Finding well approximating lattices for a finite set of points

Hajdu, András and Hajdu, Lajos and Tijdeman, R. (2019) Finding well approximating lattices for a finite set of points. MATHEMATICS OF COMPUTATION, 88 (315). pp. 369-387. ISSN 0025-5718

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Abstract

In this paper we address the task of finding well approximating lattices for a given finite set A of points in R-n motivated by practical texture analytic problems. More precisely, we search for o, d(1),..., d(n) is an element of R-n such that a - o is close to Lambda = d(1)Z + ... + d(n)Z for every a is an element of A. First we deal with the one-dimensional case, where we show that in a sense the results are almost the best possible. These results easily extend to the multi-dimensional case where the directions of the axes are given, too. Thereafter we treat the general multidimensional case. Our method relies on the LLL algorithm. Finally, we apply the least squares algorithm to optimize the results. We give several examples to illustrate our approach.

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