21,431 research outputs found
Histogram Tomography
In many tomographic imaging problems the data consist of integrals along
lines or curves. Increasingly we encounter "rich tomography" problems where the
quantity imaged is higher dimensional than a scalar per voxel, including
vectors tensors and functions. The data can also be higher dimensional and in
many cases consists of a one or two dimensional spectrum for each ray. In many
such cases the data contain not just integrals along rays but the distribution
of values along the ray. If this is discretized into bins we can think of this
as a histogram. In this paper we introduce the concept of "histogram
tomography". For scalar problems with histogram data this holds the possibility
of reconstruction with fewer rays. In vector and tensor problems it holds the
promise of reconstruction of images that are in the null space of related
integral transforms. For scalar histogram tomography problems we show how bins
in the histogram correspond to reconstructing level sets of function, while
moments of the distribution are the x-ray transform of powers of the unknown
function. In the vector case we give a reconstruction procedure for potential
components of the field. We demonstrate how the histogram longitudinal ray
transform data can be extracted from Bragg edge neutron spectral data and
hence, using moments, a non-linear system of partial differential equations
derived for the strain tensor. In x-ray diffraction tomography of strain the
transverse ray transform can be deduced from the diffraction pattern the full
histogram transverse ray transform cannot. We give an explicit example of
distributions of strain along a line that produce the same diffraction pattern,
and characterize the null space of the relevant transform.Comment: Small corrections from last versio
The Baker-Akhiezer function and factorization of the Chebotarev-Khrapkov matrix
A new technique is proposed for the solution of the Riemann-Hilbert problem
with the Chebotarev-Khrapkov matrix coefficient
, ,
is a zero-trace polynomial matrix, and is the unit matrix. This
problem has numerous applications in elasticity and diffraction theory. The
main feature of the method is the removal of the essential singularities of the
solution to the associated homogeneous scalar Riemann-Hilbert problem on the
hyperelliptic surface of an algebraic function by means of the Baker-Akhiezer
function. The consequent application of this function for the derivation of the
general solution to the vector Riemann-Hilbert problem requires finding of the
zeros of the Baker-Akhiezer function ( is the genus of the
surface). These zeros are recovered through the solution to the associated
Jacobi problem of inversion of abelian integrals or, equivalently, the
determination of the zeros of the associated degree- polynomial and
solution of a certain linear algebraic system of equations.Comment: 17 pages, 1 figur
- …