Lajos Hajdu

Lattice of generalized neighbourhood sequences in nD

The neighbourhood sequences have got a very important role in the digital picture processing. In this paper we give some new results from this area. We describe an ordering relation on the set of generalized nD-neighbourhood sequences. As this ordering does not have nice properties, we introduce another relation and prove that a complete distributive lattice can be obtained in this wax. Our investigations generalize the previous results of Das (1990) and Fazekas (1999) in 2D and 3D, respectively.

An algorithm for discrete tomography

Abstract There are many algorithms in the literature for the approximating reconstruction of a binary matrix from its line sums. In this paper we provide an algorithm which starts from the line sums of an unknown binary matrix f , and outputs an integer matrix S with small entries in absolute values such that the line sums of f and S coincide. We also give the results of some experiments with the algorithm.

Powers from Products of Consecutive Terms in Arithmetic Progression

We show that if

Approximating the Euclidean distance using non-periodic neighbourhood sequences

k

On the Diophantine Equation n(n+d) ... (n+(k-1)d) = byl

is a positive integer, then there are, under certain technical hypotheses, only finitely many coprime positive

Perfect powers from products of consecutive terms in arithmetic progression

k

General neighborhood sequences in Zn

-term arithmetic progressions whose product is a perfect power. If

Bounds on the difference between reconstructions in binary tomography

4 \leq k \leq 11

Generalizing the Majority Voting Scheme to Spatially Constrained Voting

, we obtain the more precise conclusion that there are, in fact, no such progressions. Our proofs exploit the modularity of Galois representations corresponding to certain Frey curves, together with a variety of results, classical and modern, on solvability of ternary Diophantine equations. As a straightforward corollary of our work, we sharpen and generalize a theorem of Sander on rational points on superelliptic curves.

Explicit Bounds for the Solutions of Elliptic Equations with Rational Coefficients

Abstract In this paper, we discuss some possibilities of approximating the Euclidean distance in Z 2 by the help of digital metrics induced by neighbourhood sequences. Contrary to the earlier approaches, we use general (non-periodic) neighbourhood sequences which allows us to derive more precise results. We determine these metrics which can be regarded as the best approximations to the Euclidean distance in some sense. We compare our results with earlier studies of Das (J. Approx. Theory 68 (1992) 155) and Mukherjee et al. (Pattern Recognition Lett. 21 (2000) 573).