5,895 research outputs found
Determining the Geometry of Boundaries of Objects from Medial Data
For 2D objects in R2 or 3D objects in R3 with (smooth) boundaries B, the Blum medial axis M [BN], or an appropriate variant, is a fundamental objct for describing shape. There has ben a significant body of work devoted to methods for computing it, including a grassfire method (Kimia et al [KTZ]), the Hamilton-Jacobi skeleton (Siddiqi at al [SB]), and Voronoi methods (Szekely et al [SN]) among others
Recommended from our members
An Ontology for Grounding Vague Geographic Terms
Many geographic terms, such as “river” and “lake”, are vague, with no clear boundaries of application. In particular, the spatial extent of such features is often vaguely carved out of a continuously varying observable domain. We present a means of defining vague terms using standpoint semantics, a refinement of the
philosophical idea of supervaluation semantics. Such definitions can be grounded in actual data by geometric analysis and segmentation of the data set. The issues
raised by this process with regard to the nature of boundaries and domains of logical quantification are discussed. We describe a prototype implementation of a system capable of segmenting attributed polygon data into geographically significant regions and evaluating queries involving vague geographic feature terms
Medial/skeletal linking structures for multi-region configurations
We consider a generic configuration of regions, consisting of a collection of
distinct compact regions in which may be
either smooth regions disjoint from the others or regions which meet on their
piecewise smooth boundaries in a generic way. We introduce a
skeletal linking structure for the collection of regions which simultaneously
captures the regions' individual shapes and geometric properties as well as the
"positional geometry" of the collection. The linking structure extends in a
minimal way the individual "skeletal structures" on each of the regions,
allowing us to significantly extend the mathematical methods introduced for
single regions to the configuration.
We prove for a generic configuration of regions the existence of a special
type of Blum linking structure which builds upon the Blum medial axes of the
individual regions. This requires proving several transversality theorems for
certain associated "multi-distance" and "height-distance" functions for such
configurations. We show that by relaxing the conditions on the Blum linking
structures we obtain the more general class of skeletal linking structures
which still capture the geometric properties.
In addition to yielding geometric invariants which capture the shapes and
geometry of individual regions, the linking structures are used to define
invariants which measure positional properties of the configuration such as:
measures of relative closeness of neighboring regions and relative significance
of the individual regions for the configuration. These invariants, which are
computed by formulas involving "skeletal linking integrals" on the internal
skeletal structures, are then used to construct a "tiered linking graph," which
identifies subconfigurations and provides a hierarchical ordering of the
regions.Comment: 135 pages, 36 figures. Version to appear in Memoirs of the Amer.
Math. So
Zoom invariant vision of figural shape: The mathematics of cores
Believing that figural zoom invariance and the cross-figural boundary linking implied by medial loci are important aspects of object shape, we present the mathematics of and algorithms for the extraction of medial loci directly from image intensities. The medial loci called cores are defined as generalized maxima in scale space of a form of medial information that is invariant to translation, rotation, and in particular, zoom. These loci are very insensitive to image disturbances, in strong contrast to previously available medial loci, as demonstrated in a companion paper. Core-related geometric properties and image object representations are laid out which, together with the aforementioned insensitivities, allow the core to be used effectively for a variety of image analysis objectives.
Designing heterogeneous porous tissue scaffolds for additive manufacturing processes
A novel tissue scaffold design technique has been proposed with controllable heterogeneous architecture design suitable for additive manufacturing processes. The proposed layer-based design uses a bi-layer pattern of radial and spiral layers consecutively to generate functionally gradient porosity, which follows the geometry of the scaffold. The proposed approach constructs the medial region from the medial axis of each corresponding layer, which represents the geometric internal feature or the spine. The radial layers of the scaffold are then generated by connecting the boundaries of the medial region and the layer's outer contour. To avoid the twisting of the internal channels, reorientation and relaxation techniques are introduced to establish the point matching of ruling lines. An optimization algorithm is developed to construct sub-regions from these ruling lines. Gradient porosity is changed between the medial region and the layer's outer contour. Iso-porosity regions are determined by dividing the subregions peripherally into pore cells and consecutive iso-porosity curves are generated using the isopoints from those pore cells. The combination of consecutive layers generates the pore cells with desired pore sizes. To ensure the fabrication of the designed scaffolds, the generated contours are optimized for a continuous, interconnected, and smooth deposition path-planning. A continuous zig-zag pattern deposition path crossing through the medial region is used for the initial layer and a biarc fitted isoporosity curve is generated for the consecutive layer with C-1 continuity. The proposed methodologies can generate the structure with gradient (linear or non-linear), variational or constant porosity that can provide localized control of variational porosity along the scaffold architecture. The designed porous structures can be fabricated using additive manufacturing processes
Recommended from our members
Shape matching and clustering in design
Generalising knowledge and matching patterns is a basic human trait in re-using past experiences. We often cluster (group) knowledge of similar attributes as a process of learning and or aid to manage the complexity and re-use of experiential knowledge [1, 2]. In conceptual design, an ill-defined shape may be recognised as more than one type. Resulting in shapes possibly being classified differently when different criteria are applied. This paper outlines the work being carried out to develop a new technique for shape clustering. It highlights the current methods for analysing shapes found in computer aided sketching systems, before a method is proposed that addresses shape clustering and pattern matching. Clustering for vague geometric models and multiple viewpoint support are explored
Applying spatial reasoning to topographical data with a grounded geographical ontology
Grounding an ontology upon geographical data has been pro-
posed as a method of handling the vagueness in the domain more effectively. In order to do this, we require methods of reasoning about the spatial relations between the regions within the data. This stage can be computationally expensive, as we require information on the location of
points in relation to each other. This paper illustrates how using knowledge about regions allows us to reduce the computation required in an efficient and easy to understand manner. Further, we show how this system can be implemented in co-ordination with segmented data to reason abou
Notions of optimal transport theory and how to implement them on a computer
This article gives an introduction to optimal transport, a mathematical
theory that makes it possible to measure distances between functions (or
distances between more general objects), to interpolate between objects or to
enforce mass/volume conservation in certain computational physics simulations.
Optimal transport is a rich scientific domain, with active research
communities, both on its theoretical aspects and on more applicative
considerations, such as geometry processing and machine learning. This article
aims at explaining the main principles behind the theory of optimal transport,
introduce the different involved notions, and more importantly, how they
relate, to let the reader grasp an intuition of the elegant theory that
structures them. Then we will consider a specific setting, called
semi-discrete, where a continuous function is transported to a discrete sum of
Dirac masses. Studying this specific setting naturally leads to an efficient
computational algorithm, that uses classical notions of computational geometry,
such as a generalization of Voronoi diagrams called Laguerre diagrams.Comment: 32 pages, 17 figure
- …