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Combining Persistent Homology and Invariance Groups for Shape Comparison

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Abstract

Persistent homology has proven itself quite efficient in the topological and qualitative comparison of filtered topological spaces, when invariance with respect to every homeomorphism is required. However, we can make the following two observations about the use of persistent homology for application purposes. On the one hand, more restricted kinds of invariance are sometimes preferable (e.g., in shape comparison). On the other hand, in several practical situations filtering functions are not just auxiliary technical tools that can be exploited to study a given topological space, but instead the main aim of our analysis. Indeed, most of the data is usually produced by measurements, whose results are quite often functions defined on a topological space. As a simple example we can consider a 3D laser scanning of a surface, where the result of each measurement can be seen as a real-valued function defined on the manifold that describes the positions of the rangefinder measuring the distances. In fact, in many applications the dataset of interest is seen as a collection \({\varPhi }\) of real-valued functions defined on a given topological space X, instead of a family of topological spaces. As a natural consequence, in these cases observers can be seen as collections of suitable operators on \({\varPhi }\). Starting from these remarks, this paper proposes a way to combine persistent homology with the use of G-invariant non-expansive operators defined on \({\varPhi }\), where G is a group of self-homeomorphisms of X. Our goal is to give a method to study \({\varPhi }\) in a way that is invariant with respect to G. Some theoretical results concerning our approach are proven, and two experiments are presented. An experiment illustrates the application of the proposed technique to compare 1D-signals, when the invariance is expressed by the group of affinities, the group of orientation-preserving affinities, the group of isometries, the group of translations and the identity group. Another experiment shows how our technique can be used for image comparison.

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Acknowledgments

The authors thank Marian Mrozek for his suggestions and advice. The research described in this article has been partially supported by GNSAGA-INdAM (Italy), and is based on the work realized by the authors within the ESF-PESC Networking Programme “Applied and Computational Algebraic Topology”. The second author is supported by National Science Centre (Poland) DEC-2013/09/N/ST6/02995 Grant and by the TOPOSYS Project FP7-ICT-318493-STREP. The first author is grateful to the city of Viareggio for its inspiring hospitality.

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Correspondence to Patrizio Frosini.

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Editors-in-Charge: Herbert Edelsbrunner and Kenneth Clarkson

This paper is dedicated to the memory of Marcello D’Orta and Jerry Essan Masslo.

Appendices

Appendix 1: Proof of Proposition 7

Proof

  1. 1.

    The value \(d_{\mathcal {F}}(F_1,F_2)\) is finite for every \(F_1,F_2\in \mathcal {F}\), because \({{\varPhi }}\) is bounded. Indeed, a finite constant L exists such that \(d_\infty (\varphi ,\mathbf {0}):=\Vert \varphi \Vert _\infty \le L\) for every \(\varphi \in {\varPhi }\). Hence \(\Vert F_1(\varphi )-F_2(\varphi )\Vert _\infty \le \Vert F_1(\varphi )\Vert _\infty +\Vert F_2(\varphi )\Vert _\infty \le 2L\) for any \(\varphi \in {\varPhi }\) and any \(F_1,F_2\in \mathcal {F}\), since \(F_1(\varphi ), F_2(\varphi )\in {\varPhi }\). This implies that \(d_{\mathcal {F}}(F_1,F_2)\le 2L<\infty \) for every \(F_1,F_2\in \mathcal {F}\).

  2. 2.

    \(d_{\mathcal {F}}\) is obviously symmetrical.

  3. 3.

    The triangle inequality holds, since

    $$\begin{aligned} \begin{aligned} d_{\mathcal {F}}(F_1,F_2)&:=\sup _{\varphi \in {{\varPhi }}}\Vert F_1(\varphi )-F_2(\varphi )\Vert _\infty \\&\qquad \le \sup _{\varphi \in {{\varPhi }}}\big (\Vert F_1(\varphi )-F_3(\varphi )\Vert _\infty + \Vert F_3(\varphi )-F_2(\varphi )\Vert _\infty \big )\\&\qquad \le \sup _{\varphi \in {{\varPhi }}}\Vert F_1(\varphi )-F_3(\varphi )\Vert _\infty + \sup _{\varphi \in {{\varPhi }}}\Vert F_3(\varphi )-F_2(\varphi )\Vert _\infty \\&\qquad = d_{\mathcal {F}}(F_1,F_3)+d_{\mathcal {F}}(F_3,F_2) \end{aligned} \end{aligned}$$

    for any \(F_1,F_2,F_3\in \mathcal {F}\).

  4. 4.

    The definition of \(d_{\mathcal {F}}\) immediately implies that \(d_{\mathcal {F}}(F,F)=0\) for any \(F\in \mathcal {F}\).

  5. 5.

    If \(d_{\mathcal {F}}(F_1,F_2)=0\), then the definition of \(d_{\mathcal {F}}\) implies that \(\Vert F_1(\varphi )-F_2(\varphi )\Vert _\infty =0\) for every \(\varphi \in {{\varPhi }}\), and hence \(F_1(\varphi )=F_2(\varphi )\) for every \(\varphi \in {{\varPhi }}\). Therefore \(F_1\equiv F_2\).

\(\square \)

Appendix 2: Remark

If X and Y are two homeomorphic spaces and \(h{:}\,Y\rightarrow X\) is a homeomorphism, then the persistent homology group with respect to the function \(\varphi {:}\,X\rightarrow \mathbb {R}\) and the persistent homology group with respect to the function \(\varphi \circ h{:}\,Y\rightarrow \mathbb {R}\) are isomorphic at each point (uv) in the domain. The isomorphism between the two persistent homology groups is the one taking each homology class \([c=\sum _{i=1}^r a_i\cdot \sigma _i]\in PH_k^\varphi (u,v)\) to the homology class \([c'=\sum _{i=1}^r a_i\cdot (h^{-1}\circ \sigma _i)]\in PH_k^{\varphi \circ h}(u,v)\), where each \(\sigma _i\) is a singular simplex involved in the representation of the cycle c.

Appendix 3: Approximation of the Non-expansive \(G_1\)-Operator \(F_{\hat{w},\hat{\gamma }}\)

The operator \(F_{\hat{w},\hat{c}}(\varphi )(x):=\sup _{r\in \mathbb {R}} \sum _{i=1}^n w_i\cdot \varphi (x+rc_i)\) can be approximated by substituting the supremum in its definition with a maximum for r belonging to a finite set.

In order to show this, first of all we observe that if \(\hat{w}=\mathbf {0}\) then \(F_{\hat{w},\hat{c}}\) is just the null operator, while if \(\hat{c}=\mathbf {0}\) then \(F_{\hat{w},\hat{c}}\) is just the operator taking each function \(\varphi \) to the function \((\sum _{i=1}^n w_i)\cdot \varphi \). Therefore, we can restrict ourselves to consider the case \(\hat{w}\ne \mathbf {0}\), \(\hat{c}\ne \mathbf {0}\).

Secondly, let us consider the function \(\psi :=\varphi \circ h\), where \(h{:}\,\mathbb {R}\rightarrow \mathbb {R}\) is the linear homeomorphism taking 0 to \(-1\) and 1 to 2 (i.e., \(h(y)=3y-1\)). Given that every function \(\varphi \in {\varPhi }_{ds}\) is assumed to be Lipschitz, with Lipschitz constant C, the function \(\psi \) is Lipschitz, with Lipschitz constant 3C. The support of \(\psi \) is the interval [0, 1]. It is easy to check that \(F_{\hat{w},\hat{c}}(\varphi )(x)=F_{\hat{w},\hat{\gamma }}(\psi )(y)\), where \(y=\frac{1}{3}x+\frac{1}{3}\) and \(\hat{\gamma }=(\gamma _1,\ldots ,\gamma _n):=\frac{1}{3}\hat{c}\).

Furthermore, we can assume that \(\gamma _p\ne \gamma _q\) for \(p<q\). Indeed, if \(\gamma _p=\gamma _q\) with \(p<q\) we can consider the new vectors \(\hat{w}':=(w_1',\ldots ,w_{n-1}')\), \(\hat{\gamma }':=(\gamma _1',\ldots ,\gamma _{n-1}')\) obtained by setting for \(1\le i\le n-1\)

$$\begin{aligned} w_i'={\left\{ \begin{array}{ll} w_i &{} \text{ if } \quad i< p,\\ w_p+w_q &{} \text{ if } \quad i=p, \\ w_{i} &{} \text{ if } \quad p<i<q, \\ w_{i+1} &{} \text{ if } \quad i\ge q, \end{array}\right. } \text{ and } \gamma _i'={\left\{ \begin{array}{ll} \gamma _i &{} \text{ if } \quad i<q,\\ \gamma _{i+1} &{} \text{ if } \quad i\ge q. \end{array}\right. } \end{aligned}$$

It is easy to check that \(F_{\hat{w}',\hat{\gamma }'}=F_{\hat{w},\hat{\gamma }}\).

Hence, we can assume \(\mu :=\min \{|\gamma _p-\gamma _q|:p\ne q\}>0\). Let us set \(M:=\max _i|\gamma _i|\).

We start by observing that if \(|y|> \frac{M}{\mu }+1\) the value \(F_{\hat{w},\hat{\gamma }}(\psi )(y)\) is easily computable, because the condition \(0\le y+r\gamma _p,y+r\gamma _q\le 1\) cannot hold for \(p\ne q\).

Indeed, if \(y<- \frac{M}{\mu }-1\), then \(0\le y+r\gamma _i\) implies \(|r\gamma _i|\ge r\gamma _i\ge -y>\frac{M+\mu }{\mu }>0\), and hence \(|r|>\frac{M+\mu }{\mu |\gamma _i|}\ge \frac{1}{\mu }\). If \(y> \frac{M}{\mu }+1\), then \(y+r\gamma _i\le 1\) implies \(r\gamma _i\le 1-y<-\frac{M}{\mu }< 0\), and hence \(|r\gamma _i|>\frac{M}{\mu }\), so that \(|r|>\frac{M}{\mu |\gamma _i|}\ge \frac{1}{\mu }\). As a consequence, in both cases \(|(y+r\gamma _p)-(y+r\gamma _q)|=|r|\cdot |\gamma _p-\gamma _q|\ge |r|\cdot \mu > 1\), for \(p\ne q\). Therefore, the condition \(0\le y+r\gamma _p,y+r\gamma _q\le 1\) cannot hold, so that at most one of the points \(y+r\gamma _p,y+r\gamma _q\) can belong to [0, 1].

Now, let us consider the value \(\sum _{i=1}^n w_i\cdot \psi (y+r\gamma _i)\) as a function of \(r\in \mathbb {R}\), under the assumption that \(|y|> \frac{M}{\mu }+1\).

When \(r=0\), for every index i we have that \(y+r\gamma _i=y\notin [0,1]\), so implying that \(\sum _{i=1}^n w_i\cdot \psi (y+r\gamma _i)=0\) because the support of \(\psi \) is contained in [0, 1].

When \(r\ne 0\) we have that at most one of the points in the set \(\{y+rc_1,\ldots ,y+rc_n\}\) can belong to the interval [0, 1]. Moreover, for every index i such that \(\gamma _i\ne 0\) and every \(\eta \in [0,1]\) exactly one value \(r\ne 0\) exists, such that \(y+rc_i=\eta \).

It follows that \(\sup _{r\in \mathbb {R}}\sum _{i=1}^n w_i\cdot \psi (y+rc_i)=\max \{0,\max (w_1\cdot \psi ),\ldots ,\max (w_n\cdot \psi )\}\).

In conclusion, if \(|y|> \frac{M}{\mu }+1\), \(F_{\hat{w},\hat{\gamma }}(\psi )(y)=\max \{0,\max (w_1\cdot \psi ),\ldots ,\max (w_n\cdot \psi )\}\).

It follows that if \(|y|> \frac{M}{\mu }+1\) we can easily approximate \(F_{\hat{w},\hat{\gamma }}(\psi )(y)\) because the values \(\max (w_i\cdot \psi )\) can be approximated with arbitrary precision. Indeed, we know that the function \(\psi \) is Lipschitz, with Lipschitz constant 3C. This implies that we can approximate \(\max (w_i\cdot \psi )\) by computing the value \(\max _s (w_i\cdot \psi (r_s))\), where \(\{r_1,\ldots ,r_m\}\) is a sufficiently dense finite subset of the interval [0, 1].

Let us now consider the case \(|y|\le \frac{M}{\mu }+1\). In this case, if \(\gamma _i\ne 0\) and \(|r|>\frac{M+2\mu }{\mu |\gamma _i|}\) then \(\psi (y+r\gamma _i)=0\). Indeed, the support of \(\psi \) is contained in [0, 1] and \(|y+r\gamma _i|\ge ||r\gamma _i|-|y||=|r\gamma _i|-|y|> 1\) because \(|r\gamma _i|>\frac{M}{\mu }+2\) and \(|y|\le \frac{M}{\mu }+1\).

Setting \(R:=\frac{M+2\mu }{\mu \cdot \min \{|\gamma _i|:\gamma _i\ne 0\}}\), it follows that \(|y|\le \frac{M}{\mu }+1\) implies

$$\begin{aligned} \sup _{|r|>R}\sum _{i=1}^n w_i\cdot \psi (y+r\gamma _i)= {\left\{ \begin{array}{ll} w_j\cdot \psi (y) &{} \text{ if }\quad \text{ an } \text{ index } \text{ j } \text{ exists } \text{ s.t. } \gamma _j= 0,\\ 0 &{} \text{ if }\quad \gamma _i\ne 0 \text{ for } \text{ every } \text{ index } i. \end{array}\right. } \end{aligned}$$

so that

$$\begin{aligned} \begin{aligned}&\sup _{r\in \mathbb {R}}\sum _{i=1}^n w_i\cdot \psi (y+r\gamma _i) \\&\quad = \max \Big \{\sup _{|r|\le R}\sum _{i=1}^n w_i\cdot \psi (y+r\gamma _i), \sup _{|r|> R}\sum _{i=1}^n w_i\cdot \psi (y+r\gamma _i)\Big \} \\&\quad ={\left\{ \begin{array}{ll} \max \big \{\sup _{|r|\le R}\sum _{i=1}^n w_i\cdot \psi \big (y+r\gamma _i\big ), w_j\cdot \psi (y)\big \} &{} \text{ if }\quad \text{ an } \text{ index } \text{ j } \text{ exists } \text{ s.t. } \gamma _j= 0,\\ \max \big \{\sup _{|r|\le R}\sum _{i=1}^n w_i\cdot \psi \big (y+r\gamma _i\big ),0\big \} &{} \text{ if } \quad \gamma _i\ne 0 \text{ for } \text{ every } \text{ index } i. \end{array}\right. } \end{aligned} \end{aligned}$$

Now, we have

$$\begin{aligned}&\Big |\sum _{i=1}^n w_i\cdot \psi \big (y+r\gamma _i\big ) - \sum _{i=1}^n w_i\cdot \psi \big (y+r'\gamma _i\big )\Big |\\&\quad = \Big |\sum _{i=1}^n w_i\cdot \big (\psi \big (y+r\gamma _i\big ) - \psi \big (y+r'\gamma _i\big )\big )\Big |\\&\quad \le \sum _{i=1}^n |w_i|\cdot \big |\psi \big (y+r\gamma _i\big ) - \psi \big (y+r'\gamma _i\big )\big |\\&\quad \le \sum _{i=1}^n |w_i|\cdot 3C\cdot |(r-r')\cdot \gamma _i|\\&\quad = \sum _{i=1}^n |w_i|\cdot 3C\cdot |r-r'|\cdot |\gamma _i|\\&\quad \le \sum _{i=1}^n |w_i|\cdot 3C\cdot |r-r'|\cdot M\\&\quad = 3C\cdot M\cdot |r-r'| \end{aligned}$$

because \(\sum _{i=1}^n |w_i|=1\) and \(\psi \) is Lipschitz, with Lipschitz constant 3C. Hence we can approximate \(\sup _{|r|\le R}\sum _{i=1}^n w_i\cdot \psi (y+r\gamma _i)\) by computing the value \(\max _s\sum _{i=1}^n w_i\cdot \psi (y+r_s\gamma _i)\), where \(\{r_1,\ldots ,r_m\}\) is a sufficiently dense finite subset of the interval \([-R,R]\). It follows that we can easily approximate \(F_{\hat{w},\hat{\gamma }}(\psi )(y)\) also in the case \(|y|\le \frac{M}{\mu }+1\).

Therefore our statement is proven.

Appendix 4: Definition of the First Five Operators Used in Sect. 5

The formal definition of our operators is

$$\begin{aligned} F_\beta (\varphi )({\varvec{x}}):=\int _{B}\varphi ({\varvec{x}}-{\varvec{y}})\cdot \beta (\Vert {\varvec{y}}\Vert _2)\ \mathrm{d}{\varvec{y}}, \end{aligned}$$

where \(\beta \) is an integrable function defined on a ball \(B\subset \mathbb {R}^2\) such that \(\int _{B}|\beta (\Vert {\varvec{y}}\Vert _2)|\ \mathrm{d}{\varvec{y}}\le 1\) (here, \(\Vert {\varvec{y}}\Vert _2\) denotes the Euclidean norm of the vector \({\varvec{y}}\)). This condition is necessary in the proof of non-expansiveness, and \(\beta \) can be considered as a kernel function. We used the following four kernel functions:

  1. 1.

    \(\beta (t) = {\left\{ \begin{array}{ll} 16/\pi &{} \text{ if } 0\le t\le 1/4, \\ 0 &{} \text{ if } t<0 \vee t>1/4. \end{array}\right. }\)

  2. 2.

    \(\beta (t) = {\left\{ \begin{array}{ll} 16/\pi &{} \text{ if } 0\le t< 1/8, \\ -16/\pi &{} \text{ if } 1/8\le t\le 1/4, \\ 0 &{} \text{ if } t<0 \vee t>1/4. \end{array}\right. }\)

  3. 3.

    \(\beta (t) = {\left\{ \begin{array}{ll} 16/\pi &{} \text{ if } 0\le t< 1/16, \\ -16/\pi &{} \text{ if } 1/16\le t< 1/8, \\ 16/\pi &{} \text{ if } 1/8\le t< 3/16, \\ -16/\pi &{} \text{ if } 3/16\le t\le 1/4, \\ 0 &{} \text{ if } t<0 \vee t>1/4. \end{array}\right. }\)

  4. 4.

    \(\beta (t) = {\left\{ \begin{array}{ll} 4/\pi &{} \text{ if } 0\le t< 1/4, \\ -4/\pi &{} \text{ if } 1/4\le t\le 1/2, \\ 0 &{} \text{ if } t<0 \vee t>1/2. \end{array}\right. }\)

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Frosini, P., Jabłoński, G. Combining Persistent Homology and Invariance Groups for Shape Comparison. Discrete Comput Geom 55, 373–409 (2016). https://doi.org/10.1007/s00454-016-9761-y

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