Transcript Document 7836027

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
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)

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
Multiple Spatial Scales
Mesh of voxels

Scale aspects



Medial atom mesh
Geometric element coverage
Inter-element communication distance
Thesis: The two measures need to be similar
Multiple scale levels
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
Figurally Relevant
Spatial 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, ...

medial atom
Figural Types and the Manifold of
Medial Atoms
M-rep
Boundary implied from interpolated
continuous manifold of medial atoms
Slab
Tube
Magnification Invariance at
All Spatial Scale Levels
 Inside boundary features
 radius of curvatureproportional distances
 Inside figural sections
 r-proportional distances
 Inside individual figures
 r-proportional distances
Magnification Invariance at
All Spatial Scale Levels
 Individual
 In
object
interface between figures
 blended
 Multiple
 Outside
r-proportional distances
objects
objects
 blended
r-proportional distances
 concavities’ effect disappear with
distance
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??
Spatial coordinates capable of
providing correspondence at any scale
 Medial
coordinates (u[,v])
integer multiples of lr
at samples, where l is scale level
 r-proportional along medial surface
 continuous,
 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
*
*
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
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
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
Figural Coordinates between Objects
 Near
boundary: via blending
 Far outside boundary
 same
convexification principle as with
single figures
 blend geodesics according to dk/rk
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

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
Figural (Medially based) Geometry
Internal points on single figure
Sails are separate (q>0)
 Both sails move with motion on medial surface

Figural (Medially based) Geometry
Branches 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
Multiscale Geometry and Probability
for a Figure
 Geometrically
 smaller scale
(1st order) finer
spacing of atoms
 Residual atom change, i.e., local
coarse, global
 Interpolate
coarse resampled
 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

fine, local