Mathematical definitions Topological skeleton

1 mathematical definitions

1.1 quench points of fire propagation model
1.2 centers of maximal disks (or balls)
1.3 centers of bi-tangent circles
1.4 ridges of distance function
1.5 other definitions

mathematical definitions

skeletons have several different mathematical definitions in technical literature; of them lead similar results in continuous spaces, yield different results in discrete spaces.

quench points of fire propagation model

in seminal paper, harry blum of air force cambridge research laboratories @ hanscom air force base, in bedford, massachusetts, defined medial axis computing skeleton of shape, using intuitive model of fire propagation on grass field, field has form of given shape. if 1 sets fire @ points on boundary of grass field simultaneously, skeleton set of quench points, i.e., points 2 or more wavefronts meet. intuitive description starting point number of more precise definitions.

centers of maximal disks (or balls)

a disk (or ball) b said maximal in set if

b
⊆
a


{\displaystyle b\subseteq a}

, and
if disc d contains b,



d
⊈
a


{\displaystyle d\not \subseteq a}

.

one way of defining skeleton of shape set of centers of maximal disks in a.

centers of bi-tangent circles

the skeleton of shape can defined set of centers of discs touch boundary of in 2 or more locations. definition assures skeleton points equidistant shape boundary , mathematically equivalent blum s medial axis transform.

ridges of distance function

many definitions of skeleton make use of concept of distance function, function returns each point x inside shape distance closest point on boundary of a. using distance function attractive because computation relatively fast.

one of definitions of skeleton using distance function ridges of distance function. there common mis-statement in literature skeleton consists of points locally maximum in distance transform. not case, cursory comparison of distance transform , resulting skeleton show.

other definitions

points no upstream segments in distance function. upstream of point x segment starting @ x follows maximal gradient path.
points gradient of distance function different 1 (or, equivalently, not defined)
smallest possible set of lines preserve topology , equidistant borders




^ harry blum (1967)
^ a. k. jain (1989), section 9.9, p. 387.
^ cite error: named reference gonzales543 invoked never defined (see page).
^ cite error: named reference jain invoked never defined (see page).