1. 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 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 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 ä The need: object-based positional, orientational, and metric correspondence among topologically figurally equivalent objects or groups of objects ä Boundary of object ä In interior of object ä Exterior to object, between objects ä Suitability for shape description implies ä Magnification invariance ä At all levels of spatial scale (locality) ä The need: object-based positional, orientational, and metric correspondence among topologically figurally equivalent objects or groups of objects ä Boundary of object ä In interior of object ä Exterior to object, between objects ä Suitability for shape description implies ä Magnification invariance ä At all levels of spatial scale (locality)

3. Definition of Spatial Scale ä Scale: There are two separate and different notions: ä Spatial coverage of each geometric element ä Distance of inter-element communication ä Scale: There are two separate and different notions: ä Spatial coverage of each geometric element ä Distance of inter-element communication Mesh of voxels Boundary atom mesh Medial atom mesh

4. Multiple Spatial Scales ä Scale aspects ä Geometric element coverage ä Inter-element communication distance ä Thesis: The two measures need to be similar Multiple scale levels Multiple scale levels ä Scale aspects ä Geometric element coverage ä Inter-element communication distance ä Thesis: The two measures need to be similar Multiple scale levels Multiple scale levels Mesh of voxels Medial atom mesh Mesh of voxels Medial atom mesh

5. Figural Geometry (position, orientation, local size) Comes from Medial Atoms ä Medial atoms (1st order medial locus)  x, F = (b,n,b  ) frame, r,  (object angle) ä b in direction of minimum dr/ds (-  x r) ä b  in level direction of r [3D] ä n is normal to medial skeleton ä Medial atoms (1st order medial locus)  x, F = (b,n,b  ) frame, r,  (object angle) ä b in direction of minimum dr/ds (-  x r) ä b  in level direction of r [3D] ä n is normal to medial skeleton

6. Figurally Relevant Spatial Scale Levels ä Multiple objects ä Individual object ä i.e., multiple figures ä Individual figure ä mesh of medial atoms ä Figural section ä i.e., multiple figural sections ä figural section centered at medial atom ä Figural section more finely spaced,.. ä Boundary section ä Boundary section more finely spaced,... ä Multiple objects ä Individual object ä i.e., multiple figures ä Individual figure ä mesh of medial atoms ä Figural section ä i.e., multiple figural sections ä figural section centered at medial atom ä Figural section more finely spaced,.. ä Boundary section ä Boundary section more finely spaced,... medial atom

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

8. Magnification Invariance at All Spatial Scale Levels ä Inside boundary features ä radius of curvature- proportional distances ä Inside figural sections ä r-proportional distances ä Inside individual figures ä r-proportional distances ä Inside boundary features ä radius of curvature- proportional distances ä Inside figural sections ä r-proportional distances ä Inside individual figures ä r-proportional distances

9. Magnification Invariance at All Spatial Scale Levels ä Individual object ä In interface between figures ä blended r-proportional distances ä Multiple objects ä Outside objects ä blended r-proportional distances ä concavities’ effect disappear with distance ä Individual object ä In interface between figures ä blended r-proportional distances ä Multiple objects ä Outside objects ä blended r-proportional distances ä concavities’ effect disappear with distance

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

11. Spatial coordinates capable of providing correspondence at any scale ä Medial coordinates (u[,v])  continuous, integer multiples of r at samples, where is scale level ä r-proportional along medial surface ä Boundary coordinates (u[,v],t) ä Spatial coordinates (u[,v],t,d/r) ä From implied boundary along geodesic of distance that at boundary is in normal direction ä Medial coordinates (u[,v])  continuous, integer multiples of r at samples, where is scale level ä r-proportional along medial surface ä Boundary coordinates (u[,v],t) ä Spatial coordinates (u[,v],t,d/r) ä From implied boundary along geodesic of distance that at boundary is in normal direction

12. ä Inside object: (u[,v],t,d/r) ä (u,v) give multiples of r ä distance on medial sheet along geodesics of r-proportional distance ä Outside object ä Near boundary (inside focal surface): (u[,v],t,d/r) ä Far outside boundary: (u[,v],t,d/r) via distance (scale) related figural convexification ä geodesics do not cross ä Inside object: (u[,v],t,d/r) ä (u,v) give multiples of r ä distance on medial sheet along geodesics of r-proportional distance ä Outside object ä Near boundary (inside focal surface): (u[,v],t,d/r) ä Far outside boundary: (u[,v],t,d/r) via distance (scale) related figural convexification ä geodesics do not cross Figural Coordinates for Single Figure

13. ä Inside figures not near hinges ä same as for single figure ä Outside object: see two slides later ä Inside figures not near hinges ä same as for single figure ä Outside object: see two slides later Figural Coordinates for Object Made From Multiple Attached Figures

14. ä Blend in hinge regions ä w=(d 1 /r 1 - d 2 /r 2 )/T ä Blended d/r when |w| <1 and u-u 0 < T ä Implicit boundary: (u,w, t) ä Implicit normals and geodesics ä Blend in hinge regions ä w=(d 1 /r 1 - d 2 /r 2 )/T ä Blended d/r when |w| <1 and u-u 0 < T ä Implicit boundary: (u,w, t) ä Implicit normals and geodesics Figural Coordinates for Object Made From Multiple Attached Figures

15. ä Near boundary: via blending ä Far outside boundary ä same convexification principle as with single figures ä blend geodesics according to d k /r k ä Near boundary: via blending ä Far outside boundary ä same convexification principle as with single figures ä blend geodesics according to d k /r k Figural Coordinates between Objects

16. Uses of Correspondence ä Geometric typicality (segment’n prior) ä by boundary point to boundary point correspondence ä Geometric representation to image match measure ä by boundary-relative correspondence ä in collar about boundary out to fixed distance via metric ä union of collar and interior of object ä For homologies used in statistical shape characterization: leads to locality ä For elements in mechanical calculations ä For comparison of segmented object to true object ä Geometric typicality (segment’n prior) ä by boundary point to boundary point correspondence ä Geometric representation to image match measure ä by boundary-relative correspondence ä in collar about boundary out to fixed distance via metric ä union of collar and interior of object ä For homologies used in statistical shape characterization: leads to locality ä For elements in mechanical calculations ä For comparison of segmented object to true object

17. Open Geometric Questions ä Full space metric ä Outside figure convexification ä Reflecting scale level ä Representing tolerance ä Controlling II medial locus, D x 2 r,  x r ä Principled means for ä Inter-figural blending of figural metrics for attached figures ä Inter-object blending of object metrics ä Full space metric ä Outside figure convexification ä Reflecting scale level ä Representing tolerance ä Controlling II medial locus, D x 2 r,  x r ä Principled means for ä Inter-figural blending of figural metrics for attached figures ä Inter-object blending of object metrics

18. Figural (Medially based) Geometry Internal points on single figure  Sails are separate (  >0) ä Both sails move with motion on medial surface  Sails are separate (  >0) ä Both sails move with motion on medial surface

19. Figural (Medially based) Geometry Branches and Ends ä Ends  Sails come together (  =0) ä Boundary is vertex (2D) or crest (3D) ä Medial disk or ball osculates ä Branches ä Medial disk or ball tritangent ä Swallowtail of medial atom ä Retrograde motion of one sail ä Ends  Sails come together (  =0) ä Boundary is vertex (2D) or crest (3D) ä Medial disk or ball osculates ä Branches ä Medial disk or ball tritangent ä Swallowtail of medial atom ä Retrograde motion of one sail

20. ä Geometrically  smaller scale ä Interpolate (1st order) finer spacing of atoms ä Residual atom change, i.e., local ä Probability ä At any scale, relates figurally homologous points ä Markov random field relating medial atom with ä its immediate neighbors at that scale ä its parent atom at the next larger scale and the corresponding position ä its children atoms ä Geometrically  smaller scale ä Interpolate (1st order) finer spacing of atoms ä Residual atom change, i.e., local ä Probability ä At any scale, relates figurally homologous points ä Markov random field relating medial atom with ä its immediate neighbors at that scale ä its parent atom at the next larger scale and the corresponding position ä its children atoms Multiscale Geometry and Probability for a Figure coarse, global coarse resampled fine, local