Lecture 5: Modeling shapes and surfaces, projections and transformations

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

https://www.youtube.com/watch?v=mR8J1ylYIIA

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.

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

https://stat.duke.edu/~sayan/auto3dgm/Bones/alignment.off























































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

.

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 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.


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.


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.

Persistence diagrams

Evolution of homology as birth-death pair.

0 2⇡

Dgm0(f)

birth

death

f�1((�1, a])

35,



0 2⇡

f�1((�1, a])

a

Dgm0(f)

birth

death

36,



0 2⇡

f�1((�1, a])

a

Dgm0(f)

birth

death

37,



0 2⇡

f�1((�1, a])

a

Dgm0(f)

birth

death

38,



0 2⇡

f�1((�1, a])

a

Dgm0(f)

birth

death

39,



0 2⇡

f�1((�1, a])

a

Dgm0(f)

birth

death

40,



0 2⇡

f�1((�1, a])

a

Dgm0(f)

birth

death

41,



0 2⇡

f�1((�1, a])

a

Dgm0(f)

birth

death

42,



0 2⇡

f�1((�1, a])

a

Dgm0(f)

birth

death

43,



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.

Height function: v1

Height function: v2

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 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)).

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 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).

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.

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

.

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

Silhouettes

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Student Version of MATLAB

Silhouettes

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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


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.

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.

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.

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,

Lecture 5: Modeling shapes and surfaces, projections and transformations

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Transcript of Lecture 5: Modeling shapes and surfaces, projections and transformations