# Discovering Metrics and Scale Space - Preliminary brittany/research/docs/fasy_prelim_ ¢  Discovering

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• 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 multiple agents are diffusing at the same time?

3 Understanding Vq: Does there exist a stability result for 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 : S 1 → Rn.

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p2

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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).

10.2 0.4 0.6 0.80

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).

10.2 0.4 0.6 0.80

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).

10.6 0.850.250

0.05 0.4 0.7 0.9

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

• 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 such that mesh(Pk) goes to zero, then

ℓ(γ) = lim k→∞

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

0 1/rx dx .

B. Fasy (Duke CS) Prelim 26 July 2010 9 / 46

• Curves and Metrics

Turning Angle

v9

v10 v5 v4

v1v6

v7

v8

v2

v3

α9

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

0 1/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

0 1/rx dx .

κ(P) = ∑

i

αi .

Lemma

If Pk is a sequence of polygons inscribed in a smooth closed curve γ such that mesh(Pk) goes to zero, then

κ(γ) = lim k→∞

κ(Pk).

B. Fasy (Duke CS) Prelim 26 July 2010 11 / 46

• Curves and Metrics

The Fréchet Distance

F(γ1, γ2) = inf α : S1→S1

max t∈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 Fréchet Distance

F(γ1, γ2) = inf α : S1→S1

max t∈S1

(γ(t) − γ(α(t)))

B. Fasy (Duke CS) Prelim 26 July 2010 14 / 46

• Curves and Metrics

The Fréchet Distance

F(γ1, γ2) = inf α : S1→S1

max t∈S1

(γ(t) − γ(α(t)))

Lemma

If Pk and Qk are sequences of polygons inscribed in smooth closed curves γ1 and γ2 such that mesh(P

k) and mesh(Qk) go to zero, then

F(γ1, γ2) = lim k→∞

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, F5́0] 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 multiple agents are diffusing at the same time?

3 Understanding Vq: Does there exist a stability result for 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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B. Fasy (Duke CS) Prelim 26 July 2010 19 / 46

• 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

−4 −2 0 2 4 0

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0.1

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0.5

G1(x , t) = 1

( √

2πt) e − x

2

2t2

0

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Gn(x , t) = 1

( √

2πt)n e −

|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 linear interpolation.

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H(·, 0)

H(·, 1)

H(·, 2)

H(x, 0.5) = 9

B. Fasy (Duke CS) Prelim 26 July 2010 23 / 46

• Sca

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