-step approach in which each projection is first smoothed to reduce noise and then additional projections are interpolated. The most successful approach entails the use of a spline-based smoothing technique in which each measured projection is fit with a cubic spline maximizing an objective function comprising a roughness-penalized Poisson-likelihood function. Additional projections are then interpolated by use of periodic spline interpolation. In helical CT, each projection is acquired at a different longitudinal position along the object being imaged. In order to reconstruct a transverse image, a complete set of projections at the appropriate longitudinal position should be interpolated from the measured ones. In this work, we develop Fourier- and spline-based approaches to this task for both single- and multi-slice helical CT. In single-slice helical CT, under certain clinically realistic conditions, the best of the approaches lead to reconstructed volumes with more uniform and isotropic resolution and noise properties than do standard approaches based on the use of linear interpolation. In multi-slice helical CT, some of the approaches lead to improvements over current approaches, but more significantly the work reveals some barriers posed by the multi-slice geometry to achieving these uniform and isotropic properties.