In previous work by Baaz and Iemhoff, a Gentzen calculus for intuitionistic logic with existence predicate is presented that satisfies partial cut elimination and Craigs interpolation property; it is ...also conjectured that interpolation fails for the implication-free fragment. In this paper an equivalent calculus is introduced that satisfies full cut elimination and allows a direct proof of interpolation via Maeharas lemma. In this way, it is possible to obtain much simpler interpolants and to better understand and (partly) overcome the failure of interpolation for the implication-free fragment.
Purpose: VMAT has been widely used due to enhanced delivery efficiency with high angular coverage. However, general VMAT optimized by single aperture at the control points cannot meet sufficient ...intensity modulation in some directions. To improve the plan quality over the limit, this work proposes to use the non‐uniform segments sampling for VMAT planning by fluence‐map based optimization approach. Methods: Reweighted L1‐minimization can optimize the fluence‐map for VMAT as it successfully eliminates unnecessary information by further simplifying the fluence‐map. To increase the connectivity of adjacent fluence‐maps for VMAT planning, this study considers the similarity of neighboring fluence‐maps by combining the simple penalized term with the reweighted L1‐minmization. Importantly, instead of taking one segment uniformly, the non‐uniform segments sampling that takes additional segment at certain directions is introduced to enhance the plan quality for VMAT plans. To choose the appropriate directions giving great benefits in plan quality from the additional segment, we sequentially quantified the respective cost at a specific field by adding extra segment to the field. The extra segments were assigned to 6 to 8 field directions with the lowest costs, which were redistributed into adjacent fields and linked by linear interpolation for VMAT delivery. Prostate patient data was employed to evaluate and compare the uniform/non‐uniform segment sampled VMAT in plan quality and treatment time. Results: Introducing the additional term into optimization effectively enhanced the similarity of adjacent fluence‐maps, which were used to form uniform/non‐uniform segments sampled VMAT plans. Significantly, non‐uniform sampling VMAT formed by assigning 8 segments to the 8 chosen fields has better dose conformity to the target (0.8245 to 0.8409) than uniform sampled VMAT plan at the small expense of the estimated treatment time (63s to 65s). Conclusion: Fluence‐map based VMAT planning incorporated by non‐uniform segments sampling can achieve efficient VMAT plan with enhanced plan quality.
In this work we construct an Hermite interpolant starting from basis functions that satisfy a Lagrange property. In fact, we extend and generalise an iterative approach, introduced by Cirillo and ...Hormann (2018) for the Floater-Hormann family of interpolants. Secondly, we apply this scheme to produce an effective barycentric rational trigonometric Hermite interpolant at general ordered nodes using as basis functions the ones of the trigonometric interpolant introduced by Berrut (1988). For an easy computational construction, we calculate analytically the differentation matrix. Finally, we conclude with various examples and a numerical study of the convergence at equidistant nodes and conformally mapped nodes.
Let Ā=(A0,A1), B̄=(B0,B1) be Banach couples, let E be a Banach space and let T be a bilinear operator such that ‖T(a,b)‖E≤Mj‖a‖Aj‖b‖Bj for a∈A0∩A1, b∈B0∩B1, j=0,1. If T:Aj∘×Bj∘⟶E compactly for j=0 ...or 1, we show that T may be uniquely extended to a compact bilinear operator from the complex interpolation spaces generated by Ā and B̄ to E. Furthermore, the corresponding result for the real method is given and we also study the case when E is replaced by a couple (E0,E1) of Banach function spaces on the same measure space.
Time-domain backprojection algorithms are widely used in state-of-the-art synthetic aperture radar (SAR) imaging systems that are designed for applications where motion error compensation is ...required. These algorithms include an interpolation procedure, under which an unknown SAR range-compressed data parameter is estimated based on complex-valued SAR data samples and backprojected into a defined image plane. However, the phase of complex-valued SAR parameters estimated based on existing interpolators does not contain correct information about the range distance between the SAR imaging system and the given point of space in a defined image plane, which affects the quality of reconstructed SAR scenes. Thus, a phase-control procedure is required. This paper introduces extensions of existing linear, cubic, and sinc interpolation algorithms to interpolate complex-valued SAR data, where the phase of the interpolated SAR data value is controlled through the assigned a priori known range time that is needed for a signal to reach the given point of the defined image plane and return back. The efficiency of the extended algorithms is tested at the Nyquist rate on simulated and real data at THz frequencies and compared with existing algorithms. In comparison to the widely used nearest-neighbor interpolation algorithm, the proposed extended algorithms are beneficial from the lower computational complexity perspective, which is directly related to the offering of smaller memory requirements for SAR image reconstruction at THz frequencies.
Let x0,x1,…,xn, be a set of n+1 distinct real numbers (i.e., xi≠xj, for i≠j) and ym,k, for m=0,1,…,n, and k=0,1,…,nm, with nm∈N, be given of real numbers, we know that there exists a unique ...polynomial pN−1 of degree N−1 where N=∑i=0n(ni+1), such that pN−1(k)(xm)=ym,k, for m=0,1,…,n and k=0,1,…,nm. pN−1 is the Hermite interpolation polynomial for the set {(xm,ym,k),m=0,1,…,n,k=0,1,…,nm}. The polynomial pN−1 can be computed by using the Lagrange generalized polynomials. Recently Messaoudi et al. (2018) presented a new algorithm for computing the Hermite interpolation polynomials, for a general case, called Generalized Recursive Polynomial Interpolation Algorithm (GRPIA), this algorithm has been developed without using the Matrix Recursive Interpolation Algorithm (Jbilou and Messaoudi, 1999). Messaoudi et al. (2017) presented also a new algorithm called Matrix Recursive Polynomial Interpolation Algorithm (MRPIA), for a particular case where nm=μ=1, for m=0,1,…,n. In this paper we will give the version of the MRPIA for a particular case nm=μ≥0, for m=0,1,…,n. We will recall the result of the existence of the polynomial pN−1 for this case, some of its properties will also be given. Using the MRPIA, a method will be proposed for the general case, where nm, for some m, are different and some examples will also be given.
Purpose
To develop an improved k‐space reconstruction method using scan‐specific deep learning that is trained on autocalibration signal (ACS) data.
Theory
Robust artificial‐neural‐networks for ...k‐space interpolation (RAKI) reconstruction trains convolutional neural networks on ACS data. This enables nonlinear estimation of missing k‐space lines from acquired k‐space data with improved noise resilience, as opposed to conventional linear k‐space interpolation‐based methods, such as GRAPPA, which are based on linear convolutional kernels.
Methods
The training algorithm is implemented using a mean square error loss function over the target points in the ACS region, using a gradient descent algorithm. The neural network contains 3 layers of convolutional operators, with 2 of these including nonlinear activation functions. The noise performance and reconstruction quality of the RAKI method was compared with GRAPPA in phantom, as well as in neurological and cardiac in vivo data sets.
Results
Phantom imaging shows that the proposed RAKI method outperforms GRAPPA at high (≥4) acceleration rates, both visually and quantitatively. Quantitative cardiac imaging shows improved noise resilience at high acceleration rates (rate 4:23% and rate 5:48%) over GRAPPA. The same trend of improved noise resilience is also observed in high‐resolution brain imaging at high acceleration rates.
Conclusion
The RAKI method offers a training database‐free deep learning approach for MRI reconstruction, with the potential to improve many existing reconstruction approaches, and is compatible with conventional data acquisition protocols.
Since the momentum interpolation method was proposed by Rhie and Chow in 1983, it has been widely used in studies of Computational Fluids Dynamics (CFD). The conventional momentum interpolation ...methods were designed across the edge between two neighboring cells. In this study, an alternative momentum interpolation method, called multi-point momentum interpolation correction (IC) method, is proposed. The proposed IC method is distinguished from the conventional cross-edge momentum interpolation methods by correcting and improving the edge velocity with the interpolated values of its surrounding edges. Examples including analytic, experimental and field cases demonstrated that the proposed IC method is generally capable of improving the convergence process and numerical accuracy.
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