Guido Gerig UNC, September 2002 1 Building of statistical models Guido Gerig Guido Gerig Department of Computer Science, UNC, Department of Computer Science, UNC, Chapel Hill Chapel Hill
Jan 15, 2016
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
Guido Gerig UNC, September 2002 2
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)
Guido Gerig UNC, September 2002 3
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).
Guido Gerig UNC, September 2002 4
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
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Segmentation by level set evolution (midag.cs.unc.edu)
Segmentation by level set evolution (midag.cs.unc.edu)
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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
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Modeling of Caudate ShapeModeling of Caudate Shape
M-rep
PDM
Surface Parametrization
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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
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Surface ParametrizationSurface Parametrization
Mapping single faces to spherical quadrilaterals
Latitude and longitude from diffusion
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Initial ParametrizationInitial Parametrization
a) Spherical parameter space with surface net, b) cylindrical projection, c) object with coordinate grid.
Problem: Distortion / Inhomogeneous distribution
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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.
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Optimization: Nonlinear / ConstraintsOptimization: Nonlinear / Constraints
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Shape Representation by Spherical Harmonics (SPHARM)
Shape Representation by Spherical Harmonics (SPHARM)
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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).
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Importance of uniform parametrizationImportance of uniform parametrization
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Parametrization with spherical harmonicsParametrization with spherical harmonics
1
3
7
12
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Correspondence through Normalization Correspondence through Normalization
Normalization using first order ellipsoid:
• Spatial alignment to major axes
• Rotation of parameter space.
Guido Gerig UNC, September 2002 22
3D Natural Shape Variability: Left Hippocampus of 90 Subjects
3D Natural Shape Variability: Left Hippocampus of 90 Subjects
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Computing the statistical model: PCAComputing the statistical model: PCA
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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.
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3D Eigenmodes of Deformation3D Eigenmodes of Deformation
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Set of Statistical Anatomical ModelsSet of Statistical Anatomical Models
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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
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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
Guido Gerig UNC, September 2002 34
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.
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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
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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
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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
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Common model generationCommon model generation
Training populationTraining population
CommonCommon modelmodel
Study population
...
Two Shape Analyses - New insights, findings
Model building
Boundary: SPHARMMedial: m-rep
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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.
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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
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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
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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
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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
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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
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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)
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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.
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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).