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A Recursive Algorithm for Calculating the Relative Convex Hull

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A Recursive Algorithm for Calculating the Relative Convex Hull Gisela Klette AUT University Computing & Mathematical Sciences Auckland, New Zealand – PowerPoint PPT presentation

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Title: A Recursive Algorithm for Calculating the Relative Convex Hull


1
A Recursive Algorithm for Calculating the
Relative Convex Hull
Gisela Klette AUT University Computing
Mathematical Sciences Auckland, New Zealand
2
Motivation
  • The calculation of relative convex hulls of
    simple polygons is
  • a special subject in computational geometry
    (shortest paths),
  • in image analysis (MLP calculation of features),
  • in robotics (shortest path of a robot in a
    constrained environment), ...

3
Motivation Image Analysis
  • MLP is a digital length estimator of the
    circumference of a digital object that is
    multigrid convergent
  • MLP is characteristic for the digital convexity
    of the shape
  • MLP is a tangent estimator

4
Motivation Robotics Path Planning
B
Informally The relative convex hull of A
relative to B is the shortest path between A and
B.
A
5
Definition 1
A subset S in R2 of points is convex iff S is
equal to the intersection of all half planes
containing S. The convex hull CH(S) of a set of
points S is the smallest (by area) convex
polygon P that contains S.
CH(S)
S
6
Definition 2
A cavity of a polygon A is the topological
closure of any connected component of CH(A) \ A.
CAV1
CAV2
CAV4
CAV3
Polygon with 4 cavities
7
Definition 3
A cover is a straight line segment in the
frontier of CH(A) that is not part of the
frontier of A.
cover4
8
Definition 4
A polygon A is B-convex iff any straight line
segment in B that has both end points in A, is
also contained in A. The convex hull of A
relatively to B is the intersection of all
B-convex polygons containing A.
9
Definition 5
The minimum length polygon (MLP) of a 2D digital
object coincides with the
relative convex hull of an inner grid polygon
relatively to an outer grid polygon,
normally defined in a way like simulating an
inner and outer Jordan digitization.
10
Properties of MLP
  • - The inner and the outer polygon of a Jordan
    digitization have the constraint that they are at
    Hausdorff distance 1.
  • Mappings exist between vertices and between
    cavities in A and in B.
  • Convex vertices of the inner polygon and concave
    vertices of the outer polygon are candidates for
    the MLP.
  • MLP is uniquely defined for a given digitized
    object

11
Algorithm, general case
Toussaint, G.T. An optimal algorithm for
computing the relative convex hull of a set of
points in a polygon. In EURASIP, Signal
processing lll Theories and Applications, Part
2, pages 853856, North-Holland, 1986.
Computation of the relative convex hull
  1. Find an extreme vertex p in A
  2. Construct a polygon B\A with one newly created
    double-oriented edge between A and B
  3. Triangulate the new polygon
  4. Find the shortest path from p to p

Time Complexity O(n log log n)
12
Algorithm, MLP
Klette, R., Kovalevsky, V.V., Yip., B. Length
estimation of digital curves. In Vision
Geometry, SPIE 3811, pages 117129,
1999. Klette, R., Rosenfeld, A. Digital
Geometry. Morgan Kaufmann, San Francisco, 2004.
Computation of MLP
  • Trace frontier of A and include concave vertices
    of B
  • Start at extreme vertex
  • Compute positive and negative sides
  • As long as next vertex is between positive and
    negative sides update the sides
  • Otherwise a new vertex for
  • CHB(A) has been found

Time Complexity O(n)
13
Arithmetic MLP (AMLP)
Provencal, X., Lachaud, J.-O. Two linear-time
algorithms for computing the minimum length
polygon of a digital contour. In DGCI 2009, LNCS
5810, pages 104117, Springer, Heidelberg, 2009.
Computation of MLP
  1. Input contour words (Freeman code)
  2. Compute tangential cover
  3. Decompose into zones (convex, concave, inflexion)
  4. Compute convex hull for each polyline per zone

Time Complexity O(n), only for polyominos
14
Combinatorial MLP (CMLP)
Provencal, X., Lachaud, J.-O. Two linear-time
algorithms for computing the minimum length
polygon of a digital contour. In DGCI 2009, LNCS
5810, pages 104117, Springer, Heidelberg, 2009.
CMLP
Digital contour
Time Complexity O(n), only for polyominos
15
Result 1
  • The B-convex hull of a simple polygon A is
    equal to
  • the convex hull of A
  • iff
  • the convex hull of A is completely contained in
    B.

16
Result 2
All vertices of the convex hull of a simple
polygon A inside a simple polygon B are
vertices of the B-convex hull of A.
17
Result 3
All the vertices of the convex hull of Inew
belong to the relative convex hull of A between
ps and pe.
Inew ps, q3, q4,q11, pe
18
New algorithm
The relative convex hull CHB(A) for simple
polygons A and B is only different from CH(A) if
there is at least one cavity in A and one in B
such that the intersection of those cavities is
not empty.
19
New recursive algorithm
  • Compute the convex hulls for the inner and the
    outer polygon (Melkman algorithm for example)
  • Copy vertices of the inner polygon one by one to
    the CHB(A) until it finds a cavity
  • Check the outer polygon for a cavity
  • Construct the new polygons and find the convex
    hulls
  • Copy all vertices of the inner polygon to the
    CHB(A)
  • Stop if the base case of the recursion is
    reached.

20
New recursive algorithm
Covers
Relative convex hull
First cavity
Second cavity
21
Discussion
  1. Recursive algorithm works for arbitrary simple
    polygons A and B in O(n2), worst case.
  2. It is a recursive procedure that is very simple
    and of low time complexity.
  3. The algorithm runs in linear time if the maximum
    depth of stacked cavities of A is limited by a
    constant. We continue to study the expected time
    complexity of the algorithm under some general
    assumptions of variations (i.e., distribution)
    for possible input polygons A and B.
  4. The depth of stacked cavities could be used as a
    shape descriptor.
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