Transcript z - MIDAG
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 MIDAG@UNC Real-World Analysis with Images as Data Shape 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 Compare log p(z|I), so log p(z) + log p(I|z) two populations Medical science: localities where p(z|healthy) & p(z|diseased) differ Diagnostic: Is particular patient’s geometry diseased? p(z|healthy, I) vs. p(z|diseased, I) MIDAG@UNC 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 MIDAG@UNC 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 MIDAG@UNC Representations z of Deformation All 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) MIDAG@UNC 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 MIDAG@UNC Standard Method of Making Big Problems Small via Statistics: Principal Component Analysis (PCA) D m e.g., D=Tn with T 3 Linear Statistics (PCA) •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. MIDAG@UNC 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 MIDAG@UNC Landmarks as Representation z First historically Kendall, Bookstein, Dryden & Mardia, Joshi Landmarks defined by special properties Won’t 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 MIDAG@UNC 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) MIDAG@UNC 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 MIDAG@UNC B-rep via F(x)’s level set: z is F Objective: to handle topological variability Run nonlinear diffusion to achieve deformation Fit to training cases easy via distance Correspondence? Topology change 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?? MIDAG@UNC 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 MIDAG@UNC 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 cf. 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 MIDAG@UNC 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 MIDAG@UNC 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?? MIDAG@UNC 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 MIDAG@UNC 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 u Tn , n local T’s, a symmetric space t v MIDAG@UNC 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) MIDAG@UNC Principal Geodesic Analysis (PGA) [Fletcher] Tn with T 3 × + × S2 × S2 Linear Statistics (PCA) Curved Statistics (PGA) Mean: closest to data in square distance. Principal direction submanifold: through mean; closest to data in square distance. MIDAG@UNC 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 MIDAG@UNC 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 MIDAG@UNC 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 MIDAG@UNC 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: <5 min. today, ~10 seconds in new version MIDAG@UNC 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 MIDAG@UNC 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? MIDAG@UNC 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 MIDAG@UNC 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 MIDAG@UNC 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. MIDAG@UNC 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. MIDAG@UNC