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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
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OODA in Image Analysis
First Generation Problems:
• Denoising
• Segmentation (find object
boundaries)
• Registration (align objects)
(all about single images)
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OODA in Image Analysis
Second Generation Problems:
• Populations of Images
– Understanding Population Variation
– Discrimination (a.k.a.
Classification)
• Complex Data Structures (& Spaces)
• HDLSS Statistics
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Image Object Representation
Major Approaches for Images:
• Landmark Representations
• Boundary Representations
• Medial Representations
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Landmark RepresentationsLandmarks for fly wing data:
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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???
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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)
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Boundary Representations
Example of triangular mesh rep’n:
From:www.geometry.caltech.edu/pubs.html
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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…
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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)
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Medial Representations2-d M-Rep Example: Corpus Callosum(Yushkevich)
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Medial Representations2-d M-Rep Example: Corpus Callosum(Yushkevich)
AtomsSpokesImpliedBoundary
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Medial Representations3-d M-Rep Example: From Ja-Yeon Jeong
Bladder – Prostate - Rectum
Atoms - Spokes - Implied Boundary
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Medial Representations3-d M-reps: there are several variations
Two choices:From Fletcher(2004)
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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
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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
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Medial RepresentationsWhat is the average (181o?) or (1o?) of:
359
,358
,4
,3
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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
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Medial RepresentationsData on Curved Manifold Toy Example:
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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
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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?
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Male Pelvis – Raw Data
One CT Slice
(in 3d image)
Tail Bone
Rectum
Prostate
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Male Pelvis – Raw Data
Prostate:
manual segmentation
Slice by slice
Reassembled
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Male Pelvis – Raw Data
Prostate:
Slices:
Reassembled in 3d
How to represent?
Thanks: Ja-Yeon Jeong
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Object Representation
• Landmarks (hard to find)
• Boundary Rep’ns (no correspondence)
• Medial representations
– Find “skeleton”
– Discretize as “atoms” called M-reps
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3-d m-reps
Bladder – Prostate – Rectum (multiple objects, J. Y. Jeong)
• Medial Atoms provide “skeleton”
• Implied Boundary from “spokes” “surface”
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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
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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
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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
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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
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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
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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
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Mildly Non-Euclidean Spaces
E.g. Fréchet Mean for data on a circle
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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
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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
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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
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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)
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Mildly Non-Euclidean Spaces
Pics from: Fletcher (2004)
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Mildly Non-Euclidean Spaces
“Exponential Map” Terminology:From Complex Exponential Function
Exponential Map:
In Tangent Space On
Manifold
ie sincos i
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Mildly Non-Euclidean Spaces
Exponential Map TerminologyMemory Trick:• Exponential Map
Tangent Plane Curved Manifold
• Log Map (Inverse)Curved Manifold Tangent Plane
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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)
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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)
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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)
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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)
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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)
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Mildly Non-Euclidean Spaces
E.g. PGA on the unit sphere:
Unit Sphere
Data
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Mildly Non-Euclidean Spaces
E.g. PGA on the unit sphere:
Unit Sphere
Data
Geodesic Mean
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Mildly Non-Euclidean Spaces
E.g. PGA on the unit sphere:
Unit Sphere
Data
Geodesic Mean
PG 1
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Mildly Non-Euclidean Spaces
E.g. PGA on the unit sphere:
Unit Sphere
Data
Geodesic Mean
PG 1
Best Fit Geodesic
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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???