Digital Convex + Unimodular Mapping =8-Connected (All Points but One 4-Connected)

03/08/2021
by   Crombez Loïc, et al.
0

In two dimensional digital geometry, two lattice points are 4-connected (resp. 8-connected) if their Euclidean distance is at most one (resp. √(2)). A set S ⊂ Z^2 is 4-connected (resp. 8-connected) if for all pair of points p_1, p_2 in S there is a path connecting p_1 to p_2 such that every edge consists of a 4-connected (resp. 8-connected) pair of points. The original definition of digital convexity which states that a set S ⊂ Z^d is digital convex if (S) ∩ Z^d= S, where (S) denotes the convex hull of S does not guarantee connectivity. However, multiple algorithms assume connectivity. In this paper, we show that in two dimensional space, any digital convex set S of n points is unimodularly equivalent to a 8-connected digital convex set C. In fact, the resulting digital convex set C is 4-connected except for at most one point which is 8-connected to the rest of the set. The matrix of SL_2(Z) defining the affine isomorphism of Z^2 between the two unimodularly equivalent lattice polytopes S and C can be computed in roughly O(n) time. We also show that no similar result is possible in higher dimension.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

12/13/2018

Peeling Digital Potatoes

The potato-peeling problem (also known as convex skull) is a fundamental...
01/15/2019

Efficient Algorithms to Test Digital Convexity

A set S ⊂Z^d is digital convex if conv(S) ∩Z^d = S, where conv(S) denote...
12/17/2017

Properties of Digital n-Dimensional Spheres and Manifolds. Separation of Digital Manifolds

In the present paper, we study basic properties of digital n-dimensional...
03/11/2015

Properties of simple sets in digital spaces. Contractions of simple sets preserving the homotopy type of a digital space

A point of a digital space is called simple if it can be deleted from th...
05/21/2019

Lonely Points in Simplices

Given a lattice L in Z^m and a subset A of R^m, we say that a point in A...
10/31/2020

Fragmented digital connectivity and security at sea

This paper explores how uneven and often unreliable digital connections ...
01/14/2014

A programme to determine the exact interior of any connected digital picture

Region filling is one of the most important and fundamental operations i...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.