Invertible affine transformations on integer coordinate system — general theory in n - dimensional space

✍ Scribed by Masahiko Shizawa

• Publisher: John Wiley and Sons
• Year: 1993
• Tongue: English
• Weight: 852 KB
• Volume: 24
• Category: Article
• ISSN: 0882-1666
• DOI: 10.1002/scj.4690240701

✦ Synopsis

Abstract

This paper considers a lattice represented by integer coordinates in n‐dimensional space (digital lattice). On the digital lattice, an automorphism is constructed by a newly proposed theory and algorithm, to enable the approximation of the arbitrarily given equivolume affine transformation. The equivolume affine transformation is a general affine transformation that satisfies the condition of volume invariance, and is characterized by the determinant of the representation matrix having an absolute value of 1. An equivolume affine transformation in n‐dimensional space can represent any combination of reflection, rotation, equivolume expansion/contraction and skew deformation. These transformations often are employed as fundamental transformations in the handling of geometrical information in computers.

This paper discusses first the geometrical characters of equivolume affine transformations. It then defines the fundamental reflection, skew and translations. It is shown that the equivolume affine transformation in n‐dimensional space can be decomposed into the product of (n^2^‐1) fundamental skew transformations, n fundamental translations and a finite number of fundamental reflections.

For the fundamental transformation, one‐to‐one integer approximation is defined, and a systematic method is proposed to evaluate the upper bound of the approximation error. With the proposed error estimation method, an algorithm is proposed to determine the decomposition suppressing the error in the overall system. One advantage of this theory is that it is no longer necessary to preserve the geometrical information before transformation in the computer, which has been commonplace up to now in computer programming.