Spectral surface quadrangulation

Abstract

Resampling raw surface meshes is one of the most fundamental operations used by nearly all digital geometry processing systems. The vast majority of this work has focused on triangular remeshing, yet quadrilateral meshes are preferred for many surface PDE problems, especially fluid dynamics, and are best suited for defining Catmull-Clark subdivision surfaces. We describe a fundamentally new approach to the quadrangulation of manifold polygon meshes using Laplacian eigenfunctions, the natural harmonics of the surface. These surface functions distribute their extrema evenly across a mesh, which connect via gradient flow into a quadrangular base mesh. An iterative relaxation algorithm simultaneously refines this initial complex to produce a globally smooth parameterization of the surface. From this, we can construct a well-shaped quadrilateral mesh with very few extraordinary vertices. The quality of this mesh relies on the initial choice of eigenfunction, for which we describe algorithms and hueristics to efficiently and effectively select the harmonic most appropriate for the intended application.

A Genealogy of Convex Solids Via Local and Global Bifurcations of Gradient Vector Fields

“…As noted earlier, in case of generic vector fields [26], topology graphs can be defined by the Morse-Smale complex associated with the vector field [10]. Now we proceed to define the metagraph G. (1)) of G. We require that v is generic except for a unique value λ = λ ∈ (0, 1), for which v(λ ) exhibits a codimension 1 bifurcation [14].…”

“…Saddle points can be removed from the primary graph without losing information [10]: first we connect sources and sinks inside each quadrangle, producing a triangulated topology graph in class T 3 ; we then remove all saddle points and edges incident to them, as in Figures 5(b,c). Since non-degenerate saddles have degree 4, the resulting graph is a 2-vertex-colored quadrangulation in class Q 2 : the quasidual topology graph (cf.…”

“…Since non-degenerate saddles have degree 4, the resulting graph is a 2-vertex-colored quadrangulation in class Q 2 : the quasidual topology graph (cf. [10]). Here we use the latter; however, in Section 4, the primary graph representation is preferable.…”

Isotropic 2D Quadrangle Meshing with Size and Orientation Control

“…Including methods devised to generate quadrangle surface tilings, the rich literature on this topic contains approaches which proceed by quadrangulation [9,3], square packing [28], advancing front [22], conversion [8], decimation [18], Morse-Smale complexes [14,33,17], clustering [4,20], local and global operators [23,19], whisker weaving [32], medial axis [24], streamlining [1,21] and parameterization [25,29,7].…”

Feature aligned quad dominant remeshing using iterative local updates

“…Dong et al [12] propose a quad remeshing method which connects extrema of Laplacian eigenfunctions via gradient flow to form a quadrangular base mesh. High quality remeshing with few extraordinary vertices is achieved this way, although features are usually not well captured.…”