Last update: Nov. 28, 1998

Publications by Ugo Montanari et al. on shape symmetry elicitation :

• A Grey-Weighted Skeleton, 1970.
• Continuous Skeletons from Digitized Images, 1969.
• A Method for Obtaining Skeletons Using a Quasi-Euclidean Distance, 1968.

BibTeX references.

The "propagation of the figure contour is simulated analytically".

An intrinsic coordinate system" and its "singularities" are defined.

A "simulation of the propagation process is performed on a discrete plane"

"The wavefront is composed of straight-line segments and arcs of circles, while the skeleton consists of straight-line segments and arcs of parabola".

"A successive wavefront can be obtained, according to Huygens' principle".

"A wavefront is a sequence of extended elements" :

• straight-line segments (sls), and
• arcs of circle (ac)

1st order shock points == suture points

• Intersections of 2 adjacent extended elements is called a "suture point".
• A suture point has a "characteristic angle" (between 0 and 180 deg.)

Higher-order shocks == breakpoints

Introduces "breakpoints" which are intersections of 2 or more non-adjacent extended elements.

A "final breakpoint" is "created by the coincidence of 3 (or more) "suture points of a vanishing wavefront" (that is like a 4th-order shock it seems).

An "intermediate breakpoint" is "created by the coincidence of 2 suture points of a nonvanishing wavefront" (this is a generic node: two branches coming together create a new one).

An "initial breakpoint" is "not created by the coincidence of suture points and "at least one of the 2 intersecting extended elements is an arc of circle" (this is the 2nd-order shock, a neck).

Some "special cases" are considered, which include the 3rd order shock (only 2 parallel sls are mentioned ... missing the 2 ac case?).

Follows a data-structure for the wavefront, which includes the parametric equation of every indefinite extended element, etc.

He then considers properties of breakpoints; in particular, the characteristic angles obey simple equations.

2 important theorems are given.

Theorem 1 : a breakpoint is either final, intermediat or initial

Theorem 2 : the total number of non-final breakpoints (i.e. initial - 2nd order shocks - and intermediate - tree nodes -) can be directly deduced from the initial wavefront (!!) (a formula is given)

An algorithm is detailed.

A formula for the outgoing speed of the "leaving branch" at an intermediate breakpoint is given (function of the speeds of the 2 incoming branches).

Then, 3 types of skeleton branches are considered:

1. Two sls meet: gives a sls skeleton branch; formula for speed along the branch is given.
2. One sls and one ac meet: gives an arc of parabola; formula for speed along this parabola is also given.
3. Two ac meet (with same radius of curvature, but opposite normals ...

Montanari does not consider 2 ac making-up a curved cylinder ... did he miss this (particular case) ? I guess he only considers generic cases ... although this is not clearly explained) the skeleton branch is an sls; speed function is given.

Page created & maintained by Frederic Leymarie, 1998.
Comments, suggestions, etc., mail to: leymarie@lems.brown.edu