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Triangle Surfaces with Discrete Equivalence Classes

Published in SIGGRAPH 2010

報告者 :丁琨桓

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Introduction

Modeling freeform shapes has many uses modeling the body of a car for

manufacturing purposes, or depicting the shape of a building for architectural applications.

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Introduction the pieces that make up these shapes require a

great deal of customization. This paper propose a technique that altered

geometry such that each polygon falls into a set of discrete equivalence classes.

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Discrete Equivalence Classes

Input mesh

Clustering

Rigid Transformation

Global Optimization

output mesh

Polygon Assignment & detect Canonical Triangles

Mesh of Canonical Triangles

Modified Geometry

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Triangle Similarity

Rigid transformation

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Triangle Similarity

(a1,a2,a3)

(b1,b2,b3), (b1,b3,b2)(b2,b1,b3), (b2,b3,b1)(b3,b1,b2), (b3,b2,b1)

permutation

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Triangle Similarity

find the best Rigid Transformation Least-squares fitting of two 3-d point sets [Arun et al. 19

87].

where , represent the polygon’s centroid ,

R is then given by R = UVT where M = UΣVT is the singular value decomposition of M

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canonical triangle

5-Point Tensile Roof1280 triangles

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canonical triangle

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canonical triangle

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canonical triangle

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Clustering

Clustering begin our optimization with a single cluster

Iteratively add a new cluster corresponding to the polygon with the worst error in the summation from similar Equation

repeat this process until n clusters have been added

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Clustering

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Clustering

1280 triangles | 10 clusters

canonical triangle

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Clustering1 5

10 20

Before Global Optimization

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Clustering

Spacing between Triangles

20 clusters Before Global Optimization

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Global Optimization Mesh Editing with Poisson-Based Gradient Field

Manipulation [2004] Solve a Poisson equation to find the new positio

ns of the vertices to match the canonical polygons.

Disconnected Triangles

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Global Optimization

Wher α and β are small constants (0.001 and 0.01)

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Global Optimization

The Poisson equation attempts to find vertex positions for the shape P such that is minimized

∇Pi is the gradient of the triangle Pi and △i is the area of Pi , Cind(i) is canonical polygon

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Global Optimization

(xi, ni) is the closest point and normal on the initial shape P0 to Pi’s centroid

xi

ni

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Global Optimization

If pℓ is a vertex on the boundary of P and y1, y2 are vertices on the boundary of P0 such that their e

dge is the closest to pℓ y1

y2

pℓ

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Global Optimization

Before Global Optimization

After Global Optimization

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Clustering & Global Optimization

1-Clusters 2-Clusters

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Clustering & Global Optimization

3-Clusters 4-Clusters

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Clustering & Global Optimization

5-Clusters 6-Clusters

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Result

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Result

1724 polygons optimized using 42 clusters