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Common Property of Shape Data Objects: Natural Feature Space is Curved I.e. a Manifold (from Differential Geometry) Shapes As Data Objects

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Common Property of Shape Data Objects:

Natural Feature Space is CurvedI.e. a Manifold (from Differential Geometry)Shapes As Data Objects1Manifold Feature Spaces

2Standard Statistical Example:

Directional Data (aka Circular Data)Idea: Angles as Data Objects Wind Directions Magnetic Compass Headings Cracks in MinesManifold Feature Spaces3Standard Statistical Example:

Directional Data (aka Circular Data)

Reasonable View:Points on Unit CircleManifold Feature Spaces4

Geodesics:Idea: March Along Manifold Without Turning(Defined in Tangent Plane)Manifold Feature Spaces5Manifold Feature Spaces6Manifold Feature Spaces

7OODA in Image AnalysisFirst Generation Problems:DenoisingSegmentationRegistration

(all about single images,still interesting challenges)8OODA in Image AnalysisSecond Generation Problems:Populations of ImagesUnderstanding Population Variation Discrimination (a.k.a. Classification)Complex Data Structures (& Spaces)HDLSS Statistics

9Image Object RepresentationMajor Approaches for Image Data Objects:Landmark RepresentationsBoundary RepresentationsMedial Representations10Medial RepresentationsMain Idea: Represent Objects as:Discretized skeletons (medial atoms)Plus spokes from center to edgeWhich imply a boundary

Very accessible early reference:Yushkevich, et al (2001)11Medial Representations2-d M-Rep Example: Corpus Callosum(Yushkevich)

12CCMsciznormRaw.aviA Challenging ExampleMale PelvisBladder Prostate RectumHow do they move over time (days)?Critical to Radiation Treatment (cancer)Work with 3-d CTVery Challenging to SegmentFind boundary of each object?Represent each Object?13Male Pelvis Raw DataOne CT Slice(in 3d image)

Tail BoneRectumBladderProstate

143-d m-reps

Bladder Prostate Rectum (multiple objects, J. Y. Jeong)Medial Atoms provide skeletonImplied Boundary from spokes surface153-d m-repsM-rep model fittingEasy, when starting from binary (blue)But very expensive (30 40 minutes technicians time)Want automatic approachChallenging, because of poor contrast, noise, Need to borrow information across training sampleUse Bayes approach: prior & likelihood posterior~Conjugate Gaussians, but there are issues:Major HLDSS challengesManifold aspect of data16Mildly Non-Euclidean SpacesStatistical Analysis of M-rep DataRecall: Many direct products of:LocationsRadiiAngles I.e. points on smooth manifoldData in non-Euclidean SpaceBut only mildly non-Euclidean17PCA for m-reps, IIPCA on non-Euclidean spaces?(i.e. on Lie Groups / Symmetric Spaces)

T. Fletcher: Principal Geodesic Analysis

Idea: replace linear summary of dataWith geodesic summary of data#

UNC, Stat & OR18PCA Extensions for Data on Manifolds

#

UNC, Stat & OR19

PCA Extensions for Data on ManifoldsFletcher (Principal Geodesic Anal.)Best fit of geodesic to dataConstrained to go through geodesic mean

Counterexample:Data on sphere, along equator#

UNC, Stat & OR20PCA Extensions for Data on ManifoldsFletcher (Principal Geodesic Anal.)Best fit of geodesic to dataConstrained to go through geodesic meanHuckemann, Hotz & Munk (Geod. PCA)Best fit of any geodesic to data#

UNC, Stat & OR21PCA Extensions for Data on ManifoldsFletcher (Principal Geodesic Anal.)Best fit of geodesic to dataConstrained to go through geodesic meanHuckemann, Hotz & Munk (Geod. PCA)Best fit of any geodesic to data

Counterexample:Data follows Tropic of Capricorn

#

UNC, Stat & OR22PCA Extensions for Data on ManifoldsFletcher (Principal Geodesic Anal.)Best fit of geodesic to dataConstrained to go through geodesic meanHuckemann, Hotz & Munk (Geod. PCA)Best fit of any geodesic to dataJung, Foskey & Marron (Princ. Arc Anal.)Best fit of any circle to data(motivated by conformal maps)#

UNC, Stat & OR23PCA Extensions for Data on Manifolds

#

UNC, Stat & OR24Principal Arc AnalysisJung, Foskey & Marron (2011)Best fit of any circle to dataCan give better fit than geodesics#

UNC, Stat & OR25Principal Arc AnalysisJung, Foskey & Marron (2011)Best fit of any circle to dataCan give better fit than geodesicsObserved for simulated m-rep example

#

UNC, Stat & OR26Challenge being addressed

#

UNC, Stat & OR27Composite Principal Nested Spheres#

UNC, Stat & OR28Composite Principal Nested Spheres

#

UNC, Stat & OR29Composite Principal Nested Spheres

#

UNC, Stat & OR30Composite Principal Nested Spheres#

UNC, Stat & OR31Composite Principal Nested Spheres#

UNC, Stat & OR32Composite Principal Nested Spheres#

UNC, Stat & OR33Composite Principal Nested Spheres#

UNC, Stat & OR34Composite Principal Nested Spheres#

UNC, Stat & OR35Composite Principal Nested Spheres#

UNC, Stat & OR36Composite Principal Nested Spheres#

UNC, Stat & OR37Composite Principal Nested Spheres#

UNC, Stat & OR38Composite Principal Nested Spheres#

UNC, Stat & OR39Composite Principal Nested Spheres#

UNC, Stat & OR40Composite Principal Nested Spheres#

UNC, Stat & OR41Composite Principal Nested Spheres#

UNC, Stat & OR42Composite Principal Nested Spheres#

UNC, Stat & OR43Composite Principal Nested Spheres#

UNC, Stat & OR44Composite Principal Nested Spheres#

UNC, Stat & OR45Composite Principal Nested Spheres#

UNC, Stat & OR46Landmark Based Shape AnalysisKendallBooksteinDryden & Mardia

(recall major monographs)#

UNC, Stat & OR47Landmark Based Shape AnalysisKendallBooksteinDryden & Mardia

Digit 3 Data#

UNC, Stat & OR48Landmark Based Shape AnalysisKendallBooksteinDryden & Mardia

Digit 3 Data

#

UNC, Stat & OR49Landmark Based Shape AnalysisKendallBooksteinDryden & Mardia

Digit 3 Data(digitized to 13 landmarks)

#

UNC, Stat & OR50Landmark Based Shape AnalysisKey Step: mod outTranslationScalingRotation#

UNC, Stat & OR51Recall Main Idea: Represent Shapes as Coordinates Mod Out Transln, Rotatn, Scale

Variation on Landmark Based Shape#

UNC, Stat & OR52Typical Viewpoint: Variation in Shape is Goal Other Variation+ is Nuisance

Recall Main Idea: Represent Shapes as Coordinates Mod Out Transln, Rotatn, Scale

Variation on Landmark Based Shape#

UNC, Stat & OR53Typical Viewpoint: Variation in Shape is Goal Other Variation+ is Nuisance

Interesting Alternative: Study Variation in Transformation Treat Shape as NuisanceVariation on Landmark Based Shape#

UNC, Stat & OR54Context: Study of Tectonic PlatesMovement of Earths Crust (over time)Take Motions as Data Objects

Interesting Alternative: Study Variation in Transformation Treat Shape as NuisanceVariation on Landmark Based Shape#

UNC, Stat & OR55Context: Study of Tectonic PlatesMovement of Earths Crust (over time)Take Motions as Data Objects

Royer & Chang (1991)

Thanks to Wikipedia

Variation on Landmark Based Shape

#

UNC, Stat & OR56Landmark Based Shape AnalysisKendallBooksteinDryden & Mardia

Digit 3 Data

#

UNC, Stat & OR57Landmark Based Shape AnalysisKey Step: mod outTranslationScalingRotationResult: Data Objects points on Manifold ( ~ S2k-4)#

UNC, Stat & OR58Landmark Based Shape AnalysisCurrently popular approaches to PCA on Sk:Early: PCA on projections

(Tangent Plane Analysis)#

UNC, Stat & OR59Landmark Based Shape AnalysisCurrently popular approaches to PCA on Sk:Early: PCA on projectionsFletcher: Geodesics through mean#

UNC, Stat & OR60Landmark Based Shape AnalysisCurrently popular approaches to PCA on Sk:Early: PCA on projectionsFletcher: Geodesics through meanHuckemann, et al: Any Geodesic#

UNC, Stat & OR61Landmark Based Shape AnalysisCurrently popular approaches to PCA on Sk:Early: PCA on projectionsFletcher: Geodesics through meanHuckemann, et al: Any Geodesic

New Approach:Principal Nested Sphere AnalysisJung, Dryden & Marron (2012)#

UNC, Stat & OR62Principal Nested Spheres AnalysisMain Goal:Extend Principal Arc Analysis (S2 to Sk)

Jung, Dryden & Marron (2012)

#

UNC, Stat & OR63Principal Nested Spheres AnalysisMain Goal:Extend Principal Arc Analysis (S2 to Sk)

#

UNC, Stat & OR64Principal Nested Spheres AnalysisTop Down Nested (small) spheres

#

UNC, Stat & OR65Digit 3 data: Principal variations of the shapePrinc. geodesics by PNSPrincipal arcs by PNS

66


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