building of statistical models

Guido Gerig UNC, September 2002 1

Building of statistical models

Building of statistical models

Guido GerigGuido Gerig

Department of Computer Science, UNC, Department of Computer Science, UNC, Chapel HillChapel Hill


Statistical Shape ModelsStatistical Shape Models

• Drive deformable Drive deformable model segmentationmodel segmentation• statistical geometric model

• statistical image boundary model

• Analysis of shape Analysis of shape deformation deformation (evolution, (evolution, development, development, degeneration, disease)degeneration, disease)


Manual Image SegmentationManual Image Segmentation

IRIS segmentation tool: Segmentation of hippocampus/amygdala from 3D MRI data.

•Manual segmentation in all three orthogonal slice orientations.

• Instant 3D display of segmented structures.

•Cursor interaction between 2D/ 3D.

•Painting and cutting in 3D display.

•Open standard s (C++, openGL, Fltk, VTK).


SNAP: Segmentation by level set evolutionSNAP: Segmentation by level set evolutionSNAP (prototype): (prototype):

• 3D level-set evolution

• Preprocessing pipeline and manual editing

• Boundary-driven and region-competition snakes


Segmentation by level set evolution (midag.cs.unc.edu)

Segmentation by level set evolution (midag.cs.unc.edu)


Extraction of anatomical models: SNAP Tool: 3D Geodesic SnakeSegmentation by 3D level set evolution:• region-competition & boundary driven snake• manual interaction for initialization and postprocessing (IRIS)

free dowload: midag.cs.unc.edu


Modeling of Caudate ShapeModeling of Caudate Shape

M-rep

PDM

Surface Parametrization


Parametrized 3D surface modelsParametrized 3D surface models

Raw 3D voxel model

Ch. Brechbuehler, G. Gerig and O. Kuebler, Parametrization of closed surfaces for 3-D shape description, CVIU, Vol. 61, No. 2, pp. 154-170, March 1995

A. Kelemen, G. Székely, and G. Gerig, Three-dimensional Model-based Segmentation, IEEE TMI, 18(10):828-839, Oct. 1999

Smoothed object Parametrized surface


Surface ParametrizationSurface Parametrization

Mapping single faces to spherical quadrilaterals

Latitude and longitude from diffusion


Initial ParametrizationInitial Parametrization

a) Spherical parameter space with surface net, b) cylindrical projection, c) object with coordinate grid.

Problem: Distortion / Inhomogeneous distribution


Parametrization after OptimizationParametrization after Optimization

a) Spherical parameter space with surface net, b) cylindrical projection, c) object with coordinate grid.

After optimization: Equal parameter area of elementary surface facets, reduced distortion.


Optimization: Nonlinear / ConstraintsOptimization: Nonlinear / Constraints


Shape Representation by Spherical Harmonics (SPHARM)

Shape Representation by Spherical Harmonics (SPHARM)

),(

),(

),(

),(

z

y

x

r

),(),(0

K

k

k

km

mk

mk Ycr

mzk

myk

mxk

mk

c

c

c

c


Reconstruction from coefficientsReconstruction from coefficients

Global shape description by expansion into spherical harmonics: Reconstruction of the partial spherical harmonic series, using coefficients up to degree 1 (a), to degree 3 (b) and 7 (c).


Importance of uniform parametrizationImportance of uniform parametrization


Parametrization with spherical harmonicsParametrization with spherical harmonics

1

3

7

12


Correspondence through Normalization Correspondence through Normalization

Normalization using first order ellipsoid:

• Spatial alignment to major axes

• Rotation of parameter space.


3D Natural Shape Variability: Left Hippocampus of 90 Subjects

3D Natural Shape Variability: Left Hippocampus of 90 Subjects


Computing the statistical model: PCAComputing the statistical model: PCA


Major Eigenmodes of Deformation by PCAMajor Eigenmodes of Deformation by PCA

PCA of parametric shapes PCA of parametric shapes Average Shape, Major Average Shape, Major EigenmodesEigenmodes

Major Eigenmodes of Major Eigenmodes of Deformation define shape Deformation define shape space space expected variability. expected variability.


3D Eigenmodes of Deformation3D Eigenmodes of Deformation


Set of Statistical Anatomical ModelsSet of Statistical Anatomical Models


Correspondence through parameter space rotation

Normalization using first order ellipsoid:

•Rotation of parameter space to align major axis

•Spatial alignment to major axes

Parameters rotated to first order ellipsoids


Correspondence ctd.Correspondence ctd.

Rhodri Davies and Chris Rhodri Davies and Chris TaylorTaylor

• MDL criterion applied to shape population

• Refinement of correspondence to yield minimal description

• 83 left and right hippocampal surfaces

• Initial correspondence via SPHARM normalization

• IEEE TMI August 2002


Correspondence ctd.Correspondence ctd.

Homologous points before (blue) and after MDL refinement (red).

MSE of reconstructed vs. original shapes using n Eigenmodes (leave one out). SPHARM vs. MDL correspondence.


Model BuildingModel Building

Medial Medial representation representation for shape for shape populationpopulation

Styner, Gerig et al. , MMBIA’00 / IPMI 2001 / MICCAI 2001 / CVPR 2001/ MEDIA 2002 / IJCV 2003 /

VSkelTool


VSkelToolPhD Martin StynerVSkelToolPhD Martin Styner

Surface

PDM

Voronoi

Voronoi+M-rep

M-rep

M-rep

M-rep+Radii

Implied Bdr

Caudate

Population models:

•PDM

•M-rep


II: Medial Models for Shape AnalysisII: Medial Models for Shape Analysis

Medial Medial representation representation for shape for shape populationpopulation

Styner and Gerig, MMBIA’00 / IPMI 2001 / MICCAI 2001 / CVPR 2001/ ICPR 2002


Common model generationCommon model generation

Training populationTraining population

CommonCommon modelmodel

Study population

...

Two Shape Analyses - New insights, findings

Model building

Boundary: SPHARMMedial: m-rep


1. Shape space from training population1. Shape space from training population

• Variability from training populationVariability from training population• Major PCA deformations define Major PCA deformations define

shape space covering 95%shape space covering 95%• Variability is smoothed Variability is smoothed • Sample objects from shape spaceSample objects from shape space

1.1.

2.2.

3.3.


2. Common medial branching topology2. Common medial branching topology

a. Compute a. Compute individual medial individual medial branching branching topologies in topologies in shape spaceshape space

b. Combine medial b. Combine medial branching branching topologies into topologies into one common one common branching branching topologytopology


2a. Single branching topology2a. Single branching topology

Fine sampling of Fine sampling of boundaryboundary

Compute inner Compute inner Voronoi diagramVoronoi diagram

Group vertices into Group vertices into medial sheets medial sheets (Naef)(Naef)

Remove Remove unimportant unimportant medial sheets medial sheets (Pruning)(Pruning)

98% vol. overlap98% vol. overlap


2b. Common branching topology 2b. Common branching topology

Define common frame for spatial Define common frame for spatial comparisoncomparison

TPS-warp objects into common TPS-warp objects into common frame using boundary frame using boundary correspondencecorrespondence

Spatial match of sheets, paired Spatial match of sheets, paired Mahalanobis distanceMahalanobis distance

No structural (graph) topology No structural (graph) topology matchmatch

Warp topology Warp topology using SPHARM using SPHARM correspondence correspondence

on boundaryon boundary

MatchMatch MatchMatch

Match Match whole whole shape shape spacespace

Initial topology Initial topology (average case)(average case)

For all For all objects in objects in shape shape spacespace

Final topologyFinal topology


3. Optimal grid sampling of medial sheets3. Optimal grid sampling of medial sheets

Appropriate sampling Appropriate sampling for modelfor model

How to sample a How to sample a sheet ?sheet ?

Compute minimal grid Compute minimal grid parameters for parameters for sampling given sampling given predefined predefined approximation error approximation error in shape spacein shape space


3a. Sampling of medial sheet3a. Sampling of medial sheet

Smoothing of Smoothing of sheet edgesheet edge

Determine medial Determine medial axis of sheetaxis of sheet

Sample axisSample axisFind grid edgeFind grid edgeInterpolate restInterpolate rest m-rep fit to object m-rep fit to object

(Joshi)(Joshi)

1

2


3b. Minimal sampling of medial sheet3b. Minimal sampling of medial sheet

2x62x6 3x63x6 3x73x7 3x123x12 4x124x12

• Find minimal sampling given a predefined approximation errorFind minimal sampling given a predefined approximation error

norm. MAD error vs sampling

0.14

0.08

0.0530.048

0.075

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

2x6 3x6 3x7 3x12 4x12

MA

D /

AV

G(r

ad

ius)


Medial models of subcortical structuresMedial models of subcortical structures

Shapes with common m-rep model and implied boundaries Shapes with common m-rep model and implied boundaries of putamen, hippocampus, and lateral ventricles. of putamen, hippocampus, and lateral ventricles.

Each structure has a single-sheet branching topology. Each structure has a single-sheet branching topology.

Medial representations calculated automatically.Medial representations calculated automatically.


Medial models of subcortical structuresMedial models of subcortical structures

Shapes with common topology: M-rep and implied boundaries of putamen, hippocampus, and lateral ventricles.

Medial representations calculated automatically (goodness of fit criterion).

building of statistical models

Documents

d shape description

d surface modelsraw

d level set evolution

d natural shape variability

d mesh points

d geodesic snakesegmentation

d mri data

d voxel modelch