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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
Vertically symmetric alternating sign matrices and a multivariate Laurent polynomial identity
In 2007, the first author gave an alternative proof of the refined
alternating sign matrix theorem by introducing a linear equation system that
determines the refined ASM numbers uniquely. Computer experiments suggest that
the numbers appearing in a conjecture concerning the number of vertically
symmetric alternating sign matrices with respect to the position of the first 1
in the second row of the matrix establish the solution of a linear equation
system similar to the one for the ordinary refined ASM numbers. In this paper
we show how our attempt to prove this fact naturally leads to a more general
conjectural multivariate Laurent polynomial identity. Remarkably, in contrast
to the ordinary refined ASM numbers, we need to extend the combinatorial
interpretation of the numbers to parameters which are not contained in the
combinatorial admissible domain. Some partial results towards proving the
conjectured multivariate Laurent polynomial identity and additional motivation
why to study it are presented as well
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