Computing the cut distance of two curves Maike Buchin, Ruhr - - PowerPoint PPT Presentation

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Computing the cut distance of two curves Maike Buchin, Ruhr - - PowerPoint PPT Presentation

Nov 05, 2022 352 likes •697 views

Computing the cut distance of two curves Maike Buchin, Ruhr University Bochum Leonie Ryvkin , Ruhr University Bochum J er ome Urhausen, Utrecht University Leonie Ryvkin 16.03.2020 Introduction Visiting W urzburg Schloss Veitsh

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Leonie Ryvkin 16.03.2020

Computing the cut distance

  • f two curves

Maike Buchin, Ruhr University Bochum Leonie Ryvkin, Ruhr University Bochum J´ erˆ

  • me Urhausen, Utrecht University
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Leonie Ryvkin 16.03.2020

Introduction

Visiting W¨ urzburg

Schloss Veitsh¨

  • chheim

Mainbr¨ ucke, Marienburg, Residenz, Dom...

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Leonie Ryvkin 16.03.2020

Introduction

Visiting W¨ urzburg

Schloss Veitsh¨

  • chheim

Mainbr¨ ucke, Marienburg, Residenz, Dom...

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SLIDE 4

Leonie Ryvkin 16.03.2020

Introduction

Visiting W¨ urzburg

Schloss Veitsh¨

  • chheim

Mainbr¨ ucke, Marienburg, Residenz, Dom...

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Leonie Ryvkin 16.03.2020

Introduction

δF(P, Q) = infτ,σ maxt∈[0,1] P(τ(t)) − Q(σ(t))

τ, σ range over all orientation-preserving homeomorphisms

The Fr´ echet distance

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Leonie Ryvkin 16.03.2020

The cut distance

δcut(k, P, Q) = infτ,σ maxt∈[0,1] P(σ(t)) − Q(τ(t))

where τ, σ range over all piecewise continuous, surjective functions with k′ < k jump discontinuities, each.

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Leonie Ryvkin 16.03.2020

The cut distance

δcut(k, P, Q) = infτ,σ maxt∈[0,1] P(σ(t)) − Q(τ(t))

where τ, σ range over all piecewise continuous, surjective functions with k′ < k jump discontinuities, each.

P Q

ε1 ≥ δH(P, Q) ε2 < δcut(2, P, Q) ε3 ≥ δcut(2, P, Q) ε4 ≥ δwF(P, Q)

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Leonie Ryvkin 16.03.2020

The cut distance

δcut(k, P, Q) = infτ,σ maxt∈[0,1] P(σ(t)) − Q(τ(t))

where τ, σ range over all piecewise continuous, surjective functions with k′ < k jump discontinuities, each. Fε(P, Q) = {(t1, t2): P(t1) − Q(t2) ≤ ε}

P Q

ε1 ≥ δH(P, Q) ε2 < δcut(2, P, Q) ε3 ≥ δcut(2, P, Q) ε4 ≥ δwF(P, Q)

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Leonie Ryvkin 16.03.2020

The cut distance

δcut(k, P, Q) = infτ,σ maxt∈[0,1] P(σ(t)) − Q(τ(t)) Observe that δH ≤ δcut ≤ δwF ≤ δF.

where τ, σ range over all piecewise continuous, surjective functions with k′ < k jump discontinuities, each.

P Q

ε1 ≥ δH(P, Q) ε2 < δcut(2, P, Q) ε3 ≥ δcut(2, P, Q) ε4 ≥ δwF(P, Q)

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Leonie Ryvkin 16.03.2020

Placing cut lines

ε P Q

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Leonie Ryvkin 16.03.2020

Placing cut lines

ε P Q

ℓ

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Leonie Ryvkin 16.03.2020

Placing cut lines

ε P Q

ℓ

h1

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Leonie Ryvkin 16.03.2020

Placing cut lines

ε P Q h2

ℓ

h1

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Leonie Ryvkin 16.03.2020

Placing cut lines

ε P Q h2

Problem: In general, there are infinitely many possible placements for cut lines. ℓ

h1

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Leonie Ryvkin 16.03.2020

Placing cut lines

ε P Q

For k = 2, it is possible to move valid cut lines such that one of them coincides with an extremal point of a component’s boundary. ℓ

h1

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Leonie Ryvkin 16.03.2020

Algorithm for k = 2

Input: polygonal curves P, Q and ε > 0

P Q ε

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Leonie Ryvkin 16.03.2020

Algorithm for k = 2

Compute FSD 1. P Q ε

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Leonie Ryvkin 16.03.2020

Algorithm for k = 2

Compute FSD 1. P Q ε

G

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Leonie Ryvkin 16.03.2020

Algorithm for k = 2

Compute FSD Identify candidates ai, bj 1. 2. P Q ε

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Leonie Ryvkin 16.03.2020

Algorithm for k = 2

Compute FSD Identify candidates Compute cut lines ℓ at extremal points ai, bj 1. 2. 3. P Q ε

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Leonie Ryvkin 16.03.2020

Algorithm for k = 2

Compute FSD Identify candidates Compute cut lines ℓ at extremal points Identify aℓ

i , bℓ j

ai, bj ℓ 1. 2. 3. 4. P Q ε

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Leonie Ryvkin 16.03.2020

Algorithm for k = 2

Compute FSD Identify candidates Compute cut lines ℓ at extremal points Identify aℓ

i , bℓ j

ai, bj ℓ 1. 2. 3. 4. P Q ε

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Leonie Ryvkin 16.03.2020

Algorithm for k = 2

Compute FSD Identify candidates Compute cut lines ℓ at extremal points Identify aℓ

i , bℓ j

ai, bj ℓ 1. 2. 3. 4. P Q ε

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Leonie Ryvkin 16.03.2020

Algorithm for k = 2

Compute FSD Identify candidates Compute cut lines ℓ at extremal points Identify aℓ

i , bℓ j

ai, bj Determine horizontal cut h h ℓ 1. 2. 3. 4. 5.

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Leonie Ryvkin 16.03.2020

Algorithm for k = 2

Compute FSD Identify candidates Compute cut lines ℓ at extremal points Identify aℓ

i , bℓ j

ai, bj Determine horizontal cut h h ℓ 1. 2. 3. 4. 5.

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Leonie Ryvkin 16.03.2020

Algorithm for k = 2

Compute FSD Identify candidates Compute cut lines ℓ at extremal points Identify aℓ

i , bℓ j

ai, bj Determine horizontal cut h h ℓ 1. 2. 3. 4. 5. 6. ( Repeat for h at extremal points)

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Leonie Ryvkin 16.03.2020

Algorithm for k = 2

Compute FSD Identify candidates Compute cut lines ℓ at extremal points Identify aℓ

i , bℓ j

ai, bj Determine horizontal cut h h ℓ Repeat for opposite quadrants if necessary) 7. 1. 2. 3. 4. 5. 6. ( ( Repeat for h at extremal points)

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Leonie Ryvkin 16.03.2020

Algorithm for k = 2

Compute FSD Identify candidates Compute cut lines ℓ at extremal points Identify aℓ

i , bℓ j

ai, bj Determine horizontal cut h h ℓ Repeat for opposite quadrants if necessary) 7. Return cut positions 1. 2. 3. 4. 5. 6. ( ( Repeat for h at extremal points) 8.

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Leonie Ryvkin 16.03.2020

Conclusion

Overview on the cut distance

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Leonie Ryvkin 16.03.2020

Conclusion

Overview on the cut distance

Deciding the cut distance is NP-hard in general

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Leonie Ryvkin 16.03.2020

Conclusion

Overview on the cut distance

Deciding the cut distance is NP-hard in general Optimizing k is APX-hard

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Leonie Ryvkin 16.03.2020

Conclusion

Overview on the cut distance

Deciding the cut distance is NP-hard in general Optimizing k is APX-hard O(n2 log n) time algorithm for k = 2

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Leonie Ryvkin 16.03.2020

Conclusion

Overview on the cut distance

Deciding the cut distance is NP-hard in general Optimizing k is APX-hard O(n2 log n) time algorithm for k = 2 Conjecture: already NP-hard for k = 3

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Leonie Ryvkin 16.03.2020

Conclusion

Overview on the cut distance

Deciding the cut distance is NP-hard in general Optimizing k is APX-hard O(n2 log n) time algorithm for k = 2 Conjecture: already NP-hard for k = 3 Thank you!