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Object Orie’d Data Analysis, Last Time
• SiZer Analysis– Zooming version, -- Dependent
version
– Mass flux data, -- Cell cycle data
• Image Analysis– 1st Generation -- 2nd Generation
• Object Representation– Landmarks
– Boundaries
– Medial
OODA in Image Analysis
First Generation Problems:
• Denoising
• Segmentation (find object
boundaries)
• Registration (align objects)
(all about single images)
OODA in Image Analysis
Second Generation Problems:
• Populations of Images
– Understanding Population Variation
– Discrimination (a.k.a.
Classification)
• Complex Data Structures (& Spaces)
• HDLSS Statistics
Image Object Representation
Major Approaches for Images:
• Landmark Representations
• Boundary Representations
• Medial Representations
Landmark RepresentationsLandmarks for fly wing data:
Landmark Representations
Major Drawback of Landmarks:
• Need to always find each landmark
• Need same relationship
• I.e. Landmarks need to correspond
• Often fails for medical images
• E.g. How many corresponding landmarks on a set of kidneys, livers or brains???
Boundary Representations
Major sets of ideas:
• Triangular Meshes– Survey: Owen (1998)
• Active Shape Models– Cootes, et al (1993)
• Fourier Boundary Representations– Keleman, et al (1997 & 1999)
Boundary Representations
Example of triangular mesh rep’n:
From:www.geometry.caltech.edu/pubs.html
Boundary RepresentationsMain Drawback:
Correspondence
• For OODA (on vectors of parameters):
Need to “match up points”
• Easy to find triangular mesh– Lots of research on this driven by gamers
• Challenge to match mesh across objects– There are some interesting ideas…
Medial RepresentationsMain Idea: Represent Objects as:• Discretized skeletons (medial atoms)• Plus spokes from center to edge• Which imply a boundary
Very accessible early reference:• Yushkevich, et al (2001)
Medial Representations2-d M-Rep Example: Corpus Callosum(Yushkevich)
Medial Representations2-d M-Rep Example: Corpus Callosum(Yushkevich)
AtomsSpokesImpliedBoundary
Medial Representations3-d M-Rep Example: From Ja-Yeon Jeong
Bladder – Prostate - Rectum
Atoms - Spokes - Implied Boundary
Medial Representations3-d M-reps: there are several variations
Two choices:From Fletcher(2004)
Medial RepresentationsStatistical Challenge
• M-rep parameters are:– Locations– Radii– Angles (not comparable)
• Stuffed into a long vector• I.e. many direct products of
these
32 , 0
Medial RepresentationsStatistical Challenge:• How to analyze angles as data?• E.g. what is the average of:
– ??? (average of the numbers)– (of course!)
• Correct View of angular data:Consider as points on the unit circle
1811
359,358,4,3
Medial RepresentationsWhat is the average (181o?) or (1o?) of:
359
,358
,4
,3
Medial RepresentationsStatistical Challenge• Many direct products of:
– Locations– Radii– Angles (not comparable)
• Appropriate View:Data Lie on Curved Manifold
Embedded in higher dim’al Eucl’n Space
32 , 0
Medial RepresentationsData on Curved Manifold Toy Example:
Medial RepresentationsData on Curved Manifold Viewpoint:• Very Simple Toy Example (last movie)• Data on a Cylinder = • Notes:
– Simplest non-Euclidean Example– 2-d data, embedded on manifold in – Can flatten the cylinder, to a plane– Have periodic representation– Movie by: Suman Sen
• Same idea for more complex direct prod’s
11 S
3R
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)
Tail Bone
Rectum
Prostate
Male Pelvis – Raw Data
Prostate:
manual segmentation
Slice by slice
Reassembled
Male Pelvis – Raw Data
Prostate:
Slices:
Reassembled in 3d
How to represent?
Thanks: Ja-Yeon Jeong
Object Representation
• Landmarks (hard to find)
• Boundary Rep’ns (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-repsM-rep model fitting
• Easy, when starting from binary (blue)
• But very expensive (30 – 40 minutes technician’s 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
Mildly Non-Euclidean Spaces
Statistical Analysis of M-rep DataRecall: Many direct products of:• Locations• Radii• Angles I.e. points on smooth manifold
Data in non-Euclidean SpaceBut only mildly non-Euclidean
Mildly Non-Euclidean Spaces
Good source for statistical analysis ofMildly non-Euclidean Data
Fletcher (2004), Fletcher, et al (2004)Main ideas:• Work with geodesic distances• I.e. distances along surface of
manifold
Mildly Non-Euclidean Spaces
What is the mean of data on a manifold?• Bad choice:
– Mean in embedded space– Since will probably leave manifold– Think about unit circle
• How to improve?• Approach study characterizations of
mean– There are many– Most fruitful: Frechét mean
Mildly Non-Euclidean Spaces
Fréchet mean of numbers:
Fréchet mean in Euclidean Space:
Fréchet mean on a manifold:Replace Euclidean by Geodesic
n
ii
xxXX
1
2minarg
d
n
ii
x
n
ii
xxXdxXX
1
2
1
2,minargminarg
d
Mildly Non-Euclidean Spaces
Fréchet Mean:• Only requires a metric (distance) space• Geodesic distance gives geodesic
mean
Well known in robust statistics:• Replace Euclidean distance• With Robust distance, e.g. with• Reduces influence of outliers• Gives another notion of robust median
2L 1L
Mildly Non-Euclidean Spaces
E.g. Fréchet Mean for data on a circle
Mildly Non-Euclidean Spaces
E.g. Fréchet Mean for data on a circle:• Not always easily interpretable
– Think about “distances along arc”– Not about “points in ”– Sum of squared distances “strongly feels the
largest”
• Not always unique– But unique “with probability one” – Non-unique requires strong symmetry– But possible to have many means
2
Mildly Non-Euclidean Spaces
E.g. Fréchet Mean for data on a circle:• Not always sensible notion of center
– Sometimes prefer “top & bottom”?– At end: farthest points from data
• Not continuous Function of Data– Jump from 1 – 2– Jump from 2 – 8
• All false for Euclidean Mean• But all happen generally for manifold data
Mildly Non-Euclidean Spaces
E.g. Fréchet Mean for data on a circle:• Also of interest is Fréchet Variance:
• Works like sample variance• Note values in movie, reflecting spread in
data• Note theoretical version:
• Useful for Laws of Large Numbers, etc.
n
iixxXd
n 1
22 ,1
min̂
22 ,min xXdEXx
Mildly Non-Euclidean Spaces
Useful Viewpoint for data on manifolds:• Tangent Space• Plane touching at one point• At which point?
Geodesic (Fréchet) Mean
Hence terminology “mildly non-Euclidean”
(pic next page)
Mildly Non-Euclidean Spaces
Pics from: Fletcher (2004)
Mildly Non-Euclidean Spaces
“Exponential Map” Terminology:From Complex Exponential Function
Exponential Map:
In Tangent Space On
Manifold
ie sincos i
Mildly Non-Euclidean Spaces
Exponential Map TerminologyMemory Trick:• Exponential Map
Tangent Plane Curved Manifold
• Log Map (Inverse)Curved Manifold Tangent Plane
Mildly Non-Euclidean Spaces
Analog of PCA?Principal geodesics (PGA):• Replace line that best fits data• By geodesic that best fits the data
(geodesic through Fréchet mean)• Implemented as PCA in tangent
space• But mapped back to surface• Fletcher (2004)
PGA for m-reps, Bladder-Prostate-Rectum
Bladder – Prostate – Rectum, 1 person, 17 days
PG 1 PG 2 PG 3
(analysis by Ja Yeon Jeong)
PGA for m-reps, Bladder-Prostate-Rectum
Bladder – Prostate – Rectum, 1 person, 17 days
PG 1 PG 2 PG 3
(analysis by Ja Yeon Jeong)
PGA for m-reps, Bladder-Prostate-Rectum
Bladder – Prostate – Rectum, 1 person, 17 days
PG 1 PG 2 PG 3
(analysis by Ja Yeon Jeong)
Mildly Non-Euclidean Spaces
Other Analogs of PCA???• Why pass through geodesic mean?• Sensible for Euclidean space• But obvious for non-Euclidean?Perhaps “geodesic that explains data as
well as possible” (no mean constraint)?
• Does this add anything?• All same for Euclidean case
(since least squares fit contains mean)
Mildly Non-Euclidean Spaces
E.g. PGA on the unit sphere:
Unit Sphere
Data
Mildly Non-Euclidean Spaces
E.g. PGA on the unit sphere:
Unit Sphere
Data
Geodesic Mean
Mildly Non-Euclidean Spaces
E.g. PGA on the unit sphere:
Unit Sphere
Data
Geodesic Mean
PG 1
Mildly Non-Euclidean Spaces
E.g. PGA on the unit sphere:
Unit Sphere
Data
Geodesic Mean
PG 1
Best Fit Geodesic
Mildly Non-Euclidean Spaces
E.g. PGA on the unit sphere:
Which is “best”?• Perhaps best fit?• What about PG2?
– Should go through geo mean?
• What about PG3?– Should cross PG1 & PG2 at same point?– Need constrained optimization
• Gaussian Distribution on Manifold???