Pablo Diaz-Gutierrez

Sketching Free-form Surfaces Using Network of Curves

This work addresses the issue of generating free-form surfaces using a 2D sketch interface. As the first step in this process, we develop a methodology to sketch 3D space curves from 2D sketches. Since the inverse projection from 2D sketches to 3D curve or surface is a one to many function, there is no unique solution. Hence we propose to interpret the given 2D curve to be the projection of the 3D curve that has minimum curvature among all the candidates in 3D. We present an algorithm to efficiently find a close approximation of this minimum curvature 3D space curve. In the second step, this network of curves along with the boundary information are given to the surface fitting method to generate free-form surfaces.

Constrained strip generation and management for efficient interactive 3D rendering

Representing a triangulated two manifold using a single triangle strip is an NP-complete problem. By introducing a few Steiner vertices, recent works find such a single-strip and hence a linear ordering of edge-connected triangles of the entire triangulation. In this paper, we highlight and exploit this linear order in efficient triangle-strip management for high-performance rendering. We present new algorithms to generate weighted single-strip representations that respect different constraint-based clustering of triangles. These functional constraints can be application dependent; for example, normal-based constraints for efficient visibility culling or spatial constraints for highly coherent vertex-caching. We also present a hierarchical single-strip-management strategy for high-performance interactive 3D rendering.

Single-strips for fast interactive rendering

Representing a triangulated two manifold using a single triangle strip is an NP-complete problem. By introducing a few Steiner vertices, recent works find such a single-strip, and hence a linear ordering of edge-connected triangles of the entire triangulation. In this paper, we extend previous results [10] that exploit this linear ordering in efficient triangle-strip management for high-performance rendering. We present new algorithms to generate single-strip representations that follow different user defined constraints or preferences in the form of edge weights. These functional constraints are application dependent. For example, normal-based constraints can be used for efficient rendering after visibility culling, or spatial constraints for highly coherent vertex-caching. We highlight the flexibility of this approach by generating single-strips with preferences as arbitrary as the orientation of the edges. We also present a hierarchical single-strip management strategy for high-performance interactive 3D rendering.

A novel page-based data structure for interactive walkthroughs

Given a data layout of a large walkthrough scene, we present a novel and simple spatial hierarchy on the disk-pages of the layout that has notable advantages over a conventional spatial hierarchy on the scene geometry. Assume that each disk-page consists of a set of triangles whose bounding boxes are computed. A spatial hierarchy of the walkthrough space is constructed, not with the given scene, but with the bounding boxes of disk-pages. The leaf nodes of the spatial-hierarchy refer directly to the page numbers of the pages of the bounding box it contains. We call this hierarchy on the pages as the disk-page hierarchy. We also propose a self-contained disk-page format that would suit this data structure well. Further, we present a new cache-oblivious graph-based data layout algorithm called the 2-factor layout that would preserve the proximity and orientation properties of the primitives in the layout. Walkthrough experiments have been conducted on a city scene consisting of over 110M triangles. Our system renders this scene on a laptop within a one pixel projection error at over 20 fps with simple texture substitution based simplification of distant objects, and with no explicit data/cache management.

Hierarchyless Simplification, Stripification and Compression of Triangulated Two-Manifolds

In this paper we explore the algorithmic space in which stripification, simplification and geometric compression of triangulated 2-manifolds overlap. Edge-collapse/uncollapse based geometric simplification algorithms develop a hierarchy of collapses such that during uncollapse the reverse order has to be maintained. We show that restricting the simplification and refinement operations only to, what we call, the collapsible edges creates hierarchyless simplification in which the operations on one edge can be performed independent of those on another. Although only a restricted set of edges is used for simplification operations, we prove topological results to show that, with minor retriangulation, any triangulated 2-manifold can be reduced to either a single vertex or a single edge using the hierarchyless simplification, resulting in extreme simplification. The set of collapsible edges helps us analyze and relate the similarities between simplification, stripification and geometric compression algorithms. We show that the maximal set of collapsible edges implicitly describes a triangle strip representation of the original model. Further, these strips can be effortlessly maintained on multiresolution models obtained through any sequence of hierarchyless simplifications on these collapsible edges. Due to natural relationship between stripification and geometric compression, these multi-resolution models can also be efficiently compressed using traditional compression algorithms. We present algorithms to find the maximal set of collapsible edges and to reorganize these edges to get the minimum number of connected components of these edges. An order-independent simplification and refinement of these edges is achieved by our novel data structure and we show the results of our implementation of view-dependent, dynamic, hierarchyless simplification. We maintain a single triangle strip across all multi-resolution models created by the view-dependent simplification process. We present a new algorithm to compress the models using the triangle strips implicitly defined by the collapsible edges.

Shape isophotic error netric controllable re-sampling for point-sampled surfaces

Shape simplification and re-sampling of underlying point-sampled surfaces under user-defined error bounds is an important and challenging issue. Based on the regular triangulation of the Gaussian sphere and the surface normals mapping onto the Gaussian sphere, a Gaussian sphere based re-sampling scheme is presented that generates a non-uniformly curvature-aware simplification of the given point-sampled model. Owing to the theoretical analysis of shape isophotic error metric for did that Gaussian sphere based sampling, the proposed simplification scheme provides a convenient way to control the re-sampling results under a user-specified error metric bound. The novel algorithm has been implemented and demonstrated on several examples.

Quadrilateral and tetrahedral mesh stripification using 2-factor partitioning of the dual graph

In order to find a 2-factor of a graph, there exists a O(n1.5) deterministic algorithm [7] and a O(n3) randomized algorithm [14]. In this paper, we propose novel O(nlog3nloglogn) algorithms to find a 2-factor, if one exists, of a graph in which all n vertices have degree 4 or less. Such graphs are actually dual graphs of quadrilateral and tetrahedral meshes. A 2-factor of such graphs implicitly defines a linear ordering of the mesh primitives in the form of strips. Further, by introducing a few additional primitives, we reduce the number of tetrahedral strips to represent the entire tetrahedral mesh and represent the entire quad surface using a single quad strip.

Curvature Aware Fundamental Cycles

We present a graph algorithm to find fundamental cycles aligned with the principal curvature directions of a surface. Specifically, we use the tree‐cotree decomposition of graphs embedded in manifolds, guided with edge weights, in order to produce these cycles. Our algorithm is very quick compared to existing methods, with a worst case running time of O(n log n+gn) where n is the number of faces and g is the surface genus. Further, its flexibility to accommodate different weighting functions and to handle boundaries may be used to produce cycles suitable for a variety of applications and models.

Streaming surface sampling using Gaussian ε -nets

We propose a robust, feature preserving and user-steerable mesh sampling algorithm, based on the one-to-many mapping of a regular sampling of the Gaussian sphere onto a given manifold surface. Most of the operations are local, and no global information is maintained. For this reason, our algorithm is amenable to a parallel or streaming implementation and is most suitable in situations when it is not possible to hold all the input data in memory at the same time. Using ε-nets, we analyze the sampling method and propose solutions to avoid shortcomings inherent to all localized sampling methods. Further, as a byproduct of our sampling algorithm, a shape approximation is produced. Finally, we demonstrate a streaming implementation that handles large meshes with a small memory footprint.

Single Triangle Strip and Loop on Manifolds with Boundaries

The single triangle-strip loop generation algorithm on a triangulated two-manifold presented by Gopi and Eppstein (2004) is based on the guaranteed existence of a perfect matching in its dual graph. However, such a perfect matching is not guaranteed in the dual graph of triangulated manifolds with boundaries. In this paper, we present algorithms that suitably modify the results of the dual graph matching to generate a single strip loop on manifolds with boundaries. Further, the algorithm presented by Gopi and Eppstein (2004) can produce only strip loops, but not linear strips. We present an algorithm that does topological surgery to construct linear strips, with user-specified start and end triangles, on manifolds with or without boundaries. The main contributions of this paper include graph algorithms to handle unmatched triangles, reduction of the number of Steiner vertices introduced to create strip loops, and finally a novel method to generate single linear strips with arbitrary start and end positions