Post on 25-Sep-2020
GEOMETRY
P Bhowmick Computational and Digital GeometryConvex Hulls and Ortho-convex Hulls
Partha BhowmickAssociate Professor
Department of Computer Science and EngineeringIndian Institute of Technology, Kharagpur
India
NIT Durgapur 23 Jan 2014
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Convex hull
Input: Point set P on xy-plane.
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Convex hull
convex hullOutput: Convex hull, CP = a sequence of vertices/edges.
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Convex hull
Edge of CP ?
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Convex hull
Left 6= empty
Right 6= empty
No, it’s not an edge of CP .
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Convex hull
Yes, it’s an edge of CP .
Right = empty
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Convex hull
|P | = n⇒ O(n2) pairs ⇒ O(n3) time
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Convex hull
|CP | = O(n): O(n3) time is quite high!
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Better observations
pL pR
Obs 1The leftmost point pL and the rightmost point pR of P formthe leftmost and the rightmost vertices of CP .
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Better observations
pL pR
Obs 2Clockwise traversal along the boundary of CP always yields aright turn at each vertex of CP .
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Better observations
The clueUse turn type to decide whether a triplet of points forms apair of consecutive edges of CP .
But how?We have O(n3) triplets of points!We can avoid checking so many triplets if we use incrementalapproach.
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Better observations
A questionLet CP,i = vertices of upper hull up to pi.Then what’s the relation between CP,i+1 and CP,i?
pRpi
pi+1
CP,i
pL
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Better observations
The answerCP,i+1 ⊆ CP,i ∪ {pi+1}.It’s a strong observation ⇒ Incremental algorithm!
pRpL pi
CP,i+1
pi+1
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
pL pR
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
After lexicographic sorting(x = primary key, y = secondary key)
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pRpL pR
1
2
4
5
6
7
8
9
10
11
12
13
14
15
3
R
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
pL pR
1
2
5
6
7
8
9
10
11
12
13
14
15
3
4
L
not in CP
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
pL pR
1
2
5
6
7
8
9
10
11
12
13
14
15
3
4
R
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
4
6
7
8
9
10
11
12
13
14
15
pL pR
1
2
6
7
8
9
10
11
12
13
14
15
3
4
55R
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
4
7
8
9
10
11
12
13
14
15
pL pR
1
2
7
8
9
10
11
12
13
14
15
3
4
66
55L
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
7
8
9
10
11
12
13
14
15
pL pR
1
2
7
8
9
10
11
12
13
14
15
3
4
66
55
L
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
7
8
9
10
11
12
13
14
15
pL pR
1
2
7
8
9
10
11
12
13
14
15
3
4
66
55
R
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
7
8
9
10
11
12
13
14
15
pL pR
1
2
7
8
9
10
11
12
13
14
15
3
4
66
55
R
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
7
8
9
10
11
12
13
14
15
pL pR
1
2
7
8
9
10
11
12
13
14
15
3
4
66
55
L
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
7
8
9
10
11
12
13
14
15
pL pR
1
2
7
8
9
10
11
12
13
14
15
3
4
66
55
R
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
7
8
9
10
11
12
13
14
15
pL pR
1
2
7
8
9
10
11
12
13
14
15
3
4
66
55
R
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
7
8
9
10
11
12
13
14
15
pL pR
1
2
7
8
9
10
11
12
13
14
15
3
4
66
55
R
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
7
8
9
11
12
13
14
15
pL pR
1
2
7
8
9
10
11
12
13
14
15
3
4
66
55
L
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
7
8
9
11
12
13
14
15
pL pR
1
2
7
8
9
10
11
12
13
14
15
3
4
66
55
L
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
7
8
9
11
12
13
14
15
pL pR
1
2
7
8
9
10
11
12
13
14
15
3
4
66
55
R
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
7
8
9
12
13
14
15
pL pR
1
2
7
8
10
11
12
13
14
15
3
4
66
55L
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
7
8
9
12
13
14
15
pL pR
1
2
7
8
10
11
12
13
14
15
3
4
66
55
L
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
7
9
12
13
14
15
pL pR
1
2
7
8
10
11
12
13
14
15
3
4
66
55
R
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
7
9
12
13
14
15
pL pR
1
2
7
8
10
11
12
13
14
15
3
4
66
55
R
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
7
9
12
14
15
pL pR
1
2
7
8
10
11
12
13
14
15
3
4
66
55
L
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
7
9
14
15
pL pR
1
2
7
8
10
11
12
13
14
15
3
4
66
55
L
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
7
9
14
15
pL pR
1
2
7
8
10
11
12
13
14
15
3
4
66
55
R
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
7
9
14
15
pL pR
1
2
7
8
10
11
12
13
14
15
3
4
66
55
R
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
7
9
14
15
pL pR
1
2
7
8
10
11
12
13
14
15
3
4
66
55
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Incremental algorithm: Graham scan
pL pR
1
2
3
7
9
14
15
pL pR
1
2
7
8
10
11
12
13
14
15
3
4
66
55
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Time complexity
pi
CP,i+1
pi+1CP,i
Let pj ∈ CP,i.If pj 6∈ CP,i+1, then pj 6∈ CP,i+2, pj 6∈ CP,i+3, . . . , pj 6∈ CP,n,since CP,i+1 ⊆ CP,i ∪ {pi+1}.So, once pj is removed from the upper hull, it’s neverreconsidered.
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Time complexity
pi
CP,i+1
pi+1CP,i
Let pj ∈ CP,i.If pj 6∈ CP,i+1, then pj 6∈ CP,i+2, pj 6∈ CP,i+3, . . . , pj 6∈ CP,n,since CP,i+1 ⊆ CP,i ∪ {pi+1}.So, once pj is removed from the upper hull, it’s neverreconsidered.
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Time complexity
pi
CP,i+1
pi+1CP,i
Let pj ∈ CP,i.If pj 6∈ CP,i+1, then pj 6∈ CP,i+2, pj 6∈ CP,i+3, . . . , pj 6∈ CP,n,since CP,i+1 ⊆ CP,i ∪ {pi+1}.So, once pj is removed from the upper hull, it’s neverreconsidered.
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Time complexity
pi
CP,i+1
pi+1CP,i
Let pj ∈ CP,i.If pj 6∈ CP,i+1, then pj 6∈ CP,i+2, pj 6∈ CP,i+3, . . . , pj 6∈ CP,n,since CP,i+1 ⊆ CP,i ∪ {pi+1}.So, once pj is removed from the upper hull, it’s neverreconsidered.
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Time complexity
pi
CP,i+1
pi+1CP,i
Let pj ∈ CP,i.If pj 6∈ CP,i+1, then pj 6∈ CP,i+2, pj 6∈ CP,i+3, . . . , pj 6∈ CP,n,since CP,i+1 ⊆ CP,i ∪ {pi+1}.So, once pj is removed from the upper hull, it’s neverreconsidered.
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Time complexity
pi
CP,i+1
pi+1CP,i
Data structure: Stack, whose top = pi.If top two vertices in stack and pi+1 do not form a right turnat pi, then pi is popped out for ever!⇒ #pushes = n and #pops < n⇒ T (n) = O(n) ← no best, average, or worst case!For lexicographic sorting, it takes O(n log n) time.
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Time complexity
pi
CP,i+1
pi+1CP,i
Data structure: Stack, whose top = pi.If top two vertices in stack and pi+1 do not form a right turnat pi, then pi is popped out for ever!⇒ #pushes = n and #pops < n⇒ T (n) = O(n) ← no best, average, or worst case!For lexicographic sorting, it takes O(n log n) time.
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Time complexity
pi
CP,i+1
pi+1CP,i
Data structure: Stack, whose top = pi.If top two vertices in stack and pi+1 do not form a right turnat pi, then pi is popped out for ever!⇒ #pushes = n and #pops < n⇒ T (n) = O(n) ← no best, average, or worst case!For lexicographic sorting, it takes O(n log n) time.
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Time complexity
pi
CP,i+1
pi+1CP,i
Data structure: Stack, whose top = pi.If top two vertices in stack and pi+1 do not form a right turnat pi, then pi is popped out for ever!⇒ #pushes = n and #pops < n⇒ T (n) = O(n) ← no best, average, or worst case!For lexicographic sorting, it takes O(n log n) time.
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Time complexity
pi
CP,i+1
pi+1CP,i
Data structure: Stack, whose top = pi.If top two vertices in stack and pi+1 do not form a right turnat pi, then pi is popped out for ever!⇒ #pushes = n and #pops < n⇒ T (n) = O(n) ← no best, average, or worst case!For lexicographic sorting, it takes O(n log n) time.
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Reference of Algorithms
1 Incremental — O(n logn) . n = #pointsR. Graham, An Efficient Algorithm for Determining the ConvexHull of a Finite Point Set, Info. Proc. Letters, 1, pp. 132–133,1972.
2 Gift wrapping — O(nh) . h = #hull verticesR. A. Jarvis, On the Identification of the Convex Hull of a FiniteSet of Points in the Plane, Info. Proc. Letters, 2, pp. 18–21, 1973.
3 Divide and Conquer — O(n logn)F. P. Preparata and S. J. Hong, Convex Hulls of Finite Sets ofPoints in Two and Three Dimensions, Commun. ACM, 20,pp. 87–93, 1977.
4 Marriage before Conquest — O(n log h)D. G. Kirkpatrick and R. Seidel, The Ultimate Planar ConvexHull Algorithm?, SIAM J. Comput., 15, pp. 287–299, 1986.
5 Simpler optimal output-sensitive — O(n log h)T. M. Chan, Optimal Output-Sensitive Convex Hull Algorithmsin Two and Three Dimensions, Discrete & ComputationalGeometry, 16, pp. 361–368, 1996.
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Convex hull of a polygon
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Linear-time algorithms
1 1979 McCallum-Avis, IPL
2 1983 Lee, Intl. J. Computers & Info. Sc.
3 1983 Graham-Yao, J. Algorithms
4 1983 ElGindy-Avis-Toussaint, Computing
5 1984 Bhattacharya-ElGindy, IEEE Trans. Info. Thy.
6 1985 Preparata-Shamos, Computational Geometry, Ch. 4
7 1985 Orlowski, Pattern Rec.
8 1986 Shin-Woo, Pattern Rec.
9 1987 Melkman, IPL
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Convex Hull versus Orthogonal Hull
Digital object(A = set/connected component of integer points)
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Convex Hull versus Orthogonal Hull
Convex hull CA
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Convex Hull versus Orthogonal Hull
Any straight line has at most one segment of intersection(a necessary property)
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Convex Hull versus Orthogonal Hull
Object A imposed on a grid G of size g = 4
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Convex Hull versus Orthogonal Hull
Orthogonal hull CA
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Convex Hull versus Orthogonal Hull
Any horizontal or vertical line has at most one segment ofintersection (a necessary property)
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Observations
R L
R R
R
R
R
RR
R
R
L
L
L
L
L
There are both left and right turns!(clockwise)
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Observations
L R
L L
L
L
L
LL
L
L
R
R
R
R
R
or, both right and left turns(anticlockwise)
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Observations
or, 900 (Type 1) and 2700 (Type 3) vertices
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Observations
But no two consecutive Type 3 vertices
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Observations
Two consecutive Type 3 vertices defy the necessary propertyof line intersection
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Algorithm
Step 1: Traverse the border of isothetic cover of A
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Algorithm
Step 2: If 33, then process to remove the concavity.
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Algorithm
Step 2: If 33, then process to remove the concavity.
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Algorithm
Step 2: If 33, then process to remove the concavity.
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Algorithm
Step 2: If 33, then process to remove the concavity.
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Combinatorial cases
13331331
Just 1331 and 1333 (= 1333+)
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Pattern 1331 (1)
0
v0v
4l
0
l
0
2
4
4v l + l
+ l
3v2
1
v2
ll
l
1v
3
Rule R11 (l1 = l3):〈v0(t0, l0), v1(1, l1), v2(3, l2), v3(3, l3), v4(1, l4)〉 →〈v0(t0, l0 + l2 + l4)〉
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Pattern 1331 (2)
v3
v2v
0
v
2
4l 2
l 4
l 0
1
v4
l 3
l − l
l +
l
1 3
l 0
v
v2
1 l 1
v0
Rule R12 (l1 > l3):〈v0(t0, l0), v1(1, l1), v2(3, l2), v3(3, l3), v4(1, l4)〉 →〈v0(t0, l0), v1(1, l1 − l3), v2(3, l2 + l4)〉
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Pattern 1331 (3)
4l
4
2
3v
1l1
v
v
v
l
2v
0
2
l + l
3l3v
4l
3 1
0
l − l4
l
0
v
0v
Rule R13 (l1 < l3):〈v0(t0, l0), v1(1, l1), v2(3, l2), v3(3, l3), v4(1, l4)〉 →〈v0(t0, l0 + l2), v3(3, l3 − l1), v4(1, l4)〉
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Pattern 1333 (1)
0
2
4
3 1 l − l
2l
2v
3l
3v
1l
0v
4l
3v4v4l
1v
0l0v
l + l
v
Rule R21 (l1 < l3):〈v0(t0, l0), v1(1, l1), v2(3, l2), v3(3, l3), v4(3, l4)〉 →〈v0(t0, l0 + l2), v3(3, l3 − l1), v4(3, l4)〉
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Pattern 1333 (2)
(ini
t)(init)
l
Vl
1l
0
3v4
v1,2
v1,1
v1,4
= l
4l
//
v2
l 2 l − 2
// l
l 3
v
v
v0
l
/l1v
4,2v
4,1v
1,3v
2v
H 1v /l
0l 0
Let v = current vertex (under traversal).lH = horizontal line thru’ v2, lV = vertical line thru’ v4.l−H ∩ l−V = region lying below lH and left of lV .if v ∈ l−H ∩ l−V , then apply R22; else traverse ahead to get v.
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Pattern 1333 (3)
(ini
t)(init)
l
Vl
1l
0
3v4
v1,2
v1,1
v1,4
= l
4l
//
v2
l 2 l − 2
// l
l 3
v
v
v0
l
/l1v
4,2v
4,1v
1,3v
2v
H 1v /l
0l 0
Rule R22 (l1 > l3 and d = d2):〈v0(t0, l0), v1(1, l1), v2(3, l2), v3(3, l3), v4(3, l4)〉 →〈v0(t0, l0), v1(1, l′), v2(3, l2 − l′′)〉d = direction from v, d2 = direction from v2.
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Pattern 1333 (4)
(ini
t)
(init)
l3v
l
= l
3
l
v
2l
Vl
4v
H
//
l
0l
2
2v1 1lv
/
0
//l
v
l
1v0l
4
v
0v
3
/
if v ∈ l−H ∩ l−V , then apply R23; else traverse ahead to get v.
Rule R23 (l1 > l3 and d = d3):〈v0(t0, l0), v1(1, l1), v2(3, l2), v3(3, l3), v4(3, l4)〉 →〈v0(t0, l0), v1(1, l1 − l3), v2(3, (l2 − l′′), v3(3, (l1 − l3 − l′)〉
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Demo
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Demo
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Demo
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Demo
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Demo
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Time Complexity
Let n = #points on object border, g = grid size.
1 Checking object containment in a cell: O(g) time.
2 #grid points visited: O(n/g)⇒ Visiting all vertices: O(n/g) ·O(g) = O(n) time.
3 Removal of a concavity (applying Rule): O(1) time.
4 Maximum #reductions: O(n/g)− 4.⇒ Total #operations: (O(n/g)− 4) ·O(1) = O(n/g).
5 Total time complexity: O(n) +O(n/g) = O(n).
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Time Complexity
Let n = #points on object border, g = grid size.
1 Checking object containment in a cell: O(g) time.
2 #grid points visited: O(n/g)⇒ Visiting all vertices: O(n/g) ·O(g) = O(n) time.
3 Removal of a concavity (applying Rule): O(1) time.
4 Maximum #reductions: O(n/g)− 4.⇒ Total #operations: (O(n/g)− 4) ·O(1) = O(n/g).
5 Total time complexity: O(n) +O(n/g) = O(n).
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Time Complexity
Let n = #points on object border, g = grid size.
1 Checking object containment in a cell: O(g) time.
2 #grid points visited: O(n/g)⇒ Visiting all vertices: O(n/g) ·O(g) = O(n) time.
3 Removal of a concavity (applying Rule): O(1) time.
4 Maximum #reductions: O(n/g)− 4.⇒ Total #operations: (O(n/g)− 4) ·O(1) = O(n/g).
5 Total time complexity: O(n) +O(n/g) = O(n).
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Time Complexity
Let n = #points on object border, g = grid size.
1 Checking object containment in a cell: O(g) time.
2 #grid points visited: O(n/g)⇒ Visiting all vertices: O(n/g) ·O(g) = O(n) time.
3 Removal of a concavity (applying Rule): O(1) time.
4 Maximum #reductions: O(n/g)− 4.⇒ Total #operations: (O(n/g)− 4) ·O(1) = O(n/g).
5 Total time complexity: O(n) +O(n/g) = O(n).
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Time Complexity
Let n = #points on object border, g = grid size.
1 Checking object containment in a cell: O(g) time.
2 #grid points visited: O(n/g)⇒ Visiting all vertices: O(n/g) ·O(g) = O(n) time.
3 Removal of a concavity (applying Rule): O(1) time.
4 Maximum #reductions: O(n/g)− 4.⇒ Total #operations: (O(n/g)− 4) ·O(1) = O(n/g).
5 Total time complexity: O(n) +O(n/g) = O(n).
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Time Complexity
Let n = #points on object border, g = grid size.
1 Checking object containment in a cell: O(g) time.
2 #grid points visited: O(n/g)⇒ Visiting all vertices: O(n/g) ·O(g) = O(n) time.
3 Removal of a concavity (applying Rule): O(1) time.
4 Maximum #reductions: O(n/g)− 4.⇒ Total #operations: (O(n/g)− 4) ·O(1) = O(n/g).
5 Total time complexity: O(n) +O(n/g) = O(n).
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Time Complexity
Let n = #points on object border, g = grid size.
1 Checking object containment in a cell: O(g) time.
2 #grid points visited: O(n/g)⇒ Visiting all vertices: O(n/g) ·O(g) = O(n) time.
3 Removal of a concavity (applying Rule): O(1) time.
4 Maximum #reductions: O(n/g)− 4.⇒ Total #operations: (O(n/g)− 4) ·O(1) = O(n/g).
5 Total time complexity: O(n) +O(n/g) = O(n).
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Result
digital object = 10541 points
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Result
Isothetic cover
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Result
Orthogonal hull
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Result
g = 4, 8, 14
#vertices = 18, 16, 16
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Result
g = 4, 8, 14
#vertices = 120, 60, 32
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
Result
g = 4, 8, 14
#vertices = 88, 44, 32
Feature analysisConcavity strength and concavity relation
Narrow mouthed concavity
Concavity complexity
GEOMETRY
P Bhowmick
Convex hull
Algorithm
Hull ofPolygon
Orthogonalhull
Observations
Algorithm
Result
References
A. Biswas, P. Bhowmick, M. Sarkar, B. B. Bhattacharya, Alinear-time combinatorial algorithm to find the orthogonal hull ofan object on the digital plane, Information Sciences, 216,pp. 176–195, 2012.
A. Biswas, P. Bhowmick, B. B. Bhattacharya, Construction ofisothetic covers of a digital object: A combinatorial approach,Journal of Visual Communication and Image Representation, 21,pp. 295–310, 2010.
Thank you