Progressive Simplification of Polygonal Curves Kevin Buchin - - PowerPoint PPT Presentation

Kevin Buchin Maximilian Konzack

Progressive Simplification

• f Polygonal Curves

Wim Reddingius

Curve Simplification

Curve Simplification

min-# Simplification problem:

• Given a polygonal curve C and an

ε > 0 as an error threshold

• Objective: minimize the number
• f vertices in a simplification S

Curve Simplification

min-# Simplification problem:

• Given a polygonal curve C and an

ε > 0 as an error threshold

• Objective: minimize the number
• f vertices in a simplification S

Curve Simplification

ε min-# Simplification problem:

• Given a polygonal curve C and an

ε > 0 as an error threshold

Curve Simplification

Upper bound [Chan and Chin, 1996] Higher dimensions [Barequet et al., 2002] A min-# simplification can be computed in O(n2) time in R2 For the L1 or L∞ metric, a min-# simplification can be computed in O(n2) time

Progressive Simplification

S1

Progressive Simplification

Zoom out S1

Progressive Simplification

Zoom out S2 S1

Progressive Simplification

Zoom out S3 S2 S1

Progressive Simplification

Zoom out S4 S3 S2 S1

Progressive Simplification

Zoom out Zoom in S4 S3 S2 S1

Progressive Simplification

Zoom out Zoom in S4 S3 S2 S1 Impose Consistency Across Many Scales

• Zoom in and out without flickering
• Require monotonicity: Sm ⊑ Sm−1 ⊑ · · · ⊑ C
• Minimize m

k=1 |Sk| (optimality)

• A sequence of m scales: 0 < ε1 < · · · < εm

Results

• An O(n3m) time algorithm for the progressive

simplification problem

S2 ⊑ S1 ⊑ C

• works with various distance measures such as Hausdorff,

Fr´ echet and area-based distances

• enables simplification for continuous scaling in O(n5) time

Shortcut Graph

ε Shortcut:

• Given a polygonal curve C, a

shortcut (pi, pj) is an ordered pair

• f vertices
• Given C and an ε > 0, (pi, pj) is

valid if ε(pi, pj) ≤ ε Validity: p1 p4

Shortcut Graph

C

Shortcut Graph

• Given a curve C and an ε > 0, the shortcut graph G(C, ε)

captures all valid shortcuts C G(C, ε)

Shortcut Graph

• Given a curve C and an ε > 0, the shortcut graph G(C, ε)

captures all valid shortcuts C G(C, ε) S

• Minimum-link path in G(C, ε) is an optimal simplification S

Minimal Progressive Simplification

Dynamic Programming

• Assign a cost value ck

i,j ∈ N for each shortcut

(pi, pj) ∈ G(C, εk) at scale εk

• ck

i,j relates to the cost of including (pi, pj) in Sk

Minimal Progressive Simplification

Dynamic Programming

• Assign a cost value ck

i,j ∈ N for each shortcut

(pi, pj) ∈ G(C, εk) at scale εk

• ck

i,j relates to the cost of including (pi, pj) in Sk

1 1 1 1 1 1 1

G(C, ε1)

• Example: ε1 < ε2 < ε3

p1 pn

Minimal Progressive Simplification

Dynamic Programming

• Assign a cost value ck

i,j ∈ N for each shortcut

(pi, pj) ∈ G(C, εk) at scale εk

• ck

i,j relates to the cost of including (pi, pj) in Sk

1 1 1 1 1 1 1 1

G(C, ε1) S1

1 1

• Example: ε1 < ε2 < ε3

p1 pn

Minimal Progressive Simplification

Dynamic Programming

• Assign a cost value ck

i,j ∈ N for each shortcut

(pi, pj) ∈ G(C, εk) at scale εk

• ck

i,j relates to the cost of including (pi, pj) in Sk

1 1 3 4 1 1 1 1 1 1

G(C, ε1) G(C, ε2) S1

1 1

⊆

• Example: ε1 < ε2 < ε3

p1 pn

Minimal Progressive Simplification

Dynamic Programming

• Assign a cost value ck

i,j ∈ N for each shortcut

(pi, pj) ∈ G(C, εk) at scale εk

• ck

i,j relates to the cost of including (pi, pj) in Sk

1 2 1 3 2 4 1 2 2 2 2 2 1 1 1 1 1

G(C, ε1) G(C, ε2) S1

1 1

⊆

• Example: ε1 < ε2 < ε3

p1 pn

Minimal Progressive Simplification

Dynamic Programming

• Assign a cost value ck

i,j ∈ N for each shortcut

(pi, pj) ∈ G(C, εk) at scale εk

• ck

i,j relates to the cost of including (pi, pj) in Sk

1 2 1 3 2 4 1 2 3 2 2 2 2 1 1 1 1 1

G(C, ε1) G(C, ε2) S2 S1

1 1 2

⊆ ⊒

• Example: ε1 < ε2 < ε3

p1 pn

Minimal Progressive Simplification

Dynamic Programming

• Assign a cost value ck

i,j ∈ N for each shortcut

(pi, pj) ∈ G(C, εk) at scale εk

• ck

i,j relates to the cost of including (pi, pj) in Sk

1 2 1 3 4 3 6 2 4 5 1 2 3 3 3 2 2 2 2 3 3 3 3 1 1 1 1 1

G(C, ε1) G(C, ε2) G(C, ε3) S2 S1

1 1 2

⊆ ⊆ ⊒

• Example: ε1 < ε2 < ε3

p1 pn

Minimal Progressive Simplification

Dynamic Programming

• Assign a cost value ck

i,j ∈ N for each shortcut

(pi, pj) ∈ G(C, εk) at scale εk

• ck

i,j relates to the cost of including (pi, pj) in Sk

1 2 1 3 4 3 6 2 4 5 1 2 3 3 3 2 2 2 2 3 3 3 3 1 1 1 1 1

G(C, ε1) G(C, ε2) G(C, ε3) S2 S1 S3

1 1 6 2

⊆ ⊆ ⊒ ⊒

• Example: ε1 < ε2 < ε3

p1 pn

Minimal Progressive Simplification

Dynamic Program ck

i,j =

     1 if k = 1 1 + min

π∈k−1

i,j

• (px,py)∈π

ck−1

x,y

if 1 < k ≤ m k

i,j denotes the set of all paths in G(C, εk) from pi to pj

Minimal Progressive Simplification

Dynamic Program ck

i,j =

     1 if k = 1 1 + min

π∈k−1

i,j

• (px,py)∈π

ck−1

x,y

if 1 < k ≤ m k

i,j denotes the set of all paths in G(C, εk) from pi to pj

Algorithm Construct simplifications from Sm down to S1

• 1. Compute costs ck

i,j at scale εk

• 2. Compute shortest path P from pi to

pj in G(C, εk) for all (pi, pj) ∈ Sk+1

• 3. Link P to obtain Sk

Minimal Progressive Simplification

Algorithm Construct simplifications from Sm down to S1

• 1. Compute costs ck

i,j at scale εk

• 2. Compute shortest path P from pi to

pj in G(C, εk) for all (pi, pj) ∈ Sk+1

• 3. Link P to obtain Sk

Running time m times

Minimal Progressive Simplification

Algorithm Construct simplifications from Sm down to S1

• 1. Compute costs ck

i,j at scale εk

• 2. Compute shortest path P from pi to

pj in G(C, εk) for all (pi, pj) ∈ Sk+1

• 3. Link P to obtain Sk

Running time m times O(n2)

• Employ the algorithm by [Chan and Chin, 1996]
• Runs in O(n2) time in the plane

Minimal Progressive Simplification

Algorithm Construct simplifications from Sm down to S1

• 1. Compute costs ck

i,j at scale εk

• 2. Compute shortest path P from pi to

pj in G(C, εk) for all (pi, pj) ∈ Sk+1

• 3. Link P to obtain Sk

Running time m times

• Run Dijkstra’s algorithm on O(n) nodes of G(C, εk)
• Dijkstra’s algorithm runs in O(n2) time on G with

integer weights O(n2) O(n3)

Minimal Progressive Simplification

Algorithm Construct simplifications from Sm down to S1

• 1. Compute costs ck

i,j at scale εk

• 2. Compute shortest path P from pi to

pj in G(C, εk) for all (pi, pj) ∈ Sk+1

• 3. Link P to obtain Sk

Running time m times O(n2) O(n3) O(n)

• Employ the algorithm by [Chan and Chin, 1996]

O(n3)

Minimal Progressive Simplification

Algorithm Construct simplifications from Sm down to S1

• 1. Compute costs ck

i,j at scale εk

• 2. Compute shortest path P from pi to

pj in G(C, εk) for all (pi, pj) ∈ Sk+1

• 3. Link P to obtain Sk

Running time m times O(n2) O(n3) O(n) O(n3) O(n) Total Running Time

• Optimal progressive simplification computable in O(n3m)

time

• Takes O(n5) time for continuous scaling

Conclusion

• An O(n3m) time algorithm for the progressive

simplification problem

S2 ⊑ S1 ⊑ C

• works with various distance measures such as Hausdorff,

Fr´ echet and area-based distances

• enables simplification for continuous scaling in O(n5) time

Conclusion

Thank you for your attention.

Further Results

• Technique to compute all shortcuts for a fixed ε in

O(n2 log n) time instead of O(n3) time

• Storage-efficient representation of the shortcut graph

allowing to find shortest paths in O(n log n) time

• Experimental evaluation on a trajectory of a migrating

vulture