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IFT 611217 - GEODESICS
http://www-labs.iro.umontreal.ca/~bmpix/teaching/6112/2018/
Mikhail Bessmeltsev
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Geodesic Distance
Extrinsically closeIntrinsically far
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Geodesic Distance
Small Euclidean distanceLarge geodesic distance
Length of a shortest path on a surface
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Geodesics
Straightest Geodesics on Polyhedral Surfaces (Polthier and Schmies)
Locally shortestNon-unique!
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Possible questions
Single source
Multi-source All-pairshttps://www.ceremade.dauphine.fr/~peyre/teaching/manifold/tp3.html http://www.sciencedirect.com/science/article/pii/S0010448511002260
Locally shortest
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Mesh ~ Graphfind shortest path?
http://www.cse.ohio-state.edu/~tamaldey/isotopic.html
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Mesh ~ Graphfind shortest path?
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Mesh ~ Graphfind shortest path?
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Mesh ~ Graphfind shortest path?
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Mesh ~ Graphfind shortest path?
May not convergeunder refinement
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Mesh ~ Graphfind shortest path?
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Mesh ~ Graphfind shortest path?
http://www.cse.ohio-state.edu/~tamaldey/isotopic.html
No, but for a good mesh, it may
be a good approximation
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How to discretize geodesic distance?
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Euclidean Space
• Globally shortest path
• Local minimizer of length
• Locally straight path
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Surfaces: choose one
• Globally shortest path
• Local minimizer of length
• Locally straight path
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Surfaces: choose one
• Globally shortest path
• Local minimizer of length
• Locally straight path
Let’s find
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Arc Length
�𝑎𝑎
𝑏𝑏‖𝛾𝛾′ 𝑡𝑡 ‖𝑑𝑑𝑡𝑡
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Wouldn’t it be nice?
�𝑎𝑎
𝑏𝑏𝛾𝛾′ 𝑡𝑡 2𝑑𝑑𝑡𝑡
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Energy of a Curve
= when parameterized by arc length.
Not the length, but turns out we can optimize it instead!
𝐿𝐿 = �𝑎𝑎
𝑏𝑏‖𝛾𝛾′ 𝑡𝑡 ‖𝑑𝑑𝑡𝑡
Note: we do not assume arclength parameterization
𝐸𝐸 =12�𝑎𝑎
𝑏𝑏𝛾𝛾′ 𝑡𝑡 2𝑑𝑑𝑡𝑡
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First Variation of Arc Length
Lemma. Let 𝛾𝛾𝑡𝑡: a, b → 𝑆𝑆 be a family of curves with fixed endpoints in surface S; assume 𝛾𝛾 is parameterized by arc length at t=0. Then,
Corollary. 𝜸𝜸: 𝐚𝐚,𝐛𝐛 → 𝑺𝑺 is a geodesic iff
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Intuition
• The only acceleration is out of the surface• No steering wheel!
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Intuition
• The only acceleration is out of the surface• No steering wheel!
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Two Local Perspectives
• Boundary value problem– Given: 𝜸𝜸 𝟎𝟎 ,𝜸𝜸(𝟏𝟏)
• Initial value problem (ODE)– Given: 𝜸𝜸 𝟎𝟎 ,𝜸𝜸𝜸(𝟎𝟎)
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Exponential Map
𝜸𝜸𝒗𝒗 𝟏𝟏 where 𝜸𝜸𝒗𝒗 is (unique) geodesic
from p with velocity v.
https://en.wikipedia.org/wiki/Exponential_map_(Riemannian_geometry)
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Instability of Geodesics
http://parametricwood2011.files.wordpress.com/2011/01/cone-with-three-geodesics.png
Locally minimizing distance is not enough to be a shortest path!
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Eikonal Equation
https://www.mathworks.com/matlabcentral/fileexchange/24827-hamilton-jacobi-solver-on-unstructured-triangular-grids/content/HJB_Solver_Package/@SolveEikonal/SolveEikonal.m
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\end{math}
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Starting Point for Algorithms
Graph shortest path algorithms arewell-understood.
Can we use them (carefully) to compute geodesics?
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Useful Principles
“Shortest path had to come from somewhere.”
“All pieces of a shortest path are optimal.”
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Dijkstra’s Algorithm
Initialization:
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Dijkstra’s Algorithm
Iteration k:
During each iteration, Sremains optimal.
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Advancing Fronts
CS 468, 2009
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Example
http://www.iekucukcay.com/wp-content/uploads/2011/09/dijkstra.gif
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Example
(Wikipedia)
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Fast Marching
Approximately solving Eikonalequation with a (modified)
Dijkstra algorithm
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Problem
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Planar Front Approximation
http://research.microsoft.com/en-us/um/people/hoppe/geodesics.pdf
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At Local Scale
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Fast Marching: Update Step
Vertex 𝑥𝑥 updated from triangle ∋ 𝑥𝑥
Distance computed from the other triangles vertices
Image from Bronstein et al., Numerical Geometry of Nonrigid Shapes
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Planar Calculations
Given:
Find:
Derivation from Bronstein et al., Numerical Geometry of Nonrigid Shapes
𝑥𝑥2
𝑥𝑥1
𝑛𝑛
𝑑𝑑 = 𝑋𝑋𝑇𝑇𝑛𝑛 + 𝑝𝑝𝟏𝟏2×1
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Planar Calculations
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Planar Calculations
Quadratic equation for p
Find:
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Two Roots
Bronstein et al., Numerical Geometry of Nonrigid Shapes
Two orientations for the normal
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Larger Root: Consistent
Two orientations for the normalBronstein et al., Numerical Geometry of Nonrigid S
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Additional Issue
Front from outside the triangle
Update should be from a different
triangle!
Bronstein et al., Numerical Geometry of Nonrigid S
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Condition for Front Direction
Front from outside the triangleBronstein et al., Numerical Geometry of Nonrigid S
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Obtuse Triangles
Must reach x3 after x1 and x2
Bronstein et al., Numerical Geometry of Nonrigid S
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Fixing the Issues
• Alternative edge-based update:
• Add connections as needed[Kimmel and Sethian 1998]
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Summary: Update Step
Bronstein, Numerical Geometry of Nonrigid S
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Fast Marching vs. Dijkstra
• Modified update step
• Update all triangles adjacent to a given vertex
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Eikonal Equation
Greek: “Image”
Solutions are geodesic distance
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STILL AN APPROXIMATION
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Modifying Fast Marching
[Novotni and Klein 2002]:Circular wavefront
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Modifying Fast Marching
Grids and parameterized surfacesBronstein, Numerical Geometry of Nonrigid Shapes
Raster scan
and/or parallelize
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Alternative to Eikonal Equation
Crane, Weischedel, and Wardetzky. “Geodesics in Heat.” TOG 2013.
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Tracing Geodesic Curves
Trace gradient of distance function
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Initial Value Problem
Trace a single geodesic exactly
Equal left and right angles
Polthier and Schmies. “Shortest Geodesics on Polyhedral Surfaces.” SIGGRAPH course notes 2006.
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Exact Geodesics
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MMP Algorithm: Big Idea
Surazhsky et al. “Fast Exact and Approximate Geodesics on Meshes.” SIGGRAPH 2005.
Dijkstra-style front with windows
explaining source.
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Practical Implementation
http://code.google.com/p/geodesic/
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Fuzzy Geodesics
Stable version of geodesic distance
Function on surface expressing difference in triangle inequality
“Intersection” by pointwise multiplication
Sun, Chen, Funkhouser. “Fuzzy geodesics and consistent sparse correspondences for deformable
shapes.” CGF2010.
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Stable Measurement
Campen and Kobbelt. “Walking On Broken Mesh: Defect-Tolerant Geodesic Distances and Parameterizations.” Eurographics 2011.
Morphological operators to fill holes rather than remeshing
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All-Pairs Distances
Xin, Ying, and He. “Constant-time all-pairs geodesic distance query on triangle meshes.” I3D 2012.
Sample points
Geodesic field
Triangulate (Delaunay) Fix edges
Query (planar
embedding)
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Geodesic Voronoi & Delaunay
From Geodesic Methods in Computer Vision and Graphics (Peyré et al., FnT 2010
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High-Dimensional Problems
Heeren et al. Time-discrete geodesics in the space of shells. SGP 2012.
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In ML: Be Careful!
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In ML: Be Careful!