Title

Title

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

Bladder Prostate Rectum How do they move over time (days)? (all within same person) A Challenging Example Male Pelvis

Bladder Prostate Rectum How do they move over time (days)? Critical to Radiation Treatment (cancer) A Challenging Example Male Pelvis

Bladder Prostate Rectum How do they move over time (days)? Critical to Radiation Treatment (cancer) Work with 3-d CT (Computed Tomography, = 3d version of X-ray) A Challenging Example Male Pelvis

Bladder Prostate Rectum How do they move over time (days)? Critical to Radiation Treatment (cancer) Work with 3-d CT

Very Challenging to Segment Find boundary of each object? Represent each Object? Male Pelvis Raw Data One CT Slice (in 3d image) Like X-ray: White = Dense

(Bone) Black = Gas Male Pelvis Raw Data One CT Slice (in 3d image) Tail Bone Male Pelvis Raw Data One CT Slice (in 3d image) Tail Bone Rectum

Male Pelvis Raw Data One CT Slice (in 3d image) Tail Bone Rectum Bladder Male Pelvis Raw Data One CT Slice (in 3d image) Tail Bone Rectum

Bladder Prostate Male Pelvis Raw Data Bladder: manual segmentation Slice by slice Reassemble d Male Pelvis Raw Data Bladder:

Slices: Reassembled in 3d How to represent? Thanks: Ja-Yeon Jeong Object Representation Landmarks (hard to find) Object Representation

Landmarks (hard to find) Boundary Repns (no correspondence) Object Representation Landmarks (hard to find)

Boundary Repns (no correspondence) Medial representations Find skeleton Discretize as atoms called M-reps

3-d m-reps Bladder Prostate Rectum (multiple objects, J. Y. Jeong) Medial Atoms provide skeleton Implied Boundary from spokes surface

3-d m-reps M-rep model fitting Easy, when starting from binary (blue) 3-d m-reps M-rep model fitting Easy, when starting from binary (blue)

But very expensive (30 40 minutes technicians time) Want automatic approach 3-d m-reps M-rep model fitting Easy, when starting from binary (blue)

But very expensive (30 40 minutes technicians time) Want automatic approach Challenging, because of poor contrast, noise,

Need to borrow information across training sample 3-d m-reps M-rep model fitting Easy, when starting from binary (blue) But very expensive (30 40 minutes technicians time)

Want automatic approach Challenging, because of poor contrast, noise, Need to borrow information across training sample

Use Bayes approach: prior & likelihood posterior (A surrogate for atomical knowledge) 3-d m-reps M-rep model fitting Easy, when starting from binary (blue) But very expensive (30 40 minutes technicians time)

Want automatic approach Challenging, because of poor contrast, noise, Need to borrow information across training sample

Use Bayes approach: prior & likelihood posterior ~Conjugate Gaussians (Embarassingly Straightforward?) 3-d m-reps M-rep model fitting

Easy, when starting from binary (blue) But very expensive (30 40 minutes technicians time) Want automatic approach Challenging, because of poor contrast, noise,

Need to borrow information across training sample Use Bayes approach: prior & likelihood posterior ~Conjugate Gaussians, but there are issues:

Major HLDSS challenges Manifold aspect of data 3-d m-reps M-rep model fitting Very Successful

Jeong (2009) 3-d m-reps M-rep model fitting Very Successful Jeong (2009)

Basis of Startup Company: Morphormics Mildly Non-Euclidean Spaces Statistical Analysis of M-rep Data Recall: Many direct products of: Locations

Radii Angles Mildly Non-Euclidean Spaces Statistical Analysis of M-rep Data Recall: Many direct products of: Locations Radii Angles I.e. points on smooth manifold

Mildly Non-Euclidean Spaces Statistical Analysis of M-rep Data Recall: Many direct products of: Locations Radii Angles I.e. points on smooth manifold Data in non-Euclidean Space But only mildly non-Euclidean PCA for m-reps, II

UNC, Stat & OR PCA on non-Euclidean spaces? (i.e. on Lie Groups / Symmetric Spaces) 28 PCA for m-reps, II UNC, Stat & OR PCA on non-Euclidean spaces? (i.e. on Lie Groups / Symmetric

Spaces) T. Fletcher: Principal Geodesic Analysis (2004 UNC CS PhD Dissertation) 29 PCA for m-reps, II UNC, Stat & OR PCA on non-Euclidean spaces? (i.e. on Lie Groups / Symmetric Spaces)

T. Fletcher: Principal Geodesic Analysis Idea: replace linear summary of data With geodesic summary of data 30 PGA for m-reps, Bladder-ProstateRectum UNC, Stat & OR Bladder Prostate Rectum, 1 person, 17 days

PG 1 PG 2 PG 3 (analysis by Ja Yeon Jeong) 31 PGA for m-reps, Bladder-ProstateRectum UNC, Stat & OR

Bladder Prostate Rectum, 1 person, 17 days PG 1 PG 2 PG 3 (analysis by Ja Yeon Jeong) 32 PGA for m-reps, Bladder-ProstateRectum

UNC, Stat & OR Bladder Prostate Rectum, 1 person, 17 days PG 1 PG 2 PG 3 (analysis by Ja Yeon Jeong) 33

PCA Extensions for Data on Manifolds UNC, Stat & OR Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data 34 PCA Extensions for Data on Manifolds UNC, Stat & OR Fletcher (Principal Geodesic Anal.)

Best fit of geodesic to data Constrained to go through geodesic mean 35 PCA Extensions for Data on Manifolds UNC, Stat & OR Geodesic Anal.) Fletcher (Principal Best fit of geodesic to data

Constrained to go through geodesic mean Happens mean Naturally contained in

in best fit line 36 PCA Extensions for Data on Manifolds UNC, Stat & OR Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic mean But Not In

Non-Euclidean Spaces 37 PCA Extensions for Data on Manifolds UNC, Stat & OR Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic mean Counterexample:

Data on sphere, along equator 38 PCA Extensions for Data on Manifolds UNC, Stat & OR Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic mean Extreme

3 Point Examples 39 PCA Extensions for Data on Manifolds UNC, Stat & OR Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic mean Huckemann, Hotz & Munk (Geod.

PCA) 40 PCA Extensions for Data on Manifolds UNC, Stat & OR Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic mean Huckemann, Hotz & Munk (Geod.

PCA) Best fit of any geodesic to data 41 PCA Extensions for Data on Manifolds UNC, Stat & OR Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic mean

Huckemann, Hotz & Munk (Geod. PCA) Best fit of any geodesic to data Counterexample: 42 PCA Extensions for Data on Manifolds UNC, Stat & OR Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic

mean Huckemann, Hotz & Munk (Geod. PCA) Best fit of any geodesic to data Jung, Foskey & Marron (Princ. Arc Anal.) 43 PCA Extensions for Data on Manifolds UNC, Stat & OR

Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic mean Huckemann, Hotz & Munk (Geod. PCA) Best fit of any geodesic to data Jung, Foskey & Marron (Princ. Arc Anal.) Best fit of any circle to data

44 PCA Extensions for Data on Manifolds UNC, Stat & OR 45 Principal Arc Analysis UNC, Stat & OR Jung, Foskey & Marron (2011) Best fit of any circle to data Can give better fit than geodesics

46 Principal Arc Analysis UNC, Stat & OR Jung, Foskey & Marron (2011) Best fit of any circle to data Can give better fit than geodesics Observed for simulated m-rep example 47

Challenge being addressed UNC, Stat & OR 48 Composite Nested Spheres UNC, Stat & OR Arc Analysis Idea: Use Principal Over Large Products of

49 Landmark Based Shape Analysis UNC, Stat & OR Kendall Bookstein Dryden & Mardia (recall major monographs) 50

Landmark Based Shape Analysis UNC, Stat & OR Kendall Bookstein Dryden & Mardia Digit 3 Data 51 Landmark Based Shape Analysis

UNC, Stat & OR Kendall Bookstein Dryden & Mardia Digit 3 Data 52 Landmark Based Shape Analysis UNC, Stat & OR

Kendall Bookstein Dryden & Mardia Digit 3 Data (digitized to 13 landmarks) 53 Landmark Based Shape Analysis UNC, Stat & OR Key Step: mod out

Translation Scaling Rotation 54 Landmark Based Shape Analysis UNC, Stat & OR Key Step: mod out Translation Scaling Rotation

Result: Data Objects points on Manifold ( ~ S2k) 4 55 Landmark Based Shape Analysis UNC, Stat & OR Currently popular approaches to PCA on Sk:

Early: PCA on projections (Tangent Plane Analysis) 56 Landmark Based Shape Analysis UNC, Stat & OR Currently popular approaches to PCA on Sk: Early: PCA on projections Fletcher: Geodesics through mean

57 Landmark Based Shape Analysis UNC, Stat & OR Currently popular approaches to PCA on Sk: Early: PCA on projections Fletcher: Geodesics through mean Huckemann, et al: Any Geodesic

58 Landmark Based Shape Analysis UNC, Stat & OR Currently popular approaches to PCA on Sk: Early: PCA on projections Fletcher: Geodesics through mean Huckemann, et al: Any Geodesic

New Approach: Principal Nested Sphere Analysis Jung, Dryden & Marron (2012) 59 Principal Nested Spheres Analysis UNC, Stat & OR Main Goal: Extend Principal Arc Analysis (S2 to Sk)

60

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