1. Analyzing Configurations of Objects in Images via
Medial/Skeletal Linking Structures
Workshop on Geometry for Anatomy
Banff International Research Station
August, 2011
James Damon
(joint with Ellen Gasparovic)

2. Overview Motivation from problems in medical imaging
Questions and Issues for “Positional and Shape Geometry” for Multi-Object Configurations:
“Medial Geometry” of Single Objects/Regions:
Relax Blum Medial structures to Skeletal structures:
Still provide mathematical tools to capture shape and
geometric properties of single objects.
Skeletal/Medial Linking Structures for Multi-Object Configurations
Linking structure extends these mathematical tools:
capture positional geometry and
shape and geometry of each object

3. Medial/Skeletal Linking Structure Multi-Object Configuration
Shape of Objects:
Local geometry
Relative geometry
Global geometry
Positional Geometry:
Neighboring objects
Relative significance
Hierarchical structure
Join shape features of objects in Rn
with their positional geometry
Medial/Skeletal
Linking Structure

4. Examples: Multi-object Structures: (MIDAG UNC)
Bladder, Prostate, Rectum
Regions of the Brain
Individual Objects modeled using discrete versions of medial structures - relations between objects involve user-based decisions.
Liver
Pelvic Region

5. II Issues for Multi-Object
Configurations

6. Positional Geometry for Collections of Points
Set of distances between each pair of points
2) Statistics (Procrustes, PCA, Clustering, etc)
3) Voronoi set (locus of points at a minimal distance from two or more of the set of points
versus issues for positional geometry of objects

7. Model for Multi-Object Configuration in Rn
Collection of regions Ωi in Rn
i) with piecewise smooth
Boundaries
ii) only meet along smooth
boundary regions (later work will allow inclusions)
iii) Medical imaging concentrates on cases
n = 2 and 3

8. Comparing Differences between Multi-object Configurations
How much of the differences are due to changes in
shape versus positional changes?
How do we numerically quantify the differences?

9. “Distance” versus “Closeness”
Closeness should measure not just the minimal distance between two objects but also “how much of the objects” are close.
“How much” in volumetric sense: single numerical value
or mathematical measure

10. Relative Positional Significance
Intrinsic shape and size does not tell us its significance Ω1
How do we
numerically measure
positional significance ? Ω1 A B

11. Hierarchical Structure
closeness criteria +
positional significance 
“tiered” Hierarchical structure
can restrict to
subconfigurations
statistical comparisons of configurations C A B G H E D F

12. III Shape and Geometry
for Single Objects

13. How Do We Capture the Shape of 3D -Objects ?

14. Blum Medial Axis for regions in R3
Medial Axis of region with generic boundary is “Whitney stratified set” exhibiting only generic singularities
Generic local forms of the Blum medial axis A3
(A1)3
A1A3
(A1)4
Blum, Yomdin, Mather, Giblin-Kimia, Bogaevsky

15. Medial Axes of 3D Generic Regions defined by B-splines
(joint with Suraj Musuvathy and Elaine Cohen)
Exactly compute stratification structure using
b-spline representations and evolution vector fields

16. Skeletal Structures
(overcoming problems
with Blum Medial Axis)

17. Small deformation of an object leaves the boundary
transverse to the radial lines along the radial vectors
of the medial axis.
Can extend or shrink radial vectors.
This will not be Blum medial structure for the deformed object.
The mean of the Blum medial structures for a collection of similar objects:
Generally will not be Blum medial structure .

18. Swept Regions and Surfaces
Represent Region W as a family of sections Wt swept-out by family of affine subspaces Pt.
Compute medial axis Mt for each section, and form the union M = t Mt.
M is not the medial axis of W

19. Skeletal Structures 1) M is a Whitney Stratified SetB
1) M is a Whitney Stratified Set
2) U is multi-valued “radial vector field” from M to points of tangency.
Blum medial axis and radial vector field have additional properties.
A “skeletal structure” (M, U) retains 1) and 2) but relaxes conditions on both M and U.

20. Simplifying Blum Medial Structure (Pizer et al)
Replace Blum medial axis:
1) with a simpler structure
2) discretized and used as deformable template
3) interpolate discrete structure to yield smooth model
Discrete Skeletal Model for Liver
(UNC MIDAG)

21. Medial Geometry: skeletal structure (M, U) in Rn
is infinitesimal form of “region”
Compute from (M, U)
smoothness properties of the region and its geometry U
Mathematically useful tools:
Radial and Edge shape operators: Srad and SE
principal radial and edge curvatures: rj and Ej .
Compatibility 1-form: hU
Medial Measure: dM = dV
Radial Flow: ft(x) = x + tU(x)

22. Regularity of Region and the Boundary
Radial and edge curvature conditions: r < 1/r,i for r,i > 0
+ compatibility condition hU = 0
radial flow is nonsingular
the level sets of radial flow parametrize W\ M .
Smoothness of B
Geometry
Compute local geometry of B using radial flow
SB = (I - r Srad)-1.Srad
Global geometry for W and B via integrals on M, using Srad , f, and medial measure

23. IV Medial/Skeletal Linking Structure

24. Discrete Medial Models for Individual Objects
in Multi-Object Configuration

25. Medial/Skeletal Linking Structure for Multi-object Configuration {Ωi} in Rn
For each region Ωi : (Mi, Ui , li)
a) skeletal structure (Mi, Ui)
with Ui = ri ui for ui the
(multivalued) unit vector field
b) linking function li on Mi,
linking vector field Li = li ui .
c) labeled refinement Si of
stratification of Mi .

26. Satisfying the conditions:
li and Li are smooth on strata of Si.
ii) “linking flow” extends radial flow
is nonsingular; each stratum Sij of Si Wij = {x + Li (x): x  Sij} is smooth.
iii) Strata {Wij} from the distinct regions match-up and form a “Whitney stratification” of the (external) linking medial axis

27. Theorem (Existence) A multi-object configuration (i. e
Theorem (Existence) A multi-object configuration (i.e. collection of disjoint regions {Ωi}) with smooth “generic” boundaries
has a “Blum medial linking structure” which
extends the Blum medial structure for each region (including exterior) and
ii) exhibits generic linking properties.
Theorem (Geometry) From a skeletal linking structure {(Mi, Ui , li)} for a multi-object configuration {Ωi}) :
we can compute the local and global geometry of:
the regions, their boundaries, and the complement of the regions.

28. Geometry from Medial/Skeletal Linking Structure
0) Still compute the local and global geometry of the regions from the linking structure.
The radial flow f extends to a linking flow  .
Radial and edge curvature conditions for the linking functions imply nonsingularity of the linking flow.
li < 1/ri,j , for ri,j > 0
Radial shape operator can be transported by the linking flow to yield the radial shape operator for the linking medial axis Slrad, i = - (I - li Srad)-1.Srad
4) Integrals of a function over a region in the complement can be computed as a sum of integrals on the medial axes of the regions .

29. Integration over a region in the complement

30. Introduce measures of closeness and significance
i j
ci,j is a volumetric/
probabilistic measure
of closeness of Ωi and Ωj
N i j
si is a volumetric/
probabilistic measure
of significance of Ωi
Properties of si, ci,j
1) 0 ≤ si, ci,j ≤ 1
2) dimensionless quantities
3) preserved under scaling and
rigid motions
Vary continuously under
small deformations
5) Computed from skeletal linking structure Ωi Ωj

31. Summary: Introduce Medial/Skeletal Linking Structures
for multi-object configurations
extend skeletal structures for individual objects
for Blum Medial linking structures, determine
generic properties using singularity theory.
compute shape, geometric properties, and positional geometry of objects
classical notions such as distance are replaced by “measure theoretic” notions such as closeness, significance.
Ongoing work: refined quantitative measures of positional properties (for statistical comparison) and
deformation properties of linking structures to analyze deformations of configurations