Towards a Multiscale Figural Geometry

1. 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

2. 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)

3. 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

4. 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

5. 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

6. medial atom 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, ...

7. Figural Types and the Manifold of Medial Atoms M-rep Boundary implied from interpolated continuous manifold of medial atoms Slab Tube

8. 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

9. 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

10. 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??

11. 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

12. 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

13. 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

14. Figural Coordinates for Object Made From Multiple Attached Figures • Blend in hinge regions • w=(d1/r1 - d2 /r2 )/T • Blended d/r when |w| <1 and u-u0 < T • Implicit boundary: (u,w, t) • Implicit normals and geodesics

15. Figural Coordinates between Objects • Near boundary: via blending • Far outside boundary • same convexification principle as with single figures • blend geodesics according to dk/rk

16. Uses of Correspondence • Geometric typicality (segment’n 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

17. 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

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

19. Figural (Medially based) GeometryBranches and Ends • Ends • Sails come together (q=0) • 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

20. 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