polycube shape space€¦ · find one optimal embedding from the polycube shape space 2. we look...
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Polycube Shape Space
Hui Zhao, Xuan Li, Wencheng Wang, Xiaoling Wang, Shaodong Wang, Na Lei, Xianfeng Gu
2019 PG2019: KOREA UNIVERSITY - SEOUL, OCTOBER 14-17, 2019
University of Science and Technology Beijing
State University of New York at Stony Brook
Dalian University of Technology
State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences
University of Chinese Academy of Sciences
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Introduction
Polycube PG2019
Given a polyhedron It is easy to extract its skeleton graph
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Introduction
Polycube PG2019
Given a graph
Three problem:
1. Is there a polyhedron whose graph is the given one?
2. If so, how to generate the polyhedron?
3. Any applications in mesh processing for this kinds of relationship?
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Introduction
Polycube PG2019
1. The problem: (graph to polyhedra)
2. The previous works
3. Polycube Shape Space
4. Polycube Embedding
5. The Applications of Polycube Mesh, Quadrangulation, Hex-meshing
Outline
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Introduction
Polycube PG2019
Steintiz’s Theorem: a graph is the skeleton graph of a convex polyhedron if and only if it is 3-connected and planar.
Branko Grünbaum call it : “ the most important and deepest known result on 3-polytopes ”
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Introduction
Polycube PG2019
Eppstein et al. : "a graph is the skeleton graph of a simple orthogonal polyhedron if and only if it is bipartite, planar, 3-regular and removal of any 2 of its vertices disconnects it into at most 2 components."
simple orthogonal polyhedral :
1) spherical topology; 2) simply-connected faces; 3) three mutually-perpendicular axis-parallel edges meeting at each vertex; 4) non-convex polyhedral
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Introduction
Polycube PG2019
“ a polycube graph is a skeleton graph of a polycube polyhedron if the patch number of the graph is bigger than 3.”
Polycube polyhedral of any genus with non-simply connected faces.
Our works:
A linear system to generate a space of polycube polyhedra from a graph.
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Introduction
Polycube PG2019
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Introduction
Polycube PG2019
Non-valid graphs for our approach
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Theory
Polycube PG2019
Polycube Shape Space
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Introduction
Polycube PG2019
Intuitive idea: every edge assigned a vector, then the sum of the vectors around any loop is
equal to zero.
We model this fact by the language of differential one-form, and results in a linear system.
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Introduction
Polycube PG2019
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Introduction
Polycube PG2019
Given an valid input graph, we build a “polycube shape space” with discrete differential one-form.
“polycube shape space” means all polycubes whose layout graphs are the input one.
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Theory
Polycube PG2019
1. arc colorization: determine the coordinate axis (X, Y or Z) for the vectors
2. Linear system: determine the size of the vectors
How to get the vectors on edges of graph?
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Introduction
Polycube PG2019
Workflow:
three steps to compute polycube shape steps
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Theory
Polycube PG2019
Input graph
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Introduction
Polycube PG2019
Input graphs
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Introduction
Polycube PG2019
Change the graph to remove Non simply connected faces
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Theory
Polycube PG2019
Arc Colorization
1. Three colors have six kinds of permutations
2. a colorization for every permutation
3. six colorizations for a polycube layout graph in total.
4. all of them are equivalent under rotation and refection transformations.
5. choose any one as our input for the next step.
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Theory
Polycube PG2019
Arc Colorization
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Theory
Polycube PG2019
1. The color label on one arc can denote the axis which the arc belongs to
2. but it can not distinguish the positive or negative directions of the axis
Polycube shape space by exact one-form
Assign a vector to every edges
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Theory
Polycube PG2019
Polycube shape space by exact one-form
The sum of edge vectors around any face is zero
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Theory
Polycube PG2019
The sum of edge vectors around homology loop is zero
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Theory
Polycube PG2019
W is a matrix, L is the set of vectors on edges
If the dimension of the kernel space K of the matrix W is zero,
it means that there is no valid polycube polyhedron.
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Theory
Polycube PG2019
Six basis of the keneral space
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Theory
Polycube PG2019
We proved:
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Theory
Polycube PG2019
Extensive experiments validate our theorem
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Theory
Polycube PG2019
Polycube Embedding
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Theory
Polycube PG2019
Find one optimal Embedding from the polycube shape space
2. We look for an optimal embedding whose node positions approximate them
1. Input a set of target positions for the nodes of the graph.
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Theory
Polycube PG2019
1. a quadratic optimization method to obtain an optimal polycube polyhedron from this space.
2. employ linear inequality constraints to remove the degenerate cases.
3. add constraints on edge lengths to adjust the shape of the polycube polyhedron.
Algorithm:
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Theory
Polycube PG2019
Polycube shape space contains degenerated embedding
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Theory
Polycube PG2019
Quadratic programming
Coordinates on the original mesh
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Theory
Polycube PG2019
Edge Length constraints
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Theory
Polycube PG2019
Quadratic programming
Interactive set the inequality constraints
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Theory
Polycube PG2019
Degenerated cases
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Experiments
Polycube PG2019
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Experiments
Polycube PG2019
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Experiments
Polycube PG2019
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Experiments
Polycube PG2019
The applications of 1) polycube mesh, 2) quadrangulation, 3) hex-meshing
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Introduction
Polycube PG2019
Related Works
construct a polycube and a cross map at the same time.
● Deformation based: Fu et al. 2016; Gregson et al. 2011;Huang et al. 2014; Zhao et al. 2017
● Graph-cut based: Livesu et al. 2013.
Limitation: Only construct a single polycube.
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Theory
Polycube PG2019
Polycube parameterization
Map every mesh patch to polyhedron face one by one .
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Introduction
Polycube PG2019
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Experiments
Polycube PG2019
Polycube Mesh
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Experiments
Polycube PG2019
Quad Meshing
Two steps:1. Compute a quadrangular polycube mesh.2. Pull back onto the original mesh by barycentric coordinates.
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Experiments
Polycube PG2019
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Experiments
Polycube PG2019
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Experiments
Polycube PG2019
All-Hex Meshing
1. Tetrahedralize the original mesh.2. Bijective map between original
volumetric mesh and polycube volumetric mesh.
3. Tessellate the volume tetrahedral polycube mesh into a hexahedral polycube mesh.
4. Pull back by barycenter coordinates.
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Experiments
Polycube PG2019
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Experiments
Polycube PG2019
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Experiments
Polycube PG2019
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Experiments
Polycube PG2019
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Experiments
Polycube PG2019
Future works
In this paper, we assume graphs are given,
How to generate the optimal graph in terms of a certain kind of application?
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Q&A
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Introduction
Polycube PG2019
Appendix
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Theory
Polycube PG2019
Arc Colorization
1. Three colors have six kinds of permutations
2. a colorization for every permutation
3. six colorizations for a polycube layout graph in total.
4. all of them are equivalent under rotation and refection transformations.
5. choose any one as our input for the next step.
![Page 56: Polycube Shape Space€¦ · Find one optimal Embedding from the polycube shape space 2. We look for an optimal embedding whose node positions approximate them 1. Input a set of target](https://reader034.vdocuments.us/reader034/viewer/2022042401/5f0fdc557e708231d4463f54/html5/thumbnails/56.jpg)
Theory
Polycube PG2019
Arc Colorization
![Page 57: Polycube Shape Space€¦ · Find one optimal Embedding from the polycube shape space 2. We look for an optimal embedding whose node positions approximate them 1. Input a set of target](https://reader034.vdocuments.us/reader034/viewer/2022042401/5f0fdc557e708231d4463f54/html5/thumbnails/57.jpg)
Theory
Polycube PG2019
Arc Colorization
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Theory
Polycube PG2019
1. The color label on one arc can denote the axis which the arc belongs to
2. but it can not distinguish the positive or negative directions of the axis
Polycube space by exact one-form
Intuitive idea: every edge assigned a vector, then the sum of the vectors around any loop is
equal to zero.
We model this fact by differential one-forms, and results in a linear system.
![Page 59: Polycube Shape Space€¦ · Find one optimal Embedding from the polycube shape space 2. We look for an optimal embedding whose node positions approximate them 1. Input a set of target](https://reader034.vdocuments.us/reader034/viewer/2022042401/5f0fdc557e708231d4463f54/html5/thumbnails/59.jpg)
Theory
Polycube PG2019
Polycube shape space by exact one-form
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Theory
Polycube PG2019
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Theory
Polycube PG2019
Polycube Shape Space
Edge length: Edge orientation: Discrete 1-form:
Closed 1-form: Exact 1-form:
Polycube shape space: The linear space of all possible closed and exact 1-forms under a fixed orientation. The dimension is |P| - 3.
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Theory
Polycube PG2019
1. assign a positive direction “d” for every colorized arc randomly
Polycube space by exact one-form
2. Modify non-simply connected faces into simply ones.
3. Define a vector-valued differential one-form by “edge length multiply direction”.
4. As the polycubes are embedded in R^3 space, therefore this one-form is exact.
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Theory
Polycube PG2019
5. Exact one-form integrate to zero on all closed loops, need only satisfy on all homology basis
loops
Polycube space by exact one-form
6. Exact one-form conditions can be expressed as a linear system W
7. The dim of the kernel space P of the linear system W is dim of the polycube shape space
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Theory
Polycube PG2019
In matrix format: W is a matrix, L is a vector of lengths
If the dimension of the kernel space K of the matrix W is zero,
it means that there is no valid polycube polyhedron.
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Theory
Polycube PG2019
We proved:
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Theory
Polycube PG2019
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Theory
Polycube PG2019
Polycube Embedding
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Theory
Polycube PG2019
Optimal Embedding from the polycube shape space
We look for an optimal embedding whose node positions approximate the input layout graph’s
node positions.
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Theory
Polycube PG2019
1. a quadratic optimization method to obtain an optimal polycube polyhedron from this space.
2. employ linear inequality constraints to remove the degenerate cases.
3. add constraints on edge lengths to adjust the shape of the polycube polyhedron.
Algorithm:
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Introduction
Polycube PG2019
Optimal Embedding from polycube shape space
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Theory
Polycube PG2019
Embedding of every basis one-form
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Theory
Polycube PG2019
Optimal Embedding from polycube shape space
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Theory
Polycube PG2019
Embedding
Well-defined 0-form on vertices:This represents the coordinates of an embedding.
Assume is the basis of the polycube space, all possible embeddings is represented by
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Theory
Polycube PG2019
Quadratic programming
Coordinates on the original mesh
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Theory
Polycube PG2019
Edge Length constraints
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Theory
Polycube PG2019
Degenerated cases
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Theory
Polycube PG2019
Quadratic programming
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Theory
Polycube PG2019
Degenerated cases