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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+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

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  1. Digital Geometry and Image Processing Dietmar Saupe Course Outline SS 2006

  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 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

  6. IntroductionColor images (pictures) • An RGB picture • Its 3 color channels • Histograms

  7. 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!

  8. 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

  9. 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

  10. IntroductionAdjacency • 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?

  11. IntroductionReplace the X´s! • Top: • X-adjacent cells • X-adjacent pixels • Bottom: • X-adjacent cells • X-adjacent pixels

  12. IntroductionSame in 3D! • X-adjacent 3-cells : • X= ? (left, middle, right) • X-adjacent voxels : • X= ? (left, right)

  13. IntroductionGrid point connectivity • Points are 4-connected? 8-connected? • Background 4-connected? 8-connected?

  14. 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)

  15. 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

  16. IntroductionImage scan sequences • Examples: • Space filling curves (Peano, Hilbert)

  17. 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

  18. 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

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

  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

  22. Selected Topics • Moments and their estimation • Other picture properties • Spatial relations

  23. 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, ...)

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