Application of Sharafutdinov?s Ray Transform in Integrated Photoelasticity

Abstract

Abstract. We explain the main concepts centered around Sharafutdinov's ray transform, its kernel, and the extent to which it can be inverted. It is shown how the ray transform emerges naturally in any attempt to reconstruct optical and stress tensors within a photoelastic medium from measurements on the state of polarization of light beams passing through the strained medium. The problem of reconstruction of stress tensors is crucially related to the fact that the ray transform has a nontrivial kernel; the latter is described by a theorem for which we provide a new proof which is simpler and shorter as in Sharafutdinov's original work, as we limit our scope to tensors which are relevant to Photoelasticity. We explain how the kernel of the ray transform is related to the decomposition of tensor fields into longitudinal and transverse components. The merits of the ray transform as a tool for tensor reconstruction are studied by walking through an explicit example of reconstructing the σ33-component of the stress tensor in a cylindrical photoelastic specimen. In order to make the paper self-contained we provide a derivation of the basic equations of Integrated Photoelasticity which describe how the presence of stress within a photoelastic medium influences the passage of polarized light through the material.

X-ray Tomography of One-forms with Partial Data

“…By duality the formula N 0 f = 2(f * |•| 1−n ) holds also for compactly supported distributions and the normal operator becomes a map N 0 : E (R n ) → D (R n ). One can invert f from its X-ray transform using the normal operator by (14) f = c 0,n (−∆) 1/2 N 0 f, where c 0,n = (2π S n−2 ) −1 is a constant depending on the dimension and (−∆) 1/2 is the fractional Laplacian of order 1/2. The inversion formula ( 14) holds for f ∈ E (R n ) and for continuous functions f decreasing rapidly enough at infinity.…”

“…The transverse ray transform of one-forms has applications in the temperature measurements of flames [4,38]. For two-tensors the applications include also diffraction tomography [24], photoelasticity [14] and polarization tomography [40]. For a more comprehensive treatment see the reviews [36,37,41] and the references therein.…”

An Investigation of Radial Basis Function Method for Strain Reconstruction by Energy-Resolved Neutron Imaging

“…Given the longitudinal ray transformation (LRT) data plane by plane, an attempt could be made to recover the strain from the ray measurements in that plane using appropriate additional a priori information [8]. The Lamé system could be represented by finite-element or finite difference methods provided that the Lamé coefficients are known.…”

Generic Local Uniqueness and Stability in Polarization Tomography

“…The physical problem of recovering electrical properties of a medium based on polarization measurements has also been studied extensively in the optics literature. In particular much work has been done in the context of the photoelastic effect, where one well known technique of inversion is integrated photoelasticity ( [1], [2], [3], [8], [7]). …”