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Algorithms to automatically quantify the geometric simliarity of anatomical surfaces

Boyer, Lipman, St.Clair, Puente, Pantel, Funkhouser, Jernval and Daubechies Ying Yin March 2018

Outline

Background Motivation General Idea Mathematical background New distances Conformal Wasserstein distances (cW) Conformal Wasserstein neighborhood dissimlarity distance (cWn) Continuous procrustes distance between surfaces (cP) Experiments Data and Perfomances Performance test Results using TLB Outline Results

Outline

Background Motivation General Idea Mathematical background New distances Conformal Wasserstein distances (cW) Conformal Wasserstein neighborhood dissimlarity distance (cWn) Continuous procrustes distance between surfaces (cP) Experiments Data and Perfomances Performance test Results using TLB Outline Results

Motivation

◮ Understand physical and biological phenomena (e.g.

speciation, evolutionary adaption, etc.) by quantifying the similarity or dissimilarity of objects affected by the phenomena.

◮ In standard morphologists’ practice, 10 to 100 points will be

identified as landmarks. By comparing these landmarks, similarity and dissimilarity between patterns of shapes can be determined.

◮ The difficulty in acquiring personal knowledge of

morphological evidence limits our understanding of the evolutionary significance of morphological diversity.

◮ Want an automatic tool to decide similarity or dissimilarity

between objects, and hence, provides more insights on the phenomenon.

Outline

Background Motivation General Idea Mathematical background New distances Conformal Wasserstein distances (cW) Conformal Wasserstein neighborhood dissimlarity distance (cWn) Continuous procrustes distance between surfaces (cP) Experiments Data and Perfomances Performance test Results using TLB Outline Results

General Idea

◮ Given two shapes S, S′ (with boundaries but not holes),

conformally map them onto D2 by applying Riemann’s uniformization theorem.

◮ Conformal geometry permits the reduction of the study of

surfaces embedded in 3D space to 2D problems

◮ By finding a coupling between the conformal factors, or by

finding a correspondence between the disks that respects the conformal factors, one may be able to define new distances that measures similarity and dissimilarity.

Outline

Background Motivation General Idea Mathematical background New distances Conformal Wasserstein distances (cW) Conformal Wasserstein neighborhood dissimlarity distance (cWn) Continuous procrustes distance between surfaces (cP) Experiments Data and Perfomances Performance test Results using TLB Outline Results

Conformal map

Definition

A map ϕ : S → S′ between two (smooth) surfaces is conformal if for any two smooth curves Γ1, Γ2 on S, the angle between their images Γ′

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

corresponding intersection point.

Definition

Two Riemannian metrics g and h on a smooth manifold M are called conformally equivalent if g = fh for some positive function f

• n M. The function f is called the conformal factor.

Remark:

• 1. the conformal factor indicates the area distortion factor

produced by the operation of conformal mapping.

• 2. the conformal factor defines a probability measure.

Disk-preserving M¨

• bius transofrmation

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

Definition

Such m is called a disk-preserving M¨

• bius transformation. And the

collection 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 let f(x, y) = [1 − (x2 + y2)]2f (x, y). Then we have fdη = fdxdy.

Push-forward and Transport Effort

Definition

Let µ be a probabilty measure, and τ be a differentiable bijection from D2 to itself, the mass distribution µ′ = τ∗µ defined by µ(u) = µ′(τ(u))Jτ(u) where Jτ is the Jacobian of τ is the transportation (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.

Definition

The 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 the Kantorovitch 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)
• ver all couplings π.

Outline

Background Motivation General Idea Mathematical background New distances Conformal Wasserstein distances (cW) Conformal Wasserstein neighborhood dissimlarity distance (cWn) Continuous procrustes distance between surfaces (cP) Experiments Data and Perfomances Performance test Results using TLB Outline Results

Conformal Wasserstein distances (cW)

Instead of comparing two surfaces S, S′, one can compare two conformal factors f, f′ obtained by conformally flattening S, S′. Let m be a disk-preserving M¨

• bius transformation, then f and

m∗f = f ◦ m−1 are both conformal factors for S. Then we define the conformal Wasserstein distance to be DcW (S, S′) = inf

m∈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 Kantorovitch

problem for every m.

Outline

Background Motivation General Idea Mathematical background New distances Conformal Wasserstein distances (cW) Conformal Wasserstein neighborhood dissimlarity distance (cWn) Continuous procrustes distance between surfaces (cP) Experiments Data and Perfomances Performance test Results using TLB Outline Results

Conformal Wasserstein neighborhood dissimlarity distance (cWn)

Instead, we quantify how dissimilar the ”landscapes” are with a measure 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 dR

f,f ′(z, z′) =

inf

m∈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 dR

f,f ′(z, z′) =

inf

m∈M,m(z)=z′

• N(z,R)

|f(w) − f(m(w))|dη(w)

• The conformal Wasserstein neighborhood dissimilarity distance

between f and f′ is DR

cWn(S, S′) =

inf

π∈(f,f ′)

• D2×D2 d R

f,f ′(z, z′)dπ(z, z′)

Remark

◮ Both cW and cWn are blind to isometric embedding of a

surface in 3D

◮ Introduce a new extrinsic distance

Outline

Background Motivation General Idea Mathematical background New distances Conformal Wasserstein distances (cW) Conformal Wasserstein neighborhood dissimlarity distance (cWn) Continuous procrustes distance between surfaces (cP) Experiments Data and Perfomances Performance test Results using TLB Outline Results

Procrustes distance between surfaces

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

R 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 of

geometric 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′ and use optimization to find the ”best” a. We require a to be area-preserving. We denote the set of all area-preserving diffeomorphisms by A(S, S′). And let d(S, S′, a)2 = min

R rigid motions

• S

|R(x) − a(x)|2dAS . Then we define the continuous Procrustes distance between S and S’ by Dp(S, S′) = inf

a∈A(S,S′) d(S, S′, a).

Continuous procrustes distance between surfaces (cP) cont.

Remarks:

• 1. There exists closed from formulas for minimizing over rigid

motions.

• 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 to conformal.

• 4. Thus it suffices to only explore a smaller space of maps
• btained 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 map that rounghly aligns high density peaks and χ be a special deformation s.t. χ ◦ ̺ ◦ m is area-preserving (up to approximation error). 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 best

possible

• 3. conpute d(S, S′, ̺ ◦ m).

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

Outline

Background Motivation General Idea Mathematical background New distances Conformal Wasserstein distances (cW) Conformal Wasserstein neighborhood dissimlarity distance (cWn) Continuous procrustes distance between surfaces (cP) Experiments Data and Perfomances Performance test Results using TLB Outline Results

Data and run time

There are three independent data sets:

• 1. 116 second mandibular molars (teeth) of prosimian primates

and non-primate close relatives

• 2. 57 proximal first metatarsals (bones behind big toe) of

prosimian primates, New and Old World monkeys

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

For each shape, geometric morphometricians collected landmarks s.t. the points are biologically and evolutionarily meaningful. Then

• ne 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

Background Motivation General Idea Mathematical background New distances Conformal Wasserstein distances (cW) Conformal Wasserstein neighborhood dissimlarity distance (cWn) Continuous procrustes distance between surfaces (cP) Experiments Data and Perfomances Performance test Results using TLB Outline Results

Mantel correlation analysis

To assess the relationship between distance matrices, they used a Mantel correlation analysis: First correlate the entries in the two square arrays, and then compute the fraction among all possible relabelings of the row/columns for one of them, that leads to a larger correlation coefficient Conclusion: cP outperforms cWn.

Distance matrix

Conclusion: cP outperforms cWn.

Leave one out

◮ Each specimen (treated as unknown) is assigned to the

taxonomic groups of its nearest neighbor among the reminder

• f the specimens in the data set (treated as known).

Outline

Background Motivation General Idea Mathematical background New distances Conformal Wasserstein distances (cW) Conformal Wasserstein neighborhood dissimlarity distance (cWn) Continuous procrustes distance between surfaces (cP) Experiments Data and Perfomances Performance test Results using TLB Outline Results

Procedure

◮ Sample uniformly from surfaces and compute local distribution ◮ Define cost of transport based on local distribution matrix ◮ Use Sinkhorn’s algorithm to find a coupling that minimizes

the transportation cost

◮ obtain the third lower bound to the Wasserstein distance

Outline

Background Motivation General Idea Mathematical background New distances Conformal Wasserstein distances (cW) Conformal Wasserstein neighborhood dissimlarity distance (cWn) Continuous procrustes distance between surfaces (cP) Experiments Data and Perfomances Performance test Results using TLB Outline Results

Distance matirx

K2000_epsParam0.005_niter44_DD0.3 10 20 30 40 50 60 70 80 90 100 110 ypm 30440 qu 10966

• q. i. 71

qu 11117 pss 7/20-8 pss 20-58 amnh 207949 amnh 212954 usnm 511930 mcz 38316 amnh 100635 usnm 83650 usnm 83652 cab 04-274 um 101958 um 101958 um 87852 amnh 100830 amnh 80072 amnh 100654 x amnh 120449 amnh 107136 amnh 203258 amnh 24958 mnhn av 4854 mnhn av 7655 mnhn ri 170 ualvp db194 ivpp 11001.1 ivpp v10591 ivpp 11994 ivpp 11001.2 amnh 19159 amnh 18696 usnm 063338 amnh 119810 amnh 241124 amnh 239438 amnh 241122 amnh 187359 amnh 187362 amnh 187360 amnh 236379 amnh 241119 usnm 063355 usnm 083668 amnh 100504 amnh 100598 amnh 100821 amnh 170743 amnh 170741 amnh 100612 amnh 100642 amnh 170576 amnh 170578 ccm 71-8 ccm 71-6 ccm 73-31 amnh 150062 amnh 240827 amnh 217303 sb-14* amnh 174530 amnh 174533 amnh 174531 amnh 174489 amnh 100832 mcz 45126 amnh 87279 amnh 101508 amnh 106650 usnm 9545 amnh 269860 amnh 31252 mcz 44953 amnh 100829 pu 14270 bmnh 84.10.20.4 ualvp 43232 ualvp 43276 amnh 35469 amnh 35462 amnh 16699 amnh 100827 usnm 257397 usnm 63349 usnm 63351 usnm 488055 usnm 488059 uminn 1504 ucmp 107406 um 90198 um 90197 ualvp 39498 amnh 109368 amnh 109366 amnh 196485 amnh 106754 amnh 106649 cl 457 cl 455 irsnb 4291 irsnb m65 irsnb m64 wl 128 wl 159 irsnb 4296 ummz 113339 ummz 123395 ummz 58984 dummont specimen amnh 100514 amnh 245092 amnh 18041 amnh 17338

Distance matirx of Boyer et al. using cP

Boyer_cP 10 20 30 40 50 60 70 80 90 100 110 ypm 30440 qu 10966

• q. i. 71

qu 11117 pss 7/20-8 pss 20-58 amnh 207949 amnh 212954 usnm 511930 mcz 38316 amnh 100635 usnm 83650 usnm 83652 cab 04-274 um 101958 um 101958 um 87852 amnh 100830 amnh 80072 amnh 100654 x amnh 120449 amnh 107136 amnh 203258 amnh 24958 mnhn av 4854 mnhn av 7655 mnhn ri 170 ualvp db194 ivpp 11001.1 ivpp v10591 ivpp 11994 ivpp 11001.2 amnh 19159 amnh 18696 usnm 063338 amnh 119810 amnh 241124 amnh 239438 amnh 241122 amnh 187359 amnh 187362 amnh 187360 amnh 236379 amnh 241119 usnm 063355 usnm 083668 amnh 100504 amnh 100598 amnh 100821 amnh 170743 amnh 170741 amnh 100612 amnh 100642 amnh 170576 amnh 170578 ccm 71-8 ccm 71-6 ccm 73-31 amnh 150062 amnh 240827 amnh 217303 sb-14* amnh 174530 amnh 174533 amnh 174531 amnh 174489 amnh 100832 mcz 45126 amnh 87279 amnh 101508 amnh 106650 usnm 9545 amnh 269860 amnh 31252 mcz 44953 amnh 100829 pu 14270 bmnh 84.10.20.4 ualvp 43232 ualvp 43276 amnh 35469 amnh 35462 amnh 16699 amnh 100827 usnm 257397 usnm 63349 usnm 63351 usnm 488055 usnm 488059 uminn 1504 ucmp 107406 um 90198 um 90197 ualvp 39498 amnh 109368 amnh 109366 amnh 196485 amnh 106754 amnh 106649 cl 457 cl 455 irsnb 4291 irsnb m65 irsnb m64 wl 128 wl 159 irsnb 4296 ummz 113339 ummz 123395 ummz 58984 dummont specimen amnh 100514 amnh 245092 amnh 18041 amnh 17338

Single Linkage Dendrogram

5 6 7 8 9 10 11 12 13 14 folivorous folivorous folivorous frugivorous frugivorous folivorous folivorous folivorous folivorous frugivorous frugivorous folivorous folivorous folivorous p30 | na frugivorous folivorous folivorous u13 | na v01 | na v02 | na folivorous folivorous v08 | na folivorous folivorous folivorous folivorous folivorous p35 | na

• mnivorous

frugivorous folivorous

• mnivorous

folivorous p31 | na p32 | na insectivorous folivorous folivorous b03 | na b01 | na b02 | na b08 | na insectivorous insectivorous q02 | na q03 | na q04 | na q13 | na q11 | na q12 | na q06 | na q10 | na

• mnivorous
• mnivorous
• mnivorous
• mnivorous

w11 | na w10 | na w09 | na insectivorous insectivorous q18 | na a19 | na insectivorous insectivorous q19 | na insectivorous w01 | na w02 | na insectivorous insectivorous q28 | na a10 | na a13 | na

• mnivorous

insectivorous insectivorous insectivorous insectivorous a15 | na

• mnivorous

frugivorous a16 | na q26 | na insectivorous b19 | na insectivorous b20 | na insectivorous insectivorous d09 | na

• mnivorous

insectivorous insectivorous

• mnivorous
• mnivorous

folivorous insectivorous insectivorous folivorous folivorous frugivorous insectivorous insectivorous insectivorous insectivorous insectivorous insectivorous frugivorous

• mnivorous

folivorous insectivorous insectivorous 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 insectivorous p30 | na p31 | na q13 | na q18 | na w02 | na folivorous folivorous w09 | na folivorous folivorous w11 | na w10 | na insectivorous folivorous q12 | na insectivorous insectivorous

• mnivorous
• mnivorous
• mnivorous

folivorous folivorous

• mnivorous

folivorous frugivorous frugivorous q03 | na q19 | na insectivorous

• mnivorous

frugivorous frugivorous folivorous frugivorous folivorous insectivorous insectivorous insectivorous insectivorous

• mnivorous

insectivorous folivorous b01 | na insectivorous u13 | na b08 | na b20 | na b19 | na insectivorous insectivorous

• mnivorous
• mnivorous
• mnivorous

q26 | na

• mnivorous

frugivorous q28 | na folivorous folivorous folivorous w01 | na folivorous insectivorous

• mnivorous

folivorous b03 | na d09 | na insectivorous insectivorous insectivorous insectivorous insectivorous frugivorous insectivorous insectivorous insectivorous insectivorous frugivorous folivorous frugivorous insectivorous insectivorous insectivorous folivorous folivorous a10 | na

• mnivorous

v08 | na insectivorous insectivorous insectivorous v02 | na folivorous folivorous folivorous folivorous folivorous v01 | na folivorous folivorous folivorous folivorous q04 | na q10 | na q11 | na a13 | na a15 | na a19 | na a16 | na insectivorous p32 | na p35 | na q02 | na q06 | na Boyer_cP