张量场二维扇束Radon变换的奇异值分解
Singular value decomposition for the 2D fan-beam Radon transform of tensor fields
摘要：
本文研究了单位圆盘中任意秩对称螺管二维张量场$m$的扇束Radon变换${\cal D}_m $，以$\mathbb D$为算符，从对象空间${\mathbf L}_{2}(\mathbb D; {\bf S}_m)$作用到数据空间$L_2([0,2π)\times[0,2π)).$利用Zernike多项式构造圆盘$\mathbb D$上螺线管张量场的正交多项式基${\bf s}^{(\pm m)}_{n,k}$，然后进行奇异值分解获得了操作员的${\cal D}_m $。扇束张量变换${\cal D}_m $的反演公式由该分解得出。由此得到的反演公式可用作层析滤波器，用于将已知张量场分裂成势场和螺线管场。数值结果。
英文摘要：
In this article we study the fan-beam Radon transform ${\cal D}_m $ of symmetrical solenoidal 2D tensor fields of arbitrary rank $m$ in a unit disc $\mathbb D$ as the operator, acting from the object space ${\mathbf L}_{2}(\mathbb D; {\bf S}_m)$ to the data space $L_2([0,2π)\times[0,2π)).$ The orthogonal polynomial basis ${\bf s}^{(\pm m)}_{n,k}$ of solenoidal tensor fields on the disc $\mathbb D$ was built with the help of Zernike polynomials and then a singular value decomposition (SVD) for the operator ${\cal D}_m $ was obtained. The inversion formula for the fan-beam tensor transform ${\cal D}_m $ follows from this decomposition. Thus obtained inversion formula can be used as a tomographic filter for splitting a known tensor field into potential and solenoidal parts. Numerical results are presented.
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