Examinando por Autor "Cadavid, Carlos A."
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Ítem 2D Shape similarity as a complement for Voronoi-Delone methods in shape reconstruction(Elsevier, 2005) Ruíz S., Óscar E.; Cadavid, Carlos A.; Granados, Miguel; Peña, Sebastián; Vásquez, Eliana; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEIn surface reconstruction from planar slices it is necessary to build surfaces between corresponding 2D regions in consecutive levels -- The problem has been traditionally attacked with (i) direct reconstruction based on local geometric proximity between the regions, and (ii) classification of topological events between the slices, which control the evolution of the cross cuts -- These approaches have been separately applied with mixed success -- In the case (i), the results may be surfaces with over-stretched or unnatural branches, resulting from a local contour proximity which does not correspond to global similarity between regions -- In (ii), the consequences from topological events upon the actual surface realization have not been drawn -- In this paper an integration of (i) and (ii) is presented, which uses a criteria of similarity between composed 2D regions in consecutive slices to: (a) decide if a surface should actually relate those regions, (b) identify the topological transitions between levels and (c) construct the local surface for the related regions -- The method implemented hinders over-stretched and unnatural branches, therefore rendering a surface which adjusts to geometrically-sound topological events -- This is a good alternative when the surface reconstructed needs to be topologically faithful (for example in flow simulation) in addition to represent the a rough geometrical space (for example in radiation planning)Ítem Boolean 2D Shape Similarity For Surface Reconstruction(2001) Ruíz, Óscar E.; Cadavid, Carlos A.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAESurface reconstruction problem (SRP) from planar samples has been traditionally approached by either (i) using local proximity between data points in adjacent layers, or by(ii) classifying the topological transitions that may explain the evolution of the cross sections -- Strategy (i) is robust in the sense that it has answers for every possible case, although in some scenarios renders counterintuitive surfaces, commented below -- Approach (ii) has mainly remained in the theoretical terrain -- The present work follows on aspect (ii), by using a Morse-based topological classification of the transitions, and complementing it with reasoning based on the geometry of the evolving cross sections to determine a high level description of the transitions from m to n contours (m:n transitions) -- This reasoning of shape similarity is performed by boolean operators -- Finally, the surface is synthesized using the m:n transitions -- This conjunction of topological and geometrical reasoning renders highly intuitive results, and allows for the incorporation of methods derived from the area of machine visionÍtem Hessian Eigenfunctions for Triangular Mesh Parameterization(SCITEPRESS, 2016) Mejía, Daniel; Ruíz, Oscar; Cadavid, Carlos A.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEHessian Locally Linear Embedding (HLLE) is an algorithm that computes the nullspace of a Hessian functional H for Dimensionality Reduction (DR) of a sampled manifold M -- This article presents a variation of classic HLLE for parameterization of 3D triangular meses -- Contrary to classic HLLE which estimates local Hessian nullspaces, the proposed approach follows intuitive ideas from Differential Geometry where the local Hessian is estimated by quadratic interpolation and a partition of unity is used to join all neighborhoods -- In addition, local average triangle normals are used to estimate the tangent plane TxM at x ∈ M instead of PCA, resulting in local parameterizations which reflect better the geometry of the surface and perform better when the mesh presents sharp features -- A high frequency dataset (Brain) is used to test our algorithm resulting in a higher rate of success (96.63%) compared to classic HLLE (76.4%)Ítem Limits of quotients of polynomial functions in three variables(ASSOC COMPUTING MACHINERY, 2017-06-01) Velez, Juan D.; Hernandez, Juan P.; Cadavid, Carlos A.; Velez, Juan D.; Hernandez, Juan P.; Cadavid, Carlos A.; Universidad EAFIT. Departamento de Ciencias; Matemáticas y AplicacionesA method for computing limits of quotients of real analytic functions in two variables was developed in [4]. In this article we generalize the results obtained in that paper to the case of quotients q = f(x, y, z)/g(x, y, z) of polynomial functions in three variables with rational coefficients. The main idea consists in examining the behavior of the function q along certain real variety X(q) (the discriminant variety associated to q). The original problem is then solved by reducing to the case of functions of two variables. The inductive step is provided by the key fact that any algebraic curve is birationally equivalent to a plane curve. Our main result is summarized in Theorem 2. In Section 4 we describe an effective method for computing such limits. We provide a high level description of an algorithm that generalizes the one developed in [4], now available in Maple as the limit/multi command.Ítem On a minimal factorization conjecture(Elsevier, 2007-08) Cadavid, Carlos A.; Vélez, Juan D.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAELet be a proper holomorphic map from a connected complex surface S onto the open unit disk D⊂C, with 0∈D as its unique singular value, and having fiber genus g>0 -- Assume that in case g⩾2, admits a deformation whose singular fibers are all of simple Lefschetz type -- It has been conjectured that the factorization of the monodromy f∈M around ϕ (0) in terms of righ-thanded Dehn twists induced by the monodromy of has the least number of factors among all possible factorizations of f as a product of righthanded Dehn twists in the mapping class group (see [M. Ishizaka, One parameter families of Riemann surfaces and presentations of elements of mapping class group by Dehn twists, J. Math. Soc. Japan 58 (2) (2006) 585–594]) -- In this article, the validity of this conjecture is established for g=1Ítem On the critical point structure of eigenfunctions belonging to the first nonzero eigenvalue of a genus two closed hyperbolic surface(2012-05-30) Cadavid, Carlos A.; Osorno, María C.; Ruíz, Óscar E.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEWe develop a method based on spectral graph theory to approximate the eigenvalues and eigenfunctions of the Laplace-Beltrami operator of a compact riemannian manifold -- The method is applied to a closed hyperbolic surface of genus two -- The results obtained agree with the ones obtained by other authors by different methods, and they serve as experimental evidence supporting the conjectured fact that the generic eigenfunctions belonging to the first nonzero eigenvalue of a closed hyperbolic surface of arbitrary genus are Morse functions having the least possible total number of critical points among all Morse functions admitted by such manifoldsÍtem Spectral-based mesh segmentation(Springer Paris, 2016) Mejía, Daniel; Cadavid, Carlos A.; Ruíz-Salguero, Óscar; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEIn design and manufacturing, mesh segmentation is required for FACE construction in boundary representation (BRep), which in turn is central for featurebased design, machining, parametric CAD and reverse engineering, among others -- Although mesh segmentation is dictated by geometry and topology, this article focuses on the topological aspect (graph spectrum), as we consider that this tool has not been fully exploited -- We preprocess the mesh to obtain a edgelength homogeneous triangle set and its Graph Laplacian is calculated -- We then produce a monotonically increasing permutation of the Fiedler vector (2nd eigenvector of Graph Laplacian) for encoding the connectivity among part feature submeshes -- Within the mutated vector, discontinuities larger than a threshold (interactively set by a human) determine the partition of the original mesh -- We present tests of our method on large complex meshes, which show results which mostly adjust to BRep FACE partition -- The achieved segmentations properly locate most manufacturing features, although it requires human interaction to avoid over segmentation -- Future work includes an iterative application of this algorithm to progressively sever features of the mesh left from previous submesh removalsÍtem Triangular mesh parameterization with trimmed surfaces(Springer Verlag, 2015) Ruíz, Óscar E.; Mejía, Daniel; Cadavid, Carlos A.; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEGiven a 2manifold triangular mesh \(M \subset {\mathbb {R}}^3\), with border, a parameterization of \(M\) is a FACE or trimmed surface \(F=\{S,L_0,\ldots, L_m\}\) -- \(F\) is a connected subset or region of a parametric surface \(S\), bounded by a set of LOOPs \(L_0,\ldots ,L_m\) such that each \(L_i \subset S\) is a closed 1manifold having no intersection with the other \(L_j\) LOOPs -- The parametric surface \(S\) is a statistical fit of the mesh \(M\) -- \(L_0\) is the outermost LOOP bounding \(F\) and \(L_i\) is the LOOP of the ith hole in \(F\) (if any) -- The problem of parameterizing triangular meshes is relevant for reverse engineering, tool path planning, feature detection, redesign, etc -- Stateofart mesh procedures parameterize a rectangular mesh \(M\) -- To improve such procedures, we report here the implementation of an algorithm which parameterizes meshes \(M\) presenting holes and concavities -- We synthesize a parametric surface \(S \subset {\mathbb {R}}^3\) which approximates a superset of the mesh \(M\) -- Then, we compute a set of LOOPs trimming \(S\), and therefore completing the FACE \(F=\ {S,L_0,\ldots ,L_m\}\) -- Our algorithm gives satisfactory results for \(M\) having low Gaussian curvature (i.e., \(M\) being quasi-developable or developable) -- This assumption is a reasonable one, since \(M\) is the product of manifold segmentation preprocessing -- Our algorithm computes: (1) a manifold learning mapping \(\phi : M \rightarrow U \subset {\mathbb {R}}^2\), (2) an inverse mapping \(S: W \subset {\mathbb {R}}^2 \rightarrow {\mathbb {R}}^3\), with \ (W\) being a rectangular grid containing and surpassing \(U\) -- To compute \(\phi\) we test IsoMap, Laplacian Eigenmaps and Hessian local linear embedding (best results with HLLE) -- For the back mapping (NURBS) \(S\) the crucial step is to find a control polyhedron \(P\), which is an extrapolation of \(M\) -- We calculate \(P\) by extrapolating radial basis functions that interpolate points inside \(\phi (M)\) -- We successfully test our implementation with several datasets presenting concavities, holes, and are extremely nondevelopable -- Ongoing work is being devoted to manifold segmentation which facilitates mesh parameterizationÍtem Usage of 2D Region Similarity For Surface Reconstruction From Planar Samples(2003) Ruíz S., Óscar E.; Cadavid, Carlos A.; Granados, Miguel; Peña, Sebastián; Vásquez, Eliana; Universidad EAFIT. Departamento de Ingeniería Mecánica; Laboratorio CAD/CAM/CAEIn surface reconstruction from planar slices it is necessary to build surfaces between corresponding 2D regions in consecutive levels -- The problem has been traditionally attacked with (i) direct reconstruction based on local geometric proximity between the regions, and (ii) classification of topological events between the slices, which control the evolution of the cross cuts -- These approaches have been separately applied with mixed success -- In the case (i), the results may be surfaces with over-stretched or unnatural branches, resulting from a local contour proximity which does not correspond to global similarity between regions -- In (ii), the consequences from topological events upon the actual surface realization have not been drawn -- In this paper an integration of (i) and (ii) is presented, which uses a criteria of similarity between composed 2D regions in consecutive slices to: (a) decide if a surface should actually relate those regions, (b) identify the topological transitions between levels and (c) construct the local surface for the related regions -- The method implemented hinders over-stretched and unnatural branches, therefore rendering a surface which adjusts to geometrically-sound topological events -- This is a good alternative when the surface reconstructed needs to be topologically faithful (for example in flow simulation) in addition to represent the a rough geometrical space (for example in radiation planning)