Lecture 5: Modeling shapes and surfaces,
projections and transformations
NSF/CBMS Conference
Sayan Mukherjee
Departments of Statistical Science, Computer Science, Mathematics
Duke University
www.stat.duke.edu/⇠sayan
May 31, 2016
Title Text• Body Level One
• Body Level Two
• Body Level Three
• Body Level Four
• Body Level Five
Sufficient statistics forsurfaces
Modeling variation in shapes
S. J. Gould
Modeling variation in shapes
D’Arcy Thompson, On Growth and Form
Variation in calcanei
D. Boyer.
Fly wings
24,
Persistent homology Bar codes Brain trees Fly wings Stratified persistence Metric perversity Future biology Stratified statistics Next steps
Fruit fly wingsNormal fly wings [photos from David Houle’s lab]:
Topologically abnormal veins:
8
Models of surfaces
(1) Shape spaces: Kendall, D. G. (1984) Shape manifolds,procrustean metrics, and complex projective spaces. Bull.Lond. Math. Soc., 16, 81121.
(2) Lie group: Dupuis, P. & Grenander, U. (1998) Variationalproblems on flows of diffeomorphisms for image matching. Q.Appl. Math., LVI, 587600.
(3) Integral geometry: Worsley, K. J. (1995) Estimating thenumber of peaks in a random field using the Hadwigercharacteristic of excursion sets, with applications to medicalimages. Ann.Stat., 23, 640669.
Models of surfaces
(1) Shape spaces: Kendall, D. G. (1984) Shape manifolds,procrustean metrics, and complex projective spaces. Bull.Lond. Math. Soc., 16, 81121.
(2) Lie group: Dupuis, P. & Grenander, U. (1998) Variationalproblems on flows of diffeomorphisms for image matching. Q.Appl. Math., LVI, 587600.
(3) Integral geometry: Worsley, K. J. (1995) Estimating thenumber of peaks in a random field using the Hadwigercharacteristic of excursion sets, with applications to medicalimages. Ann.Stat., 23, 640669.
Models of surfaces
(1) Shape spaces: Kendall, D. G. (1984) Shape manifolds,procrustean metrics, and complex projective spaces. Bull.Lond. Math. Soc., 16, 81121.
(2) Lie group: Dupuis, P. & Grenander, U. (1998) Variationalproblems on flows of diffeomorphisms for image matching. Q.Appl. Math., LVI, 587600.
(3) Integral geometry: Worsley, K. J. (1995) Estimating thenumber of peaks in a random field using the Hadwigercharacteristic of excursion sets, with applications to medicalimages. Ann.Stat., 23, 640669.
Variation in calcanei
D. Boyer.
The data
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Four bones of a new species of Homo from South Africa.
w See all the bones of the newly described Homo naledi
w Read the published article
The data
From distances to trees
Shape statistics
Shape is geometrical information modulo location, scale, androtation.
Landmarks
Landmarks are points of correspondence on each object thatmatches 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) = minT2T
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}
⌃`d := S`
d/SO(d) :=n
[x ] : x 2 S`d such that [x ] = {gx : g 2 SO(d)}
o
.
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) = minT2T
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}
⌃`d := S`
d/SO(d) :=n
[x ] : x 2 S`d such that [x ] = {gx : g 2 SO(d)}
o
.
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) = minT2T
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}
⌃`d := S`
d/SO(d) :=n
[x ] : x 2 S`d such that [x ] = {gx : g 2 SO(d)}
o
.
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) = minT2T
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}
⌃`d := S`
d/SO(d) :=n
[x ] : x 2 S`d such that [x ] = {gx : g 2 SO(d)}
o
.
Kendall’s triangles
Morphology and sufficiency
Distance between heel bones across primates for evolutionaryanalysis.
Algorithms to automatically quantify the geometric similarity ofanatomical surfaces, Boyer et. al. PNAS 2011.
Geometric algorithm
index, and orientation patch count (details in SI Appendix,Materials).
Comparing the Correspondence Maps. Morphometric analyses arebased on the identification of corresponding landmark pointson each of S and S0; the cP algorithm constructs a correspon-dence map a from S to S0. (The correspondence induced bycWn is less smooth and will not be considered here.) For eachlandmark point L on S, we can compare the location on S0 ofits images aðLÞ with the location of the corresponding landmarkpoints L0. Fig. 2 shows that the “propagated” landmarks aðLÞtypically turn out to be very close to those of the observer-deter-mined landmarks L0 (more in SI Appendix, Materials).
An Application. These comparisons show our algorithms capturebiologically informative shape variation. But scientists are inter-ested in more than overall shape! We illustrate how correspon-dence maps could be used to analyze more specific features. Incomparative morphological and phylogenetic studies, anatomicalidentification of certain features (e.g., particular cusps on teeth)
is controversial in some cases; an example of this is the distolin-gual corner of sportive lemur (Lepilemur) lower molars in Dataset(A) (2, 41), illustrated in Fig. 3.
In such controversial cases, transformational homology (42)hypotheses are usually supported by a specific comparative sam-ple or inferred morphocline (2, 43, 44). Lepilemur is thought bysome researchers to lack a cusp known as an entoconid (Fig. 3)but to have a hypertrophied metastylid cusp that “takes the place”of the entoconid (2) in other taxa. Yet, in comparing a Lepilemurtooth to a more “standard” primate tooth, like that ofMicrocebus,both seem to have the same basic cusps; alternatives to the view-point of ref. 2 have therefore also been argued in the literature(41). However, another lemur, Megaladapis (now extinct), argu-ably a closer relative of Lepilemur than Microcebus, has anentoconid that is very small and a metastylid that is rather large,thus providing an evolutionary argument supporting the originalhypothesis. (For more details, see SI Appendix, Materials.) Sucharguments can now be made more precise. We can propagate(as in Fig. 2) landmarks from the Microcebus to the Lepilemurmolar; this direct propagation matches the entoconid cusp ofMicrocebus with the controversial cusp of Lepilemur (Fig. 3,path 1), supporting ref. 41. In contrast, when we propagate land-marks in different steps, either from Microcebus to Megaladapisand then to Lepilemur (Fig. 3, path 2), or through the extinctAdapis and extant Lemur (Fig. 3, path 3), the Lepilemur metas-tylid takes the place of the Microcebus entoconid, supportingref. 2. Automatic propagation of landmarks via mathematicalalgorithms recenters the controversy on the (different) discussionof which propagation channel is most suitable.
Summary and Conclusion. New distances between 2D surfaces,with fast numerical implementations, were shown to lead to fast,landmark-free algorithms that map anatomical surfaces automa-tically to other instances of anatomically equivalent surfaces, in a
Table 2. Success rates (percentage) of leave-one-out classification, based on the cP, cWn, and ODLP distances
Dataset Teeth First metatarsal Radius
Classification No. N cP Obs. 1 cWn No. N Obs. 1 cP Obs. 2 cWn No. N cP Obs. 1 cWn
Genera 24 99 90.9 91.9 68 13 59 76.3 79.9 88.1 50.8 4 45 84.4 77.8 68.9Family 17 106 92.5 94.3 75.1 9 61 83.6 91.8 93.4 68.9 not applicableAbove family 5 116 94.8 95.7 83.3 2 61 100 100 100 98.4 not applicable
Fig. 2. Observer-placed landmarks can be propagated from structure S (I)using cP-determined correspondence maps (II) to another specimen S0 (III).The similarity between propagated landmarks in III and observer placed land-marks in IV on S0 shows the success of the method and makes explicit thegeometric basis for the observer determinations.
Fig. 3. Observer-placed landmarks on a tooth ofMicrocebus are propagatedusing cP-determined correspondence maps to a tooth of Lepilemur. Path 1 isdirect, paths 2 and 3 have intermediate steps, representing stepwise propa-gation between teeth of other taxa.
Boyer et al. PNAS Early Edition ∣ 5 of 6
EVOLU
TION
APP
LIED
MAT
HEMAT
ICS
Conformal maps
A map between two surfaces � : S �! S0 takes p 2 S to �(p) 2 S0.
A map � is conformal if for any two smooth curves �1
and �2
on S
intersecting at s
\s(�1, �2) = \�(s)(�(�1),�(�2)),
�(�1
),�(�2
) 2 S0.
Conformal maps
A map between two surfaces � : S �! S0 takes p 2 S to �(p) 2 S0.
A map � is conformal if for any two smooth curves �1
and �2
on S
intersecting at s
\s(�1, �2) = \�(s)(�(�1),�(�2)),
�(�1
),�(�2
) 2 S0.
Conformal map
Continuous Procrustes distance
Define A(S, S0) as the set of area preserving di↵eomorphisms,maps a : S �! S0 such that for any measurable subset ⌦ ⇢ S
Z
⌦
dAS =
Z
a(⌦)
dAS0 .
The continuous procrustes distance is
dp(S, S0) = inf
a2A(S,S0)min
R2rigid motion
Z
S
|R(x)� a(x)|2dAS.
Near optimal a are “almost” conformal, so simply above tosearching near conformal maps.
Continuous Procrustes distance
Define A(S, S0) as the set of area preserving di↵eomorphisms,maps a : S �! S0 such that for any measurable subset ⌦ ⇢ S
Z
⌦
dAS =
Z
a(⌦)
dAS0 .
The continuous procrustes distance is
dp(S, S0) = inf
a2A(S,S0)min
R2rigid motion
Z
S
|R(x)� a(x)|2dAS.
Near optimal a are “almost” conformal, so simply above tosearching near conformal maps.
Continuous Procrustes distance
Define A(S, S0) as the set of area preserving di↵eomorphisms,maps a : S �! S0 such that for any measurable subset ⌦ ⇢ S
Z
⌦
dAS =
Z
a(⌦)
dAS0 .
The continuous procrustes distance is
dp(S, S0) = inf
a2A(S,S0)min
R2rigid motion
Z
S
|R(x)� a(x)|2dAS.
Near optimal a are “almost” conformal, so simplify above tosearching near conformal maps.
Continuous Procrustes distance
dcW (S, S0) = infm2M
inf⇡2⇧(m,f,f0)
Z
D⇥Dd(z , z 0)d⇡(z , z 0)
�
,
m(z) = e i✓(z � a)(1� za⇤)�1 2 M disk preserving Mobiustransforms.
Continuous Procrustes distance
dcW (S, S0) = infm2M
inf⇡2⇧(m,f,f0)
Z
D⇥Dd(z , z 0)d⇡(z , z 0)
�
,
m(z) = e i✓(z � a)(1� za⇤)�1 2 M disk preserving Mobiustransforms.
Topological methods
What happens when the shapes are not isomorphic ?
23,
Fly wings
24,
Persistent homology Bar codes Brain trees Fly wings Stratified persistence Metric perversity Future biology Stratified statistics Next steps
Fruit fly wingsNormal fly wings [photos from David Houle’s lab]:
Topologically abnormal veins:
8
Our objective
Transform the data/object into a representation that can bemodeled using standard methods.
Desired properties(1) The transformation is injective, ideally the summary statistic is
sufficient.
(2) The transformed space is nice (may be a matrix). Cancompute distances or place probability models in thetransformed space.
Our objective
Transform the data/object into a representation that can bemodeled using standard methods.
Desired properties(1) The transformation is injective, ideally the summary statistic is
sufficient.
(2) The transformed space is nice (may be a matrix). Cancompute distances or place probability models in thetransformed space.
Our objective
Transform the data/object into a representation that can bemodeled using standard methods.
Desired properties(1) The transformation is injective, ideally the summary statistic is
sufficient.
(2) The transformed space is nice (may be a matrix). Cancompute distances or place probability models in thetransformed space.
Persistence diagrams
Evolution of homology as birth-death pair.
0 2⇡
Dgm0(f)
birth
death
f�1((�1, a])
35,
Persistence diagrams
Evolution of homology as birth-death pair.
0 2⇡
f�1((�1, a])
a
Dgm0(f)
birth
death
36,
Persistence diagrams
Evolution of homology as birth-death pair.
0 2⇡
f�1((�1, a])
a
Dgm0(f)
birth
death
37,
Persistence diagrams
Evolution of homology as birth-death pair.
0 2⇡
f�1((�1, a])
a
Dgm0(f)
birth
death
38,
Persistence diagrams
Evolution of homology as birth-death pair.
0 2⇡
f�1((�1, a])
a
Dgm0(f)
birth
death
39,
Persistence diagrams
Evolution of homology as birth-death pair.
0 2⇡
f�1((�1, a])
a
Dgm0(f)
birth
death
40,
Persistence diagrams
Evolution of homology as birth-death pair.
0 2⇡
f�1((�1, a])
a
Dgm0(f)
birth
death
41,
Persistence diagrams
Evolution of homology as birth-death pair.
0 2⇡
f�1((�1, a])
a
Dgm0(f)
birth
death
42,
Persistence diagrams
Evolution of homology as birth-death pair.
0 2⇡
f�1((�1, a])
a
Dgm0(f)
birth
death
43,
Persistence diagrams
Evolution of homology as birth-death pair.
0 2⇡
f�1((�1, a])
a
Dgm0(f)
birth
death
44,
Euler characteristic
Given a shape M the Euler characteristic is
�(M) =dX
i=0
(�1)i�i = #vertices �#edges + #faces.
Euler characteristic
Given a shape M the Euler characteristic is
�(M) =dX
i=0
(�1)i�i = #vertices �#edges + #faces.
Height function: v1
Height function: v2
Height function
v0
v1
v2
v3
v5
v4
v0
Height function
v0
v1
v2
v3
v5
v4
v0
Height function
v0
v1
v2
v3
v5
v4
v0
✓v0
v1
Height function
v0
v1
v2
v3
v5
v4
v0
✓v0
v1
✓v0
v1
v2
Height function
v0
v1
v2
v3
v5
v4
v0
✓v0
v1
✓v0
v1
v2 ✓
v0
v1
v2
v4
✓v0
v1
v2
v3
v4
Height function
v0
v1
v2
v3
v5
v4
v0
✓v0
v1
✓v0
v1
v2 ✓
v0
v1
v2
v4
✓v0
v1
v2
v3
v4
✓v0
v1
v2
v3
v5
v4
Euler characteristic curve
(a) (b) (c)
Figure 1: (a) sublevel set of a mouse embryo head; (b) EC curve of the 2D contour of a hand and(c) the associated (centered) cumulative EC-curve.
characteristic transform (ECT). Leveraging these topological transforms, we propose to developmethodology based on several variants of the ECT to address a range of problems involving shapesand networks.
2.2 Reformulation of the ECT and EC-Signatures
Euler characteristic curves are piecewise constant functions with numerous discontinuities – seeFigure 1(b). In order to obtain a numerically stable representation of the ECT, we propose thefollowing reformulation. Let �K
� be the mean value of �K� over [a�, b�]. The centered EC curve in
the direction � is the function ZK� : R � R given by ZK
� (x) = �K� (x)� �K
� , for x � [a�, b�], extendedto zero outside this interval. We define the (centered) cumulative Euler characteristic curve asFK
� (x) =� x�� ZK
� (y) dy, for any x � R. By construction, the cumulant FK� is a continuous,
piecewise linear function that vanishes outside the interval [a�, b�] – see Figure 1(c). Henceforth, tosimplify notation, we drop the superscript K if there is no ambiguity. Note that, for any x � [a�, b�],��(x) = F �
�(x) + ��, so that we can recover �� from the pair (��, F�), which we refer to as the EC-signature in the direction �. Since F� is continuous with compact support, F� may be viewed as anelement of the Hilbert space L2 of square integrable functions on the real line. Thus, we may treatthe EC-signature (��, F�) as an element of R � L2. We assemble this family of signatures of K,indexed by directions in Sn�1, into a mapping �K : Sn�1 � R � L2, where �K(�) = (��, F�). Themapping �K gives a computationally robust formulation of the ECT. Note that the Hilbert spacestructure on R � L2 let us define an L2-space of mappings Sn�1 � R � L2 with metric
��K � �L�22 =
�
Sn�1��K(�) � �L(�)�2 d�n(�) . (1)
2.3 Shape-Preserving Transformations
Simplicial complexes in Rn that di�er by scale and rigid transformations are typically viewed ashaving the same shape. There are simple ways of removing translational and size e�ects. Acommonly used method is to center a shape (i.e., translate it so that the centroid lies at the origin)and scale it to have a fixed centroid size (cf. [15, 22]). (Centroid size is the root-mean-squaredistance to the centroid.) Accounting for rotations or orthogonal transformations is more delicate.Let O(n) be the group of orthogonal transformation of Rn. The action of U � O(n) on a simplicialcomplex K (that simply applies U to all simplices of K) induces an action on �K given by
U · �K(�) = (��, F� � U�) , (2)
4
Mao Li
Euler characteristic transform (ECT)
M is simplicial complex in IRd and v 2 Sd�1 is a unit vector.�(M, v) captures changes in topology of
M(v)r = {� 2 M : x · v r for all x 2 �}.
DefinitionThe Euler characteristic transform of M 2 IRd is the function
ECT(M) : Sd�1 ! L2(R)
v 7! �(M, v).
Euler characteristic transform (ECT)
M is simplicial complex in IRd and v 2 Sd�1 is a unit vector.�(M, v) captures changes in topology of
M(v)r = {� 2 M : x · v r for all x 2 �}.
DefinitionThe Euler characteristic transform of M 2 IRd is the function
ECT(M) : Sd�1 ! L2(R)
v 7! �(M, v).
Euler characteristic curve
(a) (b) (c)
Figure 1: (a) sublevel set of a mouse embryo head; (b) EC curve of the 2D contour of a hand and(c) the associated (centered) cumulative EC-curve.
characteristic transform (ECT). Leveraging these topological transforms, we propose to developmethodology based on several variants of the ECT to address a range of problems involving shapesand networks.
2.2 Reformulation of the ECT and EC-Signatures
Euler characteristic curves are piecewise constant functions with numerous discontinuities – seeFigure 1(b). In order to obtain a numerically stable representation of the ECT, we propose thefollowing reformulation. Let �K
� be the mean value of �K� over [a�, b�]. The centered EC curve in
the direction � is the function ZK� : R � R given by ZK
� (x) = �K� (x)� �K
� , for x � [a�, b�], extendedto zero outside this interval. We define the (centered) cumulative Euler characteristic curve asFK
� (x) =� x�� ZK
� (y) dy, for any x � R. By construction, the cumulant FK� is a continuous,
piecewise linear function that vanishes outside the interval [a�, b�] – see Figure 1(c). Henceforth, tosimplify notation, we drop the superscript K if there is no ambiguity. Note that, for any x � [a�, b�],��(x) = F �
�(x) + ��, so that we can recover �� from the pair (��, F�), which we refer to as the EC-signature in the direction �. Since F� is continuous with compact support, F� may be viewed as anelement of the Hilbert space L2 of square integrable functions on the real line. Thus, we may treatthe EC-signature (��, F�) as an element of R � L2. We assemble this family of signatures of K,indexed by directions in Sn�1, into a mapping �K : Sn�1 � R � L2, where �K(�) = (��, F�). Themapping �K gives a computationally robust formulation of the ECT. Note that the Hilbert spacestructure on R � L2 let us define an L2-space of mappings Sn�1 � R � L2 with metric
��K � �L�22 =
�
Sn�1��K(�) � �L(�)�2 d�n(�) . (1)
2.3 Shape-Preserving Transformations
Simplicial complexes in Rn that di�er by scale and rigid transformations are typically viewed ashaving the same shape. There are simple ways of removing translational and size e�ects. Acommonly used method is to center a shape (i.e., translate it so that the centroid lies at the origin)and scale it to have a fixed centroid size (cf. [15, 22]). (Centroid size is the root-mean-squaredistance to the centroid.) Accounting for rotations or orthogonal transformations is more delicate.Let O(n) be the group of orthogonal transformation of Rn. The action of U � O(n) on a simplicialcomplex K (that simply applies U to all simplices of K) induces an action on �K given by
U · �K(�) = (��, F� � U�) , (2)
4
Mao Li
Persistence homology transform (PHT)
M is simplicial complex in IRd and v 2 Sd�1 is a unit vector.Xk (M, v) captures changes in topology of
M(v)r = {� 2 M : x · v r for all x 2 �}.
DefinitionThe persistent homology transform of M 2 IRd is the function
PHT(M) : Sd�1 ! Dd�1
v 7! (X0(M, v),X1, (M, v) . . . ,Xd�1(M, v)).
Persistence homology transform (PHT)
M is simplicial complex in IRd and v 2 Sd�1 is a unit vector.Xk (M, v) captures changes in topology of
M(v)r = {� 2 M : x · v r for all x 2 �}.
DefinitionThe persistent homology transform of M 2 IRd is the function
PHT(M) : Sd�1 ! Dd�1
v 7! (X0(M, v),X1, (M, v) . . . ,Xd�1(M, v)).
Distances
Md is the space of finite simplicial complexes in IRd .
The distance between two surfaces M1,M2 is
dMd (M1,M2) :=dX
k=0
Z
Sd�1d(Xk (M1, v),Xk (M2, v))dv .
Distances
Md is the space of finite simplicial complexes in IRd .
The distance between two surfaces M1,M2 is
dMd (M1,M2) :=dX
k=0
Z
Sd�1d(Xk (M1, v),Xk (M2, v))dv .
Injectivty of the PHT
Theorem (Turner-M-Boyer)The persistent homology transform is injective when the domain isMd for d = 2, 3.
Reconstruction I
We state the proof as an algorithm.
(1) Given a the function PHT(M) : S2 ! D3 we state aprocedure to find all the vertices in one of the simplestrepresentation of M.
(2) We then determine the link of each vertex.
(3) Since M is assumed to be piecewise linear computing thevertices and links is enough for reconstruction.
Reconstruction I
We state the proof as an algorithm.
(1) Given a the function PHT(M) : S2 ! D3 we state aprocedure to find all the vertices in one of the simplestrepresentation of M.
(2) We then determine the link of each vertex.
(3) Since M is assumed to be piecewise linear computing thevertices and links is enough for reconstruction.
Reconstruction I
We state the proof as an algorithm.
(1) Given a the function PHT(M) : S2 ! D3 we state aprocedure to find all the vertices in one of the simplestrepresentation of M.
(2) We then determine the link of each vertex.
(3) Since M is assumed to be piecewise linear computing thevertices and links is enough for reconstruction.
Reconstruction II
The following facts allow for the reconstruction of M fromPHT(M):
(1) Changes in homology of sublevel sets of height functions inany direction can only occur at the heights of vertices of M.
(2) Every vertex x determines a critical point for an open ball inall directions in S2, that is to say this point causes a birth ordeath of a homology class. The homology class is alsoconsistent inside the ball. We claim that there is some ball of{v} ⇢ S2 with points (a
v
, bv
) in the corresponding diagramsthat continuously change with either a
v
= x · v or bv
= x · v .
Reconstruction II
The following facts allow for the reconstruction of M fromPHT(M):
(1) Changes in homology of sublevel sets of height functions inany direction can only occur at the heights of vertices of M.
(2) Every vertex x determines a critical point for an open ball inall directions in S2, that is to say this point causes a birth ordeath of a homology class. The homology class is alsoconsistent inside the ball. We claim that there is some ball of{v} ⇢ S2 with points (a
v
, bv
) in the corresponding diagramsthat continuously change with either a
v
= x · v or bv
= x · v .
Sufficient statistic
Given X ⇠ f✓ 2 F , a statistic T = T(X) is sufficient if for theparameter ✓ if for all sets B the probability IP[X 2 B | T(X) = t ]does not depend on ✓
IP[X | T(X) = t , ✓] = IP[X | T(X) = t ].
For the normal distribution with known variance µ = 1nP
i xi is asufficient statistic.
Fisher-Neyman factorization
If the probability density function is f✓(x) then T is a su�cient for✓ if and only if nonnegative functions g and h can be found suchthat
f✓(x) = h(x) g✓(T (x)).
Bernoulli example
fp
(x) =n
Y
i=1
pxi (1� p)1�x
i = 1⇥ pP
i
x
i (1� p)n�P
i
x
i
= h(x) gp
(T (x)),
h(x) = 1, T (x) =P
i
xi
, gp
(T (x)) = pT (x)(1� p)n�T (x).
Fisher-Neyman factorization
If the probability density function is f✓(x) then T is a su�cient for✓ if and only if nonnegative functions g and h can be found suchthat
f✓(x) = h(x) g✓(T (x)).
Bernoulli example
fp
(x) =n
Y
i=1
pxi (1� p)1�x
i = 1⇥ pP
i
x
i (1� p)n�P
i
x
i
= h(x) gp
(T (x)),
h(x) = 1, T (x) =P
i
xi
, gp
(T (x)) = pT (x)(1� p)n�T (x).
Sufficiency of the PHT
Corollary (Turner-M-Boyer)Consider the subspace of shapesMN
k (for k = 2 or 3), piecewiselinear simplicial complexes with at most N vertices. Let f(x; ✓) be adensity function overMk with parameters ✓ 2 ⇥ and x 2Mkwhose support is contained in someMN
k . The persistencehomology transform t(X) 2 C(S2,D3) is a sufficient statistic.
Uses Radon-Nikodym construction for sufficient statistics byHalmos and Savage, 1949.
Su�cient statistics and the Radon-Nikodym theorem
Theorem (Halmos-Savage)
A necessary and su�cient condition that the statistic T besu�cient for a dominated set M of measures on S is that thereexist a measure � on S such that M ⌘ � and such thatdµ/d� 2 T�1(T) for every µ 2 M.
Sufficiency of the ECT
Theorem (Turner-M-Boyer)The Euler characteristic transform is injective when the domain isMd for d = 2, 3.
Corollary (Turner-M-Boyer)Consider the subspace of shapesMN
k (for k = 2 or 3), piecewiselinear simplicial complexes with at most N vertices. Let f(x; ✓) be adensity function overMk with parameters ✓ 2 ⇥ and x 2Mkwhose support is contained in someMN
k . The Euler characteristictransform t(X) 2 C(S2,D3) is a sufficient statistic.
Sufficiency of the ECT
Theorem (Turner-M-Boyer)The Euler characteristic transform is injective when the domain isMd for d = 2, 3.
Corollary (Turner-M-Boyer)Consider the subspace of shapesMN
k (for k = 2 or 3), piecewiselinear simplicial complexes with at most N vertices. Let f(x; ✓) be adensity function overMk with parameters ✓ 2 ⇥ and x 2Mkwhose support is contained in someMN
k . The Euler characteristictransform t(X) 2 C(S2,D3) is a sufficient statistic.
A sampling theory for shapes
How many directions to sample ?
(1) For 2-D: 162 directions(2) For 3-D: Over 700 directions.
A sampling theory for shapes — complexity metric for familiesshapes in terms of directions required.
A sampling theory for shapes
How many directions to sample ?(1) For 2-D: 162 directions(2) For 3-D: Over 700 directions.
A sampling theory for shapes — complexity metric for familiesshapes in terms of directions required.
A sampling theory for shapes
How many directions to sample ?(1) For 2-D: 162 directions(2) For 3-D: Over 700 directions.
A sampling theory for shapes — complexity metric for familiesshapes in terms of directions required.
Calculus of snakes
V.I. Arnol’d, The calculus of snakes and the combinatorics ofBernoulli, Euler and Springer numbers of Coxeter groups.
Exponential family and ECT
Denote the Euler characteristic curve for each direction:f(y) = �(M, v) Define the integral of f(y) as F(x) =
R x0 f(y)dy.
This results in K smooth curves {F1, ...,FK }.
Exponential family model
p✓(x) = a(✓) h(x) exp
0
B
B
B
B
B
B
@
�KX
k=1
h✓,Fk (x)i1
C
C
C
C
C
C
A
.
Exponential family and ECT
Denote the Euler characteristic curve for each direction:f(y) = �(M, v) Define the integral of f(y) as F(x) =
R x0 f(y)dy.
This results in K smooth curves {F1, ...,FK }.
Exponential family model
p✓(x) = a(✓) h(x) exp
0
B
B
B
B
B
B
@
�KX
k=1
h✓,Fk (x)i1
C
C
C
C
C
C
A
.
Exponential family and ECT
Denote the Euler characteristic curve for each direction:f(y) = �(M, v) Define the integral of f(y) as F(x) =
R x0 f(y)dy.
This results in K smooth curves {F1, ...,FK }.
Exponential family model
p✓(x) = a(✓) h(x) exp
0
B
B
B
B
B
B
@
�KX
k=1
h✓,Fk (x)i1
C
C
C
C
C
C
A
.
The matrix variate normal
Define F = [F1F2 · · ·FK ] as a K ⇥ T matrix and
p(F | A,U,V) =exp⇣
�12 tr[V�1(F � A)T
U
�1(F � A)]⌘
(2⇡)KT/2|V|L/2|U|K/2 ,
A models meanU models covariance between curvesV models covariance between points in a curve.
The given n meshes (M1, ...,Mn) we can define a likelihood model
Lik(M1, ...,Mn | A,U,V) =nY
i=1
p(F(Mi) | A,U,V), (1)
The matrix variate normal
Define F = [F1F2 · · ·FK ] as a K ⇥ T matrix and
p(F | A,U,V) =exp⇣
�12 tr[V�1(F � A)T
U
�1(F � A)]⌘
(2⇡)KT/2|V|L/2|U|K/2 ,
A models meanU models covariance between curvesV models covariance between points in a curve.
The given n meshes (M1, ...,Mn) we can define a likelihood model
Lik(M1, ...,Mn | A,U,V) =nY
i=1
p(F(Mi) | A,U,V), (1)
Exponential family model for shapes
A shape is transformed into collection of the curves {�(M, v`)}L`=1.
A natural exponential family model for the collection of thesecurves is a multivariate Gaussian process
X =
2
64�(M, v1)
...�(M, vL)
3
75 ⇠ GPL(µ, k).
A natural exponential family model for the collection of thesecurves takes the form
f (x | ✓) = g(x) exp(⌘(✓) · T (x) � A(✓)).
106,
Distances without alignment
Theorem (Turner-M)Let f : S2 ! L2(IR) and g : S2 ! L2(IR) be the ECT for two finitesimplicial complexes Mf and Mg respectively. Both f and g aregenerically injective. Let µ be the measure on S2. If f⇤(µ) = g⇤(µ),the push forwards of the measure are equal, then there is someX 2 O(3) such that Mg = X(Mf ).
The distributions of the Euler characteristic curves are sufficientstatistics.
Shapes
Shapes
Shapes
Silhouettes
Silhouettes
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.3
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0
0.1
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0.5
0.6 frkfrkfrkfrkfrkfrk
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bobobobobobobobobobobo
axe
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hrthrt
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hrt
gls
gls
glsglsglsglsglsglsglsglsglsgls
glsglsglsglsglsglsglsgls
boneheartglassfountainkeyforkaxe
Student Version of MATLAB
Silhouettes
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−10
1−0.3
−0.2
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0
0.1
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0.5
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frkfrkfrk
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axe
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axeaxe
ftn
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bo
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bo
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frk
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hrt
gls
glsglsglsglsglsglsglsglsglsglsglsglsgls
glsglsglsglsglsgls
Student Version of MATLAB
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−ayeRing tail
Howler
Spider
Saki
Squirrel
MacaqueBaboon
Gorilla
Gibbon
Chimp
Orang
Meso
Omomyid
Tetonius
Student Version of MATLAB
Primate calcanei
Figure: Phenetic clustering of phylogenetic groups of primate calcanei (n = 106). 67 genera are represented.Asterisks indicate groups of extinct taxa. Abbreviations: Str, Strepsirrhines; Plat, platyrrhines; Cerc, Cercopithecoids; Om,Omomyiforms; Adp, Adapiforms; Pp, parapithecids; Hmn, Hominoids. Note that more primitive prosimian taxa clusterseparately from simians (Om, Adp, Str.). Also note that monkeys (Plat, Cerc, Pp) cluster mainly separately from apes(Hmn).
Comment from Doug
”In at least one way the method matched shapes with familygroups better than any of the other previous methods... it linked aHylobates specimen with the the other ape specimens (pan,gorilla, pongo, and oreopithecus). Previous both hylobatids (whichARE apes) always ended up closest to some Alouatta specimens.”
Comparing methods
Dictionary learning
Given a dictionary of signals D 2 IRm⇥p, a set of coe�cientsA 2 IRp⇥n, and signals Y = (y1, ..., yn) 2 IRm⇥n
minA,D
kY �DAk22 + �kAk1.
Dictionary learning
Given a dictionary of signals D 2 IRm⇥p, a set of coe�cientsA 2 IRp⇥n, and signals Y = (y1, ..., yn) 2 IRm⇥n
minA,D
kY �DAk22 + �kAk1.
A dictionary
Dictionary learning on shapes, a dual view
Given a dictionary of p prototype shapes di 2 D compute the ECTof each prototype shape
{�(di ; v1, ..., vL)}pj=1 .
M = (M1, ...,Mn) ! Y = (Y1, ...,Yn) 2 IRp⇥n projection ofshapes onto dictionary elements
Yi 2 IRp, Yij = h�(Mi ; v1, ...., vL),�(sj ; v1, ..., vL)i, 8j = 1, .., p.
Compute the matrix of inner products between two prototypeshapes
eD 2 IRp⇥p, eD = h�(di ; v1, ...., vL),�(dj ; v1, ..., vL)i 8i , j = 1, .., p.
Solvemin
A2IRp⇥n,eDkY � eDAk22 + �kAk1.
Dictionary learning on shapes, a dual view
Given a dictionary of p prototype shapes di 2 D compute the ECTof each prototype shape
{�(di ; v1, ..., vL)}pj=1 .
M = (M1, ...,Mn) ! Y = (Y1, ...,Yn) 2 IRp⇥n projection ofshapes onto dictionary elements
Yi 2 IRp, Yij = h�(Mi ; v1, ...., vL),�(sj ; v1, ..., vL)i, 8j = 1, .., p.
Compute the matrix of inner products between two prototypeshapes
eD 2 IRp⇥p, eD = h�(di ; v1, ...., vL),�(dj ; v1, ..., vL)i 8i , j = 1, .., p.
Solvemin
A2IRp⇥n,eDkY � eDAk22 + �kAk1.
Dictionary learning on shapes, a dual view
Given a dictionary of p prototype shapes di 2 D compute the ECTof each prototype shape
{�(di ; v1, ..., vL)}pj=1 .
M = (M1, ...,Mn) ! Y = (Y1, ...,Yn) 2 IRp⇥n projection ofshapes onto dictionary elements
Yi 2 IRp, Yij = h�(Mi ; v1, ...., vL),�(sj ; v1, ..., vL)i, 8j = 1, .., p.
Compute the matrix of inner products between two prototypeshapes
eD 2 IRp⇥p, eD = h�(di ; v1, ...., vL),�(dj ; v1, ..., vL)i 8i , j = 1, .., p.
Solvemin
A2IRp⇥n,eDkY � eDAk22 + �kAk1.
Dictionary learning on shapes, a dual view
Given a dictionary of p prototype shapes di 2 D compute the ECTof each prototype shape
{�(di ; v1, ..., vL)}pj=1 .
M = (M1, ...,Mn) ! Y = (Y1, ...,Yn) 2 IRp⇥n projection ofshapes onto dictionary elements
Yi 2 IRp, Yij = h�(Mi ; v1, ...., vL),�(sj ; v1, ..., vL)i, 8j = 1, .., p.
Compute the matrix of inner products between two prototypeshapes
eD 2 IRp⇥p, eD = h�(di ; v1, ...., vL),�(dj ; v1, ..., vL)i 8i , j = 1, .., p.
Solvemin
A2IRp⇥n,eDkY � eDAk22 + �kAk1.
ECT for graphs
Is there an analog for directions on graphs ?
(1) For each graph label nodes using L directions correspondingto eigenvectors v1, ...vl ordered by �1, ...., �L of the graphLaplacian.
(2) Compute the ECT for each graph.
Proceed as the shape case.
ECT for graphs
Is there an analog for directions on graphs ?
(1) For each graph label nodes using L directions correspondingto eigenvectors v1, ...vl ordered by �1, ...., �L of the graphLaplacian.
(2) Compute the ECT for each graph.
Proceed as the shape case.
ECT for graphs
Is there an analog for directions on graphs ?
(1) For each graph label nodes using L directions correspondingto eigenvectors v1, ...vl ordered by �1, ...., �L of the graphLaplacian.
(2) Compute the ECT for each graph.
Proceed as the shape case.
ECT for graphs
Is there an analog for directions on graphs ?
(1) For each graph label nodes using L directions correspondingto eigenvectors v1, ...vl ordered by �1, ...., �L of the graphLaplacian.
(2) Compute the ECT for each graph.
Proceed as the shape case.
Can you hear the shape of a drum ?
Mao Li
Association studies of shape phenotypes
Quantitative genetics
Open questions and problems
(1) Phylogenies of surfaces and traits: we have a likelihoodmodel, we need a null substitution model.
(2) Localized transforms: The PHT and ECT can be generalizedas Euler integration
Z
Xh d�, h is a (localized) basis function.
(3) Generalization to graphs and networks.
Open questions and problems
(1) Phylogenies of surfaces and traits: we have a likelihoodmodel, we need a null substitution model.
(2) Localized transforms: The PHT and ECT can be generalizedas Euler integration
Z
Xh d�, h is a (localized) basis function.
(3) Generalization to graphs and networks.
Open questions and problems
(1) Phylogenies of surfaces and traits: we have a likelihoodmodel, we need a null substitution model.
(2) Localized transforms: The PHT and ECT can be generalizedas Euler integration
Z
Xh d�, h is a (localized) basis function.
(3) Generalization to graphs and networks.
Acknowledgements
Many people.
Funding:
I Center for Systems Biology at Duke
I NSF DMS and CCF
I AFOSR, DARPA
I NIH
204,
New journal
195,
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