Enhanced Correspondence and Statistics for Structural Shape Analysis: Current Research Martin Styner Department of Computer Science and Psychiatry.

Enhanced Correspondence and Statistics for Structural

Shape Analysis: Current Research

Martin StynerDepartment of Computer Science and Psychiatry

2

Concept: Shape Analysis

• Traditional analysis: Regional volume

• Our view: Analysis of local shape

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Binary Segmentation

Volumetric analysis: Size, Growth

Shape Representation Statistical analysis

3

Geometric Correspondence

• Template/Model fit– Fit a model to the data, model bias– m-rep, deformation fields

• Pair-wise optimization– Template/Model bias– Many PDM based analysis methods

• Object inherent– No bias, fully independent– SPHARM

• Population-wise optimization– No template, population vs. single object– MDL, DetCovar

4

SPHARM: Spherical Harmonics

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

),(

),(

),(

),(

φθ

φθ

φθ

φθ

z

y

x

r

),(),(0

φθφθ ∑∑= −=

=K

k

k

km

mk

mk Ycr

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=mzk

myk

mxk

mk

c

c

c

c

1

10

3

6

1. Surface & Parameterization2. Fit coefficients of parameterized

basis functions to surface3. Sample parameterization and

reconstruct object• Hierarchical description

5

Correspondence: SPHARM

• Correspondence by same parameterization– Area ratio preserving through optimization– Location of meridian and equator ill-defined

• Poles and Axis of first order ellipsoid• Object specific, independent, good initial correspondence

Surface

Parametrization

SPHARM

6

Parameterization based Correspondence • SPHARM

– Can also be used as initialization of other methods

• Optimization of spherical parametrization– Optimize over (φ,), evaluate on surface – Template matching

• Surface geometry: Curvature + Location• Meier, Medical Image Analysis 02

– Population based:• Optimization of location/coordinate distribution• Davies, TMI 02• Our current research (Ipek Oguz)

– Fusion with SPHARM and surface geometry, fusion of all 3 methods

7

Population Based – Davies

• Optimization using parameterization• Initialization with SPHARM parameterization

8

• Population Criterions: MDL & DetCov• MDL = Minimum Description Length

– In terms of shape modeling: Cost of transmitting the coded point location model (in number of bits)

• DetCov = log determinant of covariance matrix– Compactness of model

• Criterions very similar• MDL expensive computation

Population Based

9

Correspondence Evaluation

• How can we evaluate correspondence?1. Comparison to manual landmarks

• Selection variability quite large• Experts disagree on landmark placement

2. Correspondence quality measurements

• Best metric for evaluation => best metric for correspondence definition

• Evaluation in Styner et al, IPMI 2003– Widely cited– Shows need for evaluation and validation

• 2 structures: Lateral ventricle, Femoral headStyner, Rajamani, Nolte, Zsemlye, Szekely, Taylor, Davies: Evaluation of 3D Correspondence Methods for Model Building, IPMI 2003, p 63-75

10

Correspondence Evaluation

• Evaluation based on derived shape space– Principal Component Analysis (PCA) model

• Generalization– Does the model describe new cases well?– Leave-one-out tests (Jack-knife)

• Select a case, remove from training, build model• Check approximation error of removed case

• Specificity– Does the model only represent valid objects? – Create new objects in shape space with Gaussian

sampling• Approximation error to closest sample in training set

11

Correspondence Evaluation

Femur

LateralVentricle

M: number of modes in model

MDL and DetCov are performing the best

MDL has strong statistical bias for shape analysis For shape analysis: optimization and analysis on same features

Styner, Rajamani, Nolte, Zsemlye, Szekely, Taylor, Davies: Evaluation of 3D Correspondence Methods for Model Building, IPMI 2003, p 63-75

12

Population Based Curvature

• Current project in correspondence• Population based better modeling• Surface Geometry no statistical bias• Use of SPHARM efficiency, noise stability• Curvature

– Shape Index S and Curvedness C– SPHARM derivatives

SPHARM first derivatives

13

Statistical Analysis

• Surfaces with– Correspondence – Pose normalized

• Analyze shape feature– Features per surface point– Univariate

• Distance to template– Template bias

• Thickness

– Multivariate• Point locations (x,y,z)• m-rep parameters• Spherical wavelets

14

Hypothesis Testing

• At each location: Hypothesis test– P-value of group mean difference

• Schizophrenia group vs Control group

– Significance map– Threshold α = 5%, 1%, 0.1%

• Parametric: Model of distribution (Gaussian)

• Non-parametric: model free– P-value directly from observed distribution– Distribution estimation via permutation tests

15

Many, Many, Too Many…

• Many local features computed independently– 1000 - 5000 features

• Even if features are pure noise, still many locations are significant

• Overly optimistic Raw p-values • Multiple comparison problem

– P-value correction• False-Positive Error control• False Detection Rate

– General Linear Mixed Modeling• Model covariance structure• Dimensionality reduction• Work with Biostatistics

– MICCAI 2003, M-rep

16

P-value Correction

• Corrected significance map– As if only one test performed

• Bonferroni correction– Global, simple, very pessimistic

– pcorr = p/n = 0.05/1000 = 0.00005

• Non-parametric permutation tests– Minimum statistic of raw p-values

– Global, still pessimistic

Pantazis, Leahy, Nichols, Styner: Statistical Surface Based Morphometry Using a Non-Parametric Approach, ISBI 2004,1283-1286

Styner, Gerig: Correction scheme for multiple correlated statistical tests in local shape analysis, SPIE Medical Imaging 2004, p. 233-240,2004

Correction

17

Ongoing Research

• False Detection Rate (FDR): more relaxed, fMRI, VBM– Currently being added to software

• Program design: Software not based on ITK statistics framework

• Next:– Covariates: No account of covariates– Age, Medication, Gender– General Linear Model, per feature at each location– multivariate analysis of fitted parameters

18

The End

• Questions?

19

Permutation Hypothesis Tests

• Estimate distribution– Permute group labels

• Na , Nb in Group A and B

• Create M permutations

• Compute feature Sj for each perm

• Histogram Distribution• p-value:

#Perms larger / #Perms total

S0

Sj

Sj

perm

#