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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

Dietmar Saupe

Course Outline

SS 2006


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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)


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Primary course text book

  • Reinhard Klette,Azriel Rosenfeld

  • Digital Geometry

  • Morgan Kaufmann(Elsevier) 2004

  • UB will have copies


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Secondary course text book

  • R.C. Gonzales, R.E. Woods

  • Digital Image Processing

  • Prentice-Hall (2nd Ed.) 2002

  • 3rd edition

  • UB has copies


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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


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IntroductionColor images (pictures)

  • An RGB picture

  • Its 3 color channels

  • Histograms


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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!


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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


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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


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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?


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IntroductionReplace the X´s!

  • Top:

    • X-adjacent cells

    • X-adjacent pixels

  • Bottom:

    • X-adjacent cells

    • X-adjacent pixels


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IntroductionSame in 3D!

  • X-adjacent 3-cells :

    • X= ? (left, middle, right)

  • X-adjacent voxels :

    • X= ? (left, right)


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IntroductionGrid point connectivity

  • Points are 4-connected? 8-connected?

  • Background 4-connected? 8-connected?


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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)


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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


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IntroductionImage scan sequences

  • Examples:

  • Space filling curves (Peano, Hilbert)


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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


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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


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Topics (Chapters)Topology

  • Topological spaces, digital topologies

  • Concepts homeomorphy, isotopy (top. equivalence)

  • Simplicial complexes, triangulations


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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


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Topics (Chapters)Curves and surfaces in grids

  • Straightness, 2D and 3D

  • Measuring arc length, curvature, corners

  • Digital planes

  • Measuring surface area, curvature


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Selected Topics

  • Moments and their estimation

  • Other picture properties

  • Spatial relations


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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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