Approximation of Digital Surfaces by a Hierarchical Set of Planar Patches

Abstract

We show that the plane-probing algorithms introduced in Lachaud et al. (J. Math. Imaging Vis., 59, 1, 23–39, 2017), which compute the normal vector of a digital plane from a starting point and a set-membership predicate, are closely related to a three-dimensional generalization of the Euclidean algorithm. In addition, we show how to associate with the steps of these algorithms generalized substitutions, i.e., rules that replace square faces by unions of square faces, to build finite sets of elements that periodically generate digital planes. This work is a first step towards the incremental computation of a hierarchy of pieces of digital plane that locally fit a digital surface.

This work has been funded by PARADIS ANR-18-CE23-0007-01 research grant.

Proofs

Proof

(of Theorem 2).

$$\begin{aligned} E^*_1(\sigma _{n \cdots 0})(\textbf{0},i^*)&= E^*_1(\sigma _{n-1 \cdots 0})\big ( E^*_1(\sigma _n)(\textbf{0},i^*) \big ) \\&= E^*_1(\sigma _{n-1 \cdots 0}) \Bigg ( \underset{(s, j) \in \mathcal {S}^i_{\sigma _n}}{\bigcup } (\textbf{M}_{\sigma _n}^{-1} l(s), j^*) \Bigg ) \\&= \underset{(s, j) \in \mathcal {S}^i_{\sigma _n}}{\bigcup } E^*_1(\sigma _{n-1 \cdots 0}) \big (\textbf{M}_{\sigma _n}^{-1} l(s), j^*) \big ) \\&= \underset{(s, j) \in \mathcal {S}^i_{\sigma _n}}{\bigcup } \big ( (\textbf{M}_{\sigma _{n-1}} \cdots \textbf{M}_{\sigma _0})^{-1} (\textbf{M}_{\sigma _n})^{-1} l(s) + E^*_1(\sigma _{n-1 \cdots 0})(\textbf{0},j^*) \big ) \\&= \underset{(s, j) \in \mathcal {S}^i_{\sigma _n}}{\bigcup } \big ( (\textbf{M}_{\sigma _n} \cdots \textbf{M}_{\sigma _0})^{-1} l(s) + E^*_1(\sigma _{n-1 \cdots 0})(\textbf{0},j^*) \big ). \end{aligned}$$

The second to last line comes from

$$\begin{aligned} E^*_1(\sigma _{n-1 \cdots 0})(\textbf{x},i^*) = (\textbf{M}_{\sigma _{n-1}} \cdots \textbf{M}_{\sigma _0})^{-1} \textbf{x}+ E^*_1(\sigma _{n-1 \cdots 0})(\textbf{0},i^*), \end{aligned}$$

since $(\textbf{M}_{\sigma _{n-1}} \cdots \textbf{M}_{\sigma _0})^{-1}$ does not depend on the union in the definition of $E^*_1$, Eq. (3) (see also [14, Proposition 1.2.4, item (2)]).    $\square$