Shape recovery for sparse‐data tomography

Abstract

A two-dimensional sparse-data tomographic problem is studied. The target is assumed to be a homogeneous object bounded by a smooth curve. A nonuniform rational basis splines (NURBS) curve is used as a computational representation of the boundary. This approach conveniently provides the result in a format readily compatible with computer-aided design software. However, the linear tomography task becomes a nonlinear inverse problem because of the NURBS-based parameterization. Therefore, Bayesian inversion with Markov chain Monte Carlo sampling is used for calculating an estimate of the NURBS control points. The reconstruction method is tested with both simulated data and measured X-ray projection data. The proposed method recovers the shape and the attenuation coefficient significantly better than the baseline algorithm (optimally thresholded total variation regularization), but at the cost of heavier computation.

Torus computed tomography

“…the space of distributions. By formal self-adjointness of I, we may define the X-ray transform on distributions f ∈ T by (7) […”

“…The most common regularization methods include Tikhonov regularization and truncated singular value decomposition (TSVD) which promote smoothness of reconstructions [22]. Other common regularization approaches include total variation (TV) regularization which promotes sparsity of reconstructions [31,8,24,7]. Another approach is to encode a priori information as a probability distribution and think the reconstruction problem as an Bayesian inverse problem for finding a posterior distribution [31,17,13,8,7].…”

“…Other common regularization approaches include total variation (TV) regularization which promotes sparsity of reconstructions [31,8,24,7]. Another approach is to encode a priori information as a probability distribution and think the reconstruction problem as an Bayesian inverse problem for finding a posterior distribution [31,17,13,8,7].…”

Probabilistic approach to limited-data computed tomography reconstruction

“…When using the spectral densities corresponding to the classical regularization methods in (21) and (22), the mean equation reduces to the classical solution (on the given basis). However, also for the classical regularization methods we can compute the variance function which gives uncertainty estimate for the solution which in the classical formulation is not available.…”

“…Statistical estimation methods play an important role in handling the illposedness of the problem by restating the inverse problem as a well-posed extension in a larger space of probability distributions [19]. Over the years there have been a lot of work on tomographic reconstruction from limited data using statistical methods (see, e.g., [14,20,21,22,23,24]).…”