A Challenging Example Male Pelvis –Bladder – Prostate – Rectum.

Title

A Challenging ExampleMale PelvisBladder Prostate Rectum

1A Challenging ExampleMale PelvisBladder Prostate RectumHow do they move over time (days)?

(all within same person)2A Challenging ExampleMale PelvisBladder Prostate RectumHow do they move over time (days)?Critical to Radiation Treatment (cancer)3A Challenging ExampleMale PelvisBladder Prostate RectumHow do they move over time (days)?Critical to Radiation Treatment (cancer)Work with 3-d CT

(Computed Tomography,= 3d version of X-ray)4A 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?5Male Pelvis Raw DataOne CT Slice(in 3d image)Like X-ray:White = Dense(Bone)Black = Gas

6Male Pelvis Raw DataOne CT Slice(in 3d image)

Tail Bone


Tail BoneRectum


Tail BoneRectumBladder


Tail BoneRectumBladderProstate

10Male Pelvis Raw DataBladder:

manual segmentation

Slice by slice

Reassembled

11Male Pelvis Raw DataBladder:

Slices:Reassembled in 3d

How to represent?

Thanks: Ja-Yeon Jeong

12Object RepresentationLandmarks (hard to find)13Object RepresentationLandmarks (hard to find)Boundary Repns (no correspondence)14Object RepresentationLandmarks (hard to find)Boundary Repns (no correspondence)Medial representationsFind skeletonDiscretize as atoms called M-reps153-d m-reps

Bladder Prostate Rectum (multiple objects, J. Y. Jeong)Medial Atoms provide skeletonImplied Boundary from spokes surface163-d m-repsM-rep model fittingEasy, when starting from binary (blue)173-d m-repsM-rep model fittingEasy, when starting from binary (blue)But very expensive (30 40 minutes technicians time)Want automatic approach183-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 sample193-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

(A surrogate for atomical knowledge)203-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

(Embarassingly Straightforward?)213-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 data223-d m-repsM-rep model fittingVery SuccessfulJeong (2009)233-d m-repsM-rep model fittingVery SuccessfulJeong (2009)Basis of Startup Company: Morphormics

24Mildly Non-Euclidean SpacesStatistical Analysis of M-rep DataRecall: Many direct products of:LocationsRadiiAngles25Mildly Non-Euclidean SpacesStatistical Analysis of M-rep DataRecall: Many direct products of:LocationsRadiiAngles I.e. points on smooth manifold26Mildly 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-Euclidean27PCA for m-reps, IIPCA on non-Euclidean spaces?(i.e. on Lie Groups / Symmetric Spaces)#

UNC, Stat & OR28PCA for m-reps, IIPCA on non-Euclidean spaces?(i.e. on Lie Groups / Symmetric Spaces)

T. Fletcher: Principal Geodesic Analysis

(2004 UNC CS PhD Dissertation)#

UNC, Stat & OR29PCA 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 & OR30PGA for m-reps, Bladder-Prostate-RectumBladder Prostate Rectum, 1 person, 17 days PG 1 PG 2 PG 3(analysis by Ja Yeon Jeong)

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UNC, Stat & OR33PCA Extensions for Data on ManifoldsFletcher (Principal Geodesic Anal.)Best fit of geodesic to data

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UNC, Stat & OR34PCA Extensions for Data on ManifoldsFletcher (Principal Geodesic Anal.)Best fit of geodesic to dataConstrained to go through geodesic mean#

UNC, Stat & OR35PCA Extensions for Data on Manifolds

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UNC, Stat & OR36PCA Extensions for Data on ManifoldsFletcher (Principal Geodesic Anal.)Best fit of geodesic to dataConstrained to go through geodesic mean

But Not InNon-EuclideanSpaces

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UNC, Stat & OR37

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 & OR38PCA Extensions for Data on ManifoldsFletcher (Principal Geodesic Anal.)Best fit of geodesic to dataConstrained to go through geodesic mean

Extreme3 PointExamples

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UNC, Stat & OR39PCA Extensions for Data on ManifoldsFletcher (Principal Geodesic Anal.)Best fit of geodesic to dataConstrained to go through geodesic meanHuckemann, Hotz & Munk (Geod. PCA)

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UNC, Stat & OR40PCA 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 & OR41PCA 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

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UNC, Stat & OR42PCA 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.)#

UNC, Stat & OR43PCA 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 & OR44PCA Extensions for Data on Manifolds

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UNC, Stat & OR45Principal Arc AnalysisJung, Foskey & Marron (2011)Best fit of any circle to dataCan give better fit than geodesics#

UNC, Stat & OR46Principal Arc AnalysisJung, Foskey & Marron (2011)Best fit of any circle to dataCan give better fit than geodesicsObserved for simulated m-rep example

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UNC, Stat & OR47Challenge being addressed

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UNC, Stat & OR48Composite Nested Spheres#

UNC, Stat & OR49Landmark Based Shape AnalysisKendallBooksteinDryden & Mardia

(recall major monographs)#


Digit 3 Data#


Digit 3 Data

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Digit 3 Data(digitized to 13 landmarks)

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UNC, Stat & OR53Landmark Based Shape AnalysisKey Step: mod outTranslationScalingRotation#

UNC, Stat & OR54Landmark Based Shape AnalysisKey Step: mod outTranslationScalingRotationResult: Data Objects points on Manifold ( ~ S2k-4)#

UNC, Stat & OR55Landmark Based Shape AnalysisCurrently popular approaches to PCA on Sk:Early: PCA on projections

(Tangent Plane Analysis)#

UNC, Stat & OR56Landmark Based Shape AnalysisCurrently popular approaches to PCA on Sk:Early: PCA on projectionsFletcher: Geodesics through mean#

UNC, Stat & OR57Landmark Based Shape AnalysisCurrently popular approaches to PCA on Sk:Early: PCA on projectionsFletcher: Geodesics through meanHuckemann, et al: Any Geodesic#

UNC, Stat & OR58Landmark 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 & OR59Principal Nested Spheres AnalysisMain Goal:Extend Principal Arc Analysis (S2 to Sk)

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UNC, Stat & OR60

A Challenging Example Male Pelvis –Bladder – Prostate – Rectum.

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