Data Analysis with the Morse-Smale Complex: The msr Package for R

In many areas, scientists deal with increasingly high-dimensional data sets. An important aspect for these scientists is to gain a qualitative understanding of the process or system from which the data is gathered. Often, both input variables and an outcome are observed and the data can be characterized as a sample from a high-dimensional scalar function. This work presents the R package msr for exploratory data analysis of multivariate scalar functions based on the Morse-Smale complex. The Morse-Smale complex provides a topologically meaningful decomposition of the domain. The msr package implements a discrete approximation of the Morse-Smale complex for data sets. In previous work this approximation has been exploited for visualization and partition-based regression, which are both supported in the msr package. The visualization combines the Morse-Smale complex with dimension-reduction techniques for a visual summary representation that serves as a guide for interactive exploration of the high-dimensional function. In a similar fashion, the regression employs a combination of linear models based on the Morse-Smale decomposition of the domain. This regression approach yields topologically accurate estimates and facilitates interpretation of general trends and statistical comparisons between partitions. In this manner, the msr package supports high-dimensional data understanding and exploration through the Morse-Smale complex

Skeletonization and Partitioning of Digital Images Using Discrete Morse Theory

We show how discrete Morse theory provides a rigorous and unifying foundation for defining skeletons and partitions of grayscale digital images. We model a grayscale image as a cubical complex with a real-valued function defined on its vertices (the voxel values). This function is extended to a discrete gradient vector field using the algorithm presented in Robins, Wood, Sheppard TPAMI 33:1646 (2011). In the current paper we define basins (the building blocks of a partition) and segments of the skeleton using the stable and unstable sets associated with critical cells. The natural connection between Morse theory and homology allows us to prove the topological validity of these constructions; for example, that the skeleton is homotopic to the initial object. We simplify the basins and skeletons via Morse-theoretic cancellation of critical cells in the discrete gradient vector field using a strategy informed by persistent homology. Simple working Python code for our algorithms for efficient vector field traversal is included. Example data are taken from micro-CT images of porous materials, an application area where accurate topological models of pore connectivity are vital for fluid-flow modelling

Human perception-oriented segmentation for triangle meshes

A segmentação de malhas é um tópico importante de investigação em computação gráfica, em particular em modelação geométrica. Isto deve-se ao facto de as técnicas de segmentaçãodemalhasteremváriasaplicações,nomeadamentenaproduçãodefilmes, animaçãoporcomputador, realidadevirtual, compressãodemalhas, assimcomoemjogosdigitais. Emconcreto, asmalhastriangularessãoamplamenteusadasemaplicações interativas, visto que sua segmentação em partes significativas (também designada por segmentação significativa, segmentação perceptiva ou segmentação perceptualmente significativa ) é muitas vezes vista como uma forma de acelerar a interação com o utilizador ou a deteção de colisões entre esses objetos 3D definidos por uma malha, bem como animar uma ou mais partes significativas (por exemplo, a cabeça de uma personagem) de um dado objeto, independentemente das restantes partes. Acontece que não se conhece nenhuma técnica capaz de segmentar correctamente malhas arbitrárias −ainda que restritas aos domínios de formas livres e não-livres− em partes significativas. Algumas técnicas são mais adequadas para objetos de forma não-livre (por exemplo, peças mecânicas definidas geometricamente por quádricas), enquanto outras são mais talhadas para o domínio dos objectos de forma livre. Só na literatura recente surgem umas poucas técnicas que se aplicam a todo o universo de objetos de forma livre e não-livre. Pior ainda é o facto de que a maioria das técnicas de segmentação não serem totalmente automáticas, no sentido de que quase todas elas exigem algum tipo de pré-requisitos e assistência do utilizador. Resumindo, estes três desafios relacionados com a proximidade perceptual, generalidade e automação estão no cerne do trabalho descrito nesta tese. Para enfrentar estes desafios, esta tese introduz o primeiro algoritmo de segmentação baseada nos contornos ou fronteiras dos segmentos, cuja técnica se inspira nas técnicas de segmentação baseada em arestas, tão comuns em análise e processamento de imagem,porcontraposiçãoàstécnicasesegmentaçãobaseadaemregiões. Aideiaprincipal é a de encontrar em primeiro lugar a fronteira de cada região para, em seguida, identificar e agrupar todos os seus triângulos internos. As regiões da malha encontradas correspondem a saliências e reentrâncias, que não precisam de ser estritamente convexas, nem estritamente côncavas, respectivamente. Estas regiões, designadas regiões relaxadamenteconvexas(ousaliências)eregiõesrelaxadamentecôncavas(oureentrâncias), produzem segmentações que são menos sensíveis ao ruído e, ao mesmo tempo, são mais intuitivas do ponto de vista da perceção humana; por isso, é designada por segmentação orientada à perceção humana (ou, human perception- oriented (HPO), do inglês). Além disso, e ao contrário do atual estado-da-arte da segmentação de malhas, a existência destas regiões relaxadas torna o algoritmo capaz de segmentar de maneira bastante plausível tanto objectos de forma não-livre como objectos de forma livre. Nesta tese, enfrentou-se também um quarto desafio, que está relacionado com a fusão de segmentação e multi-resolução de malhas. Em boa verdade, já existe na literatura uma variedade grande de técnicas de segmentação, bem como um número significativo de técnicas de multi-resolução, para malhas triangulares. No entanto, não é assim tão comum encontrar estruturas de dados e algoritmos que façam a fusão ou a simbiose destes dois conceitos, multi-resolução e segmentação, num único esquema multi-resolução que sirva os propósitos das aplicações que lidam com malhas simples e segmentadas, sendo que neste contexto se entende que uma malha simples é uma malha com um único segmento. Sendo assim, nesta tese descreve-se um novo esquema (entenda-seestruturasdedadosealgoritmos)demulti-resoluçãoesegmentação,designado por extended Ghost Cell (xGC). Este esquema preserva a forma das malhas, tanto em termos globais como locais, ou seja, os segmentos da malha e as suas fronteiras, bem como os seus vincos e ápices são preservados, não importa o nível de resolução que usamos durante a/o simplificação/refinamento da malha. Além disso, ao contrário de outros esquemas de segmentação, tornou-se possível ter segmentos adjacentes com dois ou mais níveis de resolução de diferença. Isto é particularmente útil em animação por computador, compressão e transmissão de malhas, operações de modelação geométrica, visualização científica e computação gráfica. Em suma, esta tese apresenta um esquema genérico, automático, e orientado à percepção humana, que torna possível a simbiose dos conceitos de segmentação e multiresolução de malhas trianguladas que sejam representativas de objectos 3D.The mesh segmentation is an important topic in computer graphics, in particular in geometric computing. This is so because mesh segmentation techniques find many applications in movies, computer animation, virtual reality, mesh compression, and games. Infact, trianglemeshesarewidelyusedininteractiveapplications, sothattheir segmentation in meaningful parts (i.e., human-perceptually segmentation, perceptive segmentationormeaningfulsegmentation)isoftenseenasawayofspeedinguptheuser interaction, detecting collisions between these mesh-covered objects in a 3D scene, as well as animating one or more meaningful parts (e.g., the head of a humanoid) independently of the other parts of a given object. It happens that there is no known technique capable of correctly segmenting any mesh into meaningful parts. Some techniques are more adequate for non-freeform objects (e.g., quadricmechanicalparts), whileothersperformbetterinthedomainoffreeform objects. Only recently, some techniques have been developed for the entire universe of objects and shapes. Even worse it is the fact that most segmentation techniques are not entirely automated in the sense that almost all techniques require some sort of pre-requisites and user assistance. Summing up, these three challenges related to perceptual proximity, generality and automation are at the core of the work described in this thesis. In order to face these challenges, we have developed the first contour-based mesh segmentation algorithm that we may find in the literature, which is inspired in the edgebased segmentation techniques used in image analysis, as opposite to region-based segmentation techniques. Its leading idea is to firstly find the contour of each region, and then to identify and collect all of its inner triangles. The encountered mesh regions correspond to ups and downs, which do not need to be strictly convex nor strictly concave, respectively. These regions, called relaxedly convex regions (or saliences) and relaxedly concave regions (or recesses), produce segmentations that are less-sensitive to noise and, at the same time, are more intuitive from the human point of view; hence it is called human perception- oriented (HPO) segmentation. Besides, and unlike the current state-of-the-art in mesh segmentation, the existence of these relaxed regions makes the algorithm suited to both non-freeform and freeform objects. In this thesis, we have also tackled a fourth challenge, which is related with the fusion of mesh segmentation and multi-resolution. Truly speaking, a plethora of segmentation techniques, as well as a number of multiresolution techniques, for triangle meshes already exist in the literature. However, it is not so common to find algorithms and data structures that fuse these two concepts, multiresolution and segmentation, into a symbiotic multi-resolution scheme for both plain and segmented meshes, in which a plainmeshisunderstoodasameshwithasinglesegment. So, weintroducesuchanovel multiresolution segmentation scheme, called extended Ghost Cell (xGC) scheme. This scheme preserves the shape of the meshes in both global and local terms, i.e., mesh segments and their boundaries, as well as creases and apices are preserved, no matter the level of resolution we use for simplification/refinement of the mesh. Moreover, unlike other segmentation schemes, it was made possible to have adjacent segments with two or more resolution levels of difference. This is particularly useful in computer animation, mesh compression and transmission, geometric computing, scientific visualization, and computer graphics. In short, this thesis presents a fully automatic, general, and human perception-oriented scheme that symbiotically integrates the concepts of mesh segmentation and multiresolution

Structure in the 3D Galaxy Distribution: I. Methods and Example Results

Three methods for detecting and characterizing structure in point data, such as that generated by redshift surveys, are described: classification using self-organizing maps, segmentation using Bayesian blocks, and density estimation using adaptive kernels. The first two methods are new, and allow detection and characterization of structures of arbitrary shape and at a wide range of spatial scales. These methods should elucidate not only clusters, but also the more distributed, wide-ranging filaments and sheets, and further allow the possibility of detecting and characterizing an even broader class of shapes. The methods are demonstrated and compared in application to three data sets: a carefully selected volume-limited sample from the Sloan Digital Sky Survey redshift data, a similarly selected sample from the Millennium Simulation, and a set of points independently drawn from a uniform probability distribution -- a so-called Poisson distribution. We demonstrate a few of the many ways in which these methods elucidate large scale structure in the distribution of galaxies in the nearby Universe.Comment: Re-posted after referee corrections along with partially re-written introduction. 80 pages, 31 figures, ApJ in Press. For full sized figures please download from: http://astrophysics.arc.nasa.gov/~mway/lss1.pd

Progressive Wasserstein Barycenters of Persistence Diagrams

This paper presents an efficient algorithm for the progressive approximation of Wasserstein barycenters of persistence diagrams, with applications to the visual analysis of ensemble data. Given a set of scalar fields, our approach enables the computation of a persistence diagram which is representative of the set, and which visually conveys the number, data ranges and saliences of the main features of interest found in the set. Such representative diagrams are obtained by computing explicitly the discrete Wasserstein barycenter of the set of persistence diagrams, a notoriously computationally intensive task. In particular, we revisit efficient algorithms for Wasserstein distance approximation [12,51] to extend previous work on barycenter estimation [94]. We present a new fast algorithm, which progressively approximates the barycenter by iteratively increasing the computation accuracy as well as the number of persistent features in the output diagram. Such a progressivity drastically improves convergence in practice and allows to design an interruptible algorithm, capable of respecting computation time constraints. This enables the approximation of Wasserstein barycenters within interactive times. We present an application to ensemble clustering where we revisit the k-means algorithm to exploit our barycenters and compute, within execution time constraints, meaningful clusters of ensemble data along with their barycenter diagram. Extensive experiments on synthetic and real-life data sets report that our algorithm converges to barycenters that are qualitatively meaningful with regard to the applications, and quantitatively comparable to previous techniques, while offering an order of magnitude speedup when run until convergence (without time constraint). Our algorithm can be trivially parallelized to provide additional speedups in practice on standard workstations. [...

Indexing and Retrieval of 3D Articulated Geometry Models

In this PhD research study, we focus on building a content-based search engine for 3D articulated geometry models. 3D models are essential components in nowadays graphic applications, and are widely used in the game, animation and movies production industry. With the increasing number of these models, a search engine not only provides an entrance to explore such a huge dataset, it also facilitates sharing and reusing among different users. In general, it reduces production costs and time to develop these 3D models. Though a lot of retrieval systems have been proposed in recent years, search engines for 3D articulated geometry models are still in their infancies. Among all the works that we have surveyed, reliability and efficiency are the two main issues that hinder the popularity of such systems. In this research, we have focused our attention mainly to address these two issues. We have discovered that most existing works design features and matching algorithms in order to reflect the intrinsic properties of these 3D models. For instance, to handle 3D articulated geometry models, it is common to extract skeletons and use graph matching algorithms to compute the similarity. However, since this kind of feature representation is complex, it leads to high complexity of the matching algorithms. As an example, sub-graph isomorphism can be NP-hard for model graph matching. Our solution is based on the understanding that skeletal matching seeks correspondences between the two comparing models. If we can define descriptive features, the correspondence problem can be solved by bag-based matching where fast algorithms are available. In the first part of the research, we propose a feature extraction algorithm to extract such descriptive features. We then convert the skeletal matching problems into bag-based matching. We further define metric similarity measure so as to support fast search. We demonstrate the advantages of this idea in our experiments. The improvement on precision is 12\% better at high recall. The indexing search of 3D model is 24 times faster than the state of the art if only the first relevant result is returned. However, improving the quality of descriptive features pays the price of high dimensionality. Curse of dimensionality is a notorious problem on large multimedia databases. The computation time scales exponentially as the dimension increases, and indexing techniques may not be useful in such situation. In the second part of the research, we focus ourselves on developing an embedding retrieval framework to solve the high dimensionality problem. We first argue that our proposed matching method projects 3D models on manifolds. We then use manifold learning technique to reduce dimensionality and maximize intra-class distances. We further propose a numerical method to sub-sample and fast search databases. To preserve retrieval accuracy using fewer landmark objects, we propose an alignment method which is also beneficial to existing works for fast search. The advantages of the retrieval framework are demonstrated in our experiments that it alleviates the problem of curse of dimensionality. It also improves the efficiency (3.4 times faster) and accuracy (30\% more accurate) of our matching algorithm proposed above. In the third part of the research, we also study a closely related area, 3D motions. 3D motions are captured by sticking sensor on human beings. These captured data are real human motions that are used to animate 3D articulated geometry models. Creating realistic 3D motions is an expensive and tedious task. Although 3D motions are very different from 3D articulated geometry models, we observe that existing works also suffer from the problem of temporal structure matching. This also leads to low efficiency in the matching algorithms. We apply the same idea of bag-based matching into the work of 3D motions. From our experiments, the proposed method has a 13\% improvement on precision at high recall and is 12 times faster than existing works. As a summary, we have developed algorithms for 3D articulated geometry models and 3D motions, covering feature extraction, feature matching, indexing and fast search methods. Through various experiments, our idea of converting restricted matching to bag-based matching improves matching efficiency and reliability. These have been shown in both 3D articulated geometry models and 3D motions. We have also connected 3D matching to the area of manifold learning. The embedding retrieval framework not only improves efficiency and accuracy, but has also opened a new area of research

Sketching-based Skeleton Extraction

Articulated character animation can be performed by manually creating and rigging a skeleton into an unfolded 3D mesh model. Such tasks are not trivial, as they require a substantial amount of training and practice. Although methods have been proposed to help automatic extraction of skeleton structure, they may not guarantee that the resulting skeleton can help to produce animations according to user manipulation. We present a sketching-based skeleton extraction method to create a user desired skeleton structure which is used in 3D model animation. This method takes user sketching as an input, and based on the mesh segmentation result of a 3D mesh model, generates a skeleton for articulated character animation. In our system, we assume that a user will properly sketch bones by roughly following the mesh model structure. The user is expected to sketch independently on different regions of a mesh model for creating separate bones. For each sketched stroke, we project it into the mesh model so that it becomes the medial axis of its corresponding mesh model region from the current viewer perspective. We call this projected stroke a “sketched bone”. After pre-processing user sketched bones, we cluster them into groups. This process is critical as user sketching can be done from any orientation of a mesh model. To specify the topology feature for different mesh parts, a user can sketch strokes from different orientations of a mesh model, as there may be duplicate strokes from different orientations for the same mesh part. We need a clustering process to merge similar sketched bones into one bone, which we call a “reference bone”. The clustering process is based on three criteria: orientation, overlapping and locality. Given the reference bones as the input, we adopt a mesh segmentation process to assist our skeleton extraction method. To be specific, we apply the reference bones and the seed triangles to segment the input mesh model into meaningful segments using a multiple-region growing mechanism. The seed triangles, which are collected from the reference bones, are used as the initial seeds in the mesh segmentation process. We have designed a new segmentation metric [1] to form a better segmentation criterion. Then we compute the Level Set Diagrams (LSDs) on each mesh part to extract bones and joints. To construct the final skeleton, we connect bones extracted from all mesh parts together into a single structure. There are three major steps involved: optimizing and smoothing bones, generating joints and forming the skeleton structure. After constructing the skeleton model, we have proposed a new method, which utilizes the Linear Blend Skinning (LBS) technique and the Laplacian mesh deformation technique together to perform skeleton-driven animation. Traditional LBS techniques may have self-intersection problem in regions around segmentation boundaries. Laplacian mesh deformation can preserve the local surface details, which can eliminate the self-intersection problem. In this case, we make use of LBS result as the positional constraint to perform a Laplacian mesh deformation. By using the Laplacian mesh deformation method, we maintain the surface details in segmentation boundary regions. This thesis outlines a novel approach to construct a 3D skeleton model interactively, which can also be used in 3D animation and 3D model matching area. The work is motivated by the observation that either most of the existing automatic skeleton extraction methods lack well-positioned joints specification or the manually generated methods require too much professional training to create a good skeleton structure. We dedicate a novel approach to create 3D model skeleton based on user sketching which specifies articulated skeleton with joints. The experimental results show that our method can produce better skeletons in terms of joint positions and topological structure

A Fuzzy Approach for Topological Data Analysis

Geometry and topology are becoming more powerful and dominant in data analysis because of their outstanding characteristics. It has emerged recently as a promising research area, known as Topological Data Analysis (TDA), for modern computer science. In recent years, the Mapper algorithm, an outstanding TDA representative, is increasingly completed with a stabilized theoretical foundation and practical applications and diverse, intuitive, user-friendly implementations. From a theoretical perspective, the Mapper algorithm is still a fuzzy clustering algorithm, with a visualization capability to extract the shape summary of data. However, its outcomes are still very sensitive to the parameter choice, including resolution and function. Therefore, there is a need to reduce the dependence on its parameters significantly. This idea is exciting and can be solved thanks to the outstanding characteristics of fuzzy clustering. The Mapper clustering ability is getting more potent by the support from well-known techniques. Therefore, this combination is expected to usefully and powerfully solve some problems encountered in many fields. The main research goal of this thesis is to approach TDA by fuzzy theory to create the interrelationships between them in terms of clustering. Explicitly speaking, the Mapper algorithm represents TDA, and the Fuzzy $C$-Means (FCM) algorithm represents fuzzy theory. They are combined to promote their advantages and overcome their disadvantages. On the one hand, the FCM algorithm helps the Mapper algorithm simplify the choice of parameters to obtain the most informative presentation and is even more efficient in data clustering. On the other hand, the FCM algorithm is equipped with the outstanding features of the Mapper algorithm in simplifying and visualizing data with qualitative analysis. This thesis focuses on conquering and achieving the following aims: (1) Summarizing the theoretical foundations and practical applications of the Mapper algorithm in the flow of literature with improved versions and various implementations. (2) Optimizing the cover choice of the Mapper algorithm in the direction of dividing the filter range automatically into irregular intervals with a random overlapping percentage by using the FCM algorithm. (3) Constructing a novel method for mining data that can exhibit the same clustering ability as the FCM algorithm and reveal some meaningful relationships by visualizing the global shape of data supplied by the Mapper algorithm.Geometrie a topologie se stávají silnějšími a dominantnějšími v analýze dat díky svým vynikajícím vlastnostem. Nedávno se objevila jako slibná výzkumná oblast, známá jako topologická analýza dat (TDA), pro moderní informatiku. V posledních letech je algoritmus Mapper, vynikající představitel TDA, stále více doplněn o stabilizovaný teoretický základ a praktické aplikace a rozmanité, intuitivní a uživatelsky přívětivé implementace. Z teoretického hlediska je algoritmus Mapper stále fuzzy shlukovací algoritmus se schopností vizualizace extrahovat souhrn tvaru dat. Jeho výsledky jsou však stále velmi citlivé na volbu parametrů, včetně rozlišení a funkce. Proto je potřeba výrazně snížit závislost na jeho parametrech. Tato myšlenka je vzrušující a lze ji vyřešit díky vynikajícím vlastnostem fuzzy shlukování. Schopnost shlukování Mapperu je stále silnější díky podpoře známých technik. Proto se očekává, že tato kombinace užitečně a účinně vyřeší některé problémy, se kterými se setkáváme v mnoha oblastech. Hlavním výzkumným cílem této práce je přiblížit TDA pomocí fuzzy teorie a vytvořit mezi nimi vzájemné vztahy z hlediska shlukování. Explicitně řečeno, algoritmus Mapper představuje TDA a algoritmus Fuzzy $C$-Means (FCM) představuje fuzzy teorii. Jsou kombinovány, aby podpořily své výhody a překonaly své nevýhody. Na jedné straně algoritmus FCM pomáhá algoritmu Mapper zjednodušit výběr parametrů pro získání nejinformativnější prezentace a je ještě efektivnější při shlukování dat. Na druhé straně je algoritmus FCM vybaven vynikajícími vlastnostmi algoritmu Mapper pro zjednodušení a vizualizaci dat pomocí kvalitativní analýzy. Tato práce se zaměřuje na dobývání a dosažení následujících cílů: (1) Shrnutí teoretických základů a praktických aplikací Mapperova algoritmu v toku literatury s vylepšenými verzemi a různými implementacemi. (2) Optimalizace volby pokrytí algoritmu Mapper ve směru automatického rozdělení rozsahu filtru do nepravidelných intervalů s náhodně se překrývajícím procentem pomocí algoritmu FCM. (3) Vytvoření nové metody pro těžbu dat, která může vykazovat stejnou schopnost shlukování jako algoritmus FCM a odhalit některé smysluplné vztahy vizualizací globálního tvaru dat poskytovaných algoritmem Mapper.460 - Katedra informatikyvyhově