120 research outputs found
Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph
A standard way to approximate the distance between any two vertices and
on a mesh is to compute, in the associated graph, a shortest path from
to that goes through one of sources, which are well-chosen vertices.
Precomputing the distance between each of the sources to all vertices of
the graph yields an efficient computation of approximate distances between any
two vertices. One standard method for choosing sources, which has been used
extensively and successfully for isometry-invariant surface processing, is the
so-called Farthest Point Sampling (FPS), which starts with a random vertex as
the first source, and iteratively selects the farthest vertex from the already
selected sources.
In this paper, we analyze the stretch factor of
approximate geodesics computed using FPS, which is the maximum, over all pairs
of distinct vertices, of their approximated distance over their geodesic
distance in the graph. We show that can be bounded in terms
of the minimal value of the stretch factor obtained using an
optimal placement of sources as , where is the ratio of the lengths of
the longest and the shortest edges of the graph. This provides some evidence
explaining why farthest point sampling has been used successfully for
isometry-invariant shape processing. Furthermore, we show that it is
NP-complete to find sources that minimize the stretch factor.Comment: 13 pages, 4 figure
Bending invariant meshes and application to groupwise correspondences
We introduce a new bending invariant representation of a triangular mesh S. The bending invariant mesh X of S is a deformation of S that has the property that the geodesic distance between each pair of vertices on S is approximated well by the Euclidean distance between the corresponding vertices on X. Furthermore, X is intersection-free. The main advantage of the bending invariant mesh compared to previous approaches is that mesh-based features on X can be used to facilitate applications such as shape recognition or shape registration. We apply bending invariant meshes to find dense point-to-point correspondences between a number of deformed surfaces corresponding to different postures of the same non-rigid object in a fully automatic way. 1
Semantic Segmentation of Human Model Using Heat Kernel and Geodesic Distance
A novel approach of 3D human model segmentation is proposed, which is based on heat kernel signature and geodesic distance. Through calculating the heat kernel signature of the point clouds of human body model, the local maxima of thermal energy distribution of the model is found, and the set of feature points of the model is obtained. Heat kernel signature has affine invariability which can be used to extract the correct feature points of the human model in different postures. We adopt the method of geodesic distance to realize the hierarchical segmentation of human model after obtaining the semantic feature points of human model. The experimental results show that the method can overcome the defect of geodesic distance feature extraction. The human body models with different postures can be obtained with the model segmentation results of human semantic characteristics
Geometric data understanding : deriving case specific features
There exists a tradition using precise geometric modeling, where uncertainties in data can be considered noise. Another tradition relies on statistical nature of vast quantity of data, where geometric regularity is intrinsic to data and statistical models usually grasp this level only indirectly. This work focuses on point cloud data of natural resources and the silhouette recognition from video input as two real world examples of problems having geometric content which is intangible at the raw data presentation.
This content could be discovered and modeled to some degree by such machine learning (ML) approaches like deep learning, but either a direct coverage of geometry in samples or addition of special geometry invariant layer is necessary. Geometric content is central when there is a need for direct observations of spatial variables, or one needs to gain a mapping to a geometrically consistent data representation, where e.g. outliers or noise can be easily discerned.
In this thesis we consider transformation of original input data to a geometric feature space in two example problems. The first example is curvature of surfaces, which has met renewed interest since the introduction of ubiquitous point cloud data and the maturation of the discrete differential geometry. Curvature spectra can characterize a spatial sample rather well, and provide useful features for ML purposes. The second example involves projective methods used to video stereo-signal analysis in swimming analytics.
The aim is to find meaningful local geometric representations for feature generation, which also facilitate additional analysis based on geometric understanding of the model. The features are associated directly to some geometric quantity, and this makes it easier to express the geometric constraints in a natural way, as shown in the thesis. Also, the visualization and further feature generation is much easier. Third, the approach provides sound baseline methods to more traditional ML approaches, e.g. neural network methods. Fourth, most of the ML methods can utilize the geometric features presented in this work as additional features.Geometriassa käytetään perinteisesti tarkkoja malleja, jolloin datassa esiintyvät epätarkkuudet edustavat melua. Toisessa perinteessä nojataan suuren datamäärän tilastolliseen luonteeseen, jolloin geometrinen säännönmukaisuus on datan sisäsyntyinen ominaisuus, joka hahmotetaan tilastollisilla malleilla ainoastaan epäsuorasti. Tämä työ keskittyy kahteen esimerkkiin: luonnonvaroja kuvaaviin pistepilviin ja videohahmontunnistukseen. Nämä ovat todellisia ongelmia, joissa geometrinen sisältö on tavoittamattomissa raakadatan tasolla.
Tämä sisältö voitaisiin jossain määrin löytää ja mallintaa koneoppimisen keinoin, esim. syväoppimisen avulla, mutta joko geometria pitää kattaa suoraan näytteistämällä tai tarvitaan neuronien lisäkerros geometrisia invariansseja varten. Geometrinen sisältö on keskeinen, kun tarvitaan suoraa avaruudellisten suureiden havainnointia, tai kun tarvitaan kuvaus geometrisesti yhtenäiseen dataesitykseen, jossa poikkeavat näytteet tai melu voidaan helposti erottaa.
Tässä työssä tarkastellaan datan muuntamista geometriseen piirreavaruuteen kahden esimerkkiohjelman suhteen. Ensimmäinen esimerkki on pintakaarevuus, joka on uudelleen virinneen kiinnostuksen kohde kaikkialle saatavissa olevan datan ja diskreetin geometrian kypsymisen takia. Kaarevuusspektrit voivat luonnehtia avaruudellista kohdetta melko hyvin ja tarjota koneoppimisessa hyödyllisiä piirteitä. Toinen esimerkki koskee projektiivisia menetelmiä käytettäessä stereovideosignaalia uinnin analytiikkaan.
Tavoite on löytää merkityksellisiä paikallisen geometrian esityksiä, jotka samalla mahdollistavat muun geometrian ymmärrykseen perustuvan analyysin. Piirteet liittyvät suoraan johonkin geometriseen suureeseen, ja tämä helpottaa luonnollisella tavalla geometristen rajoitteiden käsittelyä, kuten väitöstyössä osoitetaan. Myös visualisointi ja lisäpiirteiden luonti muuttuu helpommaksi. Kolmanneksi, lähestymistapa suo selkeän vertailumenetelmän perinteisemmille koneoppimisen lähestymistavoille, esim. hermoverkkomenetelmille. Neljänneksi, useimmat koneoppimismenetelmät voivat hyödyntää tässä työssä esitettyjä geometrisia piirteitä lisäämällä ne muiden piirteiden joukkoon
Inferring Human Pose and Motion from Images
As optical gesture recognition technology advances, touchless human computer interfaces of the future will soon become a reality. One particular technology, markerless motion capture, has gained a large amount of attention, with widespread application in diverse disciplines, including medical science, sports analysis, advanced user interfaces, and virtual arts. However, the complexity of human anatomy makes markerless motion capture a non-trivial problem: I) parameterised pose configuration exhibits high dimensionality, and II) there is considerable ambiguity in surjective inverse mapping from observation to pose configuration spaces with a limited number of camera views. These factors together lead to multimodality in high dimensional space, making markerless motion capture an ill-posed problem. This study challenges these difficulties by introducing a new framework. It begins with automatically modelling specific subject template models and calibrating posture at the initial stage. Subsequent tracking is accomplished by embedding naturally-inspired global optimisation into the sequential Bayesian filtering framework. Tracking is enhanced by several robust evaluation improvements. Sparsity of images is managed by compressive evaluation, further accelerating computational efficiency in high dimensional space
The Necessary Structure of the All-pervading Aether: Discrete or Continuous? Simple or Symmetric?
In this book I investigate the necessary structure of the aether – the stuff that fills the whole universe. Some of my conclusions are. 1. There is an enormous variety of structures that the aether might, for all we know, have. 2. Probably the aether is point-free. 3. In that case, it should be distinguished from Space-time, which is either a fiction or a construct. 4. Even if the aether has points, we should reject the orthodoxy that all regions are grounded in points by summation. 5. If the aether is point-free but not continuous, its most likely structure has extended atoms that are not simples. 6. Space-time is symmetric if and only if the aether is continuous. 7. If the aether is continuous, we should reject the standard interpretation of General Relativity, in which geometry determines gravity. 8. Contemporary physics undermines an objection to discrete aether based on scale invariance, but does not offer much positive support
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