Directional field synthesis, design, and processing

We present a fast algorithm for low‐distortion locally injective harmonic mappings of genus 0 triangle meshes with and without cone singularities. The algorithm consists of two portions, a linear subspace analysis and construction, and a nonlinear non‐convex optimization for determination of a mapping within the reduced subspace. The subspace is the space of solutions to the Harmonic Global Parametrization (HGP) linear system [BCW17], and only vertex positions near cones are utilized, decoupling the variable count from the mesh density. A key insight shows how to construct the linear subspace at a cost comparable to that of a linear solve, extracting a very small set of elements from the inverse of the matrix without explicitly calculating it. With a variable count on the order of the number of cones, a tangential alternating projection method [HCW17] and a subsequent Newton optimization [CW17] are used to quickly find a low‐distortion locally injective mapping. This mapping determination is typically much faster than the subspace construction. Experiments demonstrating its speed and efficacy are shown, and we find it to be an order of magnitude faster than HGP and other alternatives.

Section: Previous Workmentioning

confidence: 99%

A Subspace Method for Fast Locally Injective Harmonic Mapping

Hefetz

Chien²,

Weber

2019

“…Various techniques have been proposed that allow taking some principal curvature alignment objective into account when generating a smooth cross field [VCD*16]. It can be beneficial to make use of these here in the construction of the cross field from which the layout nodes are derived.…”

Section: Layout Structure Determinationmentioning

confidence: 99%

“…One can consider using a cross field (which also provides two orthogonal directions everywhere on the surface) instead [PPM*16, KLF15], as it is significantly easier to construct for a given set of nodes (i.e. field singularities) [VCD*16]. However, not for all cross fields do the separatrices (the field's integral curves emanating from singularities) form a quad layout, and the resulting candidate arcs may not admit any quad layout either, no matter how the thresholds are chosen.…”

Section: Layout Structure Determinationmentioning

confidence: 99%

Partitioning Surfaces Into Quadrilateral Patches: A Survey

Campen

2017

The efficient and practical representation and processing of geometrically or topologically complex shapes often demands a partitioning into simpler patches. Possibilities range from unstructured arrangements of arbitrarily shaped patches on the one end, to highly structured conforming networks of all‐quadrilateral patches on the other end of the spectrum. Due to its regularity, this latter extreme of conforming partitions with quadrilateral patches, called quad layouts, is most beneficial in many application scenarios, for instance enabling the use of tensor‐product representations based on splines or Bézier patches, grid‐based multi‐resolution techniques and discrete pixel‐based map representations. However, this type of partition is also most complicated to create due to the strict inherent structural restrictions. Traditionally often performed manually in a tedious and demanding process, research in computer graphics and geometry processing has led to a number of computer‐assisted, semi‐automatic, as well as fully automatic approaches to address this problem more efficiently. This survey provides a detailed discussion of this range of methods, treats their strengths and weaknesses and outlines open problems in this field of research.

“…So‐called cross or frame fields are commonly used to A) determine a suitable set of parametrization singularities or cones which keep the overall distortion at a low level, and B) directionally influence the parametrization by optimizing its isolines for alignment with the field directions via a Poisson formulation [RLL*06, KNP07, BZK09]. Most of the methods for the construction of such guiding fields offer possibilities to (hard or soft) align the field to prescribed directions [VCD*16], which then carries over to an indirect alignment objective for the parametrization.…”

Section: Related Workmentioning

confidence: 99%

Scale‐Invariant Directional Alignment of Surface Parametrizations

Campen

Ibing

Ebke

et al. 2016

Self Cite

Various applications of global surface parametrization benefit from the alignment of parametrization isolines with principal curvature directions. This is particularly true for recent parametrization‐based meshing approaches, where this directly translates into a shape‐aware edge flow, better approximation quality, and reduced meshing artifacts. Existing methods to influence a parametrization based on principal curvature directions suffer from scale‐dependence, which implies the necessity of parameter variation, or try to capture complex directional shape features using simple 1D curves. Especially for non‐sharp features, such as chamfers, fillets, blends, and even more for organic variants thereof, these abstractions can be unfit. We present a novel approach which respects and exploits the 2D nature of such directional feature regions, detects them based on coherence and homogeneity properties, and controls the parametrization process accordingly. This approach enables us to provide an intuitive, scale‐invariant control parameter to the user. It also allows us to consider non‐local aspects like the topology of a feature, enabling further improvements. We demonstrate that, compared to previous approaches, global parametrizations of higher quality can be generated without user intervention.