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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

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digital geometry and image processing

Digital Geometry and Image Processing

Dietmar Saupe

Course Outline

SS 2006

digital geometry and image processing 3v 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)
primary course text book
Primary course text book
  • Reinhard Klette,Azriel Rosenfeld
  • Digital Geometry
  • Morgan Kaufmann(Elsevier) 2004
  • UB will have copies
secondary course text book
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
Digital GeometryGeometric 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
IntroductionColor images (pictures)
  • An RGB picture
  • Its 3 color channels
  • Histograms
introduction early digital pictures
IntroductionEarly 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
IntroductionDigital 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 Recognitionof 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
IntroductionGrid 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 finiterectangular 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
IntroductionReplace the X´s!
  • Top:
    • X-adjacent cells
    • X-adjacent pixels
  • Bottom:
    • X-adjacent cells
    • X-adjacent pixels
introduction same in 3d
IntroductionSame in 3D!
  • X-adjacent 3-cells :
    • X= ? (left, middle, right)
  • X-adjacent voxels :
    • X= ? (left, right)
introduction grid point connectivity
IntroductionGrid point connectivity
  • Points are 4-connected? 8-connected?
  • Background 4-connected? 8-connected?
introduction equivalent classes
IntroductionEquivalent 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
IntroductionComponent labelling
  • Assume 4-adjacency of pixels
  • Frequent task: label the 4-connected components of the equivalence classes
  • Some algorithms
    • Fill algorithm:
    • Rosenfeld-Pfaltzlabelling scheme
introduction image scan sequences
IntroductionImage scan sequences
  • Examples:
  • Space filling curves (Peano, Hilbert)
topics chapters metrics
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
Topics (Chapters)Adjacency graphs
  • Graphs and connectedness, basic graph theory, Euler characteristic and planarity
  • Boundaries, cycles, frontiers in incidence pseudographs
  • Inner (gray) pixelborder (black) pixelco-border (gray) pixel
topics chapters topology
Topics (Chapters)Topology
  • Topological spaces, digital topologies
  • Concepts homeomorphy, isotopy (top. equivalence)
  • Simplicial complexes, triangulations
topics chapters curves and surfaces topology geometry
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
Topics (Chapters)Curves and surfaces in grids
  • Straightness, 2D and 3D
  • Measuring arc length, curvature, corners
  • Digital planes
  • Measuring surface area, curvature
selected topics
Selected Topics
  • Moments and their estimation
  • Other picture properties
  • Spatial relations
selected topics not covered
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, ...)