Harmonic Functions for Quadrilateral Remeshing of Arbitrary Manifolds

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Harmonic Functions for QuadrilateralRemeshing of
Arbitrary Manifolds

• Shen Dong, Scott Kircher, Michael Garland

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Motivation

• Fair Morse functions for extracting the
  topological structure of a surface mesh
  SIGGRAPH 2004

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Motivation

• Use these smooth scalar fields to generate a
  quad-dominant remesh

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Why Quads

• Preferred primitive for many objects such as
  characters, buildings, etc.
• Preferred primitive for finite element simulation
• Subdivision surfaces, which prevails in graphical
  design and animation, are quad meshes

VS
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Adv. of Our Method

• High quality

Anisotropic polygonal remeshing SIGGRAPH 03
Surface simplification using quadric error
metrics SIGGRAPH 97
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Adv. of Our Method

• Work on arbitrary manifold

Genus 6
Genus 1
Genus 3
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Adv. of Our Method

• User control

Or
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Adv. of Our Method

• Fast Less than 1 min for models with 100,000
  vertices
• Easy LOD control

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Method Overview
3a. Gradient flow
2. Laplacian field
4. Overlaid
5. Output remesh
1. Input mesh
3b. Iso-parametric flow
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Laplacian Field Construction

• Harmonic function with
  Dirichlet boundary condition
• Constrained vertices are the only possible
  extrema
• If all constrained to the same value, all are
  guaranteed to be extrema
• Field very smooth elsewhere

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Laplacian Field Construction

• Discretize scalar value assigned to each
  vertex, piecewise linear in each triangle
• Laplacian operator becomes
• Use either Discrete Conformal weights or Mean
  Value weights
• Amount to solving a sparse linear system

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Vector Field Construction

• Two orthogonal vector fields
• Gradient field
• Orthogonal field 90 degree rotation cc

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Tracing A Single Flow

• Start from a seed point
• A piecewise linear integral line
• Intersect the original mesh on flow nodes
• Smoothness simplifies the tracing

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Tracing A Gradient Flow

• From a seed point, ascend to a maximum, then
  descend to a minimum
• Regular case cut through an edge and follow the
  vector field exactly
• Vertex special case look over the 1-ring to
  find the best direction
• Edge special case follow the edge direction

Regular case
Vertex case
Edge case
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Tracing A Iso-parametric Flow

• Follow the iso-parametric value across edges
• Closed curve if not encountering an open boundary
• Free of numerical error

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Placing Flow Lines

• Determines how the surface is sampled
• Sampling distance functions decide the
  spacing of gradient (iso-parametric) flows
  everywhere on the surface
• Flow-flow, seed-flow distance measured along
  orthogonal vector field direction
• Some initial seeds placed near critical points
  and features

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Placing Flow Lines

• After tracing a flow, place seeds along its two
  sides, their distance based on , for
  generation of more flows later
• Start tracing a flow from a seed when no existing
  flow nearby
• Stop tracing a flow when it gets too close
  to an existing flow
• Use Octree to accelerate the detection of nearby
  flow lines.

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Sampling Distance Functions

• A heuristic function
• underlying isotropic density
• the degree of curvature sensitivity
• dependent on , normal curvature along
  direction

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Features

• Organized into chains sequences of feature
  edges between corners or darts
• When tracing flows, mark a flow node as feature
  when it intersects a feature edge
• Neighboring feature flow nodes on the same
  chain are connected

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Crossing Detection

• A crossing the intersection of a unique
  gradient and iso-parametric flow, or a feature
  flow node is a vertex of the output mesh
• Crossing detected in each input mesh triangle
• Create sorted list of crossings of each flow,
  then each crossing would have 4 neighbors
• Erasing dangling or close-to-feature crossings

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Polygon Generation

• Walk out all the face ring of the connectivity
  graph counter-clockwise
• Additional polygons around each extremal point
• Conforming vs. non-conforming

Or
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Semi-Automatic Placement of Extremum

• Specify one minimum point P
• Solve a Poisson equation
• Maxima came out from the solution, P guaranteed
  to be the only minimum
• Perform clustering of nearby maxima

Laplacian field
Poisson field
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Results
Curve constraints
Loop constraints
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Results
Input mesh
Laplacian field
Non-conforming remesh
Conforming remesh
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Results