topological data analysis...topological data analysis introduction to urban data science lecture 2...
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Topological Data Analysis
Introduction to Urban Data ScienceLecture 2
Harish Doraiswamy
New York University
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Scalar Function
• f: S → ℝ
• S – spatial domain
• d-Manifold• Neighbourhood of every point resembles ℝd
• 2-Manifold
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Scalar Function
• f: S → ℝ
• S – spatial domain
• d-Manifold• Neighbourhood of every point resembles ℝd
• 1D
• 2D
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• f: S → ℝ
• S – spatial domain
• d-Manifold• Neighbourhood of every point resembles ℝd
• 1D
• 2D
• 3D
Scalar Function
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Level Set
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Level Set
• f -1(r)
• r – real number
The set of all points having the same function value
• Isocontour• 2D surfaces
• Isosurface• 3D volumes
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Level Sets
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Level Set
[Image: Contour Spectrum, Bajaj et. al.]
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Scalar functions
• f: S → R
• Gradient• 𝛻𝑓
• Critical Points• 𝛻𝑓 = 0
• Smooth function• 𝛻2𝑓 exists
f
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Example: 1D
• Critical Points
• Types of critical points• Stability
• 2 types of stability
• Degeneracy• Test second derivative
f
Maximum
Minimum
f is a Morse function if it has no degenerate critical points
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Example: 2D
• Critical Points• 𝛻𝑓 = 0
• 3 types of stability• Maximum
• Minimum
• Saddle
• Degeneracy• Hessian
• 𝛻2𝑓 = 0
𝜕2𝑓
𝜕𝑥2(𝑝)
𝜕2𝑓
𝜕𝑦𝜕𝑥(𝑝)
𝜕2𝑓
𝜕𝑥𝜕𝑦(𝑝)
𝜕2𝑓
𝜕𝑦2(𝑝)
f is a Morse function if it has no degenerate critical points
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Functions on Manifolds
•Gradient is based on local coordinates
•How to ensure function is Morse• Perturbation
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Functions on Manifolds
•Gradient is based on local coordinates
•How to ensure function is Morse• Perturbation• Can be as small as necessary
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Morse Lemma
• Example: scalar function defined on a surface
• p be the critical point
• f = f(p) ± x2 ± y2
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Morse Index
• How many types of instability are there?
• Classify the types of critical points• Based on Morse lemma
Morse Index 𝟏𝐃 2𝐃 3𝐃
0 𝑥2 𝑥2 + 𝑦2 𝑥2 + 𝑦2 + 𝑧2
1 −𝑥2 −𝑥2 + 𝑦2 −𝑥2 + 𝑦2
+ 𝑧2
2 −𝑥2 − 𝑦2 −𝑥2 − 𝑦2
+ 𝑧2
3 −𝑥2 − 𝑦2
− 𝑧2
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Critical Points
• Classification based on level sets
• 1D
Before 𝑓 − 𝜀
After 𝑓 + 𝜀
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Critical Points
• Classification based on level sets
• 2D
Before 𝑓 − 𝜀
After 𝑓 + 𝜀
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Critical Points
• Classification based on level sets
• 3D
After 𝑓 + 𝜀
Before 𝑓 − 𝜀
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Critical Points
• Classify based on gradient vectors
1D
2D
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Critical Points
• Classify based on gradient vectors
3D
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Mesh
• Structured Grid• Function values defined on vertices
• Bilinear / trilinear interpolation within cells
• Simplicial complex• Function values defined on vertices
• Linearly interpolated within simplicies
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Simplicial Complex
• k-Simplex• Convex hull of k+1 affinely independent points
• Face of a simplex σ• non-empty subset of the simplex
• τ ≤ σ
0-simplex 1-simplex 2-simplex 3-simplex
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Simplicial Complex
A finite collection of simplicies K such that
• σ in K and τ ≤ σ => τ in K
• σ1 and σ2 in K => σ1 ∩ σ2 is either empty or a face in both
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PL Functions
• Domain• Simplicial complex
• Function defined on vertices
• Linearly interpolated with the simplices
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Critical Points: 1D
• How to define neighborhood?
• Adjacent vertices
• Function value < vertex - maximum
• Function value > vertex - maximum
< <
> >
What about 2D / 3D?
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Neighborhood of a Point: 2D
• Star of v• Set of cofaces of v
• {σ | v ≤ σ}
• Link of v• All faces of simplicies in the star(v) that is disjoint from v
• Mesh induced by adjacent vertices
• Upper Link
• Lower Link
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Neighborhood of a Point: 3D
• Star of v
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Neighborhood of a Point: 3D
• Star of v
• Link
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Neighborhood of a Point: 3D
• Star of v
• Link
• Upper Link
• Lower Link
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Critical Points
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Multi-Saddle
• Split into multiple simple saddles
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Degenerate Critical Points
• Simulation of simplicity• Edelsbrunner [1, Sec 1.4]
• Perturbation
• Given n points with same function value 𝑓 𝑥
• 𝑓 𝑣𝑖 = 𝑓 𝑥 + 𝜀2𝑖
• 𝜀 is a infinitesimally small value
• Ensures no 2 points have the same function value
• Total ordering
𝑓(𝑥)
𝑓 𝑥 + 𝜀20
𝑓 𝑥 + 𝜀21
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Degenerate Critical Points
• Multiple critical points
CANCELLING HANDLES THEOREMUnder certain conditions, a smoother Morse function g can be obtained from f by canceling two critical points that differ in index by 1.
How to choose the appropriate pairs to cancel?
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Topological Persistence
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Intuition in 1D
f
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Intuition in 1D
f
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Intuition in 1D
f
Persistent components
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Intuition in 1D
f
Upward sweep
empty
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Intuition in 1D
f
Upward sweep
emptybirth
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Intuition in 1D
f
Upward sweep
emptybirthbirth
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Intuition in 1D
f
Upward sweep
emptybirthbirthbirth
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Intuition in 1D
f
Upward sweep
emptybirthbirthbirthdeath
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Intuition in 1D
f
Upward sweep
emptybirthbirthbirthdeathdeath
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Intuition in 1D
f
Upward sweep
emptybirthbirthbirthdeathdeath
Pair creators with destroyers
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Intuition in 2D
Upward sweep
emptym1 birthm2 births1 deaths2 birthM death
[m2, s1]
[s2, M]
Pairs:
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k-cycle
0-cycle Set of points that form a connected component
1-cycle
2-cycle
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Topological Persistence
• Analogy: Life time of a cycle
• Each cycle has a creator• positive simplex
• It is “alive” until it is destroyed• negative simplex
• Persistence = age of this cycle
• Pairing the creators with destroyers• Youngest creator
f
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f
How to go from this to this?
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Persistence-based simplification
CANCELLING HANDLES THEOREMUnder certain conditions, a smoother Morse function g can be obtained from f by canceling two critical points that differ in index by 1.
Cancel critical point pairs having low persistence
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Topology of Level Sets
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Reeb Graphs
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Reeb Graphs
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Reeb Graphs
Minimum
Maximum
Critical Points
Saddle
Saddle
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Reeb Graphs
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Reeb Graphs
• Tracks the evolution of level-sets with changing function value
• Contract each connected component of a level set to a point• Quotient space under an equivalence relation that
identifies all points within a connected component
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Reeb Graphs
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Contour Tree
• Simply connected domain
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Contour TreesCarr et al. 2003
Join Tree Split Tree
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Join Tree
• Sweep from highest function value to lowest
• Track super-level set components• {𝑣|𝑓 𝑣 ≥ 𝛼}
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Join Tree
• Sweep from highest function value to lowest
• Track super-level set components• {𝑣|𝑓 𝑣 ≥ 𝛼}
![Page 62: Topological Data Analysis...Topological Data Analysis Introduction to Urban Data Science Lecture 2 Harish Doraiswamy New York University Scalar Function •f: S →ℝ •S –spatial](https://reader034.vdocuments.mx/reader034/viewer/2022042310/5ed795bf67b53e06555d2ed8/html5/thumbnails/62.jpg)
Join Tree
• Sweep from highest function value to lowest
• Track super-level set components• {𝑣|𝑓 𝑣 ≥ 𝛼}
![Page 63: Topological Data Analysis...Topological Data Analysis Introduction to Urban Data Science Lecture 2 Harish Doraiswamy New York University Scalar Function •f: S →ℝ •S –spatial](https://reader034.vdocuments.mx/reader034/viewer/2022042310/5ed795bf67b53e06555d2ed8/html5/thumbnails/63.jpg)
Join Tree
• Sweep from highest function value to lowest
• Track super-level set components• {𝑣|𝑓 𝑣 ≥ 𝛼}
![Page 64: Topological Data Analysis...Topological Data Analysis Introduction to Urban Data Science Lecture 2 Harish Doraiswamy New York University Scalar Function •f: S →ℝ •S –spatial](https://reader034.vdocuments.mx/reader034/viewer/2022042310/5ed795bf67b53e06555d2ed8/html5/thumbnails/64.jpg)
Split Tree
• Sweep from lowest function value to highest
• Track sub-level set components• {𝑣|𝑓 𝑣 ≤ 𝛼}
![Page 65: Topological Data Analysis...Topological Data Analysis Introduction to Urban Data Science Lecture 2 Harish Doraiswamy New York University Scalar Function •f: S →ℝ •S –spatial](https://reader034.vdocuments.mx/reader034/viewer/2022042310/5ed795bf67b53e06555d2ed8/html5/thumbnails/65.jpg)
f
How to go from this to this?
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Persistence-based simplification
CANCELLING HANDLES THEOREMUnder certain conditions, a smoother Morse function g can be obtained from f by canceling two critical points that differ in index by 1.
Cancel critical point pairs having low persistence
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Using Contour trees for Simplification
• Branch• Monotone path in a graph• Defined by the scalar function
• Branch decomposition• If every edge in a graph appears in exactly one branch
• Hierarchical branch decomposition• Exactly 1 branch connecting 2 leaves• All other branches connect 1 leaf with an internal node
• Goal• Hierarchical branch decomposition of the contour tree• End points of each branch are persistence critical point pairs
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Branch Decomposition
1
2
3
4
56
7 8
9
10 10
98
76
54
3
21
Join Tree Split Tree
8
9
3
2
6
4
5
1
10
7
8
9
3
2
6
4
5
1
10
7
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Simplification
8
9
3
2
6
4
5
1
10
7
• Parent - Child hierarchy
• Branch – critical point pair• creator-destroyer
• Persistence = difference in function value
• Keep removing childless branches• until persistence of smallest branch is greater than
threshold
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Example
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Example: Reeb graph
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Example: Reeb graph
Simplification threshold = 0.005 Simplification threshold = 0.01
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Example: Reeb graph
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Persistence Diagram
Robust to noise
Stability of Persistence Diagrams, Cohen-Steiner et al.
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PL Functions
• Domain• Area of a city defined as a mesh
• Function defined on vertices
• Linearly interpolated with the simplices
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PL Functions on Graphs
• Domain• Road network defined as a graph
• Set of 0- and 1-simplices
• Function defined on vertices
• Linearly interpolated with the edges
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References
1. Geometry and Topology for Mesh Generation, Edelsbrunner.
2. Topology for computing, Zomorodian. Chapter 5http://www.caam.rice.edu/~yad1/miscellaneous/References/Math/Topology/Computational/Topology_for_Computing_-_ZOMORODIAN.pdf
3. Reeb Graphs: Computation, Visualization and Applications, Harish D. http://vgl.csa.iisc.ac.in/pdf/pub/HarishThesis.pdfChapter 2
4. Topological Persistence and Simplification, Edelsbrunner et al.
5. Flexible isosurfaces: Simplifying and displaying scalar topology using the contour tree. Carr et al.
6. Multi-Resolution computation and presentation of Contour Trees. Pascucci et al.
7. Extreme Elevation on a 2-Manifold. Agarwal et al. 2006.