On surfaces in digital topology

In R. Ayala, E. Domínguez, A.R. Francés, A. Quintero, J. Rubio. A Polyhedral Approach to Digital Topology a new framework for digital topology has been proposed. This framework offers the possibility of transfering, in an easy way, definitions, statements and proofs from continuous topology to digital topology. In particular, it provides a straightforward definition of n-dimensional digital manifold. In this paper we prove that the class of digital 2-manifolds without boundary in the grid Z3 agrees with the class of (26, 6)-surfaces defined by Kong-Roscoe and other authors. As a consequence, the separation theorem for digital surfaces stated in D.G. Morgenthaler, A. Rosenfeld. Surfaces in threedimensional digital images. Information and Control, 51 (1981), 227-247] and G.M. Reed. On the Characterization of Simple Closed Surfaces in Three-dimensional Digital Images. Computer Graphics and Image Processing, 25 (1984), 226-235 is obtained.Dirección General de Investigación Científica y TécnicaUniversidad de La Rioj

3D mesh metamorphosis from spherical parameterization for conceptual design

Engineering product design is an information intensive decision-making process that consists of several phases including design specification definition, design concepts generation, detailed design and analysis, and manufacturing. Usually, generating geometry models for visualization is a big challenge for early stage conceptual design. Complexity of existing computer aided design packages constrains participation of people with various backgrounds in the design process. In addition, many design processes do not take advantage of the rich amount of legacy information available for new concepts creation. The research presented here explores the use of advanced graphical techniques to quickly and efficiently merge legacy information with new design concepts to rapidly create new conceptual product designs. 3D mesh metamorphosis framework 3DMeshMorpher was created to construct new models by navigating in a shape-space of registered design models. The framework is composed of: i) a fast spherical parameterization method to map a geometric model (genus-0) onto a unit sphere; ii) a geometric feature identification and picking technique based on 3D skeleton extraction; and iii) a LOD controllable 3D remeshing scheme with spherical mesh subdivision based on the developedspherical parameterization. This efficient software framework enables designers to create numerous geometric concepts in real time with a simple graphical user interface. The spherical parameterization method is focused on closed genus-zero meshes. It is based upon barycentric coordinates with convex boundary. Unlike most existing similar approaches which deal with each vertex in the mesh equally, the method developed in this research focuses primarily on resolving overlapping areas, which helps speed the parameterization process. The algorithm starts by normalizing the source mesh onto a unit sphere and followed by some initial relaxation via Gauss-Seidel iterations. Due to its emphasis on solving only challenging overlapping regions, this parameterization process is much faster than existing spherical mapping methods. To ensure the correspondence of features from different models, we introduce a skeleton based feature identification and picking method for features alignment. Unlike traditional methods that align single point for each feature, this method can provide alignments for complete feature areas. This could help users to create more reasonable intermediate morphing results with preserved topological features. This skeleton featuring framework could potentially be extended to automatic features alignment for geometries with similar topologies. The skeleton extracted could also be applied for other applications such as skeleton-based animations. The 3D remeshing algorithm with spherical mesh subdivision is developed to generate a common connectivity for different mesh models. This method is derived from the concept of spherical mesh subdivision. The local recursive subdivision can be set to match the desired LOD (level of details) for source spherical mesh. Such LOD is controllable and this allows various outputs with different resolutions. Such recursive subdivision then follows by a triangular correction process which ensures valid triangulations for the remeshing. And the final mesh merging and reconstruction process produces the remeshing model with desired LOD specified from user. Usually the final merged model contains all the geometric details from each model with reasonable amount of vertices, unlike other existing methods that result in big amount of vertices in the merged model. Such multi-resolution outputs with controllable LOD could also be applied in various other computer graphics applications such as computer games

Automatic Reconstruction of Textured 3D Models

Three dimensional modeling and visualization of environments is an increasingly important problem. This work addresses the problem of automatic 3D reconstruction and we present a system for unsupervised reconstruction of textured 3D models in the context of modeling indoor environments. We present solutions to all aspects of the modeling process and an integrated system for the automatic creation of large scale 3D models

Homeomorphisms of the Sierpinski Carpet

The Sierpinski carpet is a fractal formed by dividing the unit square into nine congruent squares, removing the center one, and repeating the process for each of the eight remaining squares, continuing infinitely many times. It is a well-known fractal with many fascinating topological properties that appears in a variety of different contexts, including as rational Julia sets. In this project, we study self-homeomorphisms of the Sierpinski carpet. We investigate the structure of the homeomorphism group, identify its finite subgroups, and attempt to define a transducer homeomorphism of the carpet. In particular, we find that the symmetry groups of platonic solids and D_n x Z_2 for positive integers n are all subgroups of the homeomorphism group of the carpet, using the theorem of Whyburn that any two S-curves are homeomorphic

Universal tools for analysing structures and interactions in geometry

This study examined symmetry and perspective in modern geometric transformations, treating them as functions that preserve specific properties while mapping one geometric figure to another. The purpose of this study was to investigate geometric transformations as a tool for analysis, to consider invariants as universal tools for studying geometry. Materials and Methods: The Erlangen ideas of F. I. Klein were used, which consider geometry as a theory of group invariants with respect to the transformation of the plane and space. Results and Discussion: Projective transformations and their extension to two-dimensional primitives were investigated. Two types of geometric correspondences, collinearity and correlation, and their properties were studied. The group of homotheties, including translations and parallel translations, and their role in the affine group were investigated. Homology with ideal line axes, such as stretching and centre stretching, was considered. Involutional homology and harmonic homology with the centre, axis, and homologous pairs of points were investigated. In this study unified geometry concepts, exploring how different geometric transformations relate and maintain properties across diverse geometric systems. Conclusions: It specifically examined Möbius transforms, including their matrix representation, trace, fixed points, and categorized them into identical transforms, nonlinear transforms, shifts, dilations, and inversions

Interference-based Investigation of Microscopic Objects Near Surfaces: a View From Below

Phenomena occurring when microscopic objects approach planar surfaces are challenging to probe directly because their dynamics cannot be resolved with a sufficiently high spatial/temporal resolution in a non-invasive way, and suitable techniques/methods involve complex instrumentation/computations of limited accessibility/applicability. Interference-based techniques can overcome these barriers. However, because most set-ups and analysis methods are ideal for planar-like geometries, their accurate application for studying microscopic objects has been difficult. Reflection interference contrast microscopy (RICM) has shown particular promise allowing objects in close proximity to a surface to be observed from below, producing interferograms that inherently embed detailed information about the objects’ topography near the substrate. Because precise extraction of this information has been challenging, this study seeks to develop analysis methods applicable to RICM to facilitate its practical implementation for accurate investigation of interfacial phenomena between microscopic objects and surfaces. The most sophisticated theory of RICM was significantly improved and coupled with a general method to simulate the interference pattern from arbitrary convex geometries. Experimental results revealed that accurate reconstruction of an object’s contour is possible by fitting its interferogram; however, this is computationally intensive and of limited applicability, motivating the formulation of a simplified and accurate RICM model. This facilitated a major breakthrough: an innovative analysis of RICM interferograms provides the inclination angles of the geometry under study and a mathematical procedure allows near-instantaneous reconstruction of the contour with nanometer-scale resolution, applicable to arbitrarily shaped convex objects under different experimental conditions. A method for extracting nanometer-scale topographic information from RICM interferograms has been proposed; in particular, microspheres can be conveniently analyzed to measure surface roughness based on fringe visibility. Also, precise and accurate measurements of microspheres’ size were performed by means of optimized and robust fringe spacing analysis. Finally, RICM’s distinctive “view-from-below” perspective was applied in simple experiments involving the deposition of microspheres on surfaces, directly revealing the existence of different scenarios depending on deposition media and unique femtoliter-scale capillary condensation dynamics underneath micron-sized glass beads. Results show that RICM has a clear potential for near real-time analysis of ensembles of objects near surfaces so that statistical/probabilistic behavior can be realistically captured