ABSTRACT We present an analytical method for calculating magnetic field gradients generated by arbitrary triangulated surfaces. Our work builds upon the results published by Pissanetzky and Xiang, who presented formulas for calculating the magnetic field of current‐carrying faceted surfaces. We show that the analytical gradient expressions can be computed considerably faster than finite field value differences. We also find that the aforementioned published expressions for the magnetic field can be simplified and optimized substantially. Closer inspection of the algorithms, for both field and gradient, reveals a number pathological parameter constellations, which require special treatment. We present a detailed discussion on this. Our results can be directly applied in the optimization of complex magnetic field coils, such as magnetic resonance gradient coils. © 2014 Wiley Periodicals, Inc. Concepts Magn Reson Part B (Magn Reson Engineering) 44B: 18–25, 2014

PURPOSE:A CT scanner measures the energy that is deposited in each channel of a detector array by x rays that have been partially absorbed on their way through the object. The measurement process is complex and quantitative measurements are always and inevitably associated with errors, so CT data must be preprocessed prior to reconstruction. In recent years, the authors have formulated CT sinogram preprocessing as a statistical restoration problem in which the goal is to obtain the best estimate of the line integrals needed for reconstruction from the set of noisy, degraded measurements. The authors have explored both penalized Poisson likelihood (PL) and penalized weighted least-squares (PWLS) objective functions. At low doses, the authors found that the PL approach outperforms PWLS in terms of resolution-noise tradeoffs, but at standard doses they perform similarly. The PWLS objective function, being quadratic, is more amenable to computational acceleration than the PL objective. In this work, the authors develop and compare two different methods for implementing PWLS sinogram restoration with the hope of improving computational performance relative to PL in the standard-dose regime. Sinogram restoration is still significant in the standard-dose regime since it can still outperform standard approaches and it allows for correction of effects that are not usually modeled in standard CT preprocessing. METHODS:The authors have explored and compared two implementation strategies for PWLS sinogram restoration: (1) A direct matrix-inversion strategy based on the closed-form solution to the PWLS optimization problem and (2) an iterative approach based on the conjugate-gradient algorithm. Obtaining optimal performance from each strategy required modifying the naive off-the-shelf implementations of the algorithms to exploit the particular symmetry and sparseness of the sinogram-restoration problem. For the closed-form approach, the authors subdivided the large matrix inversion into smaller coupled problems and exploited sparseness to minimize matrix operations. For the conjugate-gradient approach, the authors exploited sparseness and preconditioned the problem to speed up convergence. RESULTS:All methods produced qualitatively and quantitatively similar images as measured by resolution-variance tradeoffs and difference images. Despite the acceleration strategies, the direct matrix-inversion approach was found to be uncompetitive with iterative approaches, with a computational burden higher by an order of magnitude or more. The iterative conjugate-gradient approach, however, does appear promising, with computation times half that of the authors' previous penalized-likelihood implementation. CONCLUSIONS:Iterative conjugate-gradient based PWLS sinogram restoration with careful matrix optimizations has computational advantages over direct matrix PWLS inversion and over penalized-likelihood sinogram restoration and can be considered a good alternative in standard-dose regimes.

This paper investigates a reconstruction method for helical computed tomography which compensates for the motion artifacts in the thorax caused by patient breathing. The method takes into account a motion vector field determined from a four-dimensional (4-D) uncompensated image data set. Surface models of the lung and the ribs are tracked through the 4-D data set to create motion information within the entire thorax. Finally, an image is reconstructed using motion compensated back-projection. The results show that due to the use of shape models for the motion estimation, the method is fast and robust. Furthermore, since the surfaces are tracked individually, reconciling the opposite motion direction of the lung and rib cage is avoided in one motion vector field.