18 research outputs found
Bounds for approximate discrete tomography solutions
In earlier papers we have developed an algebraic theory of discrete
tomography. In those papers the structure of the functions
and having given line sums in certain directions have
been analyzed. Here was a block in with sides parallel to
the axes. In the present paper we assume that there is noise in the
measurements and (only) that is an arbitrary or convex finite set in
. We derive generalizations of earlier results. Furthermore we
apply a method of Beck and Fiala to obtain results of he following type: if the
line sums in directions of a function are known, then
there exists a function such that its line sums differ by at
most from the corresponding line sums of .Comment: 16 page
Bounds for discrete tomography solutions
We consider the reconstruction of a function on a finite subset of
if the line sums in certain directions are prescribed. The real
solutions form a linear manifold, its integer solutions a grid. First we
provide an explicit expression for the projection vector from the origin onto
the linear solution manifold in the case of only row and column sums of a
finite subset of . Next we present a method to estimate the
maximal distance between two binary solutions. Subsequently we deduce an upper
bound for the distance from any given real solution to the nearest integer
solution. This enables us to estimate the stability of solutions. Finally we
generalize the first mentioned result to the torus case and to the continuous
case
Representing integers as linear combinations of power products
Let P be a finite set of at least two prime numbers, and A the set of
positive integers that are products of powers of primes from P. Let F(k) denote
the smallest positive integer which cannot be presented as sum of less than k
terms of A. In a recent paper Nathanson asked to determine the properties of
the function F(k), in particular to estimate its growth rate. In this paper we
derive several results on F(k) and on the related function which denotes the
smallest positive integer which cannot be presented as sum of less than k terms
from the union of A and -A.Comment: 7 pages, paper submitte
Representing integers as linear combinations of powers
At a conference in Debrecen in October 2010 Nathanson announced some results
concerning the arithmetic diameters of certain sets. He proposed some related
results on the representation of integers by sums or differences of powers of 2
and 3. In this note we prove some results on this problem and the more general
problem about the representation by linear combinations of powers of some fixed
integers.Comment: 8 pages, paper will appear in Publ. Math. Debrece
Linear time reconstruction by discrete tomography in three dimensions
The goal of discrete tomography is to reconstruct an unknown function via
a given set of line sums. In addition to requiring accurate reconstructions, it
is favourable to be able to perform the task in a timely manner. This is
complicated by the presence of switching functions, or ghosts, which allow many
solutions to exist in general. In this paper we consider the case of a function
where is a finite grid in . Previous
work has shown that in the two-dimensional case it is possible to determine all
solutions in parameterized form in linear time (with respect to the number of
directions and the grid size) regardless of whether the solution is unique. In
this work, we show that a similar linear method exists in three dimensions
under the condition of nonproportionality. This is achieved by viewing the
three-dimensional grid along each 2D coordinate plane, effectively solving the
problem with a series of 2D linear algorithms. We show that the condition of
nonproportionality is fulfilled in the case of three-dimensional boundary
ghosts, which motivated this research.Comment: 22 pages, 8 figures; submitted to Discrete Applied Mathematic
Two applications of number theory to discrete tomography
Tomography is the theory behind scans, e.g. MRI-scans. Most common is continuous tomography where an object is reconstructed from numerous projections. In some cases this is not applicable, because the object changes too quickly or is damaged by making hundreds of projections (by X-rays). In such cases discrete tomography may apply where only few projections are made. The present paper shows how number theory helps to provide insight in the application and structure of discrete tomography