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?

    50 100 150 200 250 300 350 400 450 500 550

    50

    100

    150

    200

    250

    300

    350

    400

    450

    500

    550

    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.

    v9

    v10 v5 v4

    v1v6

    v7

    v8

    v2

    v3p1

    p2

    p8

    p5

    p4

    p9

    p3

    p6

    p10p7

    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

    50 100 150 200 250 300 350 400 450 500 550

    50

    100

    150

    200

    250

    300

    350

    400

    450

    500

    550

    B. Fasy (Duke CS) Prelim 26 July 2010 19 / 46

  • Scale Space

    Scale Space

    50 100 150 200 250 300 350 400 450 500 550

    50

    100

    150

    200

    250

    300

    350

    400

    450

    500

    550

    H(x , 0)

    50 100 150 200 250 300 350 400 450 500 550

    50

    100

    150

    200

    250

    300

    350

    400

    450

    500

    550

    H(x , 100)

    50 100 150 200 250 300 350 400 450 500 550

    50

    100

    150

    200

    250

    300

    350

    400

    450

    500

    550

    H(x , 1000)

    B. Fasy (Duke CS) Prelim 26 July 2010 20 / 46

  • Scale Space

    The Gaussian

    −4 −2 0 2 4 0

    0.05

    0.1

    0.15

    0.2

    0.25

    0.3

    0.35

    0.4

    0.45

    0.5

    G1(x , t) = 1

    ( √

    2πt) e − x

    2

    2t2

    0

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    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)

    15

    76

    9

    410

    12

    212

    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.

    8

    4

    12

    212

    13

    76

    10

    H(·, 0)

    H(·, 1)

    H(·, 2)

    H(x, 0.5) = 9

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

  • Sca