A box spline calculus for the discretization of computed tomography reconstruction problems

Exact gram filtering and efficient back projection for iterative CT reconstruction

PURPOSE

Forward and back-projections are the basis of all model-based iterative reconstruction (MBIR) methods. However, computing these accurately is time consuming. In this paper, we present a method for model-based iterative reconstruction in parallel X-ray beam geometry that utilizes a Gram filter to efficiently implement forward and back projection.

METHODS

We propose using voxel-basis and modeling its footprint in a box spline framework to calculate the Gram filter exactly and improve the performance of back-projection. In the special case of parallel X-ray beam geometry, the forward and back-projection can be implemented by an estimated Gram filter efficiently if the sinogram signal is bandlimited. In this paper, a specialized sinogram interpolation method is proposed to eliminate the bandlimited prerequisite and thus improve the reconstruction accuracy. We build on this idea by utilizing the continuity of the voxel-basis' footprint, which provides a more accurate sinogram interpolation and further improves the efficiency and quality of back-projection. In addition, the detector blur effect can be efficiently accounted for in our method to better handle realistic scenarios.

RESULTS

The proposed method is tested on both phantom and real CT images under different resolutions, sinogram sampling steps, and noise levels. The proposed method consistently outperforms other state-of-the-art projection models in terms of speed and accuracy for both back-projection and reconstruction.

CONCLUSIONS

We proposed a iterative reconstruction methodology for 3D parallel-beam X-ray CT reconstruction. Our experimental results demonstrate that the proposed methodology is accurate, fast, and reproducible, and outperforms alternative state-of-the-art projection models on both back-projection and reconstruction results significantly. This article is protected by copyright. All rights reserved.

Numerically robust tetrahedron-based tomographic forward and backward projectors on parallel architectures

X-ray tomographic reconstruction typically uses voxel basis functions to represent volumetric images. Due to the structure in voxel basis representations, efficient ray-tracing methods exist allowing fast, GPU accelerated implementations. Tetrahedral mesh basis functions are a valuable alternative to voxel based image representations as they provide flexible, inhomogeneous partitions which can be used to provide reconstructions with reduced numbers of elements or with arbitrarily fine object surface representations. We thus present a robust parallelizable ray-tracing method for volumetric tetrahedral domains developed specifically for Computed Tomography image reconstruction. Tomographic image reconstruction requires algorithms that are robust to numerical errors in floating point arithmetic whilst typical data sizes encountered in tomography require the algorithm to be parallelisable in GPUs which leads to additional constraints on algorithm choices. Based on these considerations, this article presents numerical solutions to the design of efficient ray-tracing algorithms for the projection and backprojection operations. Initial reconstruction results using CAD data to define a triangulation of the domain demonstrate the advantages of our method and contrast tetrahedral mesh based reconstructions to voxel based methods.

A Look-Up Table-Based Ray Integration Framework for 2-D/3-D Forward and Back Projection in X-Ray CT

Iterative algorithms have become increasingly popular in computed tomography (CT) image reconstruction, since they better deal with the adverse image artifacts arising from low radiation dose image acquisition. But iterative methods remain computationally expensive. The main cost emerges in the projection and back projection operations, where accurate CT system modeling can greatly improve the quality of the reconstructed image. We present a framework that improves upon one particular aspect-the accurate projection of the image basis functions. It differs from current methods in that it substitutes the high computational complexity associated with accurate voxel projection by a small number of memory operations. Coefficients are computed in advance and stored in look-up tables parameterized by the CT system's projection geometry. The look-up tables only require a few kilobytes of storage and can be efficiently accelerated on the GPU. We demonstrate our framework with both numerical and clinical experiments and compare its performance with the current state-of-the-art scheme-the separable footprint method.

High-Quality Parallel-Ray X-Ray CT Back Projection Using Optimized Interpolation

We propose a new, cost-efficient method for computing back projections in parallel-ray X-ray CT. Forward and back projections are the basis of almost all X-ray CT reconstruction methods, but computing these accurately is costly. In the special case of parallel-ray geometry, it turns out that reconstruction requires back projection only. One approach to accelerate the back projection is through interpolation: fit a continuous representation to samples of the desired signal, then sample it at the required locations. Instead, we propose applying a prefilter that has the effect of orthogonally projecting the underlying signal onto the space spanned by the interpolator, which can significantly improve the quality of the interpolation. We then build on this idea by using oblique projection, which simplifies the computation while giving effectively the same improvement in quality. Our experiments on analytical phantoms show that this refinement can improve the reconstruction quality for both filtered back projection and iterative reconstruction in the high-quality regime, i.e., with low noise and many measurements.

Fast 3D reconstruction method for differential phase contrast X-ray CT

We present a fast algorithm for fully 3D regularized X-ray tomography reconstruction for both traditional and differential phase contrast measurements. In many applications, it is critical to reduce the X-ray dose while producing high-quality reconstructions. Regularization is an excellent way to do this, but in the differential phase contrast case it is usually applied in a slice-by-slice manner. We propose using fully 3D regularization to improve the dose/quality trade-off beyond what is possible using slice-by-slice regularization. To make this computationally feasible, we show that the two computational bottlenecks of our iterative optimization process can be expressed as discrete convolutions; the resulting algorithms for computation of the X-ray adjoint and normal operator are fast and simple alternatives to regridding. We validate this algorithm on an analytical phantom as well as conventional CT and differential phase contrast measurements from two real objects. Compared to the slice-by-slice approach, our algorithm provides a more accurate reconstruction of the analytical phantom and better qualitative appearance on one of the two real datasets.

Isosurface Visualization of Data with Nonparametric Models for Uncertainty

The problem of isosurface extraction in uncertain data is an important research problem and may be approached in two ways. One can extract statistics (e.g., mean) from uncertain data points and visualize the extracted field. Alternatively, data uncertainty, characterized by probability distributions, can be propagated through the isosurface extraction process. We analyze the impact of data uncertainty on topology and geometry extraction algorithms. A novel, edge-crossing probability based approach is proposed to predict underlying isosurface topology for uncertain data. We derive a probabilistic version of the midpoint decider that resolves ambiguities that arise in identifying topological configurations. Moreover, the probability density function characterizing positional uncertainty in isosurfaces is derived analytically for a broad class of nonparametric distributions. This analytic characterization can be used for efficient closed-form computation of the expected value and variation in geometry. Our experiments show the computational advantages of our analytic approach over Monte-Carlo sampling for characterizing positional uncertainty. We also show the advantage of modeling underlying error densities in a nonparametric statistical framework as opposed to a parametric statistical framework through our experiments on ensemble datasets and uncertain scalar fields.

Sparse stochastic processes and discretization of linear inverse problems

We present a novel statistically-based discretization paradigm and derive a class of maximum a posteriori (MAP) estimators for solving ill-conditioned linear inverse problems. We are guided by the theory of sparse stochastic processes, which specifies continuous-domain signals as solutions of linear stochastic differential equations. Accordingly, we show that the class of admissible priors for the discretized version of the signal is confined to the family of infinitely divisible distributions. Our estimators not only cover the well-studied methods of Tikhonov and l1-type regularizations as particular cases, but also open the door to a broader class of sparsity-promoting regularization schemes that are typically nonconvex. We provide an algorithm that handles the corresponding nonconvex problems and illustrate the use of our formalism by applying it to deconvolution, magnetic resonance imaging, and X-ray tomographic reconstruction problems. Finally, we compare the performance of estimators associated with models of increasing sparsity.

Fast iterative reconstruction of differential phase contrast X-ray tomograms

Differential phase-contrast is a recent technique in the context of X-ray imaging. In order to reduce the specimen's exposure time, we propose a new iterative algorithm that can achieve the same quality as FBP-type methods, while using substantially fewer angular views. Our approach is based on 1) a novel spline-based discretization of the forward model and 2) an iterative reconstruction algorithm using the alternating direction method of multipliers. Our experimental results on real data suggest that the method allows to reduce the number of required views by at least a factor of four.