Teeth, Bones and Manifolds: a meeting of mathematical and biological minds Ingrid Daubechies Inaugural Conference, IMSA, 2019
Collaborators Rima Alaifari Doug Boyer Ingrid Daubechies Tingran Gao ETH Z¨ urich Duke Duke Duke Yaron Lipman Roi Poranne Jes´ us Puente Robert Ravier Weizmann ETH Z¨ urich J.P. Morgan Duke
Panchali Nag Chen-Yun Lin Shan Shan Shahar Kovalsky I.D. : mostly cheerleader
Panchali Nag Chen-Yun Lin Shan Shan Shahar Kovalsky I.D. : mostly cheerleader
It all started with a conversation with biologists.... Jukka Jernvall More Precisely: biological morphologists � Study Teeth & Bones of extant & extinct animals � � still live today fossils Doug Boyer
First: project on “complexity” of teeth
First: project on “complexity” of teeth Then: find automatic way to compute Procrustes distances between surfaces — without landmarks
Data Acquisition Surface reconstructed from µ CT-scanned voxel data
Geometric Morphometrics • Manually put k landmarks second mandibular molar of a Philippine flying lemur
Geometric Morphometrics • Manually put k landmarks p 1 , p 2 , · · · , p k • Use spatial coordinates of the landmarks as features • Represent a shape in R 3 × k second mandibular molar of a Philippine flying lemur
Geometric Morphometrics • Manually put k landmarks p 1 , p 2 , · · · , p k • Use spatial coordinates of the landmarks as features p j = ( x j , y j , z j ) , j = 1 , · · · , k • Represent a shape in R 3 × k second mandibular molar of a Philippine flying lemur
Geometric Morphometrics • Manually put k landmarks p 1 , p 2 , · · · , p k • Use spatial coordinates of the landmarks as features p j = ( x j , y j , z j ) , j = 1 , · · · , k • Represent a shape in R 3 × k second mandibular molar of a Philippine flying lemur
The Shape Space of k landmarks in R 3
Geometric Morphometrics: Limitation of Landmarks
Geometric Morphometrics: Limitation of Landmarks • Landmark Placement: tedious and time-consuming
Geometric Morphometrics: Limitation of Landmarks • Landmark Placement: tedious and time-consuming • Fixed Number of Landmarks: lack of flexibility
Geometric Morphometrics: Limitation of Landmarks • Landmark Placement: tedious and time-consuming • Fixed Number of Landmarks: lack of flexibility • Domain Knowledge: high degree of expertise needed, not easily accessible
Geometric Morphometrics: Limitation of Landmarks • Landmark Placement: tedious and time-consuming • Fixed Number of Landmarks: lack of flexibility • Domain Knowledge: high degree of expertise needed, not easily accessible • Subjectivity: debates exist even among experts
First: project on “complexity” of teeth Then: find automatic way to compute Procrustes distances between surfaces — without landmarks Landmarked Teeth − → J � � R ( x j ) − y j � 2 d 2 Procrustes ( S 1 , S 2 ) = min R rigid tr. j =1
First: project on “complexity” of teeth Then: find automatic way to compute Procrustes distances between surfaces — without landmarks Landmarked Teeth − → J � � R ( x j ) − y j � 2 d 2 Procrustes ( S 1 , S 2 ) = min R rigid tr. j =1 Find way to compute a distance that does as well, for biological purposes, as Procrustes distance, based on expert-placed landmarks, automatically?
First: project on “complexity” of teeth Then: find automatic way to compute Procrustes distances between surfaces — without landmarks Landmarked Teeth − → J � � R ( x j ) − y j � 2 d 2 Procrustes ( S 1 , S 2 ) = min R rigid tr. j =1 Find way to compute a distance that does as well, for biological purposes, as Procrustes distance, based on expert-placed landmarks, automatically? examples: finely discretized triangulated surfaces
conformal Wasserstein neighborhood distance
Continuous Procrustes Distance (cPD) � 1 � � 2 x − C ( x ) � 2 d vol S 1 ( x ) D cP ( S 1 , S 2 ) = � , S 1 where C : S 1 → S 2 is an area-preserving diffeomorphism. so as to wrap into the next line
Continuous Procrustes Distance (cPD) � 1 � � 2 � R ( x ) − C ( x ) � 2 d vol S 1 ( x ) D cP ( S 1 , S 2 ) = inf , R ∈ E (3) S 1 where C : S 1 → S 2 is an area-preserving diffeomorphism, and E 3 is the Euclidean group on R 3 .
Continuous Procrustes Distance (cPD) � 1 � � 2 � R ( x ) − C ( x ) � 2 d vol S 1 ( x ) D cP ( S 1 , S 2 ) = inf inf , C∈A ( S 1 , S 2 ) R ∈ E (3) S 1 where A ( S 1 , S 2 ) is the set of area-preserving diffeomorphisms between S 1 and S 2 , and E 3 is the Euclidean group on R 3 .
Continuous Procrustes Distance (cPD) � 1 / 2 �� � R ( x ) − C ( x ) � 2 d vol S 1 ( x ) d cP ( S 1 , S 2 ) = inf inf C∈ A R ∈ E 3 S 1 d 12 − − − →
We defined 2 different distances d cWn ( S 1 , S 2 ): conformal flattening comparison of neighborhood geometry optimal mass transport d cP ( S 1 , S 2 ): continuous Procrustes distance
Bypass Explicit Feature Extraction S 1 S 2
Multi-Dimensional Scaling (MDS) for cPD Matrix
Diffusion Maps: “Knit together” local geometry to get “better” distances Small distances are much more reliable!
Diffusion Maps: “knitting together” local geometry Small distances are much more reliable!
Diffusion Maps: “knitting together” local geometry Small distances are much more reliable!
Diffusion Maps: “knitting together” local geometry Small distances are much more reliable!
Diffusion Maps: “knitting together” local geometry Small distances are much more reliable!
Diffusion Maps: “knitting together” local geometry
Diffusion Maps: “knitting together” local geometry Small distances are much more reliable!
Diffusion Maps: “knitting together” local geometry • P = D − 1 W defines a random S i walk on the graph d ij S j
Diffusion Maps: “knitting together” local geometry • P = D − 1 W defines a random S i walk on the graph • Solve eigen-problem d ij Pu j = λ j u j , j = 1 , 2 , · · · , m S j
Diffusion Maps: “knitting together” local geometry • P = D − 1 W defines a random S i walk on the graph • Solve eigen-problem d ij Pu j = λ j u j , j = 1 , 2 , · · · , m S j and represent each individual shape S j as an m -vector � � λ t / 2 1 u 1 ( j ) , · · · , λ t / 2 m u m ( j )
Diffusion Distance (DD) Fix 1 ≤ m ≤ N , t ≥ 0, � m � 1 2 � k ( u k ( i ) − u k ( j )) 2 D t λ t m ( S i , S j ) = k =1
Diffusion Distance (DD) Fix 1 ≤ m ≤ N , t ≥ 0, � m � 1 2 � k ( u k ( i ) − u k ( j )) 2 D t λ t m ( S i , S j ) = k =1
MDS for cPD & DD cPD DD
Even better can be obtained! HBDD DD
9/60
Horizontal Random Walk on a Fibre Bundle Fibre Bundle E = ( E , M , F , π ) ◮ E : total manifold ◮ M : base manifold ◮ π : E → M : smooth surjective map ( bundle projection ) ◮ F : fibre manifold
Horizontal Random Walk on a Fibre Bundle Fibre Bundle E = ( E , M , F , π ) ◮ E : total manifold ◮ M : base manifold ◮ π : E → M : smooth surjective map ( bundle projection ) ◮ F : fibre manifold ◮ local triviality : for “small” open set U ⊂ M , π − 1 ( U ) is diffeomorphic to U × F
16/60
Horizontal Random Walk on a Fibre Bundle Fibre Bundle E = ( E , M , F , π ) ◮ E : total manifold ◮ M : base manifold ◮ π : E → M : smooth surjective map ( bundle projection ) ◮ F : fibre manifold P = D − 1 W ◮ local triviality : for “small” open set U ⊂ M , π − 1 ( U ) is diffeomorphic to U × F S 3 S 3 S 1 S 1 S 0 S 0 S 2 S 2 M M
Horizontal Random Walk on a Fibre Bundle Fibre Bundle E = ( E , M , F , π ) ◮ E : total manifold ◮ M : base manifold ◮ π : E → M : smooth surjective map ( bundle projection ) ◮ F : fibre manifold P = D − 1 W ◮ local triviality : for “small” open set U ⊂ M , π − 1 ( U ) is diffeomorphic to U × F S 3 S 3 S 1 S 1 S 0 S 0 S 2 S 2 M M
Horizontal Random Walk on a Fibre Bundle Fibre Bundle E = ( E , M , F , π ) ◮ E : total manifold ◮ M : base manifold ◮ π : E → M : smooth surjective map ( bundle projection ) ◮ F : fibre manifold P = D − 1 W ◮ local triviality : for “small” open set U ⊂ M , π − 1 ( U ) is diffeomorphic to U × F S 3 S 3 S 1 S 1 S 0 S 0 S 2 S 2 M M
Horizontal Random Walk on a Fibre Bundle Fibre Bundle E = ( E , M , F , π ) ◮ E : total manifold ◮ M : base manifold ◮ π : E → M : smooth surjective map ( bundle projection ) ◮ F : fibre manifold P = D − 1 W ◮ local triviality : for “small” open set U ⊂ M , π − 1 ( U ) is diffeomorphic to U × F S 3 S 3 S 1 S 1 S 0 S 0 S 2 S 2 M M
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