44 research outputs found
Toward Parallel Computation of Dense Homotopy Skeletons for nD Digital Objects
An appropriate generalization of the classical notion of
abstract cell complex, called primal-dual abstract cell complex (pACC
for short) is the combinatorial notion used here for modeling and analyzing
the topology of nD digital objects and images. Let D ⊂ I be a set of
n-xels (ROI) and I be a n-dimensional digital image.We design a theoretical
parallel algorithm for constructing a topologically meaningful asymmetric
pACC HSF(D), called Homological Spanning Forest of D (HSF
of D, for short) starting from a canonical symmetric pACC associated
to I and based on the application of elementary homotopy operations
to activate the pACC processing units. From this HSF-graph representation
of D, it is possible to derive complete homology and homotopy
information of it. The preprocessing procedure of computing HSF(I) is
thoroughly discussed. In this way, a significant advance in understanding
how the efficient HSF framework for parallel topological computation of
2D digital images developed in [2] can be generalized to higher dimension
is made.Ministerio de EconomĂa y Competitividad TEC2016-77785-PMinisterio de EconomĂa y Competitividad MTM2016-81030-
Homological Region Adjacency Tree for a 3D Binary Digital Image via HSF Model
Given a 3D binary digital image I, we define and compute
an edge-weighted tree, called Homological Region Tree (or Hom-Tree,
for short). It coincides, as unweighted graph, with the classical Region
Adjacency Tree of black 6-connected components (CCs) and white 26-
connected components of I. In addition, we define the weight of an edge
(R, S) as the number of tunnels that the CCs R and S “share”. The
Hom-Tree structure is still an isotopic invariant of I. Thus, it provides
information about how the different homology groups interact between
them, while preserving the duality of black and white CCs.
An experimentation with a set of synthetic images showing different
shapes and different complexity of connected component nesting is performed
for numerically validating the method.Ministerio de EconomĂa y Competitividad MTM2016-81030-
Generating Second Order (Co)homological Information within AT-Model Context
In this paper we design a new family of relations between
(co)homology classes, working with coefficients in a field and starting
from an AT-model (Algebraic Topological Model) AT(C) of a finite cell
complex C These relations are induced by elementary relations of type
“to be in the (co)boundary of” between cells. This high-order connectivity
information is embedded into a graph-based representation model,
called Second Order AT-Region-Incidence Graph (or AT-RIG) of C. This
graph, having as nodes the different homology classes of C, is in turn,
computed from two generalized abstract cell complexes, called primal
and dual AT-segmentations of C. The respective cells of these two complexes
are connected regions (set of cells) of the original cell complex C,
which are specified by the integral operator of AT(C). In this work in
progress, we successfully use this model (a) in experiments for discriminating
topologically different 3D digital objects, having the same Euler
characteristic and (b) in designing a parallel algorithm for computing
potentially significant (co)homological information of 3D digital objects.Ministerio de EconomĂa y Competitividad MTM2016-81030-PMinisterio de EconomĂa y Competitividad TEC2012-37868-C04-0
Parallel Image Processing Using a Pure Topological Framework
Image processing is a fundamental operation
in many real time applications, where lots of parallelism
can be extracted. Segmenting the image into different
connected components is the most known operations, but
there are many others like extracting the region adjacency
graph (RAG) of these regions, or searching for features
points, being invariant to rotations, scales, brilliant
changes, etc. Most of these algorithms part from the basis
of Tracing-type approaches or scan/raster methods. This
fact necessarily implies a data dependence between the
processing of one pixel and the previous one, which
prevents using a pure parallel approach. In terms of time
complexity, this means that linear order O(N) (N being the
number of pixels) cannot be cut down. In this paper, we
describe a novel approach based on the building of a pure
Topological framework, which allows to implement fully
parallel algorithms. Concerning topological analysis, a first
stage is computed in parallel for every pixel, thus
conveying the local neighboring conditions. Then, they are
extended in a second parallel stage to the necessary global
relations (e.g. to join all the pixels of a connected
component). This combinatorial optimization process can
be seen as the compression of the whole image to just one
pixel. Using this final representation, every region can be
related with the rest, which yields to pure topological
construction of other image operations. Besides, complex
data structures can be avoided: all the processing can be
done using matrixes (with the same indexation as the
original image) and element-wise operations. The time
complexity order of our topological approach for a mĂ—n
pixel image is near O(log(m+n)), under the assumption that
a processing element exists for each pixel. Results for a
multicore processor show very good scalability until the
memory bandwidth bottleneck is reached, both for bigger
images and for much optimized implementations. The
inherent parallelism of our approach points to the
direction that even better results will be obtained in other
less classical computing architectures.1Ministerio de EconomĂa y Competitividad (España) TEC2012-37868-C04-02AEI/FEDER (UE) MTM2016-81030-PVPPI of the University of Sevill
Skeletonization methods for image and volume inpainting
Image and shape restoration techniques are increasingly important in computer graphics. Many types of restoration techniques have been proposed in the 2D image-processing and according to our knowledge only one to volumetric data. Well-known examples of such techniques include digital inpainting, denoising, and morphological gap filling. However efficient and effective, such methods have several limitations with respect to the shape, size, distribution, and nature of the defects they can find and eliminate. We start by studying the use of 2D skeletons for the restoration of two-dimensional images. To this end, we show that skeletons are useful and efficient for volumetric data reconstruction. To explore our hypothesis in the 3D case, we first overview the existing state-of-the-art in 3D skeletonization methods, and conclude that no such method provides us with the features required by efficient and effective practical usage. We next propose a novel method for 3D skeletonization, and show how it complies with our desired quality requirements, which makes it thereby suitable for volumetric data reconstruction context. The joint results of our study show that skeletons are indeed effective tools to design a variety of shape restoration methods. Separately, our results show that suitable algorithms and implementations can be conceived to yield high end-to-end performance and quality of skeleton-based restoration methods. Finally, our practical applications can generate competitive results when compared to application areas such as digital hair removal and wire artifact removal
Visual analytics of multidimensional time-dependent trails:with applications in shape tracking
Lots of data collected for both scientific and non-scientific purposes have similar characteristics: changing over time with many different properties. For example, consider the trajectory of an airplane travelling from one location to the other. Not only does the airplane itself move over time, but its heading, height and speed are changing at the same time. During this research, we investigated different ways to collect and visualze data with these characteristics. One practical application being for an automated milking device which needs to be able to determine the position of a cow's teats. By visualizing all data which is generated during the tracking process we can acquire insights in the working of the tracking system and identify possibilites for improvement which should lead to better recognition of the teats by the machine. Another important result of the research is a method which can be used to efficiently process a large amount of trajectory data and visualize this in a simplified manner. This has lead to a system which can be used to show the movement of all airplanes around the world for a period of multiple weeks
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
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Geometric and Topological Combinatorics
The 2007 Oberwolfach meeting “Geometric and Topological Combinatorics” presented a great variety of investigations where topological and algebraic methods are brought into play to solve combinatorial and geometric problems, but also where geometric and combinatorial ideas are applied to topological questions
Exploring 3D Shapes through Real Functions
This thesis lays in the context of research on representation, modelling and coding knowledge related to digital shapes, where by shape it is meant any individual object having a visual appareance which exists in some two-, three- or higher dimensional space. Digital shapes are digital representations of either physically existing or virtual objects that can be processed by computer applications. While the technological advances in terms of hardware and software have made available plenty of tools for using and interacting with the geometry of shapes, to manipulate and retrieve huge amount of data it is necessary to define methods able to effectively code them. In this thesis a conceptual model is proposed which represents a given 3D object through the coding of its salient features and defines an abstraction of the object, discarding irrelevant details. The approach is based on the shape descriptors defined with respect to real functions, which provide a very useful shape abstraction method for the analysis and structuring of the information contained in the discrete shape model. A distinctive feature of these shape descriptors is their capability of combining topological and geometrical information properties of the shape, giving an abstraction of the main shape features. To fully develop this conceptual model, both theoretical and computational aspects have been considered, related to the definition and the extension of the different shape descriptors to the computational domain. Main emphasis is devoted to the application of these shape descriptors in computational settings; to this aim we display a number of application domains that span from shape retrieval, to shape classification and to best view selection.Questa tesi si colloca nell\u27ambito di ricerca riguardante la rappresentazione, la modellazione e la codifica della conoscenza connessa a forme digitali, dove per forma si intende l\u27aspetto visuale di ogni oggetto che esiste in due, tre o pi? dimensioni. Le forme digitali sono rappresentazioni di oggetti sia reali che virtuali, che possono essere manipolate da un calcolatore. Lo sviluppo tecnologico degli ultimi anni in materia di hardware e software ha messo a disposizione una grande quantit? di strumenti per acquisire, rappresentare e processare la geometria degli oggetti; tuttavia per gestire questa grande mole di dati ? necessario sviluppare metodi in grado di fornirne una codifica efficiente. In questa tesi si propone un modello concettuale che descrive un oggetto 3D attraverso la codifica delle caratteristiche salienti e ne definisce una bozza ad alto livello, tralasciando dettagli irrilevanti. Alla base di questo approccio ? l\u27utilizzo di descrittori basati su funzioni reali in quanto forniscono un\u27astrazione della forma molto utile per analizzare e strutturare l\u27informazione contenuta nel modello discreto della forma. Una peculiarit? di tali descrittori di forma ? la capacit? di combinare propriet? topologiche e geometriche consentendo di astrarne le principali caratteristiche. Per sviluppare questo modello concettuale, ? stato necessario considerare gli aspetti sia teorici che computazionali relativi alla definizione e all\u27estensione in ambito discreto di vari descrittori di forma. Particolare attenzione ? stata rivolta all\u27applicazione dei descrittori studiati in ambito computazionale; a questo scopo sono stati considerati numerosi contesti applicativi, che variano dal riconoscimento alla classificazione di forme, all\u27individuazione della posizione pi? significativa di un oggetto