Enhanced Correspondence and Statistical Analysis Methods for Structural Shape Analysis

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

2. Concept: Shape Analysis • Traditional analysis: Regional volume • Our view: Analysis of local shape Volumetric analysis: Size, Growth Statistical analysis Shape Representation Binary Segmentation

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 • Surface & Parameterization • Fit coefficients of parameterized basis functions to surface • Sample parameterization and reconstruct object • Hierarchical description 1 3 6 10

5. Surface SPHARM Parametrization 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

6. Parameterization based Correspondence • SPHARM • Can also be used as initialization of other methods • Optimization of spherical parametrization • Optimize over (,), 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 Based • 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

9. Correspondence Evaluation • How can we evaluate correspondence? • Comparison to manual landmarks • Selection variability quite large • Experts disagree on landmark placement • 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 head Styner, 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 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 Femur Lateral Ventricle 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. Correction 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

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. S0 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 Sj # perm Sj