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Towards a Towards a Multiscale Figural Multiscale Figural Geometry Geometry Stephen Pizer Stephen Pizer Andrew Thall, Paul Yushkevich Andrew Thall, Paul Yushkevich www.cs.unc.edu/Research/Image www.cs.unc.edu/Research/Image Medical Image Display & Analysis Group Medical Image Display & Analysis Group University of North Carolina, Chapel Hill University of North Carolina, Chapel Hill Acknowledgements: James Chen, Guido Gerig, and P. Thomas Fletcher for figures, Acknowledgements: James Chen, Guido Gerig, and P. Thomas Fletcher for figures, NIH grant P01 CA47982, NSF grant CCR-9910419, and Intel for a computer NIH grant P01 CA47982, NSF grant CCR-9910419, and Intel for a computer grant grant

Towards a Multiscale Figural Geometry

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Towards a Multiscale Figural Geometry. Stephen Pizer Andrew Thall, Paul Yushkevich www.cs.unc.edu/Research/Image Medical Image Display & Analysis Group University of North Carolina, Chapel Hill - PowerPoint PPT Presentation

Text of Towards a Multiscale Figural Geometry

  • Towards aMultiscale Figural Geometry Stephen PizerAndrew Thall, Paul Yushkevichwww.cs.unc.edu/Research/Image

    Medical Image Display & Analysis GroupUniversity of North Carolina, Chapel Hill

    Acknowledgements: James Chen, Guido Gerig, and P. Thomas Fletcher for figures, NIH grant P01 CA47982, NSF grant CCR-9910419, and Intel for a computer grant

  • Intrinsic Object-Based Geometry Suitable for Shape Description The need: object-based positional, orientational, and metric correspondence among topologically figurally equivalent objects or groups of objectsBoundary of objectIn interior of objectExterior to object, between objects

    Suitability for shape description impliesMagnification invarianceAt all levels of spatial scale (locality)

  • Definition of Spatial ScaleScale: There are two separate and different notions:Spatial coverage of each geometric elementDistance of inter-element communicationMesh of voxels Boundary atom mesh Medial atom mesh

  • Multiple Spatial ScalesScale aspects Geometric element coverageInter-element communication distanceThesis: The two measures need to be similar Multiple scale levels Mesh of voxels Medial atom mesh

  • Figural Geometry (position, orientation, local size) Comes from Medial AtomsMedial atoms (1st order medial locus) x, F = (b,n,b) frame, r, q (object angle)b in direction of minimum dr/ds (-xr)b in level direction of r [3D]n is normal to medial skeleton

    EMBED Equation.3

    EMBED Equation.3

    EMBED Equation.3

    EMBED Equation.3

    EMBED Equation.3

    EMBED Equation.3

    EMBED Equation.3

    EMBED Equation.3

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  • Figurally RelevantSpatial Scale LevelsMultiple objectsIndividual objecti.e., multiple figuresIndividual figuremesh of medial atomsFigural sectioni.e., multiple figural sectionsfigural section centered at medial atomFigural section more finely spaced, ..Boundary sectionBoundary section more finely spaced, ...

  • Figural Types and the Manifold of Medial Atoms

    Slab

    TubeM-repBoundary implied from interpolated continuous manifold of medial atoms

  • Magnification Invariance at All Spatial Scale LevelsInside boundary featuresradius of curvature-proportional distancesInside figural sectionsr-proportional distancesInside individual figuresr-proportional distances

  • Magnification Invariance at All Spatial Scale LevelsIndividual objectIn interface between figuresblended r-proportional distances Multiple objectsOutside objectsblended r-proportional distancesconcavities effect disappear with distance

  • Figural (Medially based) Geometry Locally magnification invariant means r-proportional distancesAlong medial skeletonAlong medial sails (implied boundary normals)

    Medially (figurally) based coordinate system provides intrinsic coordinatesAlong medial skeletonAlong medial sails (implied boundary normals)Overall metric??

  • Spatial coordinates capable of providing correspondence at any scaleMedial coordinates (u[,v])continuous, integer multiples of lr at samples, where l is scale levelr-proportional along medial surfaceBoundary coordinates (u[,v],t)Spatial coordinates (u[,v],t,d/r)From implied boundary along geodesic of distance that at boundary is in normal direction

    *

    *

  • Figural Coordinates for Single Figure Inside object: (u[,v],t,d/r)(u,v) give multiples of rdistance on medial sheet along geodesics of r-proportional distanceOutside objectNear boundary (inside focal surface): (u[,v],t,d/r)Far outside boundary: (u[,v],t,d/r) via distance (scale) related figural convexificationgeodesics do not cross

  • Figural Coordinates for Object Made From Multiple Attached Figures Inside figures not near hingessame as for single figureOutside object: see two slides later

  • Figural Coordinates for Object Made From Multiple Attached Figures Blend in hinge regionsw=(d1/r1 - d2 /r2 )/TBlended d/r when |w|
  • Figural Coordinates between Objects

    Near boundary: via blendingFar outside boundarysame convexification principle as with single figuresblend geodesics according to dk/rk

  • Uses of CorrespondenceGeometric typicality (segmentn prior)by boundary point to boundary point correspondenceGeometric representation to image match measureby boundary-relative correspondencein collar about boundary out to fixed distance via metricunion of collar and interior of objectFor homologies used in statistical shape characterization: leads to localityFor elements in mechanical calculationsFor comparison of segmented object to true object

  • Open Geometric QuestionsFull space metricOutside figure convexificationReflecting scale levelRepresenting toleranceControlling IImedial locus, Dx2r, xrPrincipled means forInter-figural blending of figural metrics for attached figuresInter-object blending of object metrics

    EMBED Equation.3

    EMBED Equation.3

    EMBED Equation.3

    EMBED Equation.3

    EMBED Equation.3

    EMBED Equation.3

    EMBED Equation.3

    EMBED Equation.3

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  • Figural (Medially based) GeometryInternal points on single figure Sails are separate (q>0)Both sails move with motion on medial surface

    EMBED Equation.3

    EMBED Equation.3

    EMBED Equation.3

    EMBED Equation.3

    EMBED Equation.3

    EMBED Equation.3

    EMBED Equation.3

    EMBED Equation.3

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  • Figural (Medially based) GeometryBranches and Ends EndsSails come together (q=0)Boundary is vertex (2D) or crest (3D)Medial disk or ball osculates

    BranchesMedial disk or ball tritangentSwallowtail of medial atomRetrograde motion of one sail

  • Multiscale Geometry and Probability for a FigureGeometrically smaller scaleInterpolate (1st order) finer spacing of atomsResidual atom change, i.e., localProbabilityAt any scale, relates figurally homologous pointsMarkov random field relating medial atom with its immediate neighbors at that scaleits parent atom at the next larger scale and the corresponding positionits children atoms

    coarse, global coarse resampledfine, local