Limited Projection 3D X-Ray Tomography Using the Maximum Entropy Method

It is well known from physics that the reconstruction of physical quantities from experimental data is often obstructed by incomplete information, the presence of noise and the ill-posed nature of the inversion problem. It was shown [1] that a Bayesian reconstruction (BR) in terms of the Maximum Entropy Method (MEM) combined with unbiased a priori knowledge, if available, is one way to overcome the difficulties. Similar problems occur while extracting useful information from incomplete data sets in technical applications. The solution of the deduced iteration procedure, if converged, gives the most probable one among all possible solutions. In case of radiographic techniques difficulties occur if there is no free access around the object or if the number of available radiographic projections is limited due to other reasons like restricted maximum exposure as often required for medical applications or economical aspects. This situation, characterized by a significant lack of data, makes it impossible to apply reconstruction algorithms which are usually used for computer tomography (CT). Other reconstruction algorithms can be found by introducing prior information (compare [1–6]) about the object and the structures of interest. Those algorithms meet practical requirements like robustness, reduction of experimental and numerical effort, or others. For NDE applications, e.g. the inspection of welds or castings, prior knowledge can be introduced from a practical point of view by assuming a binary or multi-material structure. This reduces significantly the number of permissible solutions and therefore the number of required radiographie projections.</p

Multi Step 3D X-Ray Tomography for a Limited Number of Projections and Views

During the last few years some attempts were made to modify the approaches for 3D reconstruction algorithms, based mainly upon Radon theory, in case of extreme lack of data, i.e. limited number of projections and views. The inverse Radon transform is not applicable in this case because of the insufficiently filled Radon space. Then the interpolation of the data, which are absent in the unfilled back projection space, is unattainable. In this case, some kind of a priori knowledge or structural constraints is required to restrict the permissible solutions [1–4]. The classical regularization procedure, also known as Tichonov-Miller regularization [5–7], can be applied for the stated problem as used in some CT applications, e.g. [1], where the a priori knowledge is introduced by a special functional type, which allows to reduce the required number of projections to about 100.</p

Bayesian reconstruction of images of objects with high-density inclusions with suppression of artifacts

The technology of three-dimensional Bayesian tomographic reconstruction of homogeneous objects with high-density inclusions is developed. The approach is based on preliminary correction of projections by extracting the data corresponding to X-rays passing through a high-density region, and replacing it with synthesized data obtained by two-dimensional interpolation. An original method for selecting interpolation points is proposed and a mathematical algorithm is described that ensures the implementation of two-dimensional interpolation correction of projections

3D X-Ray Reconstruction from Strongly Incomplete Noisy Data

Recently we reported [1–3] on the theory and the technique of reconstructing three- dimensional images of flaws and inclusions from an extremely limited number of cone- beam X-ray projections. The number of projections is chosen between two and seven and they are achieved in an observation angle smaller than 180 degrees. We introduced an approach using the Bayesian reconstruction (BR) with Gibbs prior in the form of mechanical models like noncausal Markov fields. As it was pointed out the convergence of the iteration reconstruction procedure depends on the chosen prior functional within a compact set of solutions. We investigated the capabilities of three types of a priori functional, which are represented by Gibbs energies. Corresponding to the supported structures, they were named (i) cluster support, (ii) plane support and (iii) phase support. While examining the phase support we made an effort to estimate the influence of Gaussian white noise on the quality of restoration. The noise was generated artificially and superimposed to the two dimensional x-ray images. It was shown that the algorithm was stable despite the disturbance of the noise. On the other hand it was observed that an increasing noise level leads to a noticeable deterioration of the quality of the restored image. The restoration of images from extremely incomplete and noisy data is a strict practical demand in many cases. This explains the effort to investigate the influence of noise to the reconstruction results.</p

Bayesian reconstruction of images of objects with high-density inclusions with suppression of artifacts

The technology of three-dimensional Bayesian tomographic reconstruction of homogeneous objects with high-density inclusions is developed. The approach is based on preliminary correction of projections by extracting the data corresponding to X-rays passing through a high-density region, and replacing it with synthesized data obtained by two-dimensional interpolation. An original method for selecting interpolation points is proposed and a mathematical algorithm is described that ensures the implementation of two-dimensional interpolation correction of projections

3D X-Ray Reconstruction from Strongly Incomplete Noisy Data

Recently we reported [1–3] on the theory and the technique of reconstructing three- dimensional images of flaws and inclusions from an extremely limited number of cone- beam X-ray projections. The number of projections is chosen between two and seven and they are achieved in an observation angle smaller than 180 degrees. We introduced an approach using the Bayesian reconstruction (BR) with Gibbs prior in the form of mechanical models like noncausal Markov fields. As it was pointed out the convergence of the iteration reconstruction procedure depends on the chosen prior functional within a compact set of solutions. We investigated the capabilities of three types of a priori functional, which are represented by Gibbs energies. Corresponding to the supported structures, they were named (i) cluster support, (ii) plane support and (iii) phase support. While examining the phase support we made an effort to estimate the influence of Gaussian white noise on the quality of restoration. The noise was generated artificially and superimposed to the two dimensional x-ray images. It was shown that the algorithm was stable despite the disturbance of the noise. On the other hand it was observed that an increasing noise level leads to a noticeable deterioration of the quality of the restored image. The restoration of images from extremely incomplete and noisy data is a strict practical demand in many cases. This explains the effort to investigate the influence of noise to the reconstruction results