visualization and graphics research group cipic january 30, 2003multiresolution (ecs 289l) - winter...

January 30, 2003 Multiresolution (ECS 289L) - Winter 2003 1

visualization and graphics research group

CIPIC

MAPS – Multiresolution Adaptive Parameterization of Surfaces

(SIGGRAPH ’98)

By Aaron W. F. Lee; Wim Sweldens; Peter

Schröder; Lawrence Cowsar; David Dobkin

Presented byNameeta Shah and Yong J. Kil



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Introduction (1/2)What does MAPS do?• Construct smooth parameterizations of irregular connectivity triangulations of

arbitrary genus 2-manifolds.• Construct hierarchy of models of different fineness in O(N log N) time and

space complexity.

Original mesh (level 14)

Irregular Original Mesh (M) Smooth Parameterized Mesh (MJ)



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Introduction (2/2)

What is a smooth parameterization?

A “nice” parameterization. E.g. having subdivision connectivity, that is, a mesh produced by 4-to-1 splitting.

4-to-1 splittingof a triangular face: (a) the initial face; (b)after one 4-to-1 split; (c) after two 4-to-1 splits.

Advantages:

Texture mapping, morphing, adaptive remeshing, and other classical multiresolution analysis.



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General Overview


M

M0

MJ

1. Take the original mesh (M)

2. Define a base (coarse) mesh (M0) with a mapping (parameterization) of all the original points.

3. Subdivide (e.g. 4-to-1) the faces of the base mesh and do a inverse mapping.



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Previous WorkMultiresolution Analysis of Arbitrary Meshes (SIGGRAPH ’95)

by Matthias Eck and Hoppe.

1. Base domain by Voronoi tiling.

2. Parameterization by sequence of local harmonic maps.

Cons:

• Time complexity.

• No explicit control over the base domain. E.g. feature edges.



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Overview of MAPS

Constructs the base mesh (M0) by using ideas based on mesh simplification. I.e. Dobkin-Kirkpatrick (DK) algorithm: Iterative vertex removal with guarantees on number of intermediate mesh levels.

Constructs the parameterization iteratively along with the vertex removal strategy.

Intermediate mesh (level 6)

Coarsest mesh (level 0)




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Features of MAPS

Features of MAPS:• Fast coarsification strategy to define the base domain

(M0), avoiding difficulties of finding Voronoi tiles.• Vertex and edge tags to constrain the

parameterization to align with selected features.• Adaptive subdivision connectivity.



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Notations• Triangular mesh is represented as a pair (P, K) where P is a set

of N point positions pi = (xi,yi,zi), 1 I N and K contains the adjacency information. ((PL, KL) is the original mesh and (P0, K0) is the base mesh)

(|K|) is the polyhedron consisting of points, edges and triangles in R3

• A set of vertices is independent if no two vertices are neighbors• A set of vertices is maximally independent if no larger

independent set contains it.• 1-ring neighborhood of a vertex {i} is the set

N (i) = { j | {i, j} K}

• The star of a vertex {i} is the set of simplices containing i.• The curvature estimate at a vertex {i} is

K(i) = |k1| + |k2|



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Hierarchical Representation• Vertex Removal

– Based on the Dobkin – Kirkpatrick (DK) algorithm• Basic idea for going from level l to l -1

– Take any maximal independent set among the vertices of degree atmost b (b=11) in Pl (l is the level in the hierarchy)

– Remove the star of all the vertices in the set– Retriangulate the hole

• Advantage – guarantees L i.e. the number of levels to be O(logN)

• Drawback - Randomly chosen vertices, not using the geometric information

• Solution– Put vertices in priority queue based on their weights calculated as

follows:w(, i) = * a(i) / amax + (1- ) * k(i) / kmax

– Intuitively, vertices with small and flat star area will be weighed less



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Flattening and Retriangulation• Conformal map za which minimizes metric distortion to map the

neighborhood of a removed vertex into the plane. • Piecewise linear approximation of za is denoted by i for the

removed vertex {i}.• Vertex {i} is at the origin and its 1-ring neighbors are mapped as

follows:i(pj

k) = rk

a exp(ika), where ki = # of 1-ring neighbors

a = 2 / ki

rk = || pi - pjk ||

• Retriangulate using Constrained Delaunay triangulation



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Mesh levels



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Initial Parameterization• Construction of a bijection from (|KL|) to (|K0|) • Want to have a mapping l from the top level L to mesh level l

which will allow us to map points between meshes at any level of the hierarchy

• Barycentric coordinates are used for parameterization• Constructing l - 1 for each vertex {i} at level L

Case 1. {i} is in the current level, nothing to do l - 1 (pi) = l (pi) = pi

Case 2. {i} just got removed in the current level l - 1 (pi) = pj + pk + pm where pi is in {j, k, m} inthe new level



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Parameterization contd.• {i} was removed before previous level

– If the triangle that contained {i} at the previous level is still in the new level, do nothing.

– Otherwise, assign barycentric coordinates based onthe new triangle that {i} is in.



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Tagging and Feature Lines• Mark important vertices

• Mark important paths



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Remeshing

• Uniform remeshing

• Smoothing the parameterization

• Adaptive Remeshing



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Xianfeng Gu and Shing-tung Yau, Computing Conformal Structures of Surfaces, Communications in Information and Systems vol. 2, no. 2, pp. 121-146, december 2002

Conformal Structures of Surfaces



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Results(1/3)



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Results(2/3)



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Results(3/3)



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Statistics

Dataset Input size Hierarchy Levels P0 size Remeshing Remesh Output size(triangles) creation (triangles) tolerance creation (triangles)

3-hole 11776 18 (s) 14 120 (NA) 8 (s) 30720fandisk 12946 23 (s) 15 168 1% 10 (s) 3430fandisk 12946 23 (s) 15 168 5% 5 (s) 1130head 100000 160 (s) 22 180 0 5% 440 (s) 74698horse 96966 163 (s) 21 254 1% 60 (s) 15684horse 96966 163 (s) 21 254 0 5% 314 (s) 63060

Table 1: Selected statistics for the examples discussed in the text. All times are in seconds on a 200 MHz PentiumPro.



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Conclusion (Pros vs. Cons)

Contribution• Describe an O(N log N) time and storage algorithm to construct

a logarithmic level hierarchy of arbitrary topology.• Construct a smooth parameterization of the original mesh within

an error tolerance.• Using the smooth parameterization, it can do adaptive,

hierarchical remeshing of arbitrary meshes into subdivision connectivity meshes.

• Allows feature preservation of vertices and edges

Useful• Multiresolution editing and compression, morphing, texture

mapping.



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Misc2

Figure 12: Example remesh of a surface with boundaries.

Figure 10: Remeshing of 3 holed torus using midpoint subdivision.The parameterization is smooth within each base domain triangle,but clearly not across base domain triangles.

visualization and graphics research group cipic january 30, 2003multiresolution (ecs 289l) - winter...

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