Analyzing Configurations of Objects in Images via Medial/Skeletal Linking Structures

James Damon (joint with Ellen Gasparovic)

Workshop on Geometry for Anatomy Banff International Research Station August, 2011

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

Medial/Skeletal Linking Structure

Shape of Objects:

Local geometry

Relative geometryGlobal geometry

Positional Geometry:

Neighboring objects Relative significance Hierarchical structure

Multi-Object Configuration

Join shape features of objects in Rn with their positional geometry

Examples: Multi-object Structures: (MIDAG UNC)

Bladder, Prostate, Rectum

Individual Objects modeled using discrete versions of medial structures - relations between objects involve user-based decisions.

Regions of the Brain

Pelvic RegionLiver

II Issues for Multi-Object Configurations

Positional Geometry for Collections of Points

1) 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

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 casesn = 2 and 3

How much of the differences are due to changes in shape versus positional changes?

How do we numerically quantify the differences?

Comparing Differences between Multi-object Configurations

Closeness should measure not just the minimal distance between two objects but also “how much of the objects” are close.

“Distance” versus “Closeness”

“How much” in volumetric sense: single numerical value or mathematical measure

Ω1

Ω1

A B

Relative Positional Significance

How do we numerically measurepositional significance ?

Intrinsic shape and size does not tell us its significance

H

A

B

C

DE

F

G

closeness criteria + positional significance

“tiered” Hierarchical structure

can restrict to subconfigurations

statistical comparisons of configurations

Hierarchical Structure

III Shape and Geometry for Single Objects

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

Blum Medial Axis for regions in R3

Generic local forms of the Blum medial axis

A3 (A1)3A1A3 (A1)4

Blum, Yomdin, Mather, Giblin-Kimia, Bogaevsky

Medial Axis of region with generic boundary is “Whitney stratified set” exhibiting only generic singularities

Medial Axes of 3D Generic Regions defined by B-splines

Exactly compute stratification structure using b-spline representations and evolution vector fields

(joint with Suraj Musuvathy and Elaine Cohen)

Skeletal Structures

(overcoming problems

with Blum Medial Axis)

Small deformation of an object leaves the boundarytransverse 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 .

Swept Regions and Surfaces

• Represent Region as a family of sections t swept-out by family of affine subspaces t. • Compute medial axis Mt for each section, and form the union M = t Mt. M is not the medial axis of

Skeletal Structures

1) M is a Whitney Stratified Set

2) U is multi-valued “radial vector field” from M to points of tangency.

B

UM

3) Blum medial axis and radial vector field have additional properties.

4) A “skeletal structure” (M, U) retains 1) and 2) but relaxes conditions on both M and U.

Simplifying Blum Medial Structure (Pizer et al)

Discrete Skeletal Model for Liver (UNC MIDAG)

Replace Blum medial axis:

1) with a simpler structure

2) discretized and used as deformable template

3) interpolate discrete structure to yield smooth model

Medial Geometry: skeletal structure (M, U) in Rn

is infinitesimal form of “region”

Mathematically useful tools:

Radial and Edge shape operators: Srad and SE

principal radial and edge curvatures: rj and Ej .

Compatibility 1-form: U

Medial Measure: dM = dV

Radial Flow: t(x) = x + tU(x)

U

Compute from (M, U)smoothness properties of the region and its geometry

Compute local geometry of B using radial flow SB = (I - r Srad)-1.Srad

Global geometry for and B via integrals on M, using Srad , and medial measure

U

•Radial and edge curvature conditions: r< 1/r,i for r,i > 0

+ compatibility condition U = 0 radial flow is nonsingular the level sets of radial flow parametrize \ MSmoothness of B

Regularity of Region and the Boundary

Geometry

IV Medial/Skeletal Linking Structure

Discrete Medial Models for Individual Objects in Multi-Object Configuration

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 .

i) 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

Satisfying the conditions:

Theorem (Existence) A multi-object configuration (i.e. collection of disjoint regions Ωi) with smooth

“generic” boundaries

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.

has a “Blum medial linking structure” whichi) extends the Blum medial structure for each

region (including exterior) and ii) exhibits generic linking properties.

0) Still compute the local and global geometry of the regions from the linking structure.

1) The radial flow extends to a linking flow .

2) Radial and edge curvature conditions for the linking functions imply nonsingularity of the linking flow.

li < 1/ri,j , for ri,j > 0

3) 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 .

Geometry from Medial/Skeletal Linking Structure

R

Integration over a region in the complement

Properties of si, ci,j 1) 0 ≤ si, ci,j ≤ 1 2) dimensionless quantities3) preserved under scaling and rigid motions4) Vary continuously under small deformations5) Computed from skeletal

linking structure

Introduce measures of closeness and significance

ci,j is a volumetric/probabilistic measure of closeness of Ωi and Ωj

i j

N i j

si is a volumetric/probabilistic measure of significance of Ωi

Ωi

Ωj

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