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23 Feb 2004 - 13 Nov 2016

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Christer Kiselman

This was a course for undergradute students and beginning graduatestudents in mathematics and related subjects.

Straight lines and planes have been studied during more than twothousand years, and curves like circles, ellipses, parabolas andhyperbolas for almost as long. Other curves, like lemniscates andcardioids, have been subjects of our curiousity for several centuries.For these studies we have been relying on drawings by pens on paper.The visualization by means of drawings has been essential for ourperception of all kinds of geometric objects.

Nowadays more and more images are created by computers and viewedon a screen rather than on paper. Then every figure consists of afinite number of pixels. This means that the role of coordinates isno longer played by real numbers but by integers, serving as addressesof the pixels. It also means that the geometry of the computer screenno longer is that of Euclid, equipped with the coordinates ofCartesius, but something very different, called digitalgeometry.

One might think that digital geometry is but an approximation ofthe Euclidean. In fact it is a geometry in its own right and quiteexact. However, it stretches our intuition. For instance, how can acurve consist of only finitely many points? What continuityproperties can such a curve have? Can it enclose points? Is that notas vane as trying to build a fence out of poles and no wires? In thiscourse I shall try to clarify and illustrate these and other aspectsof digital geometry.

Mathematical morphology can be described as the scienceof transforming images. Perhaps one could say that it serves forimages as Fourier analysis serves for sounds. Using Fourier analysisone can analyze and manipulate sounds, e.g., remove noise. Usingmathematical morphology one can in a similar way analyze andmanipulate images. Why are two different techniques relevant forsounds and images? The reason is very simple: our eyes behavedifferently from our ears. The foundations of this technique willalso be studied during the course.

Thecourse plan has been approvedby the Faculty of Science and Technology on May 13, 2002 (revised June02, 2003).

The meetings were devoted to the following topics.

1. February 17. Introduction. Ola Weistrand introduces the labassignments. Why digital geometry? Why mathematical morphology?Morphological operations on sets and functions. Minkowski addition ofsets. Infimal convolution (beginning).
2. February 20. Infimal convolution (cont'd). Dilations anderosions.
3. March 02. Duality of dilations and erosions. Preordered sets, ordered sets, equivalence relations, closings and openings in ordered sets. Closings and openings as compositions of dilations and erosions.
4. March 05. Matheron's first structure theorem: erosions anddilations as building blocks for more general mappings. The smallestand largest extensions of an increasing mapping from a subset ofP(X) toP(X). An openingg as thesmallest extension of the identity on theg-invariant elements.Matheron's second structure theorem: the elementary openings andclosings as building blocks for all openings and closings. Definitionof distance transforms.
5. March 17. Lipschitz continuity of distance transforms: theLipschitz constant is 2, but the positive and negative parts haveLipschitz constant 1. In a normed space the Lipschitz constant is 1.Distance transforms as infimal convolutions. The sublevel sets ofdistance transforms.
6. March 18. Chamfer distances on an abelian group. How toconstruct them using infimal convolution.
7. March 23. Comparing distances: three criteria for comparing adistance with the Euclidean metric (Borgefors 1984, Verwer 1991, and anew measure). The calculus of balls: when is a ball contained inanother ball? Remarks on the calculus of balls in a space such thatall open balls are closed.
8. March 24. Distance transforms in normed vector spaces. Thesupporting function of a set in a vector space. The Fenchel transformof a function defined on a vector space. Relations between distancetransforms and supporting functions. Skeletons: what do we want fromthem?
9. March 30. Definition of skeletons. Zorn's lemma. Existence ofskeletons inZⁿ andRⁿ. Properties of skeletons. Acharacterization of skeletons.
10. April 01. Lattices: definitions and simple properties. Completelattices. The notions of sublattice and sub-complete-lattice.Examples: the lattices of compact (respectively convex and compact,respectively closed) subsets ofRⁿ. Upperand lower inverses of mappings between lattices.
11. April 15. Notions of topology: open sets, closed sets,neighborhoods, topological closure, interior. To pull back atopology; to push forward a topology. The set of integersZviewed as a subspace ofR yields the discrete topology; if weview it instead as a quotient space ofR, we get a much moreinteresting topology.
12. April 22. Connectedness. Quotient topologies onZ making it a connected topological space. Separation axioms:T₀ (Kolmogorov's axiom),T₁,T₂ (Hausdorff's axiom).Smallest-neighborhood spaces.
13. April 23. Fixed-point theorems for topological spaces and orderedsets. Fixed-point theorems for the Khalimsky topology (beginning).
14. April 28. The continuous dependence of fixed points onparameters. Khalimsky intervals, Khalimsky circles. Digital Jordancurves as homeomorphic images of Khalimsky circles. The digitalJordan curve theorem (with an idea of the proof).
15. April 29. Digitization. Voronoi cells. Digital lines. AzrielRosenfeld's theorem: The digitization of a line segment possesses thechord property. Conversely, a finite subset ofZ²possessing the chord property is the digitization of a straight linesegment.
16. May 06. Hania Uscka-Wehlou and Erik Melin present their latestresults on digitization of straight lines.
17. May 11. Ola Weistrand discusses the lab assignments. Division ofmappings between lattices. Structure theorems in lattice morphology.Inf-filters, sup-filters and strong filters in lattices.

I have written lecture notes entitled , 95 pages, and distributed them to allparticipants.

Det finns en på svenska om digital geometri.

Ekzistas enesperanto pri di^gita geometrio.

There are two constructed by Ola Weistrand: 1. Mathematical morphology and 2. Distance transformations.
They can be downloaded fromthe address indicated.

There is also be a set of after each chapter in the lecture notes.

To be approved, a student must complete successfully both labassignments as well as a reasonable number of the exercise problems,with a fair distributions over the chapters.

Christer