论文标题
椭圆形和双曲线ra radys;微局部特性和注入性
Ellipsoidal and hyperbolic Radon transforms; microlocal properties and injectivity
论文作者
论文摘要
我们提出了椭圆形和倍曲面ra晶体的新型微局部和注射率分析。我们引入了一种新的ra thronforment，$ r $，该变换定义了紧凑型$ l^2 $功能，$ f $的积分，而椭圆形和倍曲面的中心在平稳的连接表面，$ s $上。 $ r $被证明是傅立叶积分运算符（FIO），在我们的主要定理中，我们证明，如果连接$ f $的支撑并且没有由任何平面连接到$ s $的飞机，则$ r $满足了Bolker条件。在某些条件下，这是一个等效性。我们举例说明我们的理论可以应用。专门针对超声反射断层扫描（URT）感兴趣的圆柱形几何形状，我们证明了注射性结果并研究了可见的奇异性。此外，我们介绍了二维图像幻象的重建，并验证我们的微局部理论。
We present novel microlocal and injectivity analyses of ellipsoid and hyperboloid Radon transforms. We introduce a new Radon transform, $R$, which defines the integrals of a compactly supported $L^2$ function, $f$, over ellipsoids and hyperboloids with centers on a smooth connected surface, $S$. $R$ is shown to be a Fourier Integral Operator (FIO) and in our main theorem we prove that $R$ satisfies the Bolker condition if the support of $f$ is connected and not intersected by any plane tangent to $S$. Under certain conditions, this is an equivalence. We give examples where our theory can be applied. Focusing specifically on a cylindrical geometry of interest in Ultrasound Reflection Tomography (URT), we prove injectivity results and investigate the visible singularities. In addition, we present example reconstructions of image phantoms in two-dimensions, and validate our microlocal theory.
