On the Compatibility of Granulometry with Wavelet Analysis

Abstract

In this paper, regular closed Jordan measurable sets are proposed as models of digital images, and it is shown, that these are precisely the sets that are well approximated by their family of granulometric openings in measure theoretic sense. Moreover, compatibility of mathematical morphology and wavelet analysis is established in the following sense: two sets, that are indistinguishable by wavelets (i.e. their symmetric difference is a Lebesgue zero set) have undistinguishable granulometric openings, and a finite resolution wavelet approximation to a Jordan measurable set is sufficient to determine its granulometric openings up to a specified margin of error.

Appendix

We show by an example that even regular closed sets can have a “large” boundary: There exists a regular closed subset A⊆I:=[0,1] whose boundary is a thick Cantor set. That means that ∂A is homeomorphic to Cantor’s discontinuum and has positive Lebesgue measure λ(∂A)>0. The granulometric openings of such a set will necessarily miss large parts of it.

For the construction we recall that Cantor’s discontinuum $\mathcal{C}\subset I$ is the set of all reals between 0 and 1 that can be represented as triadic expansion not using the digit 1. Notice that in some cases the digit 1 can be avoided by turning a finite sequence into a periodic infinite one, like 0.021=0.020222222… , such a number belongs to Cantor’s discontinuum. Let U _even,U _odd⊂I denote the open sets of all reals, whose triadic expansion necessarily contains the digit 1 (i.e., 1 cannot be removed by a conversion like above) and the first occurrence of it is at an even (resp. odd) position; evidently we have partitioned the unit segment into three disjoint sets $I=\mathcal{C}\cup U_{\mathrm{even}}\cup U_{\mathrm{odd}}$. Every element of $\mathcal{C}$ can be approximated arbitrarily closely by a finite sequence, which can be picked from U _even as well as of U _odd by appending 101 or 0101 if necessary. Hence every element of $\mathcal{C}$ can be approximated arbitrarily closely by elements of U _even as well as of U _odd, and we conclude $\partial U_{\mathrm{even}}=\partial U_{\mathrm{odd}}=\mathcal{C}$. Then $A:= \overline{U_{\mathrm{even}}} =U_{\mathrm{even}}\cup \mathcal{C}=I\setminus \partial U_{\mathrm{odd}}$ is closed with interior U _even, i.e. A is a regular closed subset of I with boundary $\mathcal{C}$.

Now pick a homeomorphism h:I≈I mapping $\mathcal{C}$ onto a thick Cantor set, i.e. such that $\lambda (h (\mathcal{C} ) )>0$ [12, p. 179]. Then h(A)⊂I is a regular closed subset whose boundary has non zero Lebesgue measure.

As open subsets of the real line the sets U _even and U _odd are countable unions of open segments, and since every element of $\mathcal{C}$ can be approximated arbitrarily closely by elements of U _even as well as of U _odd a closed segment [a,b] with a<b is contained in $A=U_{\mathrm{even}}\cup \mathcal{C}$ if and only if the open segment ]a,b[ ⊆U _even. Hence, if A _ω denotes the union of all closed segments of positive length contained in A, then A _ω =⋃ _t>0 Ψ _t A is the union of the granulometric openings of A with respect to the unit segment, and the granulometric remainder $A\setminus \bigcup_{t>0}\varPsi_{t}A =\mathcal{C}\setminus (A_{\omega}\setminus U_{\mathrm{even}} )$ contains nearly the entire Cantor set except the countable subset A _ω ∖U _even. This property persists under the homeomorphism h, thus leading to an example of a compact set whose granulometric remainder has strictly positive Lesbegue measure.