Anatomic Geometry

1
Anatomic Geometry Deformations and Their
Population Statistics(Or Making Big Problems
Small)

• Stephen M. Pizer, Kenan Professor
• Medical Image Display Analysis Group
• University of North Carolina
• website midag.cs.unc.edu
• Co-authors P. Thomas Fletcher, Sarang Joshi,
  Conglin Lu, and numerous others in MIDAG

2
Real-World Analysis with Images as DataShape of
Objects in Populations via representations z

• Uses for probability density p(z)
• Sampling p(z) to communicate anatomic
  variability in atlases
• Log prior in posterior optimizing deformable
  model segmentation
• Optimize log p(zI), so log p(z) log
  p(Iz)
• Compare two populations
• Medical science localities where
  p(zhealthy) p(zdiseased) differ
• Diagnostic Is particular patients geometry
  diseased? p(zhealthy, I) vs.
  p(zdiseased, I)

3
Plan of Talk

• Needs of geometric object(s) representations z
• The menagerie of geometric object(s)
  representations
• How to make big problems in statistics small PCA
• Properties of the representations and of forming
  probability densities on them
• Making the problem small via interior models with
  natural deformations and statistical analysis
  suited to interior models (PGA)
• Summary Research strategy in image analysis

4
Needs of Geometric Representation z
Probability Representation p(z)

• Accurate p(z) estimation with limited samples,
  i.e., beat High Dimension Low Sample Size (HDLSS
  many features, few training cases)
• Primitives positional correspondence cases
  alignment
• Easy fit of z to each training segmentation or
  image
• Handle multi-object complexes
• Rich geometric representation
• Local twist, bend, swell?
• Make geometric effects intuitive
• Null probabilities for geometrically
  illegal objects
• Localization Multiscale framework

5
Representations z of DeformationAll but
landmarks look initially big

• Landmarks
• Boundary of objects (b-reps)
• Points spaced along boundary
• or Coefficients of expansion in basis
  functions
• or Function in 3D with level set as
  object boundary
• Deformation velocity seq. per voxel
• Medial representation of objects interiors
  (m-reps)

6
Plan of Talk

• Needs of geometric object(s) representations z
• The menagerie of geometric object(s)
  representations
• How to make big problems in statistics small PCA
• Properties of the representations and of forming
  probability densities on them
• Making the problem small via interior models with
  natural deformations and statistical analysis
  suited to interior models (PGA)
• Summary Research strategy in image analysis

7
Standard Method of Making Big Problems Small via
Statistics Principal Component Analysis (PCA)
D ? ?m e.g., DTn with T ? ?3

• New features are components in each of first few
  principal directions
• Each describes a global deformation of the mean

Mean closest to data in square distance.
Principal direction submanifold through mean
closest to data in square distance.
8
Plan of Talk

• Needs of geometric object(s) representations z
• The menagerie of geometric object(s)
  representations
• How to make big problems in statistics small PCA
• Properties of the representations and of forming
  probability densities on them
• Making the problem small via interior models with
  natural deformations and statistical analysis
  suited to interior models (PGA)
• Summary Research strategy in image analysis

9
Landmarks as Representation z

• First historically
• Kendall, Bookstein, Dryden Mardia, Joshi
• Landmarks defined by special properties
• Wont find many accurately in 3D
• Alignment via Procrustes
• p(z) via PCA on pt. displacements
• Avoid foldings via PDE-based interpolation
• Unintuitive principal warps
• Global only, and spatially inaccurate between
  landmarks

10
B-reps as Representation z

• Point samples like landmarks popular
• Fit to training objects pretty easy
• Handles multi-object complexes
• HDLSS? Typically no not enough stable principal
  directions
• Positional correspondence of primitives
• Expensive reparametrization to optimize p(z)
  tightness
• Only characterization of local translations of
  shell
• Weak re null probabilities for geometric illegals
• Basis function coefficients
• Achieves legality fit correspondence imperfect
• Implicit, questionable positional correspondence
• Global, Unintuitive
• Level set of F(x)

11
Probability p(z) for B-reps

• PCA on point displacement Cootes Taylor
• Global, i.e., no localization
• PCA on basis function coefficients Gerig
• Also, global

12
B-rep via F(x)s level set z is FObjective to
handle topological variability

• Run nonlinear diffusion to achieve deformation
• Fit to training cases easy via distance
• Correspondence?
• Rich characterization of geometric
  effects? Yes, but objects unexplicit
• Unintuitive
• Stats inadequately developed
• Null probabilities for
  geometrically illegal objects
• No if statistics on function
• Yes if statistics on PDE
• Localization via spatially varying PDE
  parameters??

13
Deformation velocity sequence for each voxel as
representation z

• Miller, Christensen, Joshi
• Labels in reference move with deformation
• Series of local interactions
• Deformation energy minimization
• Fluid flow
• Hard to fit to training cases if nonlocal
• Alignment and mean by centroid that
  minimizes deformation energies Davis
  Joshi
• A B via A m then (B m)-1

14
Probability p(z) for deformation velocity
sequence per voxel

• PCA on collection of point displacements
  Csernansky (better on velocity sequence)
• Global
• Large problem of HDLSS
• Few PCA coefficients are stable
• Intuitive characterization of geometric
  effects warp
• Very local translations, but also very local
  rotations, magnifications
• Can produce geometrically illegal objects
• Benefit from p(nonlinear transformations)?
• Need many-voxel tests done via permutations

cf.
15
Plan of Talk

• Needs of geometric object(s) representations z
• The menagerie of geometric object(s)
  representations
• How to make big problems in statistics small PCA
• Properties of the representations and of forming
  probability densities on them
• Making the problem small via interior models with
  natural deformations and statistical analysis
  suited to interior models (PGA)
• Summary Research strategy in image analysis

16
M-reps as Representation z Represent the Egg,
not the Eggshell

• The eggshell object boundary primitives
• The egg object interior primitives
• M-reps
• Transformations of primitives local
  displacement, local bending twisting
  (rotations), local swelling/contraction
• Handles multifigure objects and multi-object
  complexes
• Interstitial space??

17
A deformable model of the object interior the
m-rep

• Object interior primitives medial atoms
• Objects, figures
• Local displacement, bending/twisting, swelling
  intuitive
• Neighbor geometry
• Represent interior sections

18
Medial atom as a geometric transformation

• Medial atoms carry position, width, 2
  orientations
• Local deformation T ? ?3 ? S2 S2
• From reference atom
• Hub translation Spoke magnification in common
  spoke1 rotation spoke2 rotation
• S2 SO(3)/SO(2)
• Represent interior section
• M-rep is n-tuple of medial atoms
• Tn , n local Ts, a symmetric space

19
Preprocessing for computing p(z) for M-reps

• Fitting m-reps into training binaries
• Edge-constrained
• Irregularity penalty
• Yields correspondence(?)
• Interpenetration avoidance
• Alignment via minimization of sum of squared
  interior distances (geodesic distances -- next
  slide)

20
Principal Geodesic Analysis (PGA)Fletcher
Tn with T ? ?3 ? S2 S2
Mean closest to data in square distance.
Principal direction submanifold through mean
closest to data in square distance.
21
Advantages of Geodesic Geometry with Rotation
Magnification

• Strikingly fewer principal components than PCA
  (LDLSS)
• Avoids geometric illegals
• Procrustes in geodesic geometry to align
  interiors
• Geodesic interpolation in time space is natural

A to B to A to C to A
22
Statistics at Any Scale Level

• Global
• By object
• By figure (atom mesh)
• By atom (interior section)
• By voxel or boundary vertex

atom level
boundary level
quad-mesh neighbor relations
23
Representation of multiple objects via residues
from global variation

• Interscale residues
• E.g., global to per-object
• Provides localization
• Inter-relations between objects (or figures)
• Augmentation via highly correlated (near)
  atoms
• Prediction of remainder via augmenting atoms

24
Does nonlinear stats on m-reps work? Ex Use of
p(m) for Segmentation

• Here, within interday changes within a patient
• Extraction of bladder, prostate, rectum via
  global, bladder, prostate, rectum sequence of
  posterior optimizations
• Speed lt5 min. today, 10 seconds in new version

25
Plan of Talk

• Needs of geometric object(s) representations z
• The menagerie of geometric object(s)
  representations
• How to make big problems in statistics small PCA
• Properties of the representations and of forming
  probability densities on them
• Making the problem small via interior models with
  natural deformations and statistical analysis
  suited to interior models (PGA)
• Summary Research strategy in image analysis

26
Summary re Estimating p(z) via stats on geometric
transformations

• Needs
• Easy, correspondent fit to training segmentations
• Accuracy using limited samples
• Choices
• Representation of object interiors vs. boundaries
• Object complexes Object by object vs. global
• Statistics of inter-object relations (canonical
  correlation?)
• Combine with voxel deformation to refine and
  extrapolate to interstitial spaces
• Physical vs. or and statistical models,
  Complexity vs. simplicity, Global vs. local
• Results so far m-reps voxel deformation hybrid
  best, but jury out more representations
  to discover functions level sets?

27
Conclusions re Strategy in Object or Image
Analysis

• How to Deal with Big Geometric Things
• By doing analyses well fit to the geometry and to
  the population, make the big things small
• These analyses require strengths of
  mathematicians, of statisticians, of computer
  scientists, and of users all working together

28
Want more info?

• This tutorial, many papers on m-reps and their
  statistics and applications can be found at
  website http//midag.cs.unc.edu
• For other representations see references on
  following 2 slides

29
References

• Kendall, D (1986). Size and Shape Spaces for
  Landmark Data in Two Dimensions. Statistical
  Science, 1 222-226.
• Bookstein, F (1991). Morphometric Tools for
  Landmark Data Geometry and Biology. Cambridge
  University Press.
• Dryden, I and K Mardia (1998). Statistical Shape
  Analysis. John Wiley Sons.
• Cootes, T and C Taylor (2001). Statistical Models
  of Appearance for Medical Image Analysis and
  Computer Vision. Proc. SPIE Medical Imaging.
• Grenander, U and MI Miller (1998), Computational
  Anatomy An Emerging Discipline, Quarterly of
  Applied Math., 56 617-694.

30
References

• Csernansky, J, S Joshi, L Wang, J Haller, M Gado,
  J Miller, U Grenander, M Miller (1998).
  Hippocampal morphometry in schizophrenia via high
  dimensional brain mapping. Proc. Natl. Acad. Sci.
  USA, 95 11406-11411.
• Caselles, V, R Kimmel, G Sapiro (1997). Geodesic
  Active Contours. International Journal of
  Computer Vision, 22(1) 61-69.
• Pizer, S, K Siddiqi, G Szekely, J Damon, S Zucker
  (2003). Multiscale Medial Loci and Their
  Properties. International Journal of Computer
  Vision - Special UNC-MIDAG issue, (O Faugeras, K
  Ikeuchi, and J Ponce, eds.), 55(2) 155-179.
• Fletcher, P, C Lu, S Pizer, S Joshi (2004).
  Principal Geodesic Analysis for the Study of
  Nonlinear Statistics of Shape. IEEE Transactions
  on Medical Imaging, 23(8) 995-1005.