Optimal Differential Filter on Hexagonal Lattice - Computational Intelligence Paradigms in Advanced Pattern Classification - page 59

Chapter 5
Optimal Differential Filter on Hexagonal Lattice

Suguru Saito, Masayuki Nakajiama, and Tetsuo Shima

Department of Computer Science, Tokyo Institute of Technology, Japan

Abstract. Digital two-dimensional images are usually sampled on square lattices,
whose adjacent pixel distances in the horizontal-perpendicular and diagonal direc-
tions are not equal. On the other hand, a hexagonal lattice, however, covers an area
with sampling points whose adjacent pixel distances are the same; therefore, it has
te potential advantage that it can be used to calculate accurate two-dimensional
gradient.
The fundamental image filter in many image processing algorithms is used to ex-
tract the gradient information. For the extraction, various gradient filters have been
proposed on square lattices, and some of them have been thoroughly optimized but
not on a hexagonal lattice.
In this chapter, consistent gradient filters on hexagonal lattices are derived, the
derived filters are compared with existing optimized filters on square lattices, and
the relationship between the derived filters and existing filters on a hexagonal lat-
tice is investigated. The results of the comparison show that the derived filters on a
hexagonal lattice achieve better signal-to-noise ratio and localization than filters on
a square lattice.

1 Introduction

Obtaining the differential image of a given input image is a fundamental operation
in image processing. In most cases, the differential image is the result of a convolu-
tion of the input image with a differential filter. Accordingly, the more accurate the
differential filter is, the better the convolution results will be.
Many discrete differential filters[8, 13, 19, 18, 21] have been proposed; however,
the gradient derived by them are not so accurate. Ando therefore proposed “consis-
tent gradient filters,” which are optimized differential filters on a square lattice[2].
These filters are derived by minimizing the difference between the ideal differen-
tial and the differential obtained with filters in the frequency domain, and they have
succeeded in obtaining more accurate differential values.
On the other hand, image processing on hexagonal lattices has also been studied
for many years. Fundamental research on hexagonal lattices, for example, research