What is Digital Morse Theory ?
Digital Morse Theory (DMT) explains the properties and behavior of families
of contour lines in n-dimensions.
It effectively explains the behavior of a SpiderWeb contour lines,
surfaces, and hypersurfaces. That is only the beginning.
What is regular Morse Theory and why do we need a Digital Version ?Dr. K's to insert usual froth and excitement.
END HYPE ALERT
What about Morse Code ?
NO NO NO !
That is Samuel F. B. Morse. Wrong guy. See http://lcweb2.loc.gov/ammem/atthtml/mrshome.html
for more about this guy.
I am referring to this guy, http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Morse.html
Harold Calvin Marston Morse, Born: 24 March 1892 in Waterville, Maine,
USA Died: 1977 in Princeton, New Jersey, USA
What is to explain about contour lines ?
Lots.
First, lets define the properties of contours as opposed to other lines.
Contours are Jordan.
They have a distinct inside and outside. No Mobius strips. They hold water.
Contours are manifold.
They have normal vectors what are defined everywhere on the surface. They have a well defined inside face and outside face.
There are certain places where Contours FAIL (!).In a nutshell, we show that if you make a consistent (i.e., lookup table) connectivity decision, you making a serious error.
The behavior of contours in the vicinity of these places has troubled computer graphics people and programmers for years. The industry has consistently avoided facing the question of ambiguous contours at saddles.
The consensus in the industry is that the connectivity of contours in saddles is arbitrary.
After all, you get nice looking Jordan Manifold contours almost no matter what you do (with respect to connectivity) in ambiguous places as long as you are consistent.These ambiguous places we are calling Morse Criticalities. Contours fail here because either the contours are created,
destroyed, or merging. The properties we require for contours fail because we posit a new class of contours that represent the border between families of contours. We call these contours border contours. Each criticalities has a Zone of Influence that is bordered.
What good is it ?
Very good.
DMT insight is that by finding all of the Criticalities, we can understand
(graph) all of the regions we wish to understand.
The fundamental operation in DMT is to do a DMT Decomposition of the
function into a Graph.
All of the data is contained in the DMT Decomposition.
We can manage Level of Detail by traversing the Criticality Graph.
The graph enables us to draw objects first by topology then by geometry.
We can cull small objects and draw large objects at any resolution
and preserve important relationship between objects.
This is important for managing detail in large multidimensional databases.
Who needs it ?
Contouring is a fundamental operation.
It is vital for rendering organs for Medical Imaging, for making molecular
skeletons for structural biochemistry.
For large terrain database visualization in real time.
For Virtual Reality rendering with frame refresh time constraints.
For multidimensional
Where can I find out more about Morse Theory and friends ?
Most of the mathematical material requires years of education to understand
the jargon. Some people are trying to
make it into east to understand art. Understanding the basic nature
of reality and proving theorems about it require
lots and lots of mathematical scaffolding. Don't let it distract you.
See here for a growing list of links and reviews of them.