Discovering Metrics and Scale Space Preliminary Examination Brittany Terese Fasy Duke University, Computer Science Department 26 July 2010 B. Fasy (Duke CS) Prelim 26 July 2010 1 / 46
Discovering Metrics and Scale Space
Preliminary Examination
Brittany Terese Fasy
Duke University, Computer Science Department
26 July 2010
B. Fasy (Duke CS) Prelim 26 July 2010 1 / 46
Questions
1 Tight Bound: Is the inequality
|ℓ1 − ℓ2| ≤4
π· (κ1 + κ2) · F(γ1, γ2)
a tight bound for curves in Rn for n > 3?
2 Simultaneous Scale Space: What happens if multipleagents are diffusing at the same time?
3 Understanding Vq: Does there exist a stability resultfor Vq?
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B. Fasy (Duke CS) Prelim 26 July 2010 2 / 46
Curves and Metrics
Part 1
B. Fasy (Duke CS) Prelim 26 July 2010 3 / 46
Curves and Metrics
My Inequality
For curves γ1 and γ2,
|ℓ1−ℓ2| ≤4
π· (κ1+κ2) ·F(γ1, γ2)
B. Fasy (Duke CS) Prelim 26 July 2010 4 / 46
Curves and Metrics
Closed Space Curves
A curve is a continuous map γi : S1 → R
n.
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B. Fasy (Duke CS) Prelim 26 July 2010 5 / 46
Curves and Metrics
Closed Space Curves
A curve is a continuous map γi : [0, 1] → Rn, such that γi (0) = γi (1).
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γ(0.8)γ(0.4)
γ(0.5)γ(0.6)
γ(0.7)
γ(0.1)
γ(0) = γ(1)
γ(0.9)γ(0.3)
γ(0.2)
B. Fasy (Duke CS) Prelim 26 July 2010 5 / 46
Curves and Metrics
Closed Space Curves
A curve is a continuous map γi : I → Rn, such that γi (0) = γi (1).
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γ(0.8)γ(0.4)
γ(0.5)γ(0.6)
γ(0.7)
γ(0.1)
γ(0) = γ(1)
γ(0.9)γ(0.3)
γ(0.2)
B. Fasy (Duke CS) Prelim 26 July 2010 5 / 46
Curves and Metrics
Inscribed Polygons
B. Fasy (Duke CS) Prelim 26 July 2010 6 / 46
Curves and Metrics
Closed Space Curves
A curve is a continuous map γi : I → Rn, such that γi (0) = γi (1).
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γ(0.4)
γ(0.6)
γ(0.9)
γ(0.05)
γ(0.7)
γ(0.25)
γ(0.85)
γ(0) = γ(1)
mesh(P) = max0≤i<m (ti+1 − ti )
B. Fasy (Duke CS) Prelim 26 July 2010 7 / 46
Curves and Metrics
Arc Length
ℓi = ℓ(γi ) =
∫ 1
0||γ′
i (t)|| dt
ℓ(P) =∑
j
ℓ(ej)
B. Fasy (Duke CS) Prelim 26 July 2010 8 / 46
Curves and Metrics
Arc Length
ℓi = ℓ(γi ) =
∫ 1
0||γ′
i (t)|| dt
ℓ(P) =∑
j
ℓ(ej)
Lemma
If Pk is a sequence of polygons inscribed in a smooth closed curve γ suchthat mesh(Pk) goes to zero, then
ℓ(γ) = limk→∞
ℓ(Pk).
B. Fasy (Duke CS) Prelim 26 July 2010 8 / 46
Curves and Metrics
Curvature
Let x ∈ I .Let rx denote the radius of the best approximating circle of γi (x).Then, the total curvature is:
κi = κ(γi ) =
∫ 1
01/rx dx .
B. Fasy (Duke CS) Prelim 26 July 2010 9 / 46
Curves and Metrics
Turning Angle
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B. Fasy (Duke CS) Prelim 26 July 2010 10 / 46
Curves and Metrics
Curvature
Let x ∈ I .Let rx denote the radius of the best approximating circle of γi (x).Then, the total curvature is:
κi = κ(γi ) =
∫ 1
01/rx dx .
κ(P) =∑
i
αi .
B. Fasy (Duke CS) Prelim 26 July 2010 11 / 46
Curves and Metrics
Curvature
Let x ∈ I .Let rx denote the radius of the best approximating circle of γi (x).Then, the total curvature is:
κi = κ(γi ) =
∫ 1
01/rx dx .
κ(P) =∑
i
αi .
Lemma
If Pk is a sequence of polygons inscribed in a smooth closed curve γ suchthat mesh(Pk) goes to zero, then
κ(γ) = limk→∞
κ(Pk).
B. Fasy (Duke CS) Prelim 26 July 2010 11 / 46
Curves and Metrics
The Frechet Distance
F(γ1, γ2) = infα : S1→S1
maxt∈S1
(γ(t) − γ(α(t)))
B. Fasy (Duke CS) Prelim 26 July 2010 12 / 46
Curves and Metrics
Man and Dog
B. Fasy (Duke CS) Prelim 26 July 2010 13 / 46
Curves and Metrics
The Frechet Distance
F(γ1, γ2) = infα : S1→S1
maxt∈S1
(γ(t) − γ(α(t)))
B. Fasy (Duke CS) Prelim 26 July 2010 14 / 46
Curves and Metrics
The Frechet Distance
F(γ1, γ2) = infα : S1→S1
maxt∈S1
(γ(t) − γ(α(t)))
Lemma
If Pk and Qk are sequences of polygons inscribed in smooth closed curvesγ1 and γ2 such that mesh(Pk) and mesh(Qk) go to zero, then
F(γ1, γ2) = limk→∞
F(Pk , Qk).
B. Fasy (Duke CS) Prelim 26 July 2010 14 / 46
Curves and Metrics
My Inequality
For curves γ1 and γ2,
|ℓ1−ℓ2| ≤4
π· (κ1+κ2) ·F(γ1, γ2)
B. Fasy (Duke CS) Prelim 26 July 2010 15 / 46
Curves and Metrics
Two Related Theorems
[Cha62, F50] For a closed curve contained in a disk of radius r in Rn,
ℓi ≤ r · κi .
[CSE07] For two closed curves in Rn,
|ℓ1 − ℓ2| ≤2 vol(Sn−1)
vol(Sn)· (κ1 + κ2 − 2π) · F(γ1, γ2).
B. Fasy (Duke CS) Prelim 26 July 2010 16 / 46
Curves and Metrics
Questions
1 Tight Bound: Is the inequality
|ℓ1 − ℓ2| ≤4
π· (κ1 + κ2) · F(γ1, γ2)
a tight bound for curves in Rn for n > 3?
2 Simultaneous Scale Space: What happens if multipleagents are diffusing at the same time?
3 Understanding Vq: Does there exist a stability resultfor Vq?
B. Fasy (Duke CS) Prelim 26 July 2010 17 / 46
Scale Space
Part 2
B. Fasy (Duke CS) Prelim 26 July 2010 18 / 46
Scale Space
Scale Space
B. Fasy (Duke CS) Prelim 26 July 2010 19 / 46
Scale Space
Scale Space
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Scale Space
Scale Space
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H(x , 0)
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H(x , 100)
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H(x , 1000)
B. Fasy (Duke CS) Prelim 26 July 2010 20 / 46
Scale Space
The Gaussian
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|x|2
2t2
This is the fundamental solution to the Heat Equation:
∂u
∂t(x , t)− △ u(x , t) = 0.
B. Fasy (Duke CS) Prelim 26 July 2010 21 / 46
Scale Space
Gaussian Convolution
Blur(y , t, h0) =
∫
x∈Rn
Gn(x − y , t)h0(x) dy
B. Fasy (Duke CS) Prelim 26 July 2010 22 / 46
Scale Space
Homotopy
H : M × I → R is a continuous function such that
H(x , 0) = f (x)
H(x , 1) = g(x)
B. Fasy (Duke CS) Prelim 26 July 2010 23 / 46
Scale Space
Discrete Homotopy
H : vert(K ) × {0, 1, . . . , τ} → R is a discrete function such that
H(x , 0) = f (x)
H(x , τ) = g(x)
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H(·, 0)
H(·, 1)
H(·, 2)
B. Fasy (Duke CS) Prelim 26 July 2010 23 / 46
Scale Space
Discrete Homotopy
H : K × Iτ → R is a discrete function such that
H(x , 0) = f (x)
H(x , τ) = g(x)
and the value at a general point (x , t) ∈ M × Iτ is determined by linearinterpolation.
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H(·, 0)
H(·, 1)
H(·, 2)
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B. Fasy (Duke CS) Prelim 26 July 2010 23 / 46
Scale Space
Heat Equation Homotopy
Let f (x), g(x) : R2 → R.
Let h0(x) = f (x) − g(x).Then:
H(y , t) = ht(y) = Blur(y , t, h0) =
∫
x∈Rn
Gn(x − y , t)h0(x) dy
is the solution to the heat equation with initial condition H(x , 0) = h0(x).
B. Fasy (Duke CS) Prelim 26 July 2010 24 / 46
Scale Space
Heat Equation Homotopy
Let f (x), g(x) : R2 → R.
Let h0(x) = f (x) − g(x).Then:
H(y , t) = ht(y) = Blur(y , t, h0) =
∫
x∈Rn
Gn(x − y , t)h0(x) dy
is the solution to the heat equation with initial condition H(x , 0) = h0(x).
We define the Heat Equation Homotopy as:
H(x , t) := ht(x) + g(x).
B. Fasy (Duke CS) Prelim 26 July 2010 24 / 46
Scale Space
Questions
1 Tight Bound: Is the inequality
|ℓ1 − ℓ2| ≤4
π· (κ1 + κ2) · F(γ1, γ2)
a tight bound for curves in Rn for n > 3?
2 Simultaneous Scale Space: What happens if multipleagents are diffusing at the same time?
3 Understanding Vq: Does there exist a stability resultfor Vq?
B. Fasy (Duke CS) Prelim 26 July 2010 25 / 46
Scale Space
Color Images
B. Fasy (Duke CS) Prelim 26 July 2010 26 / 46
Scale Space
Color Images
t = 0
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t = 100
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t = 1000
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B. Fasy (Duke CS) Prelim 26 July 2010 26 / 46
Scale Space
Proportion Set
A proportion set for the RGB image is the set of pixels that have the sameratios of colors. For example, the boundaries in the following imagesdepict where 4*blue = 3*green:
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t = 0
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B. Fasy (Duke CS) Prelim 26 July 2010 27 / 46
Scale Space
Questions
1 Tight Bound: Is the inequality
|ℓ1 − ℓ2| ≤4
π· (κ1 + κ2) · F(γ1, γ2)
a tight bound for curves in Rn for n > 3?
2 Simultaneous Scale Space: What happens if multipleagents are diffusing at the same time?
3 Understanding Vq: Does there exist a stability resultfor Vq?
B. Fasy (Duke CS) Prelim 26 July 2010 28 / 46
Vineyard Distance
Part 3
B. Fasy (Duke CS) Prelim 26 July 2010 29 / 46
Vineyard Distance
Persistence Diagrams
A set of points in R2 that
describe the changing homologyof the sublevel sets of a function
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B. Fasy (Duke CS) Prelim 26 July 2010 30 / 46
Vineyard Distance
Stacking the Persistence Diagrams
We stack the diagrams so that Dgmp(ht) is drawn at height z = t.
B. Fasy (Duke CS) Prelim 26 July 2010 31 / 46
Vineyard Distance
Stacking the Persistence Diagrams
Then, we match the diagrams using a linear time algorithm [CSEM].
B. Fasy (Duke CS) Prelim 26 July 2010 32 / 46
Vineyard Distance
Vineyards
The path of an off-diagonal point is called a vine. A vine isrepresented by a function s : Iτ → R
3.
The collection of vines is called a vineyard.
Matching of Dgmp(f ) and Dgmp(g) is obtained by looking at theendpoints of the vines.
B. Fasy (Duke CS) Prelim 26 July 2010 33 / 46
Vineyard Distance
Another Representation of a Vineyard
[Movie]
B. Fasy (Duke CS) Prelim 26 July 2010 34 / 46
Vineyard Distance
Total Movement in a Vineyard
For a vine s, we can compute the weighted distance traveled by a point inthe persistence diagrams. Then, we sum this distance over all vines in thevineyard V .
Ds =
∫ 1
0ωs(t) ·
∂s(t)
∂tdt
Vq(H) =
(
∑
s∈V
Dqs
)1/q
B. Fasy (Duke CS) Prelim 26 July 2010 35 / 46
Vineyard Distance
Total Movement in a Vineyard
For a vine s, we can compute the weighted distance traveled by a point inthe persistence diagrams. Then, we sum this distance over all vines in thevineyard V .
Ds =∑
i∈{1,2,...,τ}
ωs(i) · ||s(ti ) − s(ti−1)||∞
Vq(H) =
(
∑
s∈V
Dqs
)1/q
B. Fasy (Duke CS) Prelim 26 July 2010 35 / 46
Vineyard Distance
Questions
1 Tight Bound: Is the inequality
|ℓ1 − ℓ2| ≤4
π· (κ1 + κ2) · F(γ1, γ2)
a tight bound for curves in Rn for n > 3?
2 Simultaneous Scale Space: What happens if multipleagents are diffusing at the same time?
3 Understanding Vq: Does there exist a stability resultfor Vq?
B. Fasy (Duke CS) Prelim 26 July 2010 36 / 46
Vineyard Distance
Related Metrics
Let A = Dgm(f ) and B = Dgm(g).We find a bijection between A and B by minimizing some quantity, suchas:
Bottleneck Distance
Wasserstein Distance
B. Fasy (Duke CS) Prelim 26 July 2010 37 / 46
Vineyard Distance
Bottleneck Matching
The bottleneck cost of a matching is the maximum L∞ distance betweenmatched points:
W∞(P) = max(a,b)∈P
||a − b||∞.
We seek to minimize the bottleneck distance over all perfect matchings:
W∞(A, B) = minP
{W∞(P)}.
B. Fasy (Duke CS) Prelim 26 July 2010 38 / 46
Vineyard Distance
Wasserstein Matching
The Wasserstein cost is measures the cumulative distance as follows:
Wq(P) =
∑
(a,b)∈P
||a − b||q∞
1/q
.
We seek to minimize the Wasserstein distance over all perfect matchings:
Wq(A, B) = minP
{Wq(P)}.
B. Fasy (Duke CS) Prelim 26 July 2010 39 / 46
Vineyard Distance
Related Stability Results
We say that the matching of persistence diagrams is stable if the cost ofthe matching is bounded by some reasonable function of ||f − g ||∞.
[CSEH] The Bottleneck Distance is stable for monotone Functionsf , g : M → R.
W∞(A, B) ≤ ||f − g ||∞[CSEHM10] The Wasserstein Distance is stable for tame LipschitzFunctions with bounded degree k total persistence.
Wq(A, B) ≤ C 1/q||f − g ||1−k/q∞
B. Fasy (Duke CS) Prelim 26 July 2010 40 / 46
Vineyard Distance
Related Stability Results
We say that the matching of persistence diagrams is stable if the cost ofthe matching is bounded by some reasonable function of ||f − g ||∞.
[CSEH] The Bottleneck Distance is stable for monotone Functionsf , g : M → R.
W∞(A, B) ≤ ||f − g ||∞[CSEHM10] The Wasserstein Distance is stable for tame LipschitzFunctions with bounded degree k total persistence.
Wq(A, B) ≤ C 1/q||f − g ||1−k/q∞
Is the Vineyard Metric stable too?
Vq(f , g) ≤ ???
B. Fasy (Duke CS) Prelim 26 July 2010 40 / 46
Vineyard Distance
Preliminary Findings
Let A = Dgm(f ) and B = Dgm(g).
W1(A, B) ≤ V1(f , g)
W∞(A, B) ≤ V∞(f , g)
B. Fasy (Duke CS) Prelim 26 July 2010 41 / 46
Vineyard Distance
Questions
1 Tight Bound: Is the inequality
|ℓ1 − ℓ2| ≤4
π· (κ1 + κ2) · F(γ1, γ2)
a tight bound for curves in Rn for n > 3?
2 Simultaneous Scale Space: What happens if multipleagents are diffusing at the same time?
3 Understanding Vq: Does there exist a stability resultfor Vq?
B. Fasy (Duke CS) Prelim 26 July 2010 42 / 46
Questions
1 Tight Bound: Is the inequality
|ℓ1 − ℓ2| ≤4
π· (κ1 + κ2) · F(γ1, γ2)
a tight bound for curves in Rn for n > 3?
2 Simultaneous Scale Space: What happens if multipleagents are diffusing at the same time?
3 Understanding Vq: Does there exist a stability resultfor Vq?
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B. Fasy (Duke CS) Prelim 26 July 2010 43 / 46
Thank You
My adviser, Herbert Edelsbrunner
My committee: Hubert Brey, John Harer, and Carlo Tomasi
Those who read my prelim document and provided comments,including Michael Kerber and Amit Patel.
Michelle Phillips (for making the graphics of dog and person)
Everyone here!
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Questions?
B. Fasy (Duke CS) Prelim 26 July 2010 45 / 46
References
G. Donald Chakerian, An Inequality for Closed Space Curves, Pacific J. Math. 12 (1962),no. 1, 53–57.
David Cohen-Steiner and Herbert Edelsbrunner, Inequalities for the Curvature of Curves
and Surfaces, Found. Comput. Math. 7 (2007), no. 4, 391–404.
David Cohen-Steiner, Herbert Edelsbrunner, and John Harer, Stability of persistence
diagrams, 2005, pp. 263–271.
David Cohen-Steiner, Herbert Edelsbrunner, John Harer, and Yuriy Mileyko, Lipschitz
Functions have Lp-Stable Persistence, Found. Comput. Math. 10 (2010), no. 2, 127–139.
David Cohen-Steiner, Herbert Edelsbrunner, and Dmitriy Morozov, Vines and Vineyards by
Updating Persistence in Linear Time, 2006, pp. 119–126.
Istvan Fary, Sur Certaines Inegalites Geometriques, Acta Sci. Math. (Szeged) 12 (1950),no. aa, 117–124.
B. Fasy (Duke CS) Prelim 26 July 2010 46 / 46