b7414fe14a85
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Benedikt Fluhr <http://bfluhr.com>
7 years ago

Reformulations

4 files changed,70insertions(+),22deletions(-) M 00_00_00_links.md M 00_01_intro.md M 00_02_ReebGraphs.md M 00_03_joinTrees.md

M 00_00_00_links.md +1 -0

@@ 8,6 8,7 @@[topology]: https://en.wikipedia.org/wiki/Topological_space#Definition_via_open_sets [proper]: https://en.wikipedia.org/wiki/Proper_map [continuous function]: https://en.wikipedia.org/wiki/Continuous_function#Continuous_functions_between_topological_spaces +[properties quotient space]: https://en.wikipedia.org/wiki/Quotient_space_%28topology%29#Properties [Reeb graph]: https://en.wikipedia.org/wiki/Reeb_graph [Morse]: https://en.wikipedia.org/wiki/Morse_theory#Fundamental_theorems [extended pseudometric]: https://en.wikipedia.org/wiki/Metric_(mathematics)#Extended_metrics

M 00_01_intro.md +9 -9

@@ 154,7 154,7 @@ such that for all $p \in X$ the estimatehold? Next we may ask, how large $\varepsilon$ needs to be -in order for the answer to *yes*. +in order for the answer to be *yes*. This is the motivation behind the following * **Definition.**@@ 175,7 175,7 @@ Next we observe that $M$ satisfies the tLet $h \colon Z \rightarrow \R$ be another continuous function, then $M(f, h) \leq M(f, g) + M(g, h)$. -Now we recall that we started with two continuous $f$ and $g$ +Now we recall that we started with two continuous functions $f$ and $g$ and each might describe the luminance of an image or the distance to a certain shape and this is all good,@@ 282,7 282,7 @@ Completely analogously to the above we hLet $h \colon Z \rightarrow \R$ be another continuous function, then $\mu(f, h) \leq \mu(f, g) + \mu(g, h)$. -Moreover we have the following relationship to the absolute distance. +Moreover we have the following relation to the absolute distance. * **Lemma.** We have $\mu(f, g) \leq M(f, g)$.@@ 374,9 374,9 @@ the *category of $\overline{\R}$-spaces** **Definition.** For two continuous functions $f \colon X \rightarrow \overline{\R}$ and -$g \colon Y \rightarrow \overline{\R}$ -a *homomorphism $\varphi$ from $f$ to $g$* also denoted by -$\varphi \colon f \rightarrow g$ is a continuous map +$g \colon Y \rightarrow \overline{\R}$, +a *homomorphism $\varphi$ from $f$ to $g$*, also denoted by +$\varphi \colon f \rightarrow g$, is a continuous map $\varphi \colon X \rightarrow Y$ such that the diagram $$ \xymatrix{@@ 396,12 396,12 @@ commutes.We define the composition of homomorphisms in the *category of $\overline{\R}$-spaces* by the composition of maps. - The *category of $\R$-spaces* is the full subcategory of + The *category of $\R$-spaces* is the [full subcategory][] of all real-valued continuous functions. * *Remark.* -With these definitions in place a reformulation of question @qAbs -is whether $f$ and $g$ are isomorphic as objects of the +With these definitions in place a reformulation of question @qAbs is, +whether $f$ and $g$ are isomorphic as objects of the category of $\overline{\R}$-spaces. ### Further Prior Art

M 00_02_ReebGraphs.md +56 -9

@@ 4,30 4,77 @@ In this section we introduce Reeb graphs* **Definition** (Reeb Graph)**.** Given a continuous function $f \colon X \rightarrow \overline{\R}$ -and $x \in X$ +and $x \in X$, let $\pi_f (x)$ be the connected component of $x$ in $f^{-1}(f(x))$. In this way we obtain a function $\pi_f \colon X \rightarrow \pi_f (X) \subset 2^X$ and we endow -$\pi_f (X)$ with the quotient topology. -By the universal property -of the quotient space there is a unique continuous function +$\pi_f (X)$ with the quotient topology[^refQuoTop]. +By the +[universal property of the quotient space][properties quotient space] +there is a unique continuous function $\mathcal{R} f \colon \pi_f (X) \rightarrow \overline{\R}$ such that +$$ +\xymatrix{ + X + \ar[r]^{\pi_f} + \ar[dr]_f + & + \pi_f (X) + \ar[d]^{\mathcal{R} f} + \\ + & + \overline{\R} +} +$$ +commutes, i.e. $\mathcal{R} f \circ \pi_f = f$. For another continuous function $g \colon Y \rightarrow \overline{\R}$ and a homomorphism $\varphi \colon f \rightarrow g$ -we may use the universal property of $\pi_f$ again +we may use the universal property of $\pi_f$ again, to obtain a unique continuous map $\mathcal{R} \varphi \colon \pi_f (X) \rightarrow \pi_g (Y)$ -with +such that +$$ +\xymatrix{ + X + \ar[d]_{\pi_f} + \ar[r]^{\varphi} + & + Y + \ar[d]^{\pi_g} + \\ + \pi_f (X) + \ar[r]_{\mathcal{R} \varphi} + & + \pi_g (Y) +} +$$ +commutes, i.e. $\mathcal{R} \varphi \circ \pi_f = \pi_g \circ \varphi$. +[^refQuoTop]: see for example [@bredon1993, definition I.13.1] + * **Lemma.** Let $f \colon X \rightarrow \overline{\R}$ and $g \colon Y \rightarrow \overline{\R}$ be continuous functions and let $\varphi \colon f \rightarrow g$ be a homomorphism, -then $\mathcal{R} g \circ \mathcal{R} \varphi = \mathcal{R} f$. +then the diagram +$$ +\xymatrix{ + \pi_f (X) + \ar@/^/[rr]^{\mathcal{R} \varphi} + \ar[dr]_{\mathcal{R} f} + & & + \pi_g (Y) + \ar[dl]^{\mathcal{R} g} + \\ + & + \overline{\R} +} +$$ +commutes. Or in other words $\mathcal{R} \varphi$ is a homomorphism from $\mathcal{R} f$ to $\mathcal{R} g$ in the category of $\overline{\R}$-spaces.@@ 101,6 148,6 @@ universal property of $\pi_f$ thatThe previous two lemmata imply that $\mathcal{R}$ is an endofunctor on the category of $\overline{\R}$-spaces. Later we will define an interleaving distance on Reeb graphs, -but first we will introduce join trees and their interleavings -as they are easier to understand and may provide us with some intuition +but first we will introduce join trees and their interleavings. +Join trees are easier to understand and may provide us with some intuition for understanding the more sophisticated interleavings of Reeb graphs.

M 00_03_joinTrees.md +4 -4

@@ 46,11 46,11 @@ is shown in the image below.Before we get to interleavings we make some auxiliary definitions. * **Definition.** -Let +We set $D^{\perp} := - \set{(a, b)}{(-\infty, \infty] \times (-\infty, \infty]}{a + b \geq 0}$ -then $D^{\perp}$ is a monoid by component-wise addition. + \set{(a, b)}{(-\infty, \infty] \times (-\infty, \infty]}{a + b \geq 0}$. +Component-wise addition yields the structure of a monoid on $D^{\perp}$. Next we define two weightings on $D^{\perp}$. * **Definition.**@@ 150,7 150,7 @@ the *relative interleaving distance ofWe note that these definitions of an interleaving distance are different from the one by @morozov2013. -The subsequent corollary shows however that +However the subsequent corollary shows that our absolute interleaving distance is equal to the distance by @morozov2013. The reason we gave a different definition is that now we can phrase the computation