The exact computation of the matching distance for multi-parameter persistence modules is an acti... more The exact computation of the matching distance for multi-parameter persistence modules is an active area of research in computational topology. Achieving an easily obtainable exact computation of this distance would allow multi-parameter persistent homology to be a viable option for data analysis. In this paper, we provide theoretical results for the computation of the matching distance in two dimensions along with a geometric interpretation of the lines through parameter space realizing this distance. The crucial point of the method we propose is that it can be easily implemented.
SIGGRAPH ASIA 2016 Symposium on Visualization on - SA '16, 2016
Figure 1: The Hurricane dataset is a multivariate dataset defined on a cubical grid. Here we are ... more Figure 1: The Hurricane dataset is a multivariate dataset defined on a cubical grid. Here we are considering two values per point describing the temperature and the pressure above ground. For each function, we are showing the function gradient computed on the original scalar values. Then, critical cells obtained with our method are collected in clusters called extrema-clusters. Using the extrema-clusters and their size (number of voxels composing each cluster) we provide an interactive method for filtering out uninteresting regions. The extrema-clusters are shown in the lower part. For each image, only the clusters bigger than the indicated size are shown. The color scheme is based on the cluster's size. Comparing the function gradients with the clusters obtained we notice that bigger clusters are created where the gradient disagree, for example in the eye of the hurricane.
The Discrete Morse Theory of Forman appeared to be useful for providing filtration-preserving red... more The Discrete Morse Theory of Forman appeared to be useful for providing filtration-preserving reductions of complexes in the study of persistent homology. So far, the algorithms computing discrete Morse matchings have only been used for one-dimensional filtrations. This paper is perhaps the first attempt in the direction of extending such algorithms to multidimensional filtrations. Initial framework related to Morse matchings for the multidimensional setting is proposed, and a matching algorithm given by King, Knudson, and Mramor is extended in this direction. The correctness of the algorithm is proved, and its complexity analyzed. The algorithm is used for establishing a reduction of a simplicial complex to a smaller but not necessarily optimal cellular complex. First experiments with filtrations of triangular meshes are presented.
This paper proves that in Size Theory the comparison of multidimensional size functions can be re... more This paper proves that in Size Theory the comparison of multidimensional size functions can be reduced to the 1-dimensional case by a suitable change of variables. Indeed, we show that a foliation in half-planes can be given, such that the restriction of a multidimensional size function to each of these half-planes turns out to be a classical size function in two scalar variables. This leads to the definition of a new distance between multidimensional size functions, and to the proof of their stability with respect to that distance.
In this paper, we derive inequalities bounding the number of critical cells in a filtered cell co... more In this paper, we derive inequalities bounding the number of critical cells in a filtered cell complex on the one hand, and the entries of the Betti tables of the multi-parameter persistence modules of such filtrations on the other hand. Using the Mayer-Vietoris spectral sequence we first obtain strong and weak Morse inequalities involving the above quantities, and then we improve the weak inequalities achieving a sharp lower bound for the number of critical cells. Furthermore, we prove a sharp upper bound for the minimal number of critical cells, expressed again in terms of the entries of Betti tables.
This paper explores the concept of reparametrization invariant norm (RPI-norm), that is any norm ... more This paper explores the concept of reparametrization invariant norm (RPI-norm), that is any norm invariant under composition with diffeomorphisms. We prove the existence of an infinite family of RPI-norms, called standard RPI-norms, for which we exhibit both an integral and a discrete characterization. Our main result states that, for every one-time differentiable piecewise monotone function with compact support, its standard RPI-norms allow us to compute the value of any other RPI-norm of the same function. This is proved by using the standard RPI-norms in order to reconstruct the function up to reparametrization and an arbitrarily small error with respect to the total variation norm.
Motivated by the problem of dealing with incomplete or imprecise acquisition of data in computer ... more Motivated by the problem of dealing with incomplete or imprecise acquisition of data in computer vision and computer graphics, we extend results concerning the stability of persistent homology with respect to function perturbations to results concerning the stability with respect to domain perturbations. Domain perturbations can be measured in a number of different ways. An important method to compare domains is the Hausdorff distance. We show that by encoding sets using the distance function, the multidimensional matching distance between rank invariants of persistent homology groups is always upperly bounded by the Hausdorff distance between sets. Moreover we prove that our construction maintains information about the original set. Other well known methods to compare sets are considered, such as the symmetric difference distance between classical sets and the sup-distance between fuzzy sets. Also in these cases we present results stating that the multidimensional matching distance...
Topological data analysis and its main method, persistent homology, provide a toolkit for computi... more Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for one-parameter persistent homology have been established to connect persistent homology with machine learning techniques. We contribute a kernel construction for multi-parameter persistence by integrating a one-parameter kernel weighted along straight lines. We prove that our kernel is stable and efficiently computable, which establishes a theoretical connection between topological data analysis and machine learning for multivariate data analysis.
We consider generic curves in R 2 , i.e. generic C 1 functions f : S 1 → R 2. We analyze these cu... more We consider generic curves in R 2 , i.e. generic C 1 functions f : S 1 → R 2. We analyze these curves through the persistent homology groups of a filtration induced on S 1 by f. In particular, we consider the question whether these persistent homology groups uniquely characterize f , at least up to re-parameterizations of S 1. We give a partially positive answer to this question. More precisely, we prove that f = g • h, where h : S 1 → S 1 is a C 1-diffeomorphism, if and only if the persistent homology groups of s • f and s • g coincide, for every s belonging to the group Σ 2 generated by reflections in the coordinate axes. Moreover, for a smaller set of generic functions, we show that f and g are close to each other in the max-norm (up to re-parameterizations) if and only if, for every s ∈ Σ 2 , the persistent Betti numbers functions of s • f and s • g are close to each other, with respect to a suitable distance.
Rank invariants are a parametrized version of Betti numbers of a space multi-filtered by a contin... more Rank invariants are a parametrized version of Betti numbers of a space multi-filtered by a continuous vector-valued function. In this note we give a sufficient condition for their finiteness. This condition is sharp for spaces embeddable in R n .
Multidimensional persistence modules do not admit a concise representation analogous to that prov... more Multidimensional persistence modules do not admit a concise representation analogous to that provided by persistence diagrams for real-valued functions. However, there is no obstruction for multidimensional persistent Betti numbers to admit one. Therefore, it is reasonable to look for a generalization of persistence diagrams concerning those properties that are related only to persistent Betti numbers. In this paper, the persistence space of a vector-valued continuous function is introduced to generalize the concept of persistence diagram in this sense. The main result is its stability under function perturbations: any change in vector-valued functions implies a not greater change in the Hausdorff distance between their persistence spaces.
In many different industrial fields, computer vision and rob otics are jointly employed to perfor... more In many different industrial fields, computer vision and rob otics are jointly employed to perform repetitive, high-speed tasks, for quality inspection , t automatically move goods, and so on. This paper presents a hybrid computer vision-robotics s ystem for the automatic creation of ceramic mosaics, i.e. reproductions of patterns, syntheti c g ometries, or, in general, images, by means of ceramic tiles (Fig. 3.1). This task is currently performed, in most cases, by an expe rt, that is, however, slow and prone to errors. The system under d evelopment should be able to create mosaics at a higher speed and with the precision typic al of computer-based systems. Several working hypotheses can be made:
Although there is no doubt that multi-parameter persistent homology is a useful tool to analyse m... more Although there is no doubt that multi-parameter persistent homology is a useful tool to analyse multi-variate data, efficient ways to compute these modules are still lacking in the available topological data analysis toolboxes. Other issues such as interpretation and visualization of the output remain difficult to solve. Software visualizing multi-parameter persistence diagrams is currently only available for 2-dimensional persistence modules. One of the simplest invariants for a multi-parameter persistence module is its rank invariant, defined as the function that counts the number of linearly independent homology classes that live in the filtration through a given pair of values of the multi-parameter. We propose a step towards interpretation and visualization of the rank invariant for persistence modules for any given number of parameters. We show how discrete Morse theory may be used to compute the rank invariant, proving that it is completely determined by its values at points ...
Mathematical Methods in the Applied Sciences, 2013
Multidimensional persistence mostly studies topological features of shapes by analyzing the lower... more Multidimensional persistence mostly studies topological features of shapes by analyzing the lower level sets of vector-valued functions, called filtering functions. As is well known, in the case of scalar-valued filtering functions, persistent homology groups can be studied through their persistent Betti numbers, i.e. the dimensions of the images of the homomorphisms induced by the inclusions of lower level sets into each other. Whenever such inclusions exist for lower level sets of vector-valued filtering functions, we can consider the multidimensional analogue of persistent Betti numbers. Varying the lower level sets, we get that persistent Betti numbers can be seen as functions taking pairs of vectors to the set of non-negative integers. In this paper we prove stability of multidimensional persistent Betti numbers. More precisely, we prove that small changes of the vector-valued filtering functions imply only small changes of persistent Betti numbers functions. This result can be obtained by assuming the filtering functions to be just continuous. Multidimensional stability opens the way to a stable shape comparison methodology based on multidimensional persistence. In order to obtain our stability theorem, some other new results are proved for continuous filtering functions. They concern the finiteness of persistent Betti numbers for vector-valued filtering functions, and the representation via persistence diagrams of persistent Betti numbers, as well as their stability, in the case of scalarvalued filtering functions. Finally, from the stability of multidimensional persistent Betti numbers we obtain a lower bound for the natural pseudo-distance.
We consider generic curves in R 2 , i.e. generic C 1 functions f : S 1 → R 2 . We analyze these c... more We consider generic curves in R 2 , i.e. generic C 1 functions f : S 1 → R 2 . We analyze these curves through the persistent homology groups of a filtration induced on S 1 by f . In particular, we consider the question whether these persistent homology groups uniquely characterize f , at least up to re-parameterizations of S 1 . We give a partially positive answer to this question. More precisely, we prove that f = g • h, where h : S 1 → S 1 is a C 1 -diffeomorphism, if and only if the persistent homology groups of s • f and s • g coincide, for every s belonging to the group Σ 2 generated by reflections in the coordinate axes. Moreover, for a smaller set of generic functions, we show that f and g are close to each other in the max-norm (up to re-parameterizations) if and only if, for every s ∈ Σ 2 , the persistent Betti numbers functions of s • f and s • g are close to each other, with respect to a suitable distance.
The theory of multidimensional persistent homology was initially developed in the discrete settin... more The theory of multidimensional persistent homology was initially developed in the discrete setting, and involved the study of simplicial complexes filtered through an ordering of the simplices. Later, stability properties of multidimensional persistence have been proved to hold when topological spaces are filtered by continuous functions, i.e. for continuous data. This paper aims to provide a bridge between the continuous setting, where stability properties hold, and the discrete setting, where actual computations are carried out. More precisely, a stability preserving method is developed to compare rank invariants of vector functions obtained from discrete data. These advances confirm that multidimensional persistent homology is an appropriate tool for shape comparison in computer vision and computer graphics applications. The results are supported by numerical tests.
The Hausdorff distance, the Gromov-Hausdorff, the Fréchet and the natural pseudo-distances are in... more The Hausdorff distance, the Gromov-Hausdorff, the Fréchet and the natural pseudo-distances are instances of dissimilarity measures widely used in shape comparison. We show that they share the property of being defined as inf ρ F (ρ) where F is a suitable functional and ρ varies in a set of correspondences containing the set of homeomorphisms. Our main result states that the set of homeomorphisms cannot be enlarged to a metric space K, in such a way that the composition in K (extending the composition of homeomorphisms) passes to the limit and, at the same time, K is compact.
We introduce the persistent homotopy type distance dHT to compare real valued functions defined o... more We introduce the persistent homotopy type distance dHT to compare real valued functions defined on possibly different homotopy equivalent topological spaces. The underlying idea in the definition of dHT is to measure the minimal shift that is necessary to apply to one of the two functions in order that the sublevel sets of the two functions become homotopically equivalent. This distance is interesting in connection with persistent homology. Indeed, our main result states that dHT still provides an upper bound for the bottleneck distance between the persistence diagrams of the intervening functions. Moreover, because homotopy equivalences are weaker than homeomorphisms, this implies a lifting of the standard stability results provided by the L-infty distance and the natural pseudo-distance dNP. From a different standpoint, we prove that dHT extends the L-infty distance and dNP in two ways. First, we show that, appropriately restricting the category of objects to which dHT applies, it...
The exact computation of the matching distance for multi-parameter persistence modules is an acti... more The exact computation of the matching distance for multi-parameter persistence modules is an active area of research in computational topology. Achieving an easily obtainable exact computation of this distance would allow multi-parameter persistent homology to be a viable option for data analysis. In this paper, we provide theoretical results for the computation of the matching distance in two dimensions along with a geometric interpretation of the lines through parameter space realizing this distance. The crucial point of the method we propose is that it can be easily implemented.
SIGGRAPH ASIA 2016 Symposium on Visualization on - SA '16, 2016
Figure 1: The Hurricane dataset is a multivariate dataset defined on a cubical grid. Here we are ... more Figure 1: The Hurricane dataset is a multivariate dataset defined on a cubical grid. Here we are considering two values per point describing the temperature and the pressure above ground. For each function, we are showing the function gradient computed on the original scalar values. Then, critical cells obtained with our method are collected in clusters called extrema-clusters. Using the extrema-clusters and their size (number of voxels composing each cluster) we provide an interactive method for filtering out uninteresting regions. The extrema-clusters are shown in the lower part. For each image, only the clusters bigger than the indicated size are shown. The color scheme is based on the cluster's size. Comparing the function gradients with the clusters obtained we notice that bigger clusters are created where the gradient disagree, for example in the eye of the hurricane.
The Discrete Morse Theory of Forman appeared to be useful for providing filtration-preserving red... more The Discrete Morse Theory of Forman appeared to be useful for providing filtration-preserving reductions of complexes in the study of persistent homology. So far, the algorithms computing discrete Morse matchings have only been used for one-dimensional filtrations. This paper is perhaps the first attempt in the direction of extending such algorithms to multidimensional filtrations. Initial framework related to Morse matchings for the multidimensional setting is proposed, and a matching algorithm given by King, Knudson, and Mramor is extended in this direction. The correctness of the algorithm is proved, and its complexity analyzed. The algorithm is used for establishing a reduction of a simplicial complex to a smaller but not necessarily optimal cellular complex. First experiments with filtrations of triangular meshes are presented.
This paper proves that in Size Theory the comparison of multidimensional size functions can be re... more This paper proves that in Size Theory the comparison of multidimensional size functions can be reduced to the 1-dimensional case by a suitable change of variables. Indeed, we show that a foliation in half-planes can be given, such that the restriction of a multidimensional size function to each of these half-planes turns out to be a classical size function in two scalar variables. This leads to the definition of a new distance between multidimensional size functions, and to the proof of their stability with respect to that distance.
In this paper, we derive inequalities bounding the number of critical cells in a filtered cell co... more In this paper, we derive inequalities bounding the number of critical cells in a filtered cell complex on the one hand, and the entries of the Betti tables of the multi-parameter persistence modules of such filtrations on the other hand. Using the Mayer-Vietoris spectral sequence we first obtain strong and weak Morse inequalities involving the above quantities, and then we improve the weak inequalities achieving a sharp lower bound for the number of critical cells. Furthermore, we prove a sharp upper bound for the minimal number of critical cells, expressed again in terms of the entries of Betti tables.
This paper explores the concept of reparametrization invariant norm (RPI-norm), that is any norm ... more This paper explores the concept of reparametrization invariant norm (RPI-norm), that is any norm invariant under composition with diffeomorphisms. We prove the existence of an infinite family of RPI-norms, called standard RPI-norms, for which we exhibit both an integral and a discrete characterization. Our main result states that, for every one-time differentiable piecewise monotone function with compact support, its standard RPI-norms allow us to compute the value of any other RPI-norm of the same function. This is proved by using the standard RPI-norms in order to reconstruct the function up to reparametrization and an arbitrarily small error with respect to the total variation norm.
Motivated by the problem of dealing with incomplete or imprecise acquisition of data in computer ... more Motivated by the problem of dealing with incomplete or imprecise acquisition of data in computer vision and computer graphics, we extend results concerning the stability of persistent homology with respect to function perturbations to results concerning the stability with respect to domain perturbations. Domain perturbations can be measured in a number of different ways. An important method to compare domains is the Hausdorff distance. We show that by encoding sets using the distance function, the multidimensional matching distance between rank invariants of persistent homology groups is always upperly bounded by the Hausdorff distance between sets. Moreover we prove that our construction maintains information about the original set. Other well known methods to compare sets are considered, such as the symmetric difference distance between classical sets and the sup-distance between fuzzy sets. Also in these cases we present results stating that the multidimensional matching distance...
Topological data analysis and its main method, persistent homology, provide a toolkit for computi... more Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for one-parameter persistent homology have been established to connect persistent homology with machine learning techniques. We contribute a kernel construction for multi-parameter persistence by integrating a one-parameter kernel weighted along straight lines. We prove that our kernel is stable and efficiently computable, which establishes a theoretical connection between topological data analysis and machine learning for multivariate data analysis.
We consider generic curves in R 2 , i.e. generic C 1 functions f : S 1 → R 2. We analyze these cu... more We consider generic curves in R 2 , i.e. generic C 1 functions f : S 1 → R 2. We analyze these curves through the persistent homology groups of a filtration induced on S 1 by f. In particular, we consider the question whether these persistent homology groups uniquely characterize f , at least up to re-parameterizations of S 1. We give a partially positive answer to this question. More precisely, we prove that f = g • h, where h : S 1 → S 1 is a C 1-diffeomorphism, if and only if the persistent homology groups of s • f and s • g coincide, for every s belonging to the group Σ 2 generated by reflections in the coordinate axes. Moreover, for a smaller set of generic functions, we show that f and g are close to each other in the max-norm (up to re-parameterizations) if and only if, for every s ∈ Σ 2 , the persistent Betti numbers functions of s • f and s • g are close to each other, with respect to a suitable distance.
Rank invariants are a parametrized version of Betti numbers of a space multi-filtered by a contin... more Rank invariants are a parametrized version of Betti numbers of a space multi-filtered by a continuous vector-valued function. In this note we give a sufficient condition for their finiteness. This condition is sharp for spaces embeddable in R n .
Multidimensional persistence modules do not admit a concise representation analogous to that prov... more Multidimensional persistence modules do not admit a concise representation analogous to that provided by persistence diagrams for real-valued functions. However, there is no obstruction for multidimensional persistent Betti numbers to admit one. Therefore, it is reasonable to look for a generalization of persistence diagrams concerning those properties that are related only to persistent Betti numbers. In this paper, the persistence space of a vector-valued continuous function is introduced to generalize the concept of persistence diagram in this sense. The main result is its stability under function perturbations: any change in vector-valued functions implies a not greater change in the Hausdorff distance between their persistence spaces.
In many different industrial fields, computer vision and rob otics are jointly employed to perfor... more In many different industrial fields, computer vision and rob otics are jointly employed to perform repetitive, high-speed tasks, for quality inspection , t automatically move goods, and so on. This paper presents a hybrid computer vision-robotics s ystem for the automatic creation of ceramic mosaics, i.e. reproductions of patterns, syntheti c g ometries, or, in general, images, by means of ceramic tiles (Fig. 3.1). This task is currently performed, in most cases, by an expe rt, that is, however, slow and prone to errors. The system under d evelopment should be able to create mosaics at a higher speed and with the precision typic al of computer-based systems. Several working hypotheses can be made:
Although there is no doubt that multi-parameter persistent homology is a useful tool to analyse m... more Although there is no doubt that multi-parameter persistent homology is a useful tool to analyse multi-variate data, efficient ways to compute these modules are still lacking in the available topological data analysis toolboxes. Other issues such as interpretation and visualization of the output remain difficult to solve. Software visualizing multi-parameter persistence diagrams is currently only available for 2-dimensional persistence modules. One of the simplest invariants for a multi-parameter persistence module is its rank invariant, defined as the function that counts the number of linearly independent homology classes that live in the filtration through a given pair of values of the multi-parameter. We propose a step towards interpretation and visualization of the rank invariant for persistence modules for any given number of parameters. We show how discrete Morse theory may be used to compute the rank invariant, proving that it is completely determined by its values at points ...
Mathematical Methods in the Applied Sciences, 2013
Multidimensional persistence mostly studies topological features of shapes by analyzing the lower... more Multidimensional persistence mostly studies topological features of shapes by analyzing the lower level sets of vector-valued functions, called filtering functions. As is well known, in the case of scalar-valued filtering functions, persistent homology groups can be studied through their persistent Betti numbers, i.e. the dimensions of the images of the homomorphisms induced by the inclusions of lower level sets into each other. Whenever such inclusions exist for lower level sets of vector-valued filtering functions, we can consider the multidimensional analogue of persistent Betti numbers. Varying the lower level sets, we get that persistent Betti numbers can be seen as functions taking pairs of vectors to the set of non-negative integers. In this paper we prove stability of multidimensional persistent Betti numbers. More precisely, we prove that small changes of the vector-valued filtering functions imply only small changes of persistent Betti numbers functions. This result can be obtained by assuming the filtering functions to be just continuous. Multidimensional stability opens the way to a stable shape comparison methodology based on multidimensional persistence. In order to obtain our stability theorem, some other new results are proved for continuous filtering functions. They concern the finiteness of persistent Betti numbers for vector-valued filtering functions, and the representation via persistence diagrams of persistent Betti numbers, as well as their stability, in the case of scalarvalued filtering functions. Finally, from the stability of multidimensional persistent Betti numbers we obtain a lower bound for the natural pseudo-distance.
We consider generic curves in R 2 , i.e. generic C 1 functions f : S 1 → R 2 . We analyze these c... more We consider generic curves in R 2 , i.e. generic C 1 functions f : S 1 → R 2 . We analyze these curves through the persistent homology groups of a filtration induced on S 1 by f . In particular, we consider the question whether these persistent homology groups uniquely characterize f , at least up to re-parameterizations of S 1 . We give a partially positive answer to this question. More precisely, we prove that f = g • h, where h : S 1 → S 1 is a C 1 -diffeomorphism, if and only if the persistent homology groups of s • f and s • g coincide, for every s belonging to the group Σ 2 generated by reflections in the coordinate axes. Moreover, for a smaller set of generic functions, we show that f and g are close to each other in the max-norm (up to re-parameterizations) if and only if, for every s ∈ Σ 2 , the persistent Betti numbers functions of s • f and s • g are close to each other, with respect to a suitable distance.
The theory of multidimensional persistent homology was initially developed in the discrete settin... more The theory of multidimensional persistent homology was initially developed in the discrete setting, and involved the study of simplicial complexes filtered through an ordering of the simplices. Later, stability properties of multidimensional persistence have been proved to hold when topological spaces are filtered by continuous functions, i.e. for continuous data. This paper aims to provide a bridge between the continuous setting, where stability properties hold, and the discrete setting, where actual computations are carried out. More precisely, a stability preserving method is developed to compare rank invariants of vector functions obtained from discrete data. These advances confirm that multidimensional persistent homology is an appropriate tool for shape comparison in computer vision and computer graphics applications. The results are supported by numerical tests.
The Hausdorff distance, the Gromov-Hausdorff, the Fréchet and the natural pseudo-distances are in... more The Hausdorff distance, the Gromov-Hausdorff, the Fréchet and the natural pseudo-distances are instances of dissimilarity measures widely used in shape comparison. We show that they share the property of being defined as inf ρ F (ρ) where F is a suitable functional and ρ varies in a set of correspondences containing the set of homeomorphisms. Our main result states that the set of homeomorphisms cannot be enlarged to a metric space K, in such a way that the composition in K (extending the composition of homeomorphisms) passes to the limit and, at the same time, K is compact.
We introduce the persistent homotopy type distance dHT to compare real valued functions defined o... more We introduce the persistent homotopy type distance dHT to compare real valued functions defined on possibly different homotopy equivalent topological spaces. The underlying idea in the definition of dHT is to measure the minimal shift that is necessary to apply to one of the two functions in order that the sublevel sets of the two functions become homotopically equivalent. This distance is interesting in connection with persistent homology. Indeed, our main result states that dHT still provides an upper bound for the bottleneck distance between the persistence diagrams of the intervening functions. Moreover, because homotopy equivalences are weaker than homeomorphisms, this implies a lifting of the standard stability results provided by the L-infty distance and the natural pseudo-distance dNP. From a different standpoint, we prove that dHT extends the L-infty distance and dNP in two ways. First, we show that, appropriately restricting the category of objects to which dHT applies, it...
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