Digital Geometry and Mathematical Geometry

# Digital Geometry and Mathematical Morphology

#### Christer Kiselman

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 Zn and Rn.

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: T0 (Kolmogorov's axiom), T1, T2 (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 Z2 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.