Ivan G. Kazantsev

Statistical detection of defects in radiographic images in nondestructive testing

In this paper, we investigate applicability of statistical techniques for defect detection in radiographic images of welds. The defect detection procedure consists in a statistical hypothesis testing using several nonparametric tests. A comparison of rules derived for image thresholding for a given level of false alarm is presented. In this work we consider circular defects such as cavities and voids. Numerical experiments with real data are performed.

Fourier-based reconstruction for fully 3-D PET: optimization of interpolation parameters

Fourier-based approaches for three-dimensional (3-D) reconstruction are based on the relationship between the 3-D Fourier transform (FT) of the volume and the two-dimensional (2-D) FT of a parallel-ray projection of the volume. The critical step in the Fourier-based methods is the estimation of the samples of the 3-D transform of the image from the samples of the 2-D transforms of the projections on the planes through the origin of Fourier space, and vice versa for forward-projection (reprojection). The Fourier-based approaches have the potential for very fast reconstruction, but their straightforward implementation might lead to unsatisfactory results if careful attention is not paid to interpolation and weighting functions. In our previous work, we have investigated optimal interpolation parameters for the Fourier-based forward and back-projectors for iterative image reconstruction. The optimized interpolation kernels were shown to provide excellent quality comparable to the ideal sinc interpolator. This work presents an optimization of interpolation parameters of the 3-D direct Fourier method with Fourier reprojection (3D-FRP) for fully 3-D positron emission tomography (PET) data with incomplete oblique projections. The reprojection step is needed for the estimation (from an initial image) of the missing portions of the oblique data. In the 3D-FRP implementation, we use the gridding interpolation strategy, combined with proper weighting approaches in the transform and image domains. We have found that while the 3-D reprojection step requires similar optimal interpolation parameters as found in our previous studies on Fourier-based iterative approaches, the optimal interpolation parameters for the main 3D-FRP reconstruction stage are quite different. Our experimental results confirm that for the optimal interpolation parameters a very good image accuracy can be achieved even without any extra spectral oversampling, which is a common practice to decrease errors caused by interpolation in Fourier reconstruction

Optimal Ordering of Projections using Permutation Matrices and Angles between Projection Subspaces

This paper addresses the topic of the order of projection views when they are iteratively processed in a view-by-view manner. We investigate the 2D reconstruction problem with several processing order strategies. Numerical experiments giving a comparison of these strategies are presented.

On the accuracy of line-, strip- and fan-based algebraic reconstruction from few projections

In this paper we investigate accuracy features of algebraic reconstruction techniques applied in line-, strip- and fan-based tomographic systems using few views. A comparison of algorithms that calculate integrals along lines, strips and fans passing through the pixel grid of a digital image is presented. Numerical experiments with test object typical for plasma diagnostics problems are performed. Some possibilities of tomographic data acquisition system design with suboptimal parameters are discussed. ( 1999 Elsevier Science B.V. All rights reserved. Zusammenfassung

Iterative tomographic image reconstruction using Fourier-based forward and back-projectors

Iterative image reconstruction algorithms play an increasingly important role in modern tomographic systems, especially in emission tomography. With the fast increase of the sizes of the tomographic data, reduction of the computation demands of the reconstruction algorithms is of great importance. Fourier-based forward and back-projection methods have the potential to considerably reduce the computation time in iterative reconstruction. Additional substantial speed-tip of those approaches can be obtained utilizing powerful and cheap off-the-shelf FFT processing hardware. The Fourier reconstruction approaches are based on the relationship between the Fourier transform or the image and Fourier transformation of the parallel-ray projections. The critical two steps are the estimations of the samples of the projection transform, on the central section through the origin of Fourier space, from the samples of the transform of the image, and vice versa for back-projection. Interpolation errors are a limitation of Fourier-based reconstruction methods. We have applied min-max optimized Kaiser-Bessel interpolation within the nonuniform Fast Fourier transform (NUFFT) framework. This approach is particularly well suited to the geometries of PET scanners. Numerical and computer simulation results show that the min-max NUFFT approach provides substantially lower approximation errors in tomographic forward and back-projection than conventional interpolation methods, and that it is a viable candidate for fast iterative image reconstruction.

System and gram matrices of 3-D planogram data

This paper addresses the topic of geometric voxel-based approaches to forward and inverse problems of image reconstruction from planogram data. We investigate 2- and 3-D (two-, three-dimensional) planar detector configurations for which we reduce complex geometrical calculations of Gram and system matrices to a series of simple transformations like shifts and parallel projections. We use a direct algebraic approach, or natural pixels approach in reconstruction. The approach needs precomputation and inversion of the Gram matrix with entries as volumes of intersection of two arbitrary natural pixels-strips or parallelepipeds. Numerical experiments with small-scale test data are presented.

Inversion of 2D planogram data for finite-length detectors

In this work we investigate the problem of inverting data acquired from finite-length linear detectors in the 2D case. Image reconstruction algorithms using planogram data are usually derived under the initial assumption that the detectors are infinitely long and complete data can be acquired. This work deals with an analytical approach taking into account the finite length of the detectors explicitly and does not require operations on huge matrices. We derive integral equations for a single pair and for two pairs of linear detectors. This approach extends our previous work that dealt with explicit formulas for the elements of system and Gram matrices involved in 3D algebraic reconstruction from planograms, to a mathematical model in terms of continuous variables and a Fredholm-Volterra integral operator. As a result, fast filtered backprojection-like (FBP) algorithms based on the Hilbert and Fourier transforms (HFBP) are proposed

Limited angle tomography and ridge functions

Over the years a number of tomographic applications have emerged in which only limited ranges of projection views are available, such as breast imaging PET scanners based on a pair of parallel planar or curved detectors. This paper addresses the topic of ridge functions and their possible usefulness for tomographic reconstruction from projections distributed in a limited range of views. In this work we derive the basis for speeding up the ridge functions procedure of reconstruction front equally spaced directions which constitute a part of the whole range of views. We describe the possibilities of fast reconstruction in terms of analytical inversion of a certain matrix, for which inversion was known only for the case of a complete range of views. A new inversion formula reducing the dimensionality of the reconstruction problem is derived. Two problems of reconstruction from truncated fan beam projections within the limited angle of view are considered with relevance to PET breast scanning geometries. Algebraic techniques for their solution are proposed. Numerical results of test reconstructions are presented and possible extensions to the 3D case are discussed.

Geometric model of single scatter in PET

We present a mathematical model of single scatter in PET for a detector system possessing excellent energy resolution. Derivation of the model exploits assumptions widely used in PET and SPECT modalities for unscattered photons forward modeling. Integral equations describing a flux of photons emanating from the same annihilation event and undergoing a single scattering with a certain angle are derived in both 2D and 3D cases. Preliminary results of numerical experiments are presented.