algorithms to automatically quantify the geometric ...€¦ · outline background motivation...

Algorithms to automatically quantify thegeometric simliarity of anatomical surfaces

Boyer, Lipman, St.Clair, Puente, Pantel, Funkhouser, Jernvaland Daubechies

Ying Yin

March 2018

Outline

BackgroundMotivationGeneral IdeaMathematical background

New distancesConformal Wasserstein distances (cW)Conformal Wasserstein neighborhood dissimlarity distance(cWn)Continuous procrustes distance between surfaces (cP)

ExperimentsData and PerfomancesPerformance test

Results using TLBOutlineResults

Motivation

I Understand physical and biological phenomena (e.g.speciation, evolutionary adaption, etc.) by quantifying thesimilarity or dissimilarity of objects affected by the phenomena.

I In standard morphologists’ practice, 10 to 100 points will beidentified as landmarks. By comparing these landmarks,similarity and dissimilarity between patterns of shapes can bedetermined.

I The difficulty in acquiring personal knowledge ofmorphological evidence limits our understanding of theevolutionary significance of morphological diversity.

I Want an automatic tool to decide similarity or dissimilaritybetween objects, and hence, provides more insights on thephenomenon.

Outline





General Idea

I Given two shapes S,S ′ (with boundaries but not holes),conformally map them onto D2 by applying Riemann’suniformization theorem.

I Conformal geometry permits the reduction of the study ofsurfaces embedded in 3D space to 2D problems

I By finding a coupling between the conformal factors, or byfinding a correspondence between the disks that respects theconformal factors, one may be able to define new distancesthat measures similarity and dissimilarity.

Outline





Conformal map

DefinitionA map ϕ : S → S ′ between two (smooth) surfaces is conformal iffor any two smooth curves Γ1, Γ2 on S, the angle between theirimages Γ′1, Γ

′2 is the same as that between Γ1, Γ2 at the

corresponding intersection point.

DefinitionTwo Riemannian metrics g and h on a smooth manifold M arecalled conformally equivalent if g = fh for some positive function fon M. The function f is called the conformal factor.

Remark:

1. the conformal factor indicates the area distortion factorproduced by the operation of conformal mapping.

2. the conformal factor defines a probability measure.

Disk-preserving Mobius transofrmation

If γ is a conformal mapping from S to S ′, and ϕ,ϕ′ are confromalmaps to the disk D2 of S,S ′, then the family of all possibleconformal mappings from S to S ′ is given by γ = ϕ′−1 ◦m ◦ ϕ,where m ranges over all the conformal bijective self-mappings ofthe unit disk D2.

DefinitionSuch m is called a disk-preserving Mobius transformation. And thecollection of such m is denoted by M.

Hyperbolic measure

Let dη(x , y) be the hyperbolic measure on the disk D2, i.e.

dη(x , y) = [1− (x2 + y2)]−2dxdy

.Let f (x , y) be a conformal factor. And letf(x , y) = [1− (x2 + y2)]2f (x , y).Then we have fdη = fdxdy .

Push-forward and Transport Effort

DefinitionLet µ be a probabilty measure, and τ be a differentiable bijectionfrom D2 to itself, the mass distribution µ′ = τ∗µ defined byµ(u) = µ′(τ(u))Jτ (u) where Jτ is the Jacobian of τ is thetransportation (or push-forward) of µ by τ .

Remark: τ∗µ = µ ◦ τ−1.Note that for any (well-behaved) function F on D2,∫D2 F (u)µ′(u)du =

∫D2 F (τ(u))µ(u)du.

DefinitionThe total transport effort ετ =

∫D2 d(u, τ(u))µ(u)du where

d(u, v) is the distance between u, v in D2.

Optimal Transport

By infimizing ετ over all measurable bijections τ from D2 to itself,we solve the Monge problem.Alternatively, since the bijections are hard to search, consider theKantorovitch problem, i.e. for all continuous functions F ,G on D2,let π be a coupling with marginals µ, ν satisfying that∫D2×D2 F (u)dπ(u, v) =

∫D2 F (u)µ(u)du and∫

D2×D2 G (v)dπ(u, v) =∫D2 G (v)ν(v)dv , we find the Wasserstein

distance by finding infimum of

Eπ =

∫D2×D2

d(u, v)dπ(u, v)

over all couplings π.

Outline





Conformal Wasserstein distances (cW)

Instead of comparing two surfaces S,S ′, one can compare twoconformal factors f , f ′ obtained by conformally flattening S,S ′.Let m be a disk-preserving Mobius transformation, then f andm∗f = f ◦m−1 are both conformal factors for S.Then we define the conformal Wasserstein distance to be

DcW (S,S ′) = infm∈M

[inf

π∈∏

(m∗f ,f ′)

∫D2×D2

d(z , z ′)dπ(z , z ′)

], where d(·, ·) is the hyperbolic distance on D2.Remark:

1. DcW is a metric.

2. However, computing DcW involves solving a Kantorovitchproblem for every m.

Outline





Conformal Wasserstein neighborhood dissimlarity distance(cWn)

Instead, we quantify how dissimilar the ”landscapes” are with ameasure of neighborhood dissimilarity.Let N(0,R) be a neighborhood at 0, i.e., N(0,R) = {z ; |z | < R}.For any m ∈M s.t. z = m(0), N(z ,R) is the image of N(0,R)under m.Then we define the dissimilarity between f at z and f ′ at z ′ by

dRf ,f ′(z , z

′) = infm∈M,m(z)=z ′

[∫N(z,R)

|f(w)− f ′(m(w))|dη(w)

]

Conformal Wasserstein neighborhood dissimlarity distance(cWn) cont.

We defined the dissimilarity between f at z and f ′ at z ′ by

dRf ,f ′(z , z

′) = infm∈M,m(z)=z ′

[∫N(z,R)

|f(w)− f(m(w))|dη(w)

]

The conformal Wasserstein neighborhood dissimilarity distancebetween f and f ′ is

DRcWn(S,S ′) = inf

π∈∏

(f ,f ′)

∫D2×D2

dRf ,f ′(z , z

′)dπ(z , z ′)

Remark

I Both cW and cWn are blind to isometric embedding of asurface in 3D

I Introduce a new extrinsic distance

Outline





Procrustes distance between surfaces

The standard Procrustes distance is between discrete sets of pointsX = (Xn)n=1,··· ,N ⊂ S and Y = (Yn)n=1,··· ,N ⊂ S ′ by

dp(X,Y) = minR rigid motions

( N∑n=1

|R(Xn)− Yn|2)1/2

where | · | is the standard Euclidean norm.Often X and Y are sets of landmarks on two surfaces.Remark:

1. dp(X,Y) depends on choices of the sets of landmarks.

2. small number of N landmarks disregards a wealth ofgeometric data

3. identifying and recording Xn,Yn requires time and expertise.

Continuous procrustes distance between surfaces (cP)

Instead, we consider a family of continuous maps a : S → S ′ anduse optimization to find the ”best” a.We require a to be area-preserving.We denote the set of all area-preserving diffeomorphisms byA(S,S ′). And let

d(S,S ′, a)2 = minR rigid motions

∫S|R(x)− a(x)|2dAS

.Then we define the continuous Procrustes distance between S andS’ by

Dp(S,S ′) = infa∈A(S,S′)

d(S,S ′, a).

Continuous procrustes distance between surfaces (cP)cont.

Remarks:

1. There exists closed from formulas for minimizing over rigidmotions.

2. But it is hard to infimize over A(S,S ′)3. For reasonable surfaces (e.g. surfaces with uniformly bounded

curvatures), transformations a close to optimal are close toconformal.

4. Thus it suffices to only explore a smaller space of mapsobtained by small deformations of conformal maps.

Continuous procrustes distance between surfaces (cP)cont.

We modify the search as follows:Let m ∈M, then m is a conformal map. Let % be a smooth mapthat rounghly aligns high density peaks and χ be a specialdeformation s.t. χ ◦ % ◦m is area-preserving (up to approximationerror).For each choice of peaks p, p′ in the conformal factors of S,S ′

1. runs through the 1-parameter family of m that maps p to p′

2. constructs a map % that aligns the other peaks, as bestpossible

3. conpute d(S,S ′, % ◦m).

Repeat for all choices of p, p′. Choose % ◦m s.t. it minimizes dand deform it to be area-preserving.Then the map a = χ ◦ % ◦m is the approximate to correspondancemap and d(S,S ′, a) is the approximate to Dp(S,S ′).

Outline





Data and run time

There are three independent data sets:

1. 116 second mandibular molars (teeth) of prosimian primatesand non-primate close relatives

2. 57 proximal first metatarsals (bones behind big toe) ofprosimian primates, New and Old World monkeys

3. 45 distal radii (bone in forearm) of apes and humans

For each shape, geometric morphometricians collected landmarkss.t. the points are biologically and evolutionarily meaningful. Thenone can compute the Procrustes distances with the landmarks,producing Observer-Determined Landmarks Procrustes (ODLP)distances.Running times for a pair of surfaces:

1. cP: ∼ 20 sec.

2. cWn: ∼ 5 min.

Outline





Mantel correlation analysis

To assess the relationship between distance matrices, they used aMantel correlation analysis:First correlate the entries in the two square arrays, and thencompute the fraction among all possible relabelings of therow/columns for one of them, that leads to a larger correlationcoefficient

Conclusion: cP outperforms cWn.

Distance matrix

Conclusion: cP outperforms cWn.

Leave one out

I Each specimen (treated as unknown) is assigned to thetaxonomic groups of its nearest neighbor among the reminderof the specimens in the data set (treated as known).

Outline





Procedure

I Sample uniformly from surfaces and compute local distribution

I Define cost of transport based on local distribution matrix

I Use Sinkhorn’s algorithm to find a coupling that minimizesthe transportation cost

I obtain the third lower bound to the Wasserstein distance

Outline





Distance matirxK2000_epsParam0.005_niter44_DD0.3

10 20 30 40 50 60 70 80 90 100 110

ypm 30440qu 10966

q. i. 71qu 11117

pss 7/20-8pss 20-58

amnh 207949amnh 212954usnm 511930

mcz 38316amnh 100635

usnm 83650usnm 83652cab 04-274um 101958um 101958

um 87852amnh 100830

amnh 80072amnh 100654

xamnh 120449amnh 107136amnh 203258

amnh 24958mnhn av 4854mnhn av 7655

mnhn ri 170ualvp db194

ivpp 11001.1ivpp v10591

ivpp 11994ivpp 11001.2amnh 19159amnh 18696

usnm 063338amnh 119810amnh 241124amnh 239438amnh 241122amnh 187359amnh 187362amnh 187360amnh 236379amnh 241119usnm 063355usnm 083668amnh 100504amnh 100598amnh 100821amnh 170743amnh 170741amnh 100612amnh 100642amnh 170576amnh 170578

ccm 71-8ccm 71-6

ccm 73-31amnh 150062amnh 240827amnh 217303

sb-14*amnh 174530amnh 174533amnh 174531amnh 174489amnh 100832

mcz 45126amnh 87279

amnh 101508amnh 106650

usnm 9545amnh 269860

amnh 31252mcz 44953

amnh 100829pu 14270

bmnh 84.10.20.4ualvp 43232ualvp 43276amnh 35469amnh 35462amnh 16699

amnh 100827usnm 257397

usnm 63349usnm 63351

usnm 488055usnm 488059

uminn 1504ucmp 107406

um 90198um 90197

ualvp 39498amnh 109368amnh 109366amnh 196485amnh 106754amnh 106649

cl 457cl 455

irsnb 4291irsnb m65irsnb m64

wl 128wl 159

irsnb 4296ummz 113339ummz 123395ummz 58984

dummont specimenamnh 100514amnh 245092

amnh 18041amnh 17338

Distance matirx of Boyer et al. using cPBoyer_cP

10 20 30 40 50 60 70 80 90 100 110

ypm 30440qu 10966

q. i. 71qu 11117

pss 7/20-8pss 20-58

amnh 207949amnh 212954usnm 511930

mcz 38316amnh 100635

usnm 83650usnm 83652cab 04-274um 101958um 101958

um 87852amnh 100830

amnh 80072amnh 100654

xamnh 120449amnh 107136amnh 203258

amnh 24958mnhn av 4854mnhn av 7655

mnhn ri 170ualvp db194

ivpp 11001.1ivpp v10591

ivpp 11994ivpp 11001.2amnh 19159amnh 18696

usnm 063338amnh 119810amnh 241124amnh 239438amnh 241122amnh 187359amnh 187362amnh 187360amnh 236379amnh 241119usnm 063355usnm 083668amnh 100504amnh 100598amnh 100821amnh 170743amnh 170741amnh 100612amnh 100642amnh 170576amnh 170578

ccm 71-8ccm 71-6

ccm 73-31amnh 150062amnh 240827amnh 217303

sb-14*amnh 174530amnh 174533amnh 174531amnh 174489amnh 100832

mcz 45126amnh 87279

amnh 101508amnh 106650

usnm 9545amnh 269860

amnh 31252mcz 44953

amnh 100829pu 14270

bmnh 84.10.20.4ualvp 43232ualvp 43276amnh 35469amnh 35462amnh 16699

amnh 100827usnm 257397

usnm 63349usnm 63351

usnm 488055usnm 488059

uminn 1504ucmp 107406

um 90198um 90197

ualvp 39498amnh 109368amnh 109366amnh 196485amnh 106754amnh 106649

cl 457cl 455

irsnb 4291irsnb m65irsnb m64

wl 128wl 159

irsnb 4296ummz 113339ummz 123395ummz 58984

dummont specimenamnh 100514amnh 245092

amnh 18041amnh 17338

Single Linkage Dendrogram

5 6 7 8 9 10 11 12 13 14 15

10 -3

folivorous folivorous folivorous

frugivorousfrugivorousfolivorous folivorous folivorous folivorous

frugivorousfrugivorousfolivorous folivorous folivorous

p30 | nafrugivorousfolivorous folivorous

u13 | nav01 | nav02 | na

folivorous folivorous

v08 | nafolivorous folivorous folivorous folivorous folivorous

p35 | naomnivorous frugivorousfolivorous

omnivorous folivorous

p31 | nap32 | na

insectivorousfolivorous folivorous

b03 | nab01 | nab02 | nab08 | na

insectivorousinsectivorous

q02 | naq03 | naq04 | naq13 | naq11 | naq12 | naq06 | naq10 | na

omnivorous omnivorous omnivorous omnivorous

w11 | naw10 | naw09 | na


q18 | naa19 | na


q19 | nainsectivorous

w01 | naw02 | na


q28 | naa10 | naa13 | na

omnivorous insectivorousinsectivorousinsectivorousinsectivorous

a15 | naomnivorous frugivorous

a16 | naq26 | na

insectivorousb19 | na

insectivorousb20 | na


d09 | naomnivorous

insectivorousinsectivorousomnivorous omnivorous

folivorous insectivorousinsectivorous


frugivorousinsectivorousinsectivorousinsectivorousinsectivorousinsectivorousinsectivorous

frugivorousomnivorous

folivorous insectivorousinsectivorous

K2000_epsParam0.005_niter44_DD0.3

Single Linkage Dendrogram for Boyer et al. using cP

0.04 0.06 0.08 0.1 0.12 0.14

insectivorousp30 | nap31 | naq13 | naq18 | naw02 | na


w09 | nafolivorous folivorous

w11 | naw10 | na

insectivorousfolivorous

q12 | nainsectivorousinsectivorousomnivorous omnivorous omnivorous


omnivorous folivorous

frugivorousfrugivorous

q03 | naq19 | na

insectivorousomnivorous frugivorousfrugivorousfolivorous

frugivorousfolivorous

insectivorousinsectivorousinsectivorousinsectivorousomnivorous

insectivorousfolivorous

b01 | nainsectivorous

u13 | nab08 | nab20 | nab19 | na

insectivorousinsectivorousomnivorous omnivorous omnivorous

q26 | naomnivorous frugivorous

q28 | nafolivorous folivorous folivorous

w01 | nafolivorous

insectivorousomnivorous

folivorous b03 | nad09 | na

insectivorousinsectivorousinsectivorousinsectivorousinsectivorous

frugivorousinsectivorousinsectivorousinsectivorousinsectivorous

frugivorousfolivorous

frugivorousinsectivorousinsectivorousinsectivorous


a10 | naomnivorous

v08 | nainsectivorousinsectivorousinsectivorous

v02 | nafolivorous folivorous folivorous folivorous folivorous

v01 | nafolivorous folivorous folivorous folivorous

q04 | naq10 | naq11 | naa13 | naa15 | naa19 | naa16 | na

insectivorousp32 | nap35 | naq02 | naq06 | na

Boyer_cP

algorithms to automatically quantify the geometric ...€¦ · outline background motivation...

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