Let $f(x)$, $x\in\mathbb R^2$, be a piecewise smooth function with a jump discontinuity across a smooth surface $\mathcal S$. Let $f_{Λε}$ denote the Lambda tomography (LT) reconstruction of $f$ from its discrete Radon data $\hat f(α_k,p_j)$. The sampling rate along each variable is $\simε$. First, we compute the limit $f_0(\check x)=\lim_{ε\to0}εf_{Λε}(x_0+ε\check x)$ for a generic $x_0\in\mathcal S$. Once the limiting function $f_0(\check x)$ is known (which we call the discrete transition behavior, or DTB for short), the resolution of reconstruction can be easily found. Next, we show that straight segments of $\mathcal S$ lead to non-local artifacts in $f_{Λε}$, and that these artifacts are of the same strength as the useful singularities of $f_{Λε}$. We also show that $f_{Λε}(x)$ does not converge to its continuous analogue $f_Λ=(-Δ)^{1/2}f$ as $ε\to0$ even if $x\not\in\mathcal S$. Results of numerical experiments presented in the paper confirm these conclusions. We also consider a class of Fourier integral operators $\mathcal{B}$ with the same canonical relation as the classical Radon transform adjoint, and a class of distributions $g\in\mathcal{E}'(Z_n)$, $Z_n:=S^{n-1}\times\mathbb R$, and obtain easy to use formulas for the DTB when $\mathcal{B} g$ is computed from discrete data $g(α_{\vec k},p_j)$. Exact and LT reconstructions are particlular cases of this more general theory.
The Fast Proximal Gradient Method (FPGM) and the Monotone FPGM (MFPGM) for minimization of nonsmooth convex functions are introduced and applied to tomographic image reconstruction. Convergence properties of the sequence of objective function values are derived, including a $O\left(1/k^{2}\right)$ non-asymptotic bound. The presented theory broadens current knowledge and explains the convergence behavior of certain methods that are known to present good practical performance. Numerical experimentation involving computerized tomography image reconstruction shows the methods to be competitive in …
A new method for solving systems of linear algebraic equations of a special type arising in solving problems of image reconstruction has been proposed. This method, due to a certain symmetry of the matrix and the choice of the voxel numeration method for two-dimensional problems, allows us to divide the initial system of algebraic equations into two independent systems, which enables us to carry out the calculation in parallel. The dimension of the matrices of the resulting systems is 4 times less than the dimension of the original matrix, and these matrices are less dispersed.
This paper proposes a new two-step procedure for sparse-view tomographic image reconstruction. It is called RISING, since it combines an early-stopped Rapid Iterative Solver with a subsequent Iteration Network-based Gaining step. So far, regularized iterative methods have widely been used for X-ray computed tomography image reconstruction from low-sampled data, since they converge to a sparse solution in a suitable domain, as upheld by the Compressed Sensing theory. Unfortunately, their use is practically limited by their high computational cost which imposes to perform only a few iterations i…
Technical University of Denmark（丹麦技术大学）；Tufts University（塔夫茨大学）
We consider tomographic reconstruction using priors in the form of a dictionary learned from training images. The reconstruction has two stages: first we construct a tensor dictionary prior from our training data, and then we pose the reconstruction problem in terms of recovering the expansion coefficients in that dictionary. Our approach differs from past approaches in that a) we use a third-order tensor representation for our images and b) we recast the reconstruction problem using the tensor formulation. The dictionary learning problem is presented as a non-negative tensor factorization pro…
Unbiased estimators are introduced for averaged Bregman divergences which generalize Stein's Unbiased (Predictive) Risk Estimator, and the minimization of these estimators is proposed as a regularization parameter selection method for regularization of inverse problems. Numerical experiments are presented in order to show the performance of the proposed technique. Experimental results indicate a useful occurence of a concentration of measure phenomena and some implications of this hypothesis are analyzed.
While Computerized Tomography (CT) images can help detect disease such as Covid-19, regular CT machines are large and expensive. Cheaper and more portable machines suffer from errors in geometry acquisition that downgrades CT image quality. The errors in geometry can be represented with parameters in the mathematical model for image reconstruction. To obtain a good image, we formulate a nonlinear least squares problem that simultaneously reconstructs the image and corrects for errors in the geometry parameters. We develop an accelerated alternating minimization scheme to reconstruct the image …
In this paper we present a new two-level iterative algorithm for tomographic image reconstruction. The algorithm uses a regularization technique, which we call edge-preserving Laplacian, that preserves sharp edges between objects while damping spurious oscillations in the areas where the reconstructed image is smooth. Our numerical simulations demonstrate that the proposed method outperforms total variation (TV) regularization and it is competitive with the combined TV-L2 penalty. Obtained reconstructed images show increased signal-to-noise ratio and visually appealing structural features. Com…
Unmatched pairs of forward and back projectors are common in X-ray CT computations for large-scale problems; they are caused by the need for fast algorithms that best utilize the computer hardware, and it is an interesting and challenging task to develop fast and easy-to-use algorithms for these cases. Our approach is to use preconditioned GMRES, in the form of the AB- and BA-GMRES algorithms, to handle the unmatched normal equations associated with an unmatched pair. These algorithms are simple to implement, they rely only on computations with the available forward and back projectors, and th…
In this paper, we consider the problem of feature reconstruction from incomplete x-ray CT data. Such problems occurs, e.g., as a result of dose reduction in the context medical imaging. Since image reconstruction from incomplete data is a severely ill-posed problem, the reconstructed images may suffer from characteristic artefacts or missing features, and significantly complicate subsequent image processing tasks (e.g., edge detection or segmentation). In this paper, we introduce a novel framework for the robust reconstruction of convolutional image features directly from CT data, without the …
Superiorization of Preconditioned Conjugate Gradient Algorithms for Tomographic Image Reconstruction
Institute of Mathematical Science and Computation, University of São Paulo（圣保罗大学数学科学与计算研究所）；Department of Computer Science, The Graduate Center, City University of New York（纽约市立大学研究生院计算机科学系）；School of Electrical Engineering, Southwest Jiaotong University（西南交通大学电气工程学院）；Program in Electrical and Computer Engineering, Federal University of Technology（联邦技术大学电气与计算机工程项目）
