论文标题
用于Sobolev空间中非周期函数的最佳正交公式及其应用于CT图像重建
Optimal quadrature formulas for non-periodic functions in Sobolev space and its application to CT image reconstruction
论文作者
论文摘要
在本文中，构建了SARD的最佳正交公式，用于在Sobolev Space $ l_2^^{Mather-in \ Mathbb {r} $中的数值整合$ \ int_a^be^{2πioming $ m $ th订单派生。在这里，使用微分运算符的离散模拟$ \ frac {d^{2m}} {d x^{2m}} $，获得了最佳系数的显式公式。获得的最佳正交公式的收敛顺序为$ O（H^m）$。作为一个应用程序，我们实现了过滤后的反射（FBP）算法，该算法是计算机断层扫描（CT）的众所周知的图像重建算法。通过使用第二个和第三阶的最佳正交公式近似傅立叶变换及其反演，我们观察到重建算法的准确性得到了提高。在数值实验中，我们比较了通过使用常规FBP的提议的最佳正交公式获得的重建图像的质量，其中使用快速傅立叶变换来计算傅立叶变换及其反转。在噪声测试中，所提出的算法比常规FBP提供了更可靠的结果。
In the present paper, optimal quadrature formulas in the sense of Sard are constructed for numerical integration of the integral $\int_a^be^{2πiωx}φ(x)d x$ with $ω\in \mathbb{R}$ in the Sobolev space $L_2^{(m)}[a,b]$ of complex-valued functions which are square integrable with $m$-th order derivative. Here, using the discrete analogue of the differential operator $\frac{d^{2m}}{d x^{2m}}$, the explicit formulas for optimal coefficients are obtained. The order of convergence of the obtained optimal quadrature formula is $O(h^m)$. As an application, we implement the filtered back-projection (FBP) algorithm, which is a well-known image reconstruction algorithm for computed tomography (CT). By approximating Fourier transforms and its inversion using the proposed optimal quadrature formula of the second and third orders, we observe that the accuracy of the reconstruction algorithm is improved. In numerical experiments, we compare the quality of the reconstructed image obtained by using the proposed optimal quadrature formulas with the conventional FBP, in which fast Fourier transform is used for the calculation of Fourier transform and its inversion. In the noise test, the proposed algorithm provides more reliable results against the noise than the conventional FBP.
