2019 Triangle Machine Learning Day - Machine Learning for 3D Imaging - Sayan Mukherjee, September 20, 2019

Machine Learning to Model 3D Shapes
Sayan Mukherjee, Duke University
Triangle Machine Learning Day
https://sayanmuk.github.io/
Euler representation of shapes — J. Curry, K. Turner, D. Boyer
Shapes as covariates — L. Crawford, A. Monod, R. Rabad´an, A. Chen
Variable selection on shapes — L. Crawford, T. Gao, H. Kirveslahti,
T. Sudijono, B. Wang

Modeling variation in shapes
S. J. Gould

Variation in calcanei
D. Boyer.

Models of surfaces
(1) Shape spaces: Kendall, D. G. (1984) Shape manifolds,
procrustean metrics, and complex projective spaces. Bull.
Lond. Math. Soc., 16, 81-121.

Models of surfaces
Lond. Math. Soc., 16, 81-121.
(2) Lie group: Dupuis, P. & Grenander, U. (1998) Variational
problems on ﬂows of di↵eomorphisms for image matching. Q.
Appl. Math., LVI, 587-600.

Models of surfaces
Lond. Math. Soc., 16, 81-121.
(2) Lie group: Dupuis, P. & Grenander, U. (1998) Variational
problems on ﬂows of di↵eomorphisms for image matching. Q.
Appl. Math., LVI, 587-600.
(3) Integral geometry: Worsley, K. J. (1995) Estimating the
number of peaks in a random ﬁeld using the Hadwiger
characteristic of excursion sets, with applications to medical
images. Ann. Stat., 23, 640-669.

Landmarks
Landmarks are points of correspondence on each object that
matches between and within populations.

Procrustes distance and shape space
For an object i: L landmarks (o`,i )L
`=1 with each o`,i 2 IRd
.

.
The procrustes distance between two points is
dp(oi , oj ) = min
T2T
1
L
LX
`=1
(o`,i To`,j )2
,
T are rotations, translations, and scalings.

.
The procrustes distance between two points is
dp(oi , oj ) = min
T2T
1
L
LX
`=1
(o`,i To`,j )2
,
T are rotations, translations, and scalings.
Kendall’s shape space: Md,` are d ⇥ ` matrices
F`
d := Md,` {0}, S`
d := {x 2 F`
d : kxk = 1}

3D image repositories
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Four bones of a new species of Homo from South Africa.
See all the bones of the newly described Homo naledi
Read the published article

Diffeomorphism based approach
Similarity between teeth: Algorithms to automatically quantify the
geometric similarity of anatomical surfaces, Boyer et. al. PNAS
2011.

Fly wings are not homeomorphic
Persistent homology Bar codes Brain trees Fly wings Stratified persistence Metric perversity Future biology Stratified statistics Nex
Fruit fly wings
Normal fly wings [photos from David Houle’s lab]:
Topologically abnormal veins:

3D mesh data
❖ Improved imaging technologies allow 3D shapes to be represented
as meshes --- a collection of vertices (V), faces (F), and edges (E).
[Boyer et al. (2011); Crawford et al. (2019)]
Ventricles
Tumor

Our objective
Model shapes without requiring landmarks or di↵eomorphisms.

Our objective
Transform the data/object into a representation that can be
modeled using standard methods.

Our objective
Desired properties
(1) The transformation is injective, ideally the summary statistic
is su cient.

Our objective
Desired properties
is su cient.
(2) The transformed space is nice (may be a matrix). Can
compute distances or place probability models in the
transformed space.

Our objective
Desired properties
is su cient.
(2) The transformed space is nice (may be a matrix). Can
compute distances or place probability models in the
transformed space.
(3) Pull back from the transformed space to positions on shapes.

Two topological summaries
(1) Euler characteristics.

Two topological summaries
(1) Euler characteristics.
(2) Persistent homology.

Euler characteristic
For a mesh M in 3 dimensions the Euler characteristic is
(M) = #vertices #edges + #faces.

Euler characteristic curve
(a) (b) (c)
1: (a) sublevel set of a mouse embryo head; (b) EC curve of the 2D contour of a han
associated (centered) cumulative EC-curve.
teristic transform (ECT). Leveraging these topological transforms, we propose to d
dology based on several variants of the ECT to address a range of problems involving
Mao Li

Euler curve
Filtration
Sublevel Sets 
(Filtration Steps)
EulerCharacteristic
[Turner et al. (2014); Crawford et al. (2019)]

Euler curve
Filtration
Sublevel Sets 
(Filtration Steps)
EulerCharacteristic
[Turner et al. (2014); Crawford et al., (2019)]

Euler curve
Filtration
Sublevel Sets 
(Filtration Steps)
EulerCharacteristic
[Turner et al. (2014); Crawford et al. (2019)]
❖ Concatenate curves over all
directions to obtain a vector
representation of the shape.
❖ End result: A matrix where each
row is the concatenated EC curve
of one shape in our dataset.

Homology
Homology is “cycles modulo boundaries”.
0 = 1, 1 = 2, 2 = 1.

Persistent homology: Morse theory
Evolution of homology as birth-death pair.
0 2⇡
Dgm0(f)
birth
death
f 1
(( 1, a])

0 2⇡
f 1
(( 1, a])
a
Dgm0(f)
birth
death

0 2⇡
f 1
(( 1, a])
a
Dgm0(f)
birth
death
1

0 2⇡
f 1
(( 1, a])
a
Dgm0(f)
birth
death
}Persistence

Euler characteristic transform (ECT)
For a ﬁxed M 2 CS(IRd
) the ECT is a map from the sphere to the
space of Euler curves
ECT(M) : Sd 1
! CF(IR)
v 7! (M, v).
where
ECT(M)(v)(t) := (M {x | x · v  t}).

Smooth Euler characteristic transform (SECT)
The smooth Euler curve for each direction is
f (y) = (M, v), F(x) =
Z x
0
f (y)dy (M, v).

Smooth Euler characteristic transform (SECT)
The smooth Euler curve for each direction is
f (y) = (M, v), F(x) =
Z x
0
f (y)dy (M, v).
Deﬁnition
The Euler characteristic transform of M 2 IRd
is the function
(M) : Sd 1
! L2(IR)
v 7! F(M, v).

Persistence homology transform (PHT)
For a ﬁxed M 2 CS(IRd
) the PHT is a map from the sphere to
persistence diagrams by ﬁltering M in the direction v
PHT(M) : Sd 1
! Dgmd
v 7! (PH0(M, v), PH1(M, v), . . . , PHd 1(M, v)).

Inversion theorem (Schapira)
Theorem (Schapira 1991)
If S ⇢ X ⇥ Y and S0 ⇢ Y ⇥ X have ﬁbers Sx and S0
x in Y
satisfying
1. (Sx S0
x ) = 1 for all x 2 X, and
2. (Sx S0
x0 ) = 2 for all x0 6= x 2 X,
then for all 2 CF(X),
(RS0 RS ) = ( 1 2) + 2
✓Z
X
d
◆
1X .

Injectivity of the ECT and PHT
Theorem (Turner-M-Boyer,Curry-M-Turner, Ghrist-Levanger-Mai)
The Euler characteristic transform CS(IRd
) ! CF(Sd 1 ⇥ IR) is
injective.
Theorem (Turner-M-Boyer,Curry-M-Turner, Ghrist-Levanger-Mai)
The persistent homology transform CS(IRd
) ! C0(Sn 1, Dgmd
) is
injective.

A sampling theory for shapes
How many directions to sample ?

(1) For 2-D: 162 directions
(2) For 3-D: Over 700 directions.

(1) For 2-D: 162 directions
(2) For 3-D: Over 700 directions.
A sampling theory for shapes — complexity metric for families
shapes in terms of directions required.

Calculus of snakes
V.I. Arnol’d, The calculus of snakes and the combinatorics of
Bernoulli, Euler and Springer numbers of Coxeter groups.

Moduli spaces of shapes
We now consider M(d, , k ) as the set of all embedded simplicial
complexes K in IRd
with the following two properties:

complexes K in IRd
(1) At every vertex x 2 K there is a lower bound on curvature .

complexes K in IRd
(1) At every vertex x 2 K there is a lower bound on curvature .
(2) For all v 2 Sd 1, the number of x 2 X for which hw (x) is an
Euler critical value of hw for some w 2 B(v, ) is bounded by
k .

Bound on the number of directions
Theorem (Curry-M-Turner)
Any shape in M(d, , k ) can be determined using the ECT or the
PHT using no more than
(d, , k ) = (d 1)k
✓
2
sin( )
◆d 1
+ 1
! ✓
1 +
2
◆d
+ O
✓
dk
d 1
◆2d
directions.

Picture of heel bone
Figure: Images of a calcaneus from two different angles.

106 primates
−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
Aye−aye
Ring tail
Howler
Spider
Saki
Squirrel
Macaque
Baboon
Gorilla
Gibbon
Chimp
Orang
Meso
Omomyid
Tetonius

Primate calcanei
Phylogenetic groups of primate calcanei with 67 genera. Asterisks
indicate groups of extinct taxa. Abbreviations: Str, Strepsirrhines;
Plat, platyrrhines; Cerc, Cercopithecoids; Om, Omomyiforms; Adp,
Adapiforms; Pp, parapithecids; Hmn, Hominoids.

Glioblastoma and radiogenomics

The data
92 patients with matched gene expression and MRI data from the
TCGA.
Gene expression: pg = 9215
Morphometric features: pm = 212
Volumetric features: pv = 5
Topological features: ps = 7200

The data
TCGA.
Response:
Disease Free Survival (DFS): The period after a successful
treatment during which there are no signs or symptoms of the
cancer that was treated.

The data
TCGA.
Response:
Disease Free Survival (DFS): The period after a successful
treatment during which there are no signs or symptoms of the
cancer that was treated.
Overall Survival (OS): The entire period after the start of
treatment during which the cancer patient is still alive.

The question I
Which of the following best explains variation in the clinical trait:
Gene expression features
Morphometric features from radiologists
Geometric summaries (5)
Topological features (SECT)

The question I
Which of the following best explains variation in the clinical trait:
Gene expression features
Morphometric features from radiologists
Geometric summaries (5)
Topological features (SECT)
Using shapes in regression models.

SECT curves for a tumor slice
0 20 40 60 80 100
3.54.04.55.05.56.0
Filtration Steps
SmoothEulerCharacteristic
0 20 40 60 80 100
3.54.04.55.05.56.0
Filtration Steps
0 20 40 60 80 100
3.54.04.55.05.56.0
Filtration Steps
⋮
Direction
ν = 1
Direction
⋮
Direction
ν = 36
ν = 72
Sublevel Sets

The model
Consider the following kernel model
Ef (X) =
nX
i=1
↵i k (X, Xi ) .
Can use standard functional data analysis.

Gaussian process regression
Deﬁnition
A stochastic process over domain X with mean function µ and
covariance kernel k is a Gaussian process if and only if for any
{x1, ..., xn} 2 X and n 2 N the following distribution holds
f =
0
B
@
f (x1)
...
f (xn)
1
C
A ⇠ N
0
B
@
2
6
4
µ(x1)
...
µ(xn)
3
7
5 ,
2
6
4
k(x1, x1) · · · k(x1, xn)
...
...
...
k(x1, xn) · · · k(xn, xn)
3
7
5
1
C
A .

Gaussian process regression
Deﬁnition
A stochastic process over domain X with mean function µ and
covariance kernel k is a Gaussian process if and only if for any
{x1, ..., xn} 2 X and n 2 N the following distribution holds
f =
0
B
@
f (x1)
...
f (xn)
1
C
A ⇠ N
0
B
@
2
6
4
µ(x1)
...
µ(xn)
3
7
5 ,
2
6
4
k(x1, x1) · · · k(x1, xn)
...
...
...
k(x1, xn) · · · k(xn, xn)
3
7
5
1
C
A .
Given observed data (X, Y) and new point(s) X⇤
f⇤
| X⇤
, X, Y ⇠ N(µ⇤
, ⌃⇤
),
µ⇤
= k(X⇤
, X)(k(X, X) + 2
I) 1
Y
⌃⇤
= k(X⇤
, X⇤
) + 2
I k(X⇤
, X)(k(X, X) + 2
I) 1
k(X, X⇤
).

Kernels: similarity measures
Linear similarity (kernel): k(U, V) =
P
t,` Ut`Vt`

Kernels: similarity measures
Linear similarity (kernel): k(U, V) =
P
t,` Ut`Vt`
Distance based kernels:
d2
(U, V) = kU V)k2
=
X
t`
|Ut` Vt`|2
Gaussian: k(U, V) = exp(  d2(U, V))
Cauchy: k(U, V) = 1
⇡(1+ d2(U,V))
.

Subimage selection: question II
What parts of the shape are most associated to variation in trait ?
This is a variable selection problem in the regression framework.

Sub-Image Analysis
using Topological
Summary Statistics
(SINATRA)

Pipeline
(a) Input 3D Shapes
Data from Species #1
Data from Species #2
(b) Derive Topological Summary Statistics
(c) Variable Selection and Reconstruction
(d) Visualize Enrichment
Enrichment in Species #1
Enrichment in Species #2
Outcome: yi = 1
Outcome: yi = 0
−0.4 −0.2 0.0 0.2 0.4
−0.50.00.51.0
Filtration Steps
TopologicalSummaryStatistics
−0.4 −0.2 0.0 0.2 0.4
−0.50.00.51.0
Filtration Steps
−0.4 −0.2 0.0 0.2 0.4
−0.50.00.51.0
Filtration Steps
−0.4 −0.2 0.0 0.2 0.4
−0.50.00.51.0
Filtration Steps
−0.4 −0.2 0.0 0.2 0.4
−0.50.00.51.0
Filtration Steps
D
irection
(i) (ii) (iii) (iv) (v)
EulerCharacteristic
EulerCharacteristic
EulerCharacteristic
EulerCharacteristic
EulerCharacteristic
Filtration Steps Filtration Steps Filtration Steps Filtration Steps Filtration Steps
Cone (θ)
(1)
(2)
(3)
(1)
(2)
(3)
Projections
Directions
× 10−3
θ : Angle
Steps in Filtration Process:
× 10−3
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
25.0
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
25.0
0 2 4 6 8 10
−4−2024
Filtration Steps
Species #1 Species #2 Evidence
Filtration Steps
EulerCharacteristic
024
logicalSummaryStatistics
4

Components of pipeline
Represent shapes via statistics summarizing
their topology/geometry; 
Use a statistical model and classify shapes
based on these summary statistics; 
Derive an “evidence of association” metric
for each topological/geometric feature; 
Project these association measures back onto
the original shape.

Linear vs nonlinear models
Linear Models
Nonlinear
20%
Additive
40%
Environ.
40%
Nonlinear
20%
Additive
40%
Environ.
40%
Nonlinear Models

E↵ect size analog
❖ A regression model is takes the form: 
 
 
❖ An effect size is the linear projection
onto the phenotype: 
 
 
❖ One standard projection operation is
uses generalized inverses:
b = Proj(X, y)<latexit sha1_base64="+e1WfXgDIIIJhPPxsSvtDIqriQs=">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</latexit><latexit sha1_base64="+e1WfXgDIIIJhPPxsSvtDIqriQs=">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</latexit><latexit sha1_base64="+e1WfXgDIIIJhPPxsSvtDIqriQs=">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</latexit><latexit sha1_base64="+e1WfXgDIIIJhPPxsSvtDIqriQs=">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</latexit>
Proj(X, y) = X†
y<latexit sha1_base64="8DoNCkCvI50ErX01d4fO7nra3UQ=">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</latexit><latexit sha1_base64="8DoNCkCvI50ErX01d4fO7nra3UQ=">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</latexit><latexit sha1_base64="8DoNCkCvI50ErX01d4fO7nra3UQ=">AAACSHicbZBLS8QwFIXT8T2+Rl26CQ6CikjrRjeC6MblCI4OTMchTW9rNE1LcisOpT/PjUt3/gY3LhRxZ2Yc8Xkh8HHOvTfJCTIpDLrug1MZGR0bn5icqk7PzM7N1xYWT02aaw5NnspUtwJmQAoFTRQooZVpYEkg4Sy4Ouz7Z9egjUjVCfYy6CQsViISnKGVurWuH0AsVMGkiNVGWfURbrBo6PSyXBtwEBWtcvMTe+U63aNfxnnhhyyOQZdfHVUfVPi5sFuru1vuoOhf8IZQJ8NqdGv3fpjyPAGFXDJj2p6bYadgGgWXYJfnBjLGr1gMbYuKJWA6xSCIkq5aJaRRqu1RSAfq94mCJcb0ksB2JgwvzG+vL/7ntXOMdjuFUFmOoPjHRVEuKaa0nyoNhQaOsmeBcS3sWym/YJpxtNlXbQje7y//hdPtLc/y8XZ9/2AYxyRZJitkjXhkh+yTI9IgTcLJLXkkz+TFuXOenFfn7aO14gxnlsiPqlTeAcUXtLI=</latexit><latexit sha1_base64="8DoNCkCvI50ErX01d4fO7nra3UQ=">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</latexit>
y = X + "<latexit sha1_base64="6YbKVsoJgQ/XkyQoq5pNnzjTtT8=">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</latexit><latexit sha1_base64="6YbKVsoJgQ/XkyQoq5pNnzjTtT8=">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</latexit><latexit sha1_base64="6YbKVsoJgQ/XkyQoq5pNnzjTtT8=">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</latexit><latexit sha1_base64="6YbKVsoJgQ/XkyQoq5pNnzjTtT8=">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</latexit>
y = f + "<latexit sha1_base64="nfST0q5jQGZw64glHtSawXRbYE8=">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</latexit><latexit sha1_base64="nfST0q5jQGZw64glHtSawXRbYE8=">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</latexit><latexit sha1_base64="nfST0q5jQGZw64glHtSawXRbYE8=">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</latexit><latexit sha1_base64="nfST0q5jQGZw64glHtSawXRbYE8=">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</latexit>
e = Proj(X, f)<latexit sha1_base64="qi14uSqxHOq6SW/RMRXlzqBHFL4=">AAACSHicbZBNSzMxEMez9b2PL1WPXoJF8Hl4kN1e9CKIXjxWsFrolpLNztZoNlmSWbUs+/G8ePTmZ/DiQRFvprWCbwMhP/4zk8z8o0wKi75/71UmJqemZ2bnqn/mFxaXassrJ1bnhkOLa6lNO2IWpFDQQoES2pkBlkYSTqOLg2H+9BKMFVod4yCDbsr6SiSCM3RSr9YLI+gLVTAp+upfWQ2vRAwoZAxFGGkZ20HqLseArCzpLg0RrrFoGn1ebo44Sop2+f8Dk/JvNQQVfzzYq9X9LX8U9CcEY6iTcTR7tbsw1jxPQSGXzNpO4GfYLZhBwSW4AXMLGeMXrA8dh4qlYLvFyIiSbjglpok27iikI/VzR8FSO9zIVaYMz+z33FD8LdfJMdnpFkJlOYLi7x8luaSo6dBVGgsDHOXAAeNGuFkpP2OGcXTeV50JwfeVf8JJYytwfNSo7+2P7Zgla2SdbJKAbJM9ckiapEU4uSEP5Ik8e7feo/fivb6XVrxxzyr5EpXKG4VttII=</latexit><latexit sha1_base64="qi14uSqxHOq6SW/RMRXlzqBHFL4=">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</latexit><latexit sha1_base64="qi14uSqxHOq6SW/RMRXlzqBHFL4=">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</latexit><latexit sha1_base64="qi14uSqxHOq6SW/RMRXlzqBHFL4=">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</latexit>
Proj(X, f) = X†
f<latexit sha1_base64="s7JfYzjlIsNaTovOq8LVQSAWdx0=">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</latexit><latexit sha1_base64="s7JfYzjlIsNaTovOq8LVQSAWdx0=">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</latexit><latexit sha1_base64="s7JfYzjlIsNaTovOq8LVQSAWdx0=">AAACSHicbZBLS8QwFIXT8T2+Rl26CQ6CikjrRjeC6MblCI4OTMchTW9rNE1LcisOpT/PjUt3/gY3LhRxZ2Yc8Xkh8HHOvTfJCTIpDLrug1MZGR0bn5icqk7PzM7N1xYWT02aaw5NnspUtwJmQAoFTRQooZVpYEkg4Sy4Ouz7Z9egjUjVCfYy6CQsViISnKGVurWuH0AsVMGkiNVGWfURbrBo6PSyXBtwEBWtcvMTo3Kd7tEv47zwQxbHoMuvjqoPKvxc2K3V3S13UPQveEOok2E1urV7P0x5noBCLpkxbc/NsFMwjYJLsMtzAxnjVyyGtkXFEjCdYhBESVetEtIo1fYopAP1+0TBEmN6SWA7E4YX5rfXF//z2jlGu51CqCxHUPzjoiiXFFPaT5WGQgNH2bPAuBb2rZRfMM042uyrNgTv95f/wun2lmf5eLu+fzCMY5IskxWyRjyyQ/bJEWmQJuHkljySZ/Li3DlPzqvz9tFacYYzS+RHVSrvhuW0jA==</latexit><latexit sha1_base64="s7JfYzjlIsNaTovOq8LVQSAWdx0=">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</latexit>
Linear Models Nonlinear Models

E↵ect size analog
 
 
 
 
Proj(X, y) = X†
Proj(X, f) = X†
 
 
 
 
Proj(X, y) = X†
Proj(X, f) = X†
 
 
❖ An effect size analog is the projection
onto the smooth nonlinear function: 
 
 
❖ We may use the same standard
projection operations:

E↵ect size analog
 
 
 
 
Proj(X, y) = X†
Proj(X, f) = X†
 
 
 
 
Proj(X, y) = X†
Proj(X, f) = X†
 
 
❖ An effect size analog is the projection
onto the smooth nonlinear function: 
 
 
❖ We can use the same standard
projection operations:

Posterior inference and sampling
Assume we have completely speciﬁed hierarchical model
y = f + ", f ⇠ N(0, K), " ⇠ N(0, ⌧2
I), ⌧2
⇠ Scale-Inv- 2
(a, b).
MCMC for this regression model includes:
1. f | ⌧2
, y ⇠ N(m⇤
, V⇤
) where m⇤
= K(K+⌧2
I) 1
y and V⇤
= K K(K+
⌧2
I) 1
K;
2. ⌧2
| f, y ⇠ Scale-Inv- 2
(a⇤
, b⇤
) where a⇤
= a + n and b⇤
= a⇤ 1
[ab + (y
f)|
(y f)];
3. e = X†
f.<latexit sha1_base64="cEFju3J7/n5EKhPNyui0BMTfEXM=">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</latexit><latexit sha1_base64="cEFju3J7/n5EKhPNyui0BMTfEXM=">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</latexit><latexit sha1_base64="cEFju3J7/n5EKhPNyui0BMTfEXM=">AAAFlHicnVRbb9owFE5b2Dp2K520l71YK5NogQp42bSL1A1NWoVadVpLKxGoHPsAVh0nsx0YyvxD97pfModbKB17mF9yfC6fz/f5xF7ImdLV6q+Nza1M9t797Qe5h48eP3m6k99tqSCSBC5IwAN55WEFnAm40ExzuAolYN/jcOndNJL45RCkYoE41+MQOj7uC9ZjBGvrus5vxa4ImKAgNPqoVOQDGgEa4CEgEvghBw18jFQIxBYBRQMGEksysAAc+QEFnnM96DMRY8764sDkXB/rgdeLxwZ9QPNNz6AScr2AUzX27Sd2h1hCqBgPhCkj93uE6VKyq5g/2dlT4lNTnEeqpjw3m2Z/UbcG918oGkfd+txxvIQ1DSSVGn7o+JuthcqxGFaMa1l360Vc9vYPcy4IuqB80jhpoF4gkR4whST0JahE8alAiAnCIwrq7VwqEFZniTVYtZgGHxWWqJNA0GkX5VTKdVR80z1YpLXsZr+ARgOQkEImKUs30UyLm6Z0R4duXKmZ9NwCwoKmWK1VrMr/wDZN4d2C+DRzQnqhwSrxdTdhuXvLlPGkO1wSs7a96b4bH9jT29grFVPgSnqc7c49N2tinaVeR4yCZpxCfGvkPNDYLE/7lbGIFPf7IE0KVZhNTXr71zt71cPqZKG7Rm1m7DmzdXad33BcGhBbLzThWKl2rRrqToylZoQn4xQpCDG5wX1oW1NgH1QnnjwTBr2yHjqZ015gf/eJd7kixr5KONnMpGm1Gkucf4u1I91704mZCCMNgkwP6kUc6QAlbw6iTALR9h2hDBPJbK+IDLDERNuXKZdzKfQmExC7ZfQTuWXLQ8DIvj8+TpznJnaZsLl2AhLNaqsK3TVa9cOatb/W944+zdTbdl44L52iU3NeO0fOF+fMuXDI1u/Mdiaf2c0+z77PNrKfp6mbG7OaZ86tlT39A2Rr5Lo=</latexit><latexit sha1_base64="cEFju3J7/n5EKhPNyui0BMTfEXM=">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</latexit>
1. f | ⌧2
, y ⇠ N(µ, ⌃);
2. ⌧2
| f, y ⇠ Scale-Inv- 2
(a, b);
3. e = X†
f<latexit sha1_base64="AfC2KGyo6xzjXCplgwA+FaVRtdw=">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</latexit>
Deterministic Step

Centrality and ranking variables
β
p(β)
Full Team
β
p(β)
Full Team
IR: Worst Player
β
p(β)
Full Team IR: Best Player

2019 Triangle Machine Learning Day - Machine Learning for 3D Imaging - Sayan Mukherjee, September 20, 2019

2019 Triangle Machine Learning Day - Machine Learning for 3D Imaging - Sayan Mukherjee, September 20, 2019

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2019 Triangle Machine Learning Day - Machine Learning for 3D Imaging - Sayan Mukherjee, September 20, 2019