18 research outputs found
Deformation of surfaces in 2D persistent homology
In the context of 2D persistent homology a new metric has been recently introduced, the coherent matching distance. In order to study this metric, the filtering function is required to present particular âregularityâ properties, based on a geometrical construction of the real plane, called extended Pareto grid. This dissertation shows a new result for modifying the extended Pareto grid associated to a filtering function defined on a smooth closed surface, with values in the real plane. In future, the technical result presented here could be used to prove the genericity of the regularity conditions assumed for the filtering function
Landscapes of data sets and functoriality of persistent homology
The aim of this article is to describe a new perspective on functoriality of
persistent homology and explain its intrinsic symmetry that is often
overlooked. A data set for us is a finite collection of functions, called
measurements, with a finite domain. Such a data set might contain internal
symmetries which are effectively captured by the action of a set of the domain
endomorphisms. Different choices of the set of endomorphisms encode different
symmetries of the data set. We describe various category structures on such
enriched data sets and prove some of their properties such as decompositions
and morphism formations. We also describe a data structure, based on coloured
directed graphs, which is convenient to encode the mentioned enrichment. We
show that persistent homology preserves only some aspects of these landscapes
of enriched data sets however not all. In other words persistent homology is
not a functor on the entire category of enriched data sets. Nevertheless we
show that persistent homology is functorial locally. We use the concept of
equivariant operators to capture some of the information missed by persistent
homology
Homotopical decompositions of simplicial and Vietoris Rips complexes
Motivated by applications in Topological Data Analysis, we consider
decompositions of a simplicial complex induced by a cover of its vertices. We
study how the homotopy type of such decompositions approximates the homotopy of
the simplicial complex itself. The difference between the simplicial complex
and such an approximation is quantitatively measured by means of the so called
obstruction complexes. Our general machinery is then specialized to clique
complexes, Vietoris-Rips complexes and Vietoris-Rips complexes of metric
gluings. For the latter we give metric conditions which allow to recover the
first and zero-th homology of the gluing from the respective homologies of the
components
View-to-Label: Multi-View Consistency for Self-Supervised 3D Object Detection
For autonomous vehicles, driving safely is highly dependent on the capability
to correctly perceive the environment in 3D space, hence the task of 3D object
detection represents a fundamental aspect of perception. While 3D sensors
deliver accurate metric perception, monocular approaches enjoy cost and
availability advantages that are valuable in a wide range of applications.
Unfortunately, training monocular methods requires a vast amount of annotated
data. Interestingly, self-supervised approaches have recently been successfully
applied to ease the training process and unlock access to widely available
unlabelled data. While related research leverages different priors including
LIDAR scans and stereo images, such priors again limit usability. Therefore, in
this work, we propose a novel approach to self-supervise 3D object detection
purely from RGB sequences alone, leveraging multi-view constraints and weak
labels. Our experiments on KITTI 3D dataset demonstrate performance on par with
state-of-the-art self-supervised methods using LIDAR scans or stereo images
Impact of Ground Motion Duration and Soil Non-Linearity on the Seismic Performance of Single Piles
Pile foundations strongly influence the performance of supported structures and bridges during an earthquake. In case of strong earthquake ground motion, soft soils may be subjected to large deformation manifesting aspects typical of the non-linear behaviour such as material yielding, gapping and cyclic degradation. Therefore, nonlinear soil-pile interaction models should be able to capture these effects and improve the prediction of the actual seismic loading transferred from the foundation to the superstructure. In this paper, a beam on nonlinear Winkler foundation (BNWF) model is used, which can simulate cyclic soil degradation/hardening, soil and structural yielding, slack zone development and radiation damping. Incremental Dynamic Analyses (IDAs) are performed to evaluate the effects of Ground Motion Duration (GMD) and soil non-linearity on the performance of single fixed-head floating piles. Various homogeneous and bilayer soil profiles are considered, including saturated clay and sand in either fully dry or saturated state and with different levels of compaction. In order to evaluate the effect of nonlinearity on the response, the results of the nonlinear analyses are compared with those obtained from linear soil-pile analysis in terms of bending moment envelope. Results show the relevance of considering the GMD on the performance of the single pile especially when founded on saturated soils. For low intensities and dry sandy soils, the linear soil-pile interaction model can be used for obtaining reliable results
Tame representations in Topological Data Analysis: decompositions, invariants and metrics
This thesis is a compilation of results that can be framed within the field of applied topology. The starting point of our study is objects presenting a possibly complex intrinsic geometry. The main goal is then to simplify, without trivializing, the geometric information characterising these objects by choosing an appropriate representation. Thus, besides being simple and compact, the chosen representation should maintain the wealth of features of the initial object.In Topological Data Analysis (TDA), this simplification process can be done by assigning to each geometric object a functor indexed by a suitable poset.The most important fact about these functors is that, under appropriate hypotheses on the geometric object, they are discretisable. Being discretisable in this context means that they can be finitely encoded by a finite poset mapping to the original indexing poset. It is then possible to make one step further by computing invariants for the representations obtained. Desirable features for such invariants are to be effectively computable and suitable to describe metrics on. Comparing them gives, in fact, a good approximation of the comparison of the underlying geometric objects which are our primary interest. Paper A studies decompositions of simplicial complexes that are induced by coverings of their vertices. These decompositions are inspired by data analysis where commonly the data is given by a distance space, to which a filtered simplicial complex can be associated. We study how the homotopy type of a decomposed complex differs from the initial one, both for generic and for metric simplicial complexes. Another model to perform data analysis from a topological perspective is given by the theory of group equivariant nonexpansive operators.In Paper B, we show that such operators form a complementary tool to persistent homology in the context of TDA. We propose a categorical structure incorporating both models and then we study the functoriality of persistence. In Paper C we investigate suitable indexing posets for tame functors. The attention is focused on upper semilattices, which are particularly well suited for this purpose. Another class of posets that have similar properties to upper semilattices is the one of realisations, which we introduce here. Their similarities are both combinatorial, in particular concerning a notion of dimension that we introduce, and related to homological algebra for the tame functors indexed by them. In Paper C we also propose a method based on Koszul complexes to compute homological invariants for tame functors indexed either by upper semilattices or realisations. This question is then expanded in Paper D, where we study homological invariants relative to a chosen class of projectives, possibly different to the standard ones. We propose a framework to translate from the relative to the standard setting, where Koszul complexes are available to perform the computations. We also identify an obstruction for such translation to be possible and characterise it for several examples of relative projectives. In Paper E we study the geometrical properties of a well-established metric in 2-parameter persistent homology, called the matching distance. Motivated by the need for effectiveness in the computation of such metric, we study its geometric properties.In particular, we show how to take advantage of the differential geometric structure of the underlying objects to understand the properties of the metric. In Paper F we study the category of discretisable functors with values in non-negative chain complexes. In this category, we are particularly interested in cofibrant indecomposables, which require a model structure to be defined. Thus, we first identify a new class of posets indexing the functors for which a projective model structure exists and give a characterisation of cofibrant indecomposables there. In the case, the indexing poset is not of this type, we outline a technique to construct arbitrarily complicated cofibrant indecomposables.Denna avhandling Ă€r en sammanstĂ€llning av resultat inom tillĂ€mpad topologi.UtgĂ„ngspunkten för vĂ„r studie Ă€r objekt som presenterar en möjligen komplex inneboende geometri.HuvudmĂ„let Ă€r dĂ„ att förenkla den geometriska informationen som kĂ€nnetecknar dessa objekt, utan att trivialisera den.SĂ„ledes, förutom att vara enkel och kompakt, bör den valda representationen bibehĂ„lla rikedomen av egenskaper hos det ursprungliga objektet.I Topologisk Data Analys (TDA) kan denna förenklingsprocess göras genom att tilldela varje geometriskt objekt en funktor indexerad av en lĂ€mplig pomĂ€ngd.Det viktigaste med dessa funktorer Ă€r att de Ă€r diskretiserbara under lĂ€mpliga antaganden om det geometriska objektet.Att vara diskretiserbar i detta sammanhang innebĂ€r att de kan Ă€ndligt kodas genom en finit pomĂ€ngd-mappning till den ursprungliga indexeringspomĂ€ngden.Det Ă€r dĂ„ möjligt att ta ytterligare steg genom att berĂ€kna invarianter av de representationer som erhĂ„lls.ĂnskvĂ€rda egenskaper för sĂ„dana invarianter Ă€r att vara effektivt berĂ€kningsbara ochlĂ€mplig att beskriva metriker pĂ„.Att jĂ€mföra invarianterna ger dĂ„ en bra approximation av jĂ€mföra de underliggande geometriska objekten, som Ă€r vĂ„rt primĂ€ra intresse. Artikel A studerar dekompositioner av simpliciala komplex som induceras av tĂ€ckningar av deras hörn.Dessa dekompositioner Ă€r inspirerade av dataanalys dĂ€r datan vanligtvis ges av ett metriskt utrymme, till vilket ett filtrerat simplicialt komplex kan associeras.Vi studerar hur homotopitypen för ett nedbrutet komplex skiljer sig frĂ„n det initiala, bĂ„de för generiska och för metriska simpliciala komplex. En annan modell för att utföra dataanalys ur ett topologiskt perspektiv ges av teorin om gruppekvivarianta icke-expansiva operatorer.I Paper B visar vi att sĂ„dana operatörer utgör ett komplementĂ€rt verktyg till ihĂ„llande homologi i samband med TDA.Vi föreslĂ„r en kategorisk struktur som inkluderar bĂ„da modellerna och sedan studerar vi funktorialiteten av persistens. I Paper C undersöker vi lĂ€mpliga indexeringspositioner för tama funktorer.Fokus ligger pĂ„ övre semigitter, som Ă€r sĂ€rskilt vĂ€l lĂ€mpade för detta Ă€ndamĂ„l.En annan klass av pomĂ€ngder som har liknande egenskaper som övre semigitter Ă€r den av realisationer, som vi introducerar hĂ€r.Deras likheter Ă€r bĂ„de kombinatoriska, sĂ€rskilt nĂ€r det gĂ€ller en dimensionsuppfattning som vi introducerar, och relaterade till homologisk algebra för de tama funktorer som indexeras av dem.I Paper C föreslĂ„r vi ocksĂ„ en metod baserad pĂ„ Koszul-komplex för att berĂ€kna homologiska invarianter för tama funktorer indexerade antingen med övre semigitter eller realisationer.Denna frĂ„ga utökas sedan i Paper D, dĂ€r vi studerar homologiska invarianter i förhĂ„llande till en vald klass av projektiva objekt, möjligen olika de vanliga.Vi föreslĂ„r ett ramverk för att översĂ€tta frĂ„n den relativa till standardfallet, dĂ€r Koszul-komplex Ă€r tillgĂ€ngliga för att utföra berĂ€kningarna.Vi identifierar ocksĂ„ ett hinder för att en sĂ„dan översĂ€ttning ska vara möjlig och karakteriserar den för flera exempel pĂ„ relativa projektiv. I Paper E studerar vi de geometriska egenskaperna hos en vĂ€letablerad metrik i 2-parameter ihĂ„llande homologi, kallad matchningsavstĂ„ndet.Motiverade av behovet av effektivitet vid berĂ€kningen av sĂ„dan metrik studerar vi dess geometriska egenskaper.I synnerhet visar vi hur man drar fördel av den differentiella geometriska strukturen hos de underliggande objekten för att förstĂ„ metrikens egenskaper. I Paper F studerar vi kategorin av diskretiserbara funktioner med vĂ€rden i icke-negativa kedjekomplex.I den hĂ€r kategorin Ă€r vi sĂ€rskilt intresserade av kofibranter odelbara, som krĂ€ver en modellstruktur för att definieras.SĂ„lunda identifierar vi först en ny klass av pomĂ€ngder som indexerar de funktioner för vilka det finns en projektiv modellstruktur och ger en karakterisering av kofibranter som Ă€r odelbara dĂ€r.Om indexeringsposen inte Ă€r av denna typ,vi skisserar en teknik för att konstruera godtyckligt komplicerade kofibranter odelbara.QC 2023-05-25</p
Corrispondenza ideali-varietĂ e criterio jacobiano per le singolaritĂ
L'elaborato ha come soggetto le varietĂ algebriche affini. I primi due capitoli vanno ad analizzare nel dettaglio la corrispondenza fra gli ideali nell'anello dei polinomi e le varietĂ , che risulta biunivoca nel caso in cui si lavori in un campo algebricamente chiuso e ci si restringa agli ideali radicali. Il terzo e ultimo capitolo Ăš dedicato allo studio di due concetti fondamentali per le varietĂ algebriche: la loro dimensione e i loro punti singolari. Vengono introdotte tre nozioni di dimensione di una varietĂ algebrica e se ne dimostra l'equivalenza. Per lo studio delle singolaritĂ , si introduce il cosiddetto criterio jacobiano, basato sullo studio della matrice jacobiana ottenuta tramite le derivate parziali dei polinomi che definiscono la varietĂ
Tame representations in Topological Data Analysis: decompositions, invariants and metrics
This thesis is a compilation of results that can be framed within the field of applied topology. The starting point of our study is objects presenting a possibly complex intrinsic geometry. The main goal is then to simplify, without trivializing, the geometric information characterising these objects by choosing an appropriate representation. Thus, besides being simple and compact, the chosen representation should maintain the wealth of features of the initial object.In Topological Data Analysis (TDA), this simplification process can be done by assigning to each geometric object a functor indexed by a suitable poset.The most important fact about these functors is that, under appropriate hypotheses on the geometric object, they are discretisable. Being discretisable in this context means that they can be finitely encoded by a finite poset mapping to the original indexing poset. It is then possible to make one step further by computing invariants for the representations obtained. Desirable features for such invariants are to be effectively computable and suitable to describe metrics on. Comparing them gives, in fact, a good approximation of the comparison of the underlying geometric objects which are our primary interest. Paper A studies decompositions of simplicial complexes that are induced by coverings of their vertices. These decompositions are inspired by data analysis where commonly the data is given by a distance space, to which a filtered simplicial complex can be associated. We study how the homotopy type of a decomposed complex differs from the initial one, both for generic and for metric simplicial complexes. Another model to perform data analysis from a topological perspective is given by the theory of group equivariant nonexpansive operators.In Paper B, we show that such operators form a complementary tool to persistent homology in the context of TDA. We propose a categorical structure incorporating both models and then we study the functoriality of persistence. In Paper C we investigate suitable indexing posets for tame functors. The attention is focused on upper semilattices, which are particularly well suited for this purpose. Another class of posets that have similar properties to upper semilattices is the one of realisations, which we introduce here. Their similarities are both combinatorial, in particular concerning a notion of dimension that we introduce, and related to homological algebra for the tame functors indexed by them. In Paper C we also propose a method based on Koszul complexes to compute homological invariants for tame functors indexed either by upper semilattices or realisations. This question is then expanded in Paper D, where we study homological invariants relative to a chosen class of projectives, possibly different to the standard ones. We propose a framework to translate from the relative to the standard setting, where Koszul complexes are available to perform the computations. We also identify an obstruction for such translation to be possible and characterise it for several examples of relative projectives. In Paper E we study the geometrical properties of a well-established metric in 2-parameter persistent homology, called the matching distance. Motivated by the need for effectiveness in the computation of such metric, we study its geometric properties.In particular, we show how to take advantage of the differential geometric structure of the underlying objects to understand the properties of the metric. In Paper F we study the category of discretisable functors with values in non-negative chain complexes. In this category, we are particularly interested in cofibrant indecomposables, which require a model structure to be defined. Thus, we first identify a new class of posets indexing the functors for which a projective model structure exists and give a characterisation of cofibrant indecomposables there. In the case, the indexing poset is not of this type, we outline a technique to construct arbitrarily complicated cofibrant indecomposables.Denna avhandling Ă€r en sammanstĂ€llning av resultat inom tillĂ€mpad topologi.UtgĂ„ngspunkten för vĂ„r studie Ă€r objekt som presenterar en möjligen komplex inneboende geometri.HuvudmĂ„let Ă€r dĂ„ att förenkla den geometriska informationen som kĂ€nnetecknar dessa objekt, utan att trivialisera den.SĂ„ledes, förutom att vara enkel och kompakt, bör den valda representationen bibehĂ„lla rikedomen av egenskaper hos det ursprungliga objektet.I Topologisk Data Analys (TDA) kan denna förenklingsprocess göras genom att tilldela varje geometriskt objekt en funktor indexerad av en lĂ€mplig pomĂ€ngd.Det viktigaste med dessa funktorer Ă€r att de Ă€r diskretiserbara under lĂ€mpliga antaganden om det geometriska objektet.Att vara diskretiserbar i detta sammanhang innebĂ€r att de kan Ă€ndligt kodas genom en finit pomĂ€ngd-mappning till den ursprungliga indexeringspomĂ€ngden.Det Ă€r dĂ„ möjligt att ta ytterligare steg genom att berĂ€kna invarianter av de representationer som erhĂ„lls.ĂnskvĂ€rda egenskaper för sĂ„dana invarianter Ă€r att vara effektivt berĂ€kningsbara ochlĂ€mplig att beskriva metriker pĂ„.Att jĂ€mföra invarianterna ger dĂ„ en bra approximation av jĂ€mföra de underliggande geometriska objekten, som Ă€r vĂ„rt primĂ€ra intresse. Artikel A studerar dekompositioner av simpliciala komplex som induceras av tĂ€ckningar av deras hörn.Dessa dekompositioner Ă€r inspirerade av dataanalys dĂ€r datan vanligtvis ges av ett metriskt utrymme, till vilket ett filtrerat simplicialt komplex kan associeras.Vi studerar hur homotopitypen för ett nedbrutet komplex skiljer sig frĂ„n det initiala, bĂ„de för generiska och för metriska simpliciala komplex. En annan modell för att utföra dataanalys ur ett topologiskt perspektiv ges av teorin om gruppekvivarianta icke-expansiva operatorer.I Paper B visar vi att sĂ„dana operatörer utgör ett komplementĂ€rt verktyg till ihĂ„llande homologi i samband med TDA.Vi föreslĂ„r en kategorisk struktur som inkluderar bĂ„da modellerna och sedan studerar vi funktorialiteten av persistens. I Paper C undersöker vi lĂ€mpliga indexeringspositioner för tama funktorer.Fokus ligger pĂ„ övre semigitter, som Ă€r sĂ€rskilt vĂ€l lĂ€mpade för detta Ă€ndamĂ„l.En annan klass av pomĂ€ngder som har liknande egenskaper som övre semigitter Ă€r den av realisationer, som vi introducerar hĂ€r.Deras likheter Ă€r bĂ„de kombinatoriska, sĂ€rskilt nĂ€r det gĂ€ller en dimensionsuppfattning som vi introducerar, och relaterade till homologisk algebra för de tama funktorer som indexeras av dem.I Paper C föreslĂ„r vi ocksĂ„ en metod baserad pĂ„ Koszul-komplex för att berĂ€kna homologiska invarianter för tama funktorer indexerade antingen med övre semigitter eller realisationer.Denna frĂ„ga utökas sedan i Paper D, dĂ€r vi studerar homologiska invarianter i förhĂ„llande till en vald klass av projektiva objekt, möjligen olika de vanliga.Vi föreslĂ„r ett ramverk för att översĂ€tta frĂ„n den relativa till standardfallet, dĂ€r Koszul-komplex Ă€r tillgĂ€ngliga för att utföra berĂ€kningarna.Vi identifierar ocksĂ„ ett hinder för att en sĂ„dan översĂ€ttning ska vara möjlig och karakteriserar den för flera exempel pĂ„ relativa projektiv. I Paper E studerar vi de geometriska egenskaperna hos en vĂ€letablerad metrik i 2-parameter ihĂ„llande homologi, kallad matchningsavstĂ„ndet.Motiverade av behovet av effektivitet vid berĂ€kningen av sĂ„dan metrik studerar vi dess geometriska egenskaper.I synnerhet visar vi hur man drar fördel av den differentiella geometriska strukturen hos de underliggande objekten för att förstĂ„ metrikens egenskaper. I Paper F studerar vi kategorin av diskretiserbara funktioner med vĂ€rden i icke-negativa kedjekomplex.I den hĂ€r kategorin Ă€r vi sĂ€rskilt intresserade av kofibranter odelbara, som krĂ€ver en modellstruktur för att definieras.SĂ„lunda identifierar vi först en ny klass av pomĂ€ngder som indexerar de funktioner för vilka det finns en projektiv modellstruktur och ger en karakterisering av kofibranter som Ă€r odelbara dĂ€r.Om indexeringsposen inte Ă€r av denna typ,vi skisserar en teknik för att konstruera godtyckligt komplicerade kofibranter odelbara.QC 2023-05-25</p
Soil-structure interaction in the seismic response of an isolated three span motorway overcrossing founded on piles
The effects of soil-structure interaction on the seismic response of an isolated three span motorway overcrossing founded on piles are investigated by considering a real bridge located along the A14 Motorway in central Italy. The dynamic and mechanical properties of the soils are obtained from a comprehensive geotechnical characterization of the sites. Ten triplets of real accelerograms, defined at the outcropping bedrock, are adopted and processed by local response analyses to capture the site amplification effects and the free-field motions within the deposits. The soil-structure interaction
effects are evaluated by means of the substructure method by comparing the seismic response of the structures with those obtained from conventional fixed base models. Analyses demonstrate that the soil-foundation dynamic compliance as well as the energy loss due to radiation damping dot not modify significantly the overall behaviour of the isolated bridges, while soil-structure interaction may increase deformations and forces on the isolation devices with respect to those obtained with fixed base models