Mean Value Coordinates for Closed Triangular Meshes Scott - - PowerPoint PPT Presentation

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Mean Value Coordinates for Closed Triangular Meshes Scott Schaefer, Tao Ju, Joe Warren (Rice University) Presented in SIGGRAPH2005 Outline Abstract Preliminaries Previous work Mean value Interpolation 3D Mean value

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“Mean Value Coordinates for Closed Triangular Meshes”

Scott Schaefer, Tao Ju, Joe Warren (Rice University) Presented in SIGGRAPH’2005

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Outline

Abstract Preliminaries Previous work Mean value Interpolation 3D Mean value coordinates for closed triangular

meshes

Applications Questions

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Abstract

Search for a function that

can interpolate a set of values at the vertices of a mesh smoothly into its interior

Mean value coordinates

have been used as an interpolant for closed 2D polygons.

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Abstract

This paper generalizes

the mean value coordinates to closed triangular meshes

Interesting applications

to surface deformation and volumetric textures

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Mean Value Theorem

Wikipedia :

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Harmonic functions and Mean Value property

A harmonic function is twice continuously

differentiable function f: U->R which satisfies the laplace’s equation

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Harmonic functions and Mean Value property

They attain there maxima/minima only at

the boundaries.

Let B(x,r) be a ball with center x and radius

r, contained totally in U,

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Harmonic functions and Mean Value property

Then,

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Barycentric Coordinates (Mobius, 1827)

Given find weights such that

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Boundary Value Interpolation

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Boundary Value Interpolation

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Boundary Value Interpolation

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Previous work: Wachpress’s solution (1975)

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Barycentric coordinates for arbitrary polygons in the plane

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Floater : Mean Value Coordinates

These weights were derived by application

  • f mean value theorem for harmonic

functions.

They depend smoothly on the vertices

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Previous Work

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Previous Work

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Previous Work

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Previous Work

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Continuous Barycentric Coordinates

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Continuous Barycentric Coordinates

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Mean Value Interpolation

Continuous form of mean value coordinates Consider evaluation of the numerator

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Mean Value Interpolation

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Mean Value Interpolation

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Mean Value Interpolation

Project the function f[x] onto the boundary

  • f this circle
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Mean Value Interpolation

Integrate the projected function divided by

(p[x]-v) over the circle Sv and then normalize.

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Mean Value Interpolation

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Mean Value Interpolation

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Mean Value Interpolation

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Relation to Discrete Coordinates

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Relation to Discrete Coordinates

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3D Mean Value Coordinates

Find weigths wi which allow us to represent

any v as a weighted combination of the vertices of a closed triangular mesh and satisfy mean value interpolation

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3D Mean Value Coordinates

Given a triangular mesh

and a vertex v in its interior

Consider a unit sphere

centered at vertex v

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3D Mean Value Coordinates

Project the mesh onto

the surface of the sphere

Planar triangles ->

spherical triangles

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3D Mean Value Coordinates

Define m as the mean

vector = integral of unit normal over spherical triangle

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3D Mean Value Coordinates

Given m, represent it

as a weighted combination of the vertex v to the vertices pk of the triangle

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3D Mean Value Coordinates

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Computing The Mean Vector

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Computing The Mean Vector

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Computing The Mean Vector

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Computing The Mean Vector

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Interpolant Computation

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Interpolant Computation

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Interpolant Computation

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Implementation Considerations

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Application: Surface Deformation

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Application: Surface Deformation

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Application: Surface Deformation

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Application: Surface Deformation

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Applications Boundary Value Problems

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Applications Solid Textures

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Applications Surface Deformation

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Applications Surface Deformation

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Applications Surface Deformation

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Summary

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Thank You

Questions?