This was a course for beginning graduate students in mathematics and related subjects. It started on March 5 and ended on June 4.

A wish was expressed to allow credit points of this course to be given also in undergraduate education. A formal course plan has been approved by the department and is now being considered by the board for undergraduate education. The proposal is available here: DVI - PS - PDF.

**1. March 5: ** Introduction. Why digital geometry? Why
mathematical morphology? Morphological operations on sets and
functions. Infimal convolution. Dilations and erosions. Ordered sets
(beginning).

**2. March 7: ** Ordered sets (cont'd). Closure operators and
openings. Compositions of dilations and erosions; closings and
openings. Matherons first structure theorem. I distributed some
(easy) excercises---they should serve mainly to make you acquainted
with dilations and erosions and stuff.

** 3. March 19: ** Ola Weistrand comments on the first lab
exercise. The smallest and largest extensions of an increasing
mapping from a subset of *P*(*X*) to *P*(*X*). An
opening *g* as the smallest extension of the identity on the
*g*-invariant elements. Matheron's second structure theorem (for
openings and closings). Definition of distance transforms. Lipschitz
continuity of distance transforms: the Lipschitz constant is 2, but
the positive and negative parts have Lipschitz constant 1. In a normed
space the Lipschitz constant is 1.

** 4. March 21: ** Distance transforms as infimal convolutions.
The sublevel sets of distance transforms. Distance transforms in
normed vector spaces. Finitely generated distance transforms
(beginning). I distributed some new exercises, sheet 2, which are
also available here (see below).

** 5. March 26: ** Finitely generated distance transforms
(cont'd).

** 6. April 9: ** Comparing distances: three criteria for
comparing a distance with the Euclidean metric (Borgefors 1984, Verwer
1991, and a new measure). Definition of skeletons. Zorn's lemma.
Existence of skeletons in **Z**^{n} and**
R**^{n}.

** 7. April 11: ** Properties of skeletons. Lattices
(definitions).

** 8. April 16: ** The calculus of balls: when is a ball
contained in another ball? A characterization of skeletons.

** 9. April 18: ** Remarks on the calculus of balls in a space
such that all open balls are closed. Morphological operations on
lattices.

** 10. April 25: ** Please note that all exercises are now
included in the lecture notes. Filters, inf-filters, and sup-filters.

** 11. May 2: ** More about morphological operations on
lattices. Notions of topology: open sets, closed sets, neighborhoods,
topological closure, interior. Connectedness. To pull back a
topology; to push forward a topology. The set of integers **Z**
viewed as a subspace of **R** yields the discrete topology; if we
view it instead as a quotient space of **R** we get a much more
interesting topology.

** 12. May 7: ** Quotient topologies on **Z** making it a
connected topological space. Separation axioms: *T*_{0}
(Kolmogorov's axiom), *T*_{1}, *T*_{2}
(Hausdorff's axiom). Smallest neighborhood spaces.
The digital Jordan curve theorem (with an idea of the proof).
Digitization. Voronoi cells. Digital lines. The digitization of a
line segment possesses the chord property (not proved). Conversely, a
finite subset of **Z**^{2} possessing the chord property is
the digitization of a straight line segment (not proved).

** 13. May 28: ** Characterization of the digitization of a
straight line segment---with complete proof.

** 14. May 30: ** Fixed-point theorems for topological spaces
and ordered sets. Fixed-point theorems for the Khalimsky topology.

** 15. June 4: ** Discussion of the excercises and the lab
work. Questions and answers.

I have written ** lecture notes **
to be distributed to all participants. (There exist improved
**lecture notes ** notes from 2004.)
I have also distributed my paper on digital Jordan curve theorems,
available from my bibliography.

Det finns även en ** populär uppsats ** på svenska om digital
geometri: DVI - PS - PDF.

** Lab assignment 1 ** by Ola Weistrand, Mathematical
morphology: PDF.

** Lab assignment
2 ** by Ola Weistrand, Distance transformations: PDF.

*Christer*

Christer Kiselman. Last update 2004-05-09.

E-mail: kiselman@math.uu.se. URL: http://www.math.uu.se/~kiselman.