|
Longitudinal Ray Transform
In mathematics the longitudinal ray transform (LRT) is a generalization of the X-ray transform In mathematics, the X-ray transform (also called ray transform or John transform) is an integral transform introduced by Fritz John in 1938 that is one of the cornerstones of modern integral geometry. It is very closely related to the Radon transfo ... to symmetric tensor fields V.A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP 1994,. Chapter on-line version/ref> Let f_ be the components of a symmetric rank-m tesnor field ( m\ge) on Euclidean space \mathbf^n ( n \ge 2). For a unit vector \xi, , \xi, =1 and a point x \in \mathbf^n the longitudinal ray transform is defined as : g(x, \xi):= If(x, \xi)= \int\limits_^\infty f_(x+ s \xi) \xi_ \cdots \xi_\, \mathrms where summation over repeated indices is implied. The transform has a null-space, assuming the components are smooth and decay at infinity any f= dg, the symmetrized derivative of a rank m-1 tensor field g, satisfies ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
X-ray Transform
In mathematics, the X-ray transform (also called ray transform or John transform) is an integral transform introduced by Fritz John in 1938 that is one of the cornerstones of modern integral geometry. It is very closely related to the Radon transform, and coincides with it in two dimensions. In higher dimensions, the X-ray transform of a function is defined by integrating over line (geometry), lines rather than over hyperplanes as in the Radon transform. The X-ray transform derives its name from X-ray tomography (used in CT scans) because the X-ray transform of a function ''ƒ'' represents the attenuation data of a tomographic scan through an inhomogeneous medium whose density is represented by the function ''ƒ''. Inverse problem, Inversion of the X-ray transform is therefore of practical importance because it allows one to reconstruct an unknown density ''ƒ'' from its known attenuation data. In detail, if ''ƒ'' is a compact support, compactly supported continuous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saint-Venant's Compatibility Condition
In the mathematical theory of elasticity, Saint-Venant's compatibility condition defines the relationship between the strain \varepsilon and a displacement field \ u by :\epsilon_ = \frac \left( \frac + \frac \right) where 1\le i,j \le 3. Barré de Saint-Venant derived the compatibility condition for an arbitrary symmetric second rank tensor field to be of this form, this has now been generalized to higher rank symmetric tensor fields on spaces of dimension n\ge 2 Rank 2 tensor fields For a symmetric rank 2 tensor field F in n-dimensional Euclidean space (n \ge 2) the integrability condition takes the form of the vanishing of the Saint-Venant's tensor W(F) defined by :W_ = \frac + \frac - \frac -\frac The result that, on a simply connected domain W=0 implies that strain is the symmetric derivative of some vector field, was first described by Barré de Saint-Venant in 1860 and proved rigorously by Beltrami in 1886. For non-simply connected domains there are finite dimens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |