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Frame Field Integration

A frame field on a surface mesh can be thought as gradient vectors of some scalar functions. Retrieving a parametrization involves "integrating" this field to recover those scalar coordinates \(u\) and \(v\).

The algorithm proceeds as follows:

  • Smooth a frame field on faces of a mesh

  • Determine the singularities and cut along seams to recover a disk topology

  • Brush the frame field: determine which branch of the cross will correspond to \(u\) and to \(v\) at each face

  • Integrate the two vector fields: find the two functions \(u\) and \(v\) whose gradient match the two vector fields in the least-square sense.

Frame Field and associated seamless parametrization on a CAD model

Left: Brushed frame field defined on the faces of a CAD model. Singularities are depicted in red (+pi/2) and blue (-pi/2). Right: seamless parametrization computed by integrating the frame field

FrameFieldIntegration

Usage

ff = M.framefield.SurfaceFrameField(mesh, "faces") 
# /!\ Integration is supported only for a frame field defined on faces
ff_param = M.parametrization.FrameFieldIntegration(ff, verbose=True)
ff_param.run()
uvs = mesh.face_corners.get_attribute("uv_coords")

Bases: BaseParametrization

Integration of a smooth vector field into a seamless parametrization

References

  • [1] Mixed-integer quadrangulation, Bommes D, Zimmer H. and Kobbelt L., ACM Transaction on Graphics, 2009

  • [2] QuadCover - Surface Parameterization using Branched Coverings, Kälberer F., Nieser M. and Polthier K., Computer Graphics Forum, 2007

Parameters:

Name Type Description Default
framefield FrameField

the frame field object to integrate.

 required
singularities Attribute

float Attribute on vertices. Gives the target angle defects of vertices

 required
use_cotan bool

if True, uses cotangents as weights in the laplacian matrix. Defaults to True.

required
verbose bool

verbose mode. Defaults to False.

 False

Other Parameters:

Name Type Description
debug bool, optional)

debug mode. Generates additionnal output. Defaults to False.
Note

It is not necessary to provide the input mesh, as it is accessed via the framefield argument

Example

https://github.com/GCoiffier/mouette/blob/main/examples/parametrization/ff_integrate.py

Raises:

Type Description
AssertionError

the frame field should be defined on faces and be of order 4

cut_graph property

Seam edges as a PolyLine

cut_mesh property

Disk-topology mesh where cuts have been performed on seams

flat_mesh property

A flat representation of the mesh where uv-coordinates are copied to xy.

Returns:

Name Type Description
SurfaceMesh SurfaceMesh

the flat mesh

seam_graph property

Returns:

Name Type Description
PolyLine PolyLine

Polyline of seam edges

export_frame_field_as_mesh()

Exports the frame field as a PolyLine for visualization

quantize()

Resolves the problem with hard constraints on the translations accross seams, which have been computed by rounding the integrated translations

References

QuadCover Mixed Integer Quadrangulation

run()

Computes the integration