Towards a Multiscale Figural Geometry

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Towards aMultiscale 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
• 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

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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 objects
• Boundary of object
• In interior of object
• Exterior to object, between objects
• Suitability for shape description implies
• Magnification invariance
• At all levels of spatial scale (locality)

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Definition of Spatial Scale

• Mesh of voxels Boundary atom mesh Medial
  atom mesh

• Scale There are two separate and different
  notions
• Spatial coverage of each geometric element
• Distance of inter-element communication

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Multiple Spatial Scales

• Mesh of voxels Medial atom mesh

• Scale aspects
• Geometric element coverage
• Inter-element communication distance
• Thesis The two measures need to be similar
• Multiple scale levels

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Figural Geometry (position, orientation, local
size) Comes from Medial Atoms

• Medial 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

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Figurally RelevantSpatial Scale Levels

• Multiple objects
• Individual object
• i.e., multiple figures
• Individual figure
• mesh of medial atoms
• Figural section
• i.e., multiple figural sections
• figural section centered at medial atom
• Figural section more finely spaced, ..
• Boundary section
• Boundary section more finely spaced, ...

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Figural Types and the Manifold of Medial Atoms
M-rep Boundary implied from interpolated
continuous manifold of medial atoms
Slab Tube
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Magnification Invariance at All Spatial Scale
Levels

• Inside boundary features
• radius of curvature-proportional distances
• Inside figural sections
• r-proportional distances
• Inside individual figures
• r-proportional distances

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Magnification Invariance at All Spatial Scale
Levels

• Individual object
• In interface between figures
• blended r-proportional distances
• Multiple objects
• Outside objects
• blended r-proportional distances
• concavities effect disappear with distance

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Figural (Medially based) Geometry

• Locally magnification invariant means
  r-proportional distances
• Along medial skeleton
• Along medial sails (implied boundary normals)
• Medially (figurally) based coordinate system
  provides intrinsic coordinates
• Along medial skeleton
• Along medial sails (implied boundary normals)
• Overall metric??

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Spatial coordinates capable of providing
correspondence at any scale

• Medial coordinates (u,v)
• continuous, integer multiples of lr at samples,
  where l is scale level
• r-proportional along medial surface
• Boundary 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

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Figural Coordinates for Single Figure

• Inside object (u,v,t,d/r)
• (u,v) give multiples of r
• distance on medial sheet along geodesics of
  r-proportional distance
• Outside object
• Near boundary (inside focal surface)
  (u,v,t,d/r)
• Far outside boundary (u,v,t,d/r) via distance
  (scale) related figural convexification
• geodesics do not cross

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Figural Coordinates for Object Made From
Multiple Attached Figures

• Inside figures not near hinges
• same as for single figure
• Outside object see two slides later

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Figural Coordinates for Object Made From
Multiple Attached Figures

• Blend in hinge regions
• w(d1/r1 - d2 /r2 )/T
• Blended d/r when w lt1 and u-u0 lt T
• Implicit boundary (u,w, t)
• Implicit normals and geodesics

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Figural Coordinates between Objects

• Near boundary via blending
• Far outside boundary
• same convexification principle as with single
  figures
• blend geodesics according to dk/rk

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Uses of Correspondence

• Geometric typicality (segmentn prior)
• by boundary point to boundary point
  correspondence
• Geometric representation to image match measure
• by boundary-relative correspondence
• in collar about boundary out to fixed distance
  via metric
• union of collar and interior of object
• For homologies used in statistical shape
  characterization leads to locality
• For elements in mechanical calculations
• For comparison of segmented object to true
  object

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Open Geometric Questions

• Full space metric
• Outside figure convexification
• Reflecting scale level
• Representing tolerance
• Controlling IImedial locus, Dx2r, ?xr
• Principled means for
• Inter-figural blending of figural metrics for
  attached figures
• Inter-object blending of object metrics

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Figural (Medially based) GeometryInternal points
on single figure

• Sails are separate (qgt0)
• Both sails move with motion on medial surface

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Figural (Medially based) GeometryBranches and
Ends

• Ends
• Sails come together (q0)
• Boundary is vertex (2D) or crest (3D)
• Medial disk or ball osculates
• Branches
• Medial disk or ball tritangent
• Swallowtail of medial atom
• Retrograde motion of one sail

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Multiscale Geometry and Probability for a Figure
coarse, global

• Geometrically ? smaller scale
• Interpolate (1st order) finer spacing of atoms
• Residual atom change, i.e., local
• Probability
• At any scale, relates figurally homologous points
• Markov random field relating medial atom with
• its immediate neighbors at that scale
• its parent atom at the next larger scale and the
  corresponding position
• its children atoms

coarse resampled
fine, local