2. Digital Geometry and Image
Processing (3V+2Ü)
Geometric methods for digital picture analysis
Scope: Graduate course
Information Engineering master and PhD students
Classes (Vorlesung), D. Saupe
Tuesdays 8:15h-11h, Z714 (preliminary)
Problem sessions (Übg.), V. Bondarenko
Thursdays 14:00h-15:30h, Z714 (preliminary)
3. Primary course text book
Reinhard Klette,
Azriel Rosenfeld
Digital Geometry
Morgan Kaufmann
(Elsevier) 2004
UB will have copies
4. Secondary course text book
R.C. Gonzales, R.E. Woods
Digital Image Processing
Prentice-Hall (2nd Ed.) 2002
3rd edition
UB has copies
5. Digital Geometry
Geometric methods for digital picture analysis
Focus is on digital image or picture
analysis
Core of the field
Related mathematical fundamenals
It is not
yet another treatment of a very broad range
of problems, algorithms, heuristics, and
„useful“ technologies
7. Introduction
Early digital pictures
A Greek pebble mosaic, detail from
“The Lion Hunt” in Pella,Macedonia,
circa 300 BC.
Pattern woven by a Jacquard loom: a
black-and-white silk portrait of Jacquard
himself, woven under the control of a
“program” consisting of about 24,000
cards (one is shown on the left).
Early 19th century, before Babbage!
8. Introduction
Digital pictures in 2005
Standard 16 Megapixel CCD cameras evolving
Specialized cameras in photogrammetry of 100 Megapixels
3D imaging modalities (CT, MRI, ...)
3D-laser range scanners
Leon Harmon of Bell Labs: picture of
Lincoln (252 pixels), “The Recognition
of Faces”, Scientific American, (Nov. 1973).
A 380 degree panoramic picture of Auckland,
New Zealand, 2002,
500 Megapixels
9. Introduction
Grid of squares versus grid of points
Two concepts for pixels (cells)
Is the value a component of the pixel?
A picture P is a mapping of a finite
rectangular grid region into the reals
Generalization to 3D: voxel
10. Introduction
Adjacency
Version 1
Cell 1-adjacency and pixel 4-adjacency (left)
Neighborhoods (right)
Version 2
Cell 0-adjacency and pixel 8-adjacency (left)
Neighborhoods (right)
In 3D:
Cells? Voxels?
14. Introduction
Equivalent classes
Equivalence relation R on finite grid
Reflexive, symmetric, transitive
Yields equivalence classes
For a picture P-equivalence:
Pixels p,q: pRq iff P(p)=P(q)
15. Introduction
Component labelling
Assume 4-adjacency of pixels
Frequent task: label the 4-connected
components of the equivalence classes
Some algorithms
Fill algorithm:
Rosenfeld-Pfaltz
labelling scheme
20. Topics (Chapters)
Curves and surfaces: topology, geometry
Jordan curves, curves in grids
Surfaces and manifolds, ... in 3D grids
Arc length, curvature, angles, areas
Surfaces and solids
Principal, gaussian, mean curvature
Tracing surfaces
21. Topics (Chapters)
Curves and surfaces in grids
Straightness, 2D and 3D
Measuring arc length, curvature, corners
Digital planes
Measuring surface area, curvature