MIDAG@UNCMIDAG@UNC

StatisticsStatistics of Anatomic Geometry of Anatomic Geometry

StatisticsStatistics of Anatomic Geometry of Anatomic Geometry

Stephen Pizer, Kenan ProfessorStephen Pizer, Kenan ProfessorMedical Image Display & Analysis GroupMedical Image Display & Analysis Group

University of North CarolinaUniversity of North CarolinaThis tutorial and other relevant papers can be found at This tutorial and other relevant papers can be found at

website: midag.cs.unc.eduwebsite: midag.cs.unc.edu

Faculty: me, Ian Dryden, P. Thomas Fletcher, Faculty: me, Ian Dryden, P. Thomas Fletcher, Xavier Pennec, Sarang Joshi, Carole TwiningXavier Pennec, Sarang Joshi, Carole Twining

Stephen Pizer, Kenan ProfessorStephen Pizer, Kenan ProfessorMedical Image Display & Analysis GroupMedical Image Display & Analysis Group

University of North CarolinaUniversity of North CarolinaThis tutorial and other relevant papers can be found at This tutorial and other relevant papers can be found at

website: midag.cs.unc.eduwebsite: midag.cs.unc.edu

Faculty: me, Ian Dryden, P. Thomas Fletcher, Faculty: me, Ian Dryden, P. Thomas Fletcher, Xavier Pennec, Sarang Joshi, Carole TwiningXavier Pennec, Sarang Joshi, Carole Twining

MIDAG@UNCMIDAG@UNC

Geometry of Geometry of Objects in Populations Objects in Populations via representations via representations zz

Geometry of Geometry of Objects in Populations Objects in Populations via representations via representations zz

Uses for probability density p(z)Sampling p(z) to communicate

anatomic variability in atlasesIssue: geometric propriety of samples?

Log prior in posterior optimizing deformable model segmentation = registration Optimizez p(z|I),

so log p(z) + log p(I|z)Or E(z|I)

Uses for probability density p(z)Sampling p(z) to communicate

anatomic variability in atlasesIssue: geometric propriety of samples?

Log prior in posterior optimizing deformable model segmentation = registration Optimizez p(z|I),

so log p(z) + log p(I|z)Or E(z|I)

MIDAG@UNCMIDAG@UNC

Geometry of Geometry of Objects in Populations Objects in Populations via representations via representations zz

Geometry of Geometry of Objects in Populations Objects in Populations via representations via representations zz

Uses for probability density p(z)Compare two populations

Medical science Hypothesis testing with null hypothesis p(z|

healthy) = p(z|diseased) If null hypothesis is not accepted, find

localities where probability densities differ and characterization of shape difference

Diagnostic: Is particular patient’s geometry diseased? p(z|healthy, I) vs. p(z|diseased, I)

Uses for probability density p(z)Compare two populations

Medical science Hypothesis testing with null hypothesis p(z|

healthy) = p(z|diseased) If null hypothesis is not accepted, find

localities where probability densities differ and characterization of shape difference

Diagnostic: Is particular patient’s geometry diseased? p(z|healthy, I) vs. p(z|diseased, I)

MIDAG@UNCMIDAG@UNC

Needs of Geometric Representation Needs of Geometric Representation zz & Probability Representation p(& Probability Representation p(zz) )

Needs of Geometric Representation Needs of Geometric Representation zz & Probability Representation p(& Probability Representation p(zz) )

Accurate p(p(zz) ) estimation with limited samples, limited samples, i.e., bi.e., beat High Dimension Low Sample Size (HDLSS: many features, few training cases) Measure of predictive strength of representation and

statistics [Muller]:

where “^” indicates projection onto training data principal space

Primitives’ positional correspondence; cases alignment Easy fit of z to each training segmentation or image

Accurate p(p(zz) ) estimation with limited samples, limited samples, i.e., bi.e., beat High Dimension Low Sample Size (HDLSS: many features, few training cases) Measure of predictive strength of representation and

statistics [Muller]:

where “^” indicates projection onto training data principal space

Primitives’ positional correspondence; cases alignment Easy fit of z to each training segmentation or image

k

trainingtestk

k

trainingtestk zzdzzd

222 ,/,ˆ

MIDAG@UNCMIDAG@UNC

Needs of Geometric Representation Needs of Geometric Representation zz & Probability Representation p(& Probability Representation p(zz) )

Needs of Geometric Representation Needs of Geometric Representation zz & Probability Representation p(& Probability Representation p(zz) )

Make significant geometric effects intuitive Null probabilities for geometrically

illegal objects Localization Handle multiple objects

and interstitial regions Speed and space

Make significant geometric effects intuitive Null probabilities for geometrically

illegal objects Localization Handle multiple objects

and interstitial regions Speed and space

MIDAG@UNCMIDAG@UNC

Schedule of TutorialSchedule of Tutorial Schedule of TutorialSchedule of Tutorial

Object representations (Pizer) PCA, ICA, hypothesis testing, landmark statistics, object-

relative intensity statistics (Dryden) Statistics on Riemannian manfolds, of m-reps & diffusion

tensors, maintaining geometric propriety (Fletcher) Statistics on Riemannian manfolds: extensions and

applications (Pennec) Statistics on diffeomorphisms, groupwise registration,

hypothesis testing on Riemannian manifolds (Joshi) Information theoretic measures on anatomy,

correspondence, ASM, AAM (Twining) Multi-object statistics & segmentation (Pizer)

Object representations (Pizer) PCA, ICA, hypothesis testing, landmark statistics, object-

relative intensity statistics (Dryden) Statistics on Riemannian manfolds, of m-reps & diffusion

tensors, maintaining geometric propriety (Fletcher) Statistics on Riemannian manfolds: extensions and

applications (Pennec) Statistics on diffeomorphisms, groupwise registration,

hypothesis testing on Riemannian manifolds (Joshi) Information theoretic measures on anatomy,

correspondence, ASM, AAM (Twining) Multi-object statistics & segmentation (Pizer)

MIDAG@UNCMIDAG@UNC

Representations Representations zz of Deformation of DeformationRepresentations Representations zz of Deformation of Deformation

LandmarksBoundary of objects (b-reps)

Points spaced along boundaryor Coefficients of expansion in

basis functionsor Function in 3D with level set as

object boundaryDeformation velocity seq. per voxelMedial representation of objects’

interiors (m-reps)

LandmarksBoundary of objects (b-reps)

Points spaced along boundaryor Coefficients of expansion in

basis functionsor Function in 3D with level set as

object boundaryDeformation velocity seq. per voxelMedial representation of objects’

interiors (m-reps)

MIDAG@UNCMIDAG@UNC

Landmarks as Representation z z = (= (pp11, , pp22, …,, …,ppNN))

Landmarks as Representation z z = (= (pp11, , pp22, …,, …,ppNN))

First historically Kendall, Bookstein, Dryden &

Mardia, Joshi Landmarks defined by

special properties Won’t find many accurately in 3D Global Alignment via minimization of

inter-case points distances2

First historically Kendall, Bookstein, Dryden &

Mardia, Joshi Landmarks defined by

special properties Won’t find many accurately in 3D Global Alignment via minimization of

inter-case points distances2

MIDAG@UNCMIDAG@UNC

B-reps as Representation B-reps as Representation zz B-reps as Representation B-reps as Representation zz

Point samples: z = (p1, p2, …,pN) Like landmarks; popular Characterization of local translations of shell Fit to training objects pretty easy Handles multi-object complexes Global Positional correspondence of primitives

Slow reparametrization optimizing p(z) tightness Problems with geometrically improper fits Mesh by adding sample neighbors list

Point, normal samples: z = ([p1,n1],…,[pN,nN]) Easier to avoid geometrically improper fits

Point samples: z = (p1, p2, …,pN) Like landmarks; popular Characterization of local translations of shell Fit to training objects pretty easy Handles multi-object complexes Global Positional correspondence of primitives

Slow reparametrization optimizing p(z) tightness Problems with geometrically improper fits Mesh by adding sample neighbors list

Point, normal samples: z = ([p1,n1],…,[pN,nN]) Easier to avoid geometrically improper fits

MIDAG@UNCMIDAG@UNC

B-reps as Representation B-reps as Representation zz B-reps as Representation B-reps as Representation zz

Basis function coefficients z = (a1, a2, …,aM) with p(u) = k=1

M ak k(u) Achieves geometric propriety Fitting to data well worked out

and programmed Implicit, questionable positional

correspondence Global, Unintuitive Alignment via first ellipsoid

Basis function coefficients z = (a1, a2, …,aM) with p(u) = k=1

M ak k(u) Achieves geometric propriety Fitting to data well worked out

and programmed Implicit, questionable positional

correspondence Global, Unintuitive Alignment via first ellipsoid

7

12

1

Representations via spherical harmonics

MIDAG@UNCMIDAG@UNC

B-rep via F(x)’s level set: zz = F, = F, an imagean image

B-rep via F(x)’s level set: zz = F, = F, an imagean image

Allows topological variability Global Unintuitive, costly in space Fit to training cases easy:

F = signed distance to boundary Modification by geometry limited diffusion Requires nonlinear statistics: not yet well developed Serious problems of geometric propriety if stats on F;

needs stats on PDE for nonlinear diffusion Correspondence? Localization: via spatially varying PDE parameters??

Allows topological variability Global Unintuitive, costly in space Fit to training cases easy:

F = signed distance to boundary Modification by geometry limited diffusion Requires nonlinear statistics: not yet well developed Serious problems of geometric propriety if stats on F;

needs stats on PDE for nonlinear diffusion Correspondence? Localization: via spatially varying PDE parameters??

Topology change

MIDAG@UNCMIDAG@UNC

Deformation velocity sequence for each voxel as representation as representation zz Deformation velocity sequence

for each voxel as representation as representation zz

z = ([v1(i.j), v2(i.j),…,vT(i.j)], (i.j) pixels) Miller, Christensen, Joshi Labels in reference move with deformation Series of local interactions Deformation energy minimization

Fluid flow; pretty slow Costly in space Slow and unsure to fit to

training cases if change from atlas is large

z = ([v1(i.j), v2(i.j),…,vT(i.j)], (i.j) pixels) Miller, Christensen, Joshi Labels in reference move with deformation Series of local interactions Deformation energy minimization

Fluid flow; pretty slow Costly in space Slow and unsure to fit to

training cases if change from atlas is large

MIDAG@UNCMIDAG@UNC

M-reps as Representation M-reps as Representation zz

Represent the Egg, not the EggshellRepresent the Egg, not the EggshellM-reps as Representation M-reps as Representation zz

Represent the Egg, not the EggshellRepresent the Egg, not the Eggshell The eggshell: object boundary primitives The egg: m-reps: object interior

primitives Poor for object that is tube, slab mix Handles multifigure objects and multi-

object complexes Interstitial space??

The eggshell: object boundary primitives The egg: m-reps: object interior

primitives Poor for object that is tube, slab mix Handles multifigure objects and multi-

object complexes Interstitial space??

MIDAG@UNCMIDAG@UNC

A deformable model of the A deformable model of the object interior: the m-repobject interior: the m-rep

A deformable model of the A deformable model of the object interior: the m-repobject interior: the m-rep

Object interior primitives: Object interior primitives: medial atomsmedial atoms

Local displacement, Local displacement,

bending/twisting, swelling: bending/twisting, swelling:

intuitiveintuitive Neighbor geometryNeighbor geometry

Objects, figures, atoms, voxelsObjects, figures, atoms, voxels

Object-relative coordinatesObject-relative coordinates Geometric Geometric

impropriety: impropriety: math math

checkcheck

Object interior primitives: Object interior primitives: medial atomsmedial atoms

Local displacement, Local displacement,

bending/twisting, swelling: bending/twisting, swelling:

intuitiveintuitive Neighbor geometryNeighbor geometry

Objects, figures, atoms, voxelsObjects, figures, atoms, voxels

Object-relative coordinatesObject-relative coordinates Geometric Geometric

impropriety: impropriety: math math

checkcheck

MIDAG@UNCMIDAG@UNC

Medial atom as a nonlinear Medial atom as a nonlinear geometric transformationgeometric transformation

Medial atom as a nonlinear Medial atom as a nonlinear geometric transformationgeometric transformation

Medial atoms carry position, width, Medial atoms carry position, width, 2 orientations2 orientations Local deformation Local deformation T T 33 × × + + × S× S22 × ×

SS22 ( (× × ++ for edge atoms) for edge atoms) From reference atomFrom reference atomHub translation Hub translation × × Spoke mSpoke magnification agnification

in common × Spokein common × Spoke11 rotation × rotation × SpokeSpoke22 rotation (× crest sharpness) rotation (× crest sharpness)

M-rep is n-tuple of medial atomsM-rep is n-tuple of medial atoms TTnn , n local T’s, a curved, symmetric space , n local T’s, a curved, symmetric space

Geodesic distance between atomsGeodesic distance between atoms Nonlinear statistics are requiredNonlinear statistics are required

Medial atoms carry position, width, Medial atoms carry position, width, 2 orientations2 orientations Local deformation Local deformation T T 33 × × + + × S× S22 × ×

SS22 ( (× × ++ for edge atoms) for edge atoms) From reference atomFrom reference atomHub translation Hub translation × × Spoke mSpoke magnification agnification

in common × Spokein common × Spoke11 rotation × rotation × SpokeSpoke22 rotation (× crest sharpness) rotation (× crest sharpness)

M-rep is n-tuple of medial atomsM-rep is n-tuple of medial atoms TTnn , n local T’s, a curved, symmetric space , n local T’s, a curved, symmetric space

Geodesic distance between atomsGeodesic distance between atoms Nonlinear statistics are requiredNonlinear statistics are required

medial atom

edgemedial atom

MIDAG@UNCMIDAG@UNC

Fitting m-reps into training binariesFitting m-reps into training binaries

Optimization penalties Distance between m-rep and

binary image boundaries Irregularity penalty: deviation of

each atom from geodesic average of its neighbors Yields correspondence(?) Avoids geometric impropriety(?)

Interpenetration avoidance Alignment via minimization of

inter-case atoms

geodesic distances2

Optimization penalties Distance between m-rep and

binary image boundaries Irregularity penalty: deviation of

each atom from geodesic average of its neighbors Yields correspondence(?) Avoids geometric impropriety(?)

Interpenetration avoidance Alignment via minimization of

inter-case atoms

geodesic distances2

MIDAG@UNCMIDAG@UNC

Schedule of TutorialSchedule of Tutorial Schedule of TutorialSchedule of Tutorial

Object representations (Pizer) PCA, ICA, hypothesis testing, landmark statistics, object-

relative intensity statistics (Dryden) Statistics on Riemannian manfolds, of m-reps & diffusion

tensors, maintaining geometric propriety (Fletcher) Statistics on Riemannian manfolds: extensions and

applications (Pennec) Statistics on diffeomorphisms, groupwise registration,

hypothesis testing on Riemannian manifolds (Joshi) Information theoretic measures on anatomy,

correspondence, ASM, AAM (Twining) Multi-object statistics & segmentation (Pizer)

Object representations (Pizer) PCA, ICA, hypothesis testing, landmark statistics, object-

relative intensity statistics (Dryden) Statistics on Riemannian manfolds, of m-reps & diffusion

tensors, maintaining geometric propriety (Fletcher) Statistics on Riemannian manfolds: extensions and

applications (Pennec) Statistics on diffeomorphisms, groupwise registration,

hypothesis testing on Riemannian manifolds (Joshi) Information theoretic measures on anatomy,

correspondence, ASM, AAM (Twining) Multi-object statistics & segmentation (Pizer)

MIDAG@UNCMIDAG@UNC

Multi-Object StatisticsMulti-Object StatisticsMulti-Object StatisticsMulti-Object Statistics

Need both Need both Object statisticsObject statistics Inter-object relation statisticsInter-object relation statistics

We choose m-reps because of We choose m-reps because of effectiveness in expressing inter-effectiveness in expressing inter-object geometryobject geometry Medial atoms as transformations of Medial atoms as transformations of

each othereach other Relative positions of boundaryRelative positions of boundary Spokes as normals Spokes as normals Object-relative coordinatesObject-relative coordinates

MIDAG@UNCMIDAG@UNC

Statistics at Any Scale LevelStatistics at Any Scale LevelStatistics at Any Scale LevelStatistics at Any Scale Level

Global: Global: zz By object By object zz11

kk

Object neighbors N(Object neighbors N(zz11kk))

By figure (atom mesh) By figure (atom mesh) zz22kk

Figure neighbors N(Figure neighbors N(zz22kk))

By atom (interior section) By atom (interior section) zz33kk

Atom neighbors N(Atom neighbors N(zz33kk))

By voxel or boundary vertexBy voxel or boundary vertex Voxel neighborsVoxel neighbors N(N(zz44

kk)) Designed for HDLSSDesigned for HDLSS quad-mesh neighbor quad-mesh neighbor

relationsrelations

atom levelatom level voxel levelvoxel level

MIDAG@UNCMIDAG@UNC

Multiscale models of spatial parcelationsMultiscale models of spatial parcelationsMultiscale models of spatial parcelationsMultiscale models of spatial parcelations

Finer parcellation Finer parcellation zzjj as j increases (scale decreases) as j increases (scale decreases) Fuzzy edged apertures Fuzzy edged apertures zzjj

kk, with fuzz (tolerance) , with fuzz (tolerance) decreasing as j increasesdecreasing as j increases

Geometric representation Geometric representation zzjjkk

We use m-reps to represent objects at moderate scale and We use m-reps to represent objects at moderate scale and diffeomorphisms to modify that representation at small scalediffeomorphisms to modify that representation at small scale

Level sets of pseudo-distance functions can represent the Level sets of pseudo-distance functions can represent the variable topology interstitial regionsvariable topology interstitial regions

Provides localization

Finer parcellation Finer parcellation zzjj as j increases (scale decreases) as j increases (scale decreases) Fuzzy edged apertures Fuzzy edged apertures zzjj

kk, with fuzz (tolerance) , with fuzz (tolerance) decreasing as j increasesdecreasing as j increases

Geometric representation Geometric representation zzjjkk

We use m-reps to represent objects at moderate scale and We use m-reps to represent objects at moderate scale and diffeomorphisms to modify that representation at small scalediffeomorphisms to modify that representation at small scale

Level sets of pseudo-distance functions can represent the Level sets of pseudo-distance functions can represent the variable topology interstitial regionsvariable topology interstitial regions

Provides localization

MIDAG@UNCMIDAG@UNC

Statistics of each entity Statistics of each entity in relation to its neighbors at its scale levelin relation to its neighbors at its scale level

Statistics of each entity Statistics of each entity in relation to its neighbors at its scale levelin relation to its neighbors at its scale level

Focus on estimating Focus on estimating p(p(zzjjkk , { , {zzjj

nn: n : n k} k}),),

via probabilities that reflect both inter-via probabilities that reflect both inter-object (region) geometric relationship object (region) geometric relationship and object themselves (also for figures)and object themselves (also for figures) Markov random fieldMarkov random field

Conditional probabilities Conditional probabilities p(p(zzjjkk | { | {zzjj

nn: n : n k} k}) = ) =

p(p(zzjjkk | { | {zzjj

nn: : N( N(zzjjkk)})}) )

Iterative Conditional Modes – convergence Iterative Conditional Modes – convergence to joint mode of to joint mode of p(p(zzjj

kk , { , {zzjjnn: n : n k} | Image k} | Image))

Focus on estimating Focus on estimating p(p(zzjjkk , { , {zzjj

nn: n : n k} k}),),

via probabilities that reflect both inter-via probabilities that reflect both inter-object (region) geometric relationship object (region) geometric relationship and object themselves (also for figures)and object themselves (also for figures) Markov random fieldMarkov random field

Conditional probabilities Conditional probabilities p(p(zzjjkk | { | {zzjj

nn: n : n k} k}) = ) =

p(p(zzjjkk | { | {zzjj

nn: : N( N(zzjjkk)})}) )

Iterative Conditional Modes – convergence Iterative Conditional Modes – convergence to joint mode of to joint mode of p(p(zzjj

kk , { , {zzjjnn: n : n k} | Image k} | Image))

MIDAG@UNCMIDAG@UNC

Representation of multiple objects via Representation of multiple objects via residues from neighbor predictionresidues from neighbor prediction

Representation of multiple objects via Representation of multiple objects via residues from neighbor predictionresidues from neighbor prediction

Inter-entity and inter-scale relation by Inter-entity and inter-scale relation by removal of conditional mean of entity on removal of conditional mean of entity on prediction of its neighbors, then prediction of its neighbors, then probability density on residueprobability density on residue p(p(zzjj

kk | { | {zzjjnn: : N( N(zzjj

kk)})}) = ) = p(p(zzjjkk interpoland interpoland

zzjjkk: from N(: from N(zzjj

kk)})}) ) Restriction of zzjj

kk to its shape space Early coarse-to-fine posterior optimization

segmentation results success- ful, but still under study

Alternative to be explored Canonical correlation

Inter-entity and inter-scale relation by Inter-entity and inter-scale relation by removal of conditional mean of entity on removal of conditional mean of entity on prediction of its neighbors, then prediction of its neighbors, then probability density on residueprobability density on residue p(p(zzjj

kk | { | {zzjjnn: : N( N(zzjj

kk)})}) = ) = p(p(zzjjkk interpoland interpoland

zzjjkk: from N(: from N(zzjj

kk)})}) ) Restriction of zzjj

kk to its shape space Early coarse-to-fine posterior optimization

segmentation results success- ful, but still under study

Alternative to be explored Canonical correlation

MIDAG@UNCMIDAG@UNC

Want more info?Want more info?Want more info?Want more info?

This tutorial, many papers on b-reps, m-reps, diffeomorphism-reps and their statistics and applications can be found at website http://midag.cs.unc.edu

This tutorial, many papers on b-reps, m-reps, diffeomorphism-reps and their statistics and applications can be found at website http://midag.cs.unc.edu

12

MIDAG@UNCMIDAG@UNC