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Unfolding 3D Laser Images of
Underground Tunnels
Po Kong Lai, School of Electrical Engineering & Computer Science, University of Ottawa
Claire Samson, Dept. of Earth Sciences, Carleton University
AGU-CGU-GAC-MAC Joint Assembly
May 4th 2015
Montreal, Québec
Motivation & Goals
• Visualization of 3D mesh models of tunnels requires:
– Multiple vantage points
– Dedicated high-performance and costly hardware in field
• Main Goal:
– Take a triangulated mesh model generated from a point
cloud of an underground tunnel and convert it into a 2D
map
– The final 2D map should be easy and intuitive to interpret
1
Dataset
• Subsection of large underground tunnel
Outside View – Double Sided Lighting
41 m
8.2 m
6.7 m
2
Background Theory
• Well studied problem in differential geometry
– Known as surface parametrization
• Very few surfaces can be mapped from 3D to 2D
without error
– Error is known as metric distortion
– Visualized as stretching and shearing of the surface
– The degree of stretching and shearing described through
the Jacobian matrix
4
Surface Parameterization
• Given:
– Surface 𝑆 ∈ ℝ3
– Parameter domain Ω ∈ ℝ2
• Find a bijective function 𝑓 such that:
– 𝑓 ∶ 𝑆 → Ω
– 𝑓−1 ∶ Ω → S
• When 𝑆 is a mesh, the problem is known as mesh
parameterization
5
Mesh Parameterization
• Let 𝑀 be a mesh with the following parametric
representation:
• The partial derivatives 𝑓𝑢 and 𝑓𝑣 are vectors which
maps movement in Ω to 𝑀
𝑓 𝑢, 𝑣 = (𝑥 𝑢, 𝑣 , 𝑦 𝑢, 𝑣 , 𝑧 𝑢, 𝑣 ), where 𝑢, 𝑣 ∈ Ω
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Metric Distortion
• Special combinations of 𝜎1 and 𝜎2:
– 𝜎1 = 𝜎2 = 1 ↔ 𝑓 is isometric (preserves length)
– 𝜎1 = 𝜎2 ↔ 𝑓 is conformal (preserves angles)
– 𝜎1𝜎2 = 1 ↔ 𝑓 is equiareal (preserves area)
• In practice: some combination of conformal and
equiareal
• Goal of mesh parameterization algorithms:
– Minimize conformal/equiareal error
12
Mesh Parameterization Methods
• Examined two state-of-the-art methods:
– Least Squares Conformal Mapping (LSCM)
– Angle Based Flattening (ABF++)
• LSCM
– Minimize conformal error per triangle: 𝐸𝑡 = 𝜎1,𝑡 − 𝜎2,𝑡2
• ABF++
– Minimize difference between 3D angles and 2D angles of
the triangles
13
Why the strange shape?
• Both methods are optimization based and thus has no
constraint for intuitive interpretation
• Any “bumps” are flattened
• To preserve the 3D surface area it must be displaced
in the 2D map
• Large number of bumps results in unintuitive final
shape
16
Mesh Deformation and Projection
Strategy
• Apply a global function, 𝐺 𝑣, 𝑑 to all vertices
– Implemented in Blender as a modifier
– Rotates the outer vertices at a greater rate than the inner
vertices
• Manually determine 𝑑 and position mesh to be
projected onto a 2D plane
17
Metric Distortion Evaluation
• Utilize measures found in differential geometry
• Stretch 𝐿2
– 𝐿2 𝑡 =𝜎1,𝑡2 +𝜎2,𝑡
2
2 where t is a triangle in mesh 𝑀
– 𝐿2 𝑀 = 𝐿2 𝑡
2𝐴𝑟𝑒𝑎 𝑡𝑡∈𝑀
𝐴𝑟𝑒𝑎 𝑡𝑡∈𝑀
• Angular distortion
– 𝐴𝑛𝑔𝐷 𝑀 = 𝜃3𝑑−𝜃2𝑑
2𝜃3𝑑,𝜃2𝑑 ∈𝐴
𝐴 where 𝐴 is the set of
angle pairs
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Metric Distortion Evaluation
Unfolding Method Stretch 𝐿2 Angular Distortion
LSCM 636554 0.0404
ABF++ 357237 0.0390
Mesh Deformation
& Projection
228 0.0814
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Summary
• Mesh parameterization algorithms:
– Automatic
– Introduces the least amount of angular distortion
– Produce unintuitive 2D maps
• Mesh deformation and projection strategy
– Requires user input
– Good compromise between angular and stretch distortion
– Produces intuitive 2D maps
28
Conclusion
• Best approach for unwrapping 3D mesh models of
tunnels:
– First apply automatic mesh parameterization
– Perform mesh deformation and projection if
parameterization is not satisfactory
29
References
1. Hormann, Kai, Bruno Lévy, and Alla Sheffer. “Mesh parameterization:
Theory and practice.” (2007).
2. Lévy, Bruno, et al. “Least squares conformal maps for automatic texture
atlas generation.” ACM Transactions on Graphics (TOG). Vol. 21. No. 3.
ACM, 2002.
3. Sheffer, Alla, et al. “ABF++: fast and robust angle based
flattening.” ACM Transactions on Graphics (TOG) 24.2 (2005): 311-330.
4. Blender. http://www.blender.org/foundation/
5. MeshLab. http://meshlab.sourceforge.net/
6. ALICE Graphite. http://alice.loria.fr/WIKI/index.php/Graphite/Graphite
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