DigitalGeometry.ppt

Digital Geometry and Image Processing
Dietmar Saupe
Course Outline
SS 2006

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)

Primary course text book
 Reinhard Klette,
Azriel Rosenfeld
 Digital Geometry
 Morgan Kaufmann
(Elsevier) 2004
 UB will have copies

Secondary course text book
 R.C. Gonzales, R.E. Woods
 Digital Image Processing
 Prentice-Hall (2nd Ed.) 2002
 3rd edition
 UB has copies

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

Introduction
Color images (pictures)
 An RGB picture
 Its 3 color channels
 Histograms

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!

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

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

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?

Introduction
Replace the X´s!
 Top:
 X-adjacent cells
 X-adjacent pixels
 Bottom:
 X-adjacent cells
 X-adjacent pixels

Introduction
Same in 3D!
 X-adjacent 3-cells :
 X= ? (left, middle, right)
 X-adjacent voxels :
 X= ? (left, right)

Introduction
Grid point connectivity
 Points are 4-connected? 8-connected?
 Background 4-connected? 8-connected?

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)

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

Introduction
Image scan sequences
 Examples:
 Space filling curves (Peano, Hilbert)

Topics (Chapters)
Metrics
 Basics: Norms, Minkowski metrics, integer valued
metrics, induced topology, Hausdorff metric
 Grid point metrics, paths, geodesics, intrinsic distances
 Metrics on pictures:
distance transforms
 medial axis

Topics (Chapters)
Adjacency graphs
 Graphs and connectedness, basic graph theory, Euler
characteristic and planarity
 Boundaries, cycles, frontiers in incidence pseudographs
 Inner (gray) pixel
border (black) pixel
co-border (gray) pixel

Topics (Chapters)
Topology
 Topological spaces, digital topologies
 Concepts homeomorphy, isotopy (top.
equivalence)
 Simplicial complexes, triangulations

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

Topics (Chapters)
Curves and surfaces in grids
 Straightness, 2D and 3D
 Measuring arc length, curvature, corners
 Digital planes
 Measuring surface area, curvature

Selected Topics
 Moments and their estimation
 Other picture properties
 Spatial relations

Selected Topics (not covered)
 Hulls and diagrams (convexity, Voronoi)
 Transformations (t. groups, symmetries,
magnification, ...)
 Morphological operators (dilation, erosion,
simplification, segmentation, ...)
 Deformations (topological-preserving def.,
shrinking, thinning, ...)

DigitalGeometry.ppt

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