论文标题
关于球形平均值ra rantrans的通用反向注射公式的精确性
On the exactness of the universal backprojection formula for the spherical means Radon transform
论文作者
论文摘要
球形均值ra transform $ \ mathcal {m} f（x，r）$是由$ \ mathbb {r}^{n} $ in sphere $ s（x，r）$ s（x，r）$ s（x，r）$ r $的积分定义的。从数据$ \ MATHCAL {m} f（x，r）$中重建$ f $的问题，其中$ x $属于Hypersurface $γ\ subset \ subset \ subset \ mathbb {r}^{n} $ r \ in（0，\ infty）$在现代成像Modal中具有重要的应用，例如Posotim podaimity，例如Posotim copotic toncoction。当$γ$与边界$ \partialΩ$重合一个有限（凸）域$ω\ subset \ mathbb {r}^{n} $的$ \partialΩ$时，可以从$γ$上已知的球形均值中独特地恢复$ω$中支持的功能。我们对这种重建的明确反转公式感兴趣。 如果$γ= \partialΩ$，则仅在$γ$是椭圆形（或其部分情况之一）的情况下才知道这种公式。这引起了自然的问题：是否可以找到其他封闭的Hypersurfaces $γ$的明确反转公式？在本文中，我们证明，对于所谓的“通用反向反向反演公式”，它们向非层状域$ω$不可能进行，因此椭圆形构成了最大的封闭凸出式悬浮面，构成了这种公式所持的。
The spherical means Radon transform $\mathcal{M}f(x,r)$ is defined by the integral of a function $f$ in $\mathbb{R}^{n}$ over the sphere $S(x,r)$ of radius $r$ centered at a $x$, normalized by the area of the sphere. The problem of reconstructing $f$ from the data $\mathcal{M}f(x,r)$ where $x$ belongs to a hypersurface $Γ\subset\mathbb{R}^{n}$ and $r \in(0,\infty)$ has important applications in modern imaging modalities, such as photo- and thermo- acoustic tomography. When $Γ$ coincides with the boundary $\partialΩ$ of a bounded (convex) domain $Ω\subset\mathbb{R}^{n}$, a function supported within $Ω$ can be uniquely recovered from its spherical means known on $Γ$. We are interested in explicit inversion formulas for such a reconstruction. If $Γ=\partialΩ$, such formulas are only known for the case when $Γ$ is an ellipsoid (or one of its partial cases). This gives rise to the natural question: can explicit inversion formulas be found for other closed hypersurfaces $Γ$? In this article we prove, for the so-called "universal backprojection inversion formulas", that their extension to non-ellipsoidal domains $Ω$ is impossible, and therefore ellipsoids constitute the largest class of closed convex hypersurfaces for which such formulas hold.
