Analyzing Configurations of Objects in Images via

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. 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 M U W B 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 Busing radial flow SB = (I - r Srad)-1.Srad Global geometry for Wand Bvia 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 = riui for ui the (multivalued) unit vector field b) linking functionlion Mi, linking vector fieldLi = liui . c)labeled refinement Siof 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. 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 axisSlrad, 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 R

30. Introduce measures of closeness and significance i j ci,j is a volumetric/ probabilistic measure of closenessof Ωiand Ωj N i j si is a volumetric/ probabilistic measure of significanceof 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