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Shape Analysis in R GM library in the light of recent methodological - PowerPoint PPT Presentation

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Outline Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics and Statistics, Comenius University, Bratislava, Slovakia Neurospin,

  1. Introduction Cubic splines TPS for shape data Acknowledgement Example 1 – shape data Data Coquerelle M, Bookstein FL, Braga J, Halazonetis DJ, Katina S , Weber GW, 2009. Visualizing mandibular shape changes of modern humans and chimpanzees (Pan troglodytes) from fetal life to the complete eruption of the deciduous dentition. The Anatomical Record (accepted) computed tomographies (CT) of 151 modern humans (78 females and 73 males) of mixed ethnicity, living in France, from birth to adulthood. [Pellegrin Hospital (Bordeaux), Necker Hospital (Paris) and Clinique Pasteur (Toulouse)]

  2. Introduction Cubic splines TPS for shape data Acknowledgement Example 1 – shape data Data each mandibular surface was reconstructed from the CT-scans via the software package Amira (Mercury Computer Systems, Chelmsford, MA) open-source software Edgewarp3D (Bookstein & Green 2002), a 3D-template of 415 landmarks and semilandmarks was created to measure the mandibular surface and was warped onto each mandible

  3. Introduction Cubic splines TPS for shape data Acknowledgement Example 1 – shape data Data each mandibular surface was reconstructed from the CT-scans via the software package Amira (Mercury Computer Systems, Chelmsford, MA) open-source software Edgewarp3D (Bookstein & Green 2002), a 3D-template of 415 landmarks and semilandmarks was created to measure the mandibular surface and was warped onto each mandible

  4. Introduction Cubic splines TPS for shape data Acknowledgement Example 1 – shape data Data

  5. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Interpolation model Consider a NCS given by k f ( x ) = c + ax + � w j φ j ( x ) , j = 1 , 2 , . . . k , j = 1 where � 3 with the x j are the knots, φ j ( x ) = φ x − x j � x − x j 1 � � � � = 12 constraints � k j = 1 w j = � k j = 1 w j x j = 0, f ′′ and f ′′′ are both zero outside the interval [ x 1 , x k ] 12 | x | 3 is a continuous function known as a function φ ( x ) = 1 radial (nodal) basis function (Jackson 1989)

  6. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Interpolation model Consider a NCS given by k f ( x ) = c + ax + � w j φ j ( x ) , j = 1 , 2 , . . . k , j = 1 where � 3 with the x j are the knots, φ j ( x ) = φ x − x j � x − x j 1 � � � � = 12 constraints � k j = 1 w j = � k j = 1 w j x j = 0, f ′′ and f ′′′ are both zero outside the interval [ x 1 , x k ] 12 | x | 3 is a continuous function known as a function φ ( x ) = 1 radial (nodal) basis function (Jackson 1989)

  7. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Interpolation model Consider a NCS given by k f ( x ) = c + ax + � w j φ j ( x ) , j = 1 , 2 , . . . k , j = 1 where � 3 with the x j are the knots, φ j ( x ) = φ x − x j � x − x j 1 � � � � = 12 constraints � k j = 1 w j = � k j = 1 w j x j = 0, f ′′ and f ′′′ are both zero outside the interval [ x 1 , x k ] 12 | x | 3 is a continuous function known as a function φ ( x ) = 1 radial (nodal) basis function (Jackson 1989)

  8. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Interpolation model � 3 , Let ( S ) ij = φ j ( x i ) = φ ( x i − x j ) = � x i − x j 1 � � 12 w = ( w 1 , . . . w k ) T constraint ( 1 k , x ) T w = 0 x j , y j � � NCS interpolation to the data  y   S 1 k x   w  1 T c  =  , 0 0 0 (1) k     x T a 0 0 0 where x k × 1 = ( x 1 , . . . x k ) T and y k × 1 = ( y 1 , y 2 , . . . y k ) T

  9. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Interpolation model � 3 , Let ( S ) ij = φ j ( x i ) = φ ( x i − x j ) = � x i − x j 1 � � 12 w = ( w 1 , . . . w k ) T constraint ( 1 k , x ) T w = 0 x j , y j � � NCS interpolation to the data  y   S 1 k x   w  1 T c  =  , 0 0 0 (1) k     x T a 0 0 0 where x k × 1 = ( x 1 , . . . x k ) T and y k × 1 = ( y 1 , y 2 , . . . y k ) T

  10. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Interpolation model � 3 , Let ( S ) ij = φ j ( x i ) = φ ( x i − x j ) = � x i − x j 1 � � 12 w = ( w 1 , . . . w k ) T constraint ( 1 k , x ) T w = 0 x j , y j � � NCS interpolation to the data  y   S 1 k x   w  1 T c  =  , 0 0 0 (1) k     x T a 0 0 0 where x k × 1 = ( x 1 , . . . x k ) T and y k × 1 = ( y 1 , y 2 , . . . y k ) T

  11. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Interpolation model � 3 , Let ( S ) ij = φ j ( x i ) = φ ( x i − x j ) = � x i − x j 1 � � 12 w = ( w 1 , . . . w k ) T constraint ( 1 k , x ) T w = 0 x j , y j � � NCS interpolation to the data  y   S 1 k x   w  1 T c  =  , 0 0 0 (1) k     x T a 0 0 0 where x k × 1 = ( x 1 , . . . x k ) T and y k × 1 = ( y 1 , y 2 , . . . y k ) T

  12. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Interpolation model Let matrix L be defined as   S 1 k x 1 T L = 0 0 k   x T 0 0 inverse of L is equal to � L 11 L 12 � k × k k × 2 L − 1 = L 21 L 22 2 × k 2 × 2

  13. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Interpolation model Let matrix L be defined as   S 1 k x 1 T L = 0 0 k   x T 0 0 inverse of L is equal to � L 11 L 12 � k × k k × 2 L − 1 = L 21 L 22 2 × k 2 × 2

  14. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Interpolation model bending energy matrix – k × k matrix B e = L 11 constrains of this matrix 1 T k B e = 0 , x T B e = 0 , so the rank of the B e is k − 2 w = B e y ( c , a ) T = L 21 y J ( f ) = w T Sw = y T B e y

  15. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Interpolation model bending energy matrix – k × k matrix B e = L 11 constrains of this matrix 1 T k B e = 0 , x T B e = 0 , so the rank of the B e is k − 2 w = B e y ( c , a ) T = L 21 y J ( f ) = w T Sw = y T B e y

  16. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Interpolation model bending energy matrix – k × k matrix B e = L 11 constrains of this matrix 1 T k B e = 0 , x T B e = 0 , so the rank of the B e is k − 2 w = B e y ( c , a ) T = L 21 y J ( f ) = w T Sw = y T B e y

  17. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Interpolation model bending energy matrix – k × k matrix B e = L 11 constrains of this matrix 1 T k B e = 0 , x T B e = 0 , so the rank of the B e is k − 2 w = B e y ( c , a ) T = L 21 y J ( f ) = w T Sw = y T B e y

  18. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Interpolation model bending energy matrix – k × k matrix B e = L 11 constrains of this matrix 1 T k B e = 0 , x T B e = 0 , so the rank of the B e is k − 2 w = B e y ( c , a ) T = L 21 y J ( f ) = w T Sw = y T B e y

  19. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Data pre-processing SVD of X c = ΓΛΓ T = � 2 j , X c = X − 1 k x T (Mardia j = 1 λ j γ j γ T et al. 2000) [ principal component analysis ] the 1th principal component of X is equal to z 1 = X c γ 1 , where γ 1 is the 1th column of Γ , and z 1 j , j = 1 , 2 , ... k are principal component scores of j th landmark ( z 1 j is j th element of k -vector z 1 ) re-ordering of the rows of X is done based on the ranks of z 1 j in z 1

  20. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Data pre-processing SVD of X c = ΓΛΓ T = � 2 j , X c = X − 1 k x T (Mardia j = 1 λ j γ j γ T et al. 2000) [ principal component analysis ] the 1th principal component of X is equal to z 1 = X c γ 1 , where γ 1 is the 1th column of Γ , and z 1 j , j = 1 , 2 , ... k are principal component scores of j th landmark ( z 1 j is j th element of k -vector z 1 ) re-ordering of the rows of X is done based on the ranks of z 1 j in z 1

  21. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Data pre-processing SVD of X c = ΓΛΓ T = � 2 j , X c = X − 1 k x T (Mardia j = 1 λ j γ j γ T et al. 2000) [ principal component analysis ] the 1th principal component of X is equal to z 1 = X c γ 1 , where γ 1 is the 1th column of Γ , and z 1 j , j = 1 , 2 , ... k are principal component scores of j th landmark ( z 1 j is j th element of k -vector z 1 ) re-ordering of the rows of X is done based on the ranks of z 1 j in z 1

  22. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Data pre-processing SVD of D dc (Gower 1966) [ principal coordinate analysis ] D 1 is k × k matrix of squared interlandmark Euklidean distances, D 2 = − 1 2 D 1 and D dc = D 2 − 1 k D 2 − 1 k + 1 k 1 k 1 T k D 2 1 k 1 T k 2 1 k 1 T k D 2 1 k 1 T k doubly centered (both row- and column-centered)

  23. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Data pre-processing SVD of D dc (Gower 1966) [ principal coordinate analysis ] D 1 is k × k matrix of squared interlandmark Euklidean distances, D 2 = − 1 2 D 1 and D dc = D 2 − 1 k D 2 − 1 k + 1 k 1 k 1 T k D 2 1 k 1 T k 2 1 k 1 T k D 2 1 k 1 T k doubly centered (both row- and column-centered)

  24. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Data pre-processing SVD of D dc (Gower 1966) [ principal coordinate analysis ] D 1 is k × k matrix of squared interlandmark Euklidean distances, D 2 = − 1 2 D 1 and D dc = D 2 − 1 k D 2 − 1 k + 1 k 1 k 1 T k D 2 1 k 1 T k 2 1 k 1 T k D 2 1 k 1 T k doubly centered (both row- and column-centered)

  25. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Modified interpolation model chordal distance d ( j ) ch of the rows j − 1 and j of ( x , y ) , j = 2 , 3 , ... k cumulative chordal distance d ( j ) cch = � j i = 2 d ( i ) ch , j = 2 , 3 , ... k d ( j ) cch = d j , j = 1 , 2 , ... k , d cch = ( d 1 , d 2 , ... d k ) T , d 1 = 0 NCS of x on d cch NCS of y on d cch

  26. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Modified interpolation model chordal distance d ( j ) ch of the rows j − 1 and j of ( x , y ) , j = 2 , 3 , ... k cumulative chordal distance d ( j ) cch = � j i = 2 d ( i ) ch , j = 2 , 3 , ... k d ( j ) cch = d j , j = 1 , 2 , ... k , d cch = ( d 1 , d 2 , ... d k ) T , d 1 = 0 NCS of x on d cch NCS of y on d cch

  27. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Modified interpolation model chordal distance d ( j ) ch of the rows j − 1 and j of ( x , y ) , j = 2 , 3 , ... k cumulative chordal distance d ( j ) cch = � j i = 2 d ( i ) ch , j = 2 , 3 , ... k d ( j ) cch = d j , j = 1 , 2 , ... k , d cch = ( d 1 , d 2 , ... d k ) T , d 1 = 0 NCS of x on d cch NCS of y on d cch

  28. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Modified interpolation model chordal distance d ( j ) ch of the rows j − 1 and j of ( x , y ) , j = 2 , 3 , ... k cumulative chordal distance d ( j ) cch = � j i = 2 d ( i ) ch , j = 2 , 3 , ... k d ( j ) cch = d j , j = 1 , 2 , ... k , d cch = ( d 1 , d 2 , ... d k ) T , d 1 = 0 NCS of x on d cch NCS of y on d cch

  29. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Modified interpolation model chordal distance d ( j ) ch of the rows j − 1 and j of ( x , y ) , j = 2 , 3 , ... k cumulative chordal distance d ( j ) cch = � j i = 2 d ( i ) ch , j = 2 , 3 , ... k d ( j ) cch = d j , j = 1 , 2 , ... k , d cch = ( d 1 , d 2 , ... d k ) T , d 1 = 0 NCS of x on d cch NCS of y on d cch

  30. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Data For the purpose of re-sampling 21 digitized semilandmarks on the symphisis X P , 2 = ( x P , 21 , x P , 22 ) , d cch , 2 (subject No . 2) NCS of y = x P , 21 on x = d cch , 2 NCS of y = x P , 22 on x = d cch , 2

  31. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Data For the purpose of re-sampling 21 digitized semilandmarks on the symphisis X P , 2 = ( x P , 21 , x P , 22 ) , d cch , 2 (subject No . 2) NCS of y = x P , 21 on x = d cch , 2 NCS of y = x P , 22 on x = d cch , 2

  32. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Data For the purpose of re-sampling 21 digitized semilandmarks on the symphisis X P , 2 = ( x P , 21 , x P , 22 ) , d cch , 2 (subject No . 2) NCS of y = x P , 21 on x = d cch , 2 NCS of y = x P , 22 on x = d cch , 2

  33. Introduction Cubic splines TPS for shape data Acknowledgement NCS for bivariate data Data 0.2 0.2 0.2 0.1 0.1 0.1 0.0 0.0 0.0 −0.1 −0.1 −0.1 −0.2 −0.2 −0.2 −0.3 −0.2 −0.1 0.0 0.1 0.2 −0.3 −0.2 −0.1 0.0 0.1 0.2 −0.3 −0.2 −0.1 0.0 0.1 0.2 Procrustes shape coordinates, symphisis Procrustes shape coordinates, symphisis Procrustes shape coordinates, symphisis subject Nr.1 subject Nr.1 subject Nr.1

  34. Introduction Cubic splines TPS for shape data Acknowledgement Outline Introduction 1 Notation and problems Cubic splines 2 Example 1 – shape data NCS for bivariate data TPS for shape data 3 TPS for shape data TPS relaxation along curves Acknowledgement 4

  35. Introduction Cubic splines TPS for shape data Acknowledgement TPS for shape data Penalized LRM Penalized linear regression model ( LRM ) + ε j , j = 1 , 2 , . . . k , � � y j = f x j where x j , y j ∈ R 2 , f = ( f 1 , f 2 ) ∈ D ( 2 ) (the class of twice-differentiable, absolutely continuous functions f with square integrable second derivative (Wahba 1990)), f m : R 2 → R , m = 1 , 2 penalized sum of squares k � 2 + λ J ( f ) S pen ( f ) = � � � � y j − f ( x j ) j = 1

  36. Introduction Cubic splines TPS for shape data Acknowledgement TPS for shape data Penalized LRM Penalized linear regression model ( LRM ) + ε j , j = 1 , 2 , . . . k , � � y j = f x j where x j , y j ∈ R 2 , f = ( f 1 , f 2 ) ∈ D ( 2 ) (the class of twice-differentiable, absolutely continuous functions f with square integrable second derivative (Wahba 1990)), f m : R 2 → R , m = 1 , 2 penalized sum of squares k � 2 + λ J ( f ) S pen ( f ) = � � � � y j − f ( x j ) j = 1

  37. Introduction Cubic splines TPS for shape data Acknowledgement TPS for shape data Penalized LRM Penalized linear regression model ( LRM ) + ε j , j = 1 , 2 , . . . k , � � y j = f x j where x j , y j ∈ R 2 , f = ( f 1 , f 2 ) ∈ D ( 2 ) (the class of twice-differentiable, absolutely continuous functions f with square integrable second derivative (Wahba 1990)), f m : R 2 → R , m = 1 , 2 penalized sum of squares k � 2 + λ J ( f ) S pen ( f ) = � � � � y j − f ( x j ) j = 1

  38. Introduction Cubic splines TPS for shape data Acknowledgement TPS for shape data Penalized LRM penalty  � 2  2 ∂ 2 f m � � �  dx ( 1 ) dx ( 2 ) J ( f ) = � � ∂ x ( i ) ∂ x ( j ) R 2 m = 1 i , j penalized least square estimator ˜ f is defined to be the minimizer of the functional S pen ( f ) over the class D ( 2 ) of f s, where f ∈D ( 2 ) S pen ( f ) ˜ f = arg min

  39. Introduction Cubic splines TPS for shape data Acknowledgement TPS for shape data Penalized LRM penalty  � 2  2 ∂ 2 f m � � �  dx ( 1 ) dx ( 2 ) J ( f ) = � � ∂ x ( i ) ∂ x ( j ) R 2 m = 1 i , j penalized least square estimator ˜ f is defined to be the minimizer of the functional S pen ( f ) over the class D ( 2 ) of f s, where f ∈D ( 2 ) S pen ( f ) ˜ f = arg min

  40. Introduction Cubic splines TPS for shape data Acknowledgement TPS for shape data Interpolation model Consider a TPS given by k f m ( x ) = c m + a T w jm φ j ( x ) � m x + j = 1 f ( x ) = c + A T x + W T s ( x ) , where c = ( c 1 , c 2 ) T , A = ( a 1 , a 2 ) , w m = ( w 1 m , w 2 m , . . . w km ) T , m = 1 , 2, W = ( w 1 , w 2 ) , s ( x ) k × 1 = [ φ 1 ( x ) , . . . φ k ( x )] T � � function φ ( x ) = � x � 2 � x � 2 2 log is a continuous function 2 known as a radial (nodal) basis function (Jackson 1989)

  41. Introduction Cubic splines TPS for shape data Acknowledgement TPS for shape data Interpolation model Consider a TPS given by k f m ( x ) = c m + a T w jm φ j ( x ) � m x + j = 1 f ( x ) = c + A T x + W T s ( x ) , where c = ( c 1 , c 2 ) T , A = ( a 1 , a 2 ) , w m = ( w 1 m , w 2 m , . . . w km ) T , m = 1 , 2, W = ( w 1 , w 2 ) , s ( x ) k × 1 = [ φ 1 ( x ) , . . . φ k ( x )] T � � function φ ( x ) = � x � 2 � x � 2 2 log is a continuous function 2 known as a radial (nodal) basis function (Jackson 1989)

  42. Introduction Cubic splines TPS for shape data Acknowledgement TPS for shape data Interpolation model Consider a TPS given by k f m ( x ) = c m + a T w jm φ j ( x ) � m x + j = 1 f ( x ) = c + A T x + W T s ( x ) , where c = ( c 1 , c 2 ) T , A = ( a 1 , a 2 ) , w m = ( w 1 m , w 2 m , . . . w km ) T , m = 1 , 2, W = ( w 1 , w 2 ) , s ( x ) k × 1 = [ φ 1 ( x ) , . . . φ k ( x )] T � � function φ ( x ) = � x � 2 � x � 2 2 log is a continuous function 2 known as a radial (nodal) basis function (Jackson 1989)

  43. Introduction Cubic splines TPS for shape data Acknowledgement TPS for shape data Interpolation model , i , j = 1 , 2 , ... k , ∀ � x � 2 > 0 � � ( S ) ij = φ j ( x i ) = φ x i − x j � T � . . constraint 1 k . X W = 0 � � TPS interpolation to the data x j , y j       Y S 1 k X W 1 T c T  =  , 0 0 0 (2) k     X T 0 0 0 A where Y k × 2 = ( y 1 , . . . y k ) T and X k × 2 = ( x 1 , . . . x k ) T

  44. Introduction Cubic splines TPS for shape data Acknowledgement TPS for shape data Interpolation model , i , j = 1 , 2 , ... k , ∀ � x � 2 > 0 � � ( S ) ij = φ j ( x i ) = φ x i − x j � T � . . constraint 1 k . X W = 0 � � TPS interpolation to the data x j , y j       Y S 1 k X W 1 T c T  =  , 0 0 0 (2) k     X T 0 0 0 A where Y k × 2 = ( y 1 , . . . y k ) T and X k × 2 = ( x 1 , . . . x k ) T

  45. Introduction Cubic splines TPS for shape data Acknowledgement TPS for shape data Interpolation model , i , j = 1 , 2 , ... k , ∀ � x � 2 > 0 � � ( S ) ij = φ j ( x i ) = φ x i − x j � T � . . constraint 1 k . X W = 0 � � TPS interpolation to the data x j , y j       Y S 1 k X W 1 T c T  =  , 0 0 0 (2) k     X T 0 0 0 A where Y k × 2 = ( y 1 , . . . y k ) T and X k × 2 = ( x 1 , . . . x k ) T

  46. Introduction Cubic splines TPS for shape data Acknowledgement TPS for shape data Interpolation model , i , j = 1 , 2 , ... k , ∀ � x � 2 > 0 � � ( S ) ij = φ j ( x i ) = φ x i − x j � T � . . constraint 1 k . X W = 0 � � TPS interpolation to the data x j , y j       Y S 1 k X W 1 T c T  =  , 0 0 0 (2) k     X T 0 0 0 A where Y k × 2 = ( y 1 , . . . y k ) T and X k × 2 = ( x 1 , . . . x k ) T

  47. Introduction Cubic splines TPS for shape data Acknowledgement TPS for shape data Interpolation model Let matrix L be defined as   S 1 k X 1 T L = 0 0 k   X T 0 0 inverse of L is equal to � L 11 L 12 � k × k k × 3 L − 1 = L 21 L 22 3 × k 3 × 3

  48. Introduction Cubic splines TPS for shape data Acknowledgement TPS for shape data Interpolation model Let matrix L be defined as   S 1 k X 1 T L = 0 0 k   X T 0 0 inverse of L is equal to � L 11 L 12 � k × k k × 3 L − 1 = L 21 L 22 3 × k 3 × 3

  49. Introduction Cubic splines TPS for shape data Acknowledgement TPS for shape data Interpolation model bending energy matrix – k × k matrix B e = L 11 constrains of this matrix 1 T k B e = 0 , X T B e = 0 , so the rank of the B e is k − 2 W = B e Y c , A T � T = L 21 Y � J ( f ) = tr ( W T SW ) = tr ( Y T B e Y )

  50. Introduction Cubic splines TPS for shape data Acknowledgement TPS for shape data Interpolation model bending energy matrix – k × k matrix B e = L 11 constrains of this matrix 1 T k B e = 0 , X T B e = 0 , so the rank of the B e is k − 2 W = B e Y c , A T � T = L 21 Y � J ( f ) = tr ( W T SW ) = tr ( Y T B e Y )

  51. Introduction Cubic splines TPS for shape data Acknowledgement TPS for shape data Interpolation model bending energy matrix – k × k matrix B e = L 11 constrains of this matrix 1 T k B e = 0 , X T B e = 0 , so the rank of the B e is k − 2 W = B e Y c , A T � T = L 21 Y � J ( f ) = tr ( W T SW ) = tr ( Y T B e Y )

  52. Introduction Cubic splines TPS for shape data Acknowledgement TPS for shape data Interpolation model bending energy matrix – k × k matrix B e = L 11 constrains of this matrix 1 T k B e = 0 , X T B e = 0 , so the rank of the B e is k − 2 W = B e Y c , A T � T = L 21 Y � J ( f ) = tr ( W T SW ) = tr ( Y T B e Y )

  53. Introduction Cubic splines TPS for shape data Acknowledgement TPS for shape data Interpolation model bending energy matrix – k × k matrix B e = L 11 constrains of this matrix 1 T k B e = 0 , X T B e = 0 , so the rank of the B e is k − 2 W = B e Y c , A T � T = L 21 Y � J ( f ) = tr ( W T SW ) = tr ( Y T B e Y )

  54. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves Data 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 −1 −1 −1 −1 0 1 2 3 4 5 −1 0 1 2 3 4 5 −1 0 1 2 3 4 5 6 6 6 4 4 4 2 2 2 0 0 0 −2 −2 −2 −4 −4 −4 −6 −4 −2 0 2 4 −6 −4 −2 0 2 4 −6 −4 −2 0 2 4

  55. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves Data For the purpose of relaxation 21 digitized semilandmarks on the symphisis from subject No . 2 its Procrustes shape coordinates Y = X P , 2 were relaxed onto Procrustes shape coordinates X = X P , 1 of subject No.1, seeking the configuration Y r

  56. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves Data For the purpose of relaxation 21 digitized semilandmarks on the symphisis from subject No . 2 its Procrustes shape coordinates Y = X P , 2 were relaxed onto Procrustes shape coordinates X = X P , 1 of subject No.1, seeking the configuration Y r

  57. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves Data

  58. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves TPS relaxation along curves Let Y k × 2 = ( y 1 , . . . y k ) T be configuration matrix with the � T � y ( 1 ) , y ( 2 ) rows y j = j j y ( r ) is free to slid away from their old position y j along the j � T � u ( 1 ) , u ( 2 ) tangent directions u j = with � u � 2 = 1 j j new position y ( r ) = y j + t j u j j y j + 1 − y j − 1 tangent directions u j = � y j + 1 − y j − 1 � 2 U is a matrix of 2 k rows and k columns in which the ( j , j ) th entry is u ( 1 ) and ( k + j , j ) th entry is u ( 2 ) , otherwise zeros j j

  59. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves TPS relaxation along curves Let Y k × 2 = ( y 1 , . . . y k ) T be configuration matrix with the � T � y ( 1 ) , y ( 2 ) rows y j = j j y ( r ) is free to slid away from their old position y j along the j � T � u ( 1 ) , u ( 2 ) tangent directions u j = with � u � 2 = 1 j j new position y ( r ) = y j + t j u j j y j + 1 − y j − 1 tangent directions u j = � y j + 1 − y j − 1 � 2 U is a matrix of 2 k rows and k columns in which the ( j , j ) th entry is u ( 1 ) and ( k + j , j ) th entry is u ( 2 ) , otherwise zeros j j

  60. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves TPS relaxation along curves Let Y k × 2 = ( y 1 , . . . y k ) T be configuration matrix with the � T � y ( 1 ) , y ( 2 ) rows y j = j j y ( r ) is free to slid away from their old position y j along the j � T � u ( 1 ) , u ( 2 ) tangent directions u j = with � u � 2 = 1 j j new position y ( r ) = y j + t j u j j y j + 1 − y j − 1 tangent directions u j = � y j + 1 − y j − 1 � 2 U is a matrix of 2 k rows and k columns in which the ( j , j ) th entry is u ( 1 ) and ( k + j , j ) th entry is u ( 2 ) , otherwise zeros j j

  61. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves TPS relaxation along curves Let Y k × 2 = ( y 1 , . . . y k ) T be configuration matrix with the � T � y ( 1 ) , y ( 2 ) rows y j = j j y ( r ) is free to slid away from their old position y j along the j � T � u ( 1 ) , u ( 2 ) tangent directions u j = with � u � 2 = 1 j j new position y ( r ) = y j + t j u j j y j + 1 − y j − 1 tangent directions u j = � y j + 1 − y j − 1 � 2 U is a matrix of 2 k rows and k columns in which the ( j , j ) th entry is u ( 1 ) and ( k + j , j ) th entry is u ( 2 ) , otherwise zeros j j

  62. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves TPS relaxation along curves Let Y k × 2 = ( y 1 , . . . y k ) T be configuration matrix with the � T � y ( 1 ) , y ( 2 ) rows y j = j j y ( r ) is free to slid away from their old position y j along the j � T � u ( 1 ) , u ( 2 ) tangent directions u j = with � u � 2 = 1 j j new position y ( r ) = y j + t j u j j y j + 1 − y j − 1 tangent directions u j = � y j + 1 − y j − 1 � 2 U is a matrix of 2 k rows and k columns in which the ( j , j ) th entry is u ( 1 ) and ( k + j , j ) th entry is u ( 2 ) , otherwise zeros j j

  63. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves TPS relaxation along curves y r = Vec ( Y r ) , B = diag ( B e , B e ) , B e depends only on some configuration X y r = y + Ut the task is now to minimize the form r By r = ( y + Ut ) T B ( y + Ut ) y T setting the gradient of this expression to zero straightforwardly generates the solution (Bookstein 1997) � − 1 U T BU U T By � t = −

  64. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves TPS relaxation along curves y r = Vec ( Y r ) , B = diag ( B e , B e ) , B e depends only on some configuration X y r = y + Ut the task is now to minimize the form r By r = ( y + Ut ) T B ( y + Ut ) y T setting the gradient of this expression to zero straightforwardly generates the solution (Bookstein 1997) � − 1 U T BU U T By � t = −

  65. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves TPS relaxation along curves y r = Vec ( Y r ) , B = diag ( B e , B e ) , B e depends only on some configuration X y r = y + Ut the task is now to minimize the form r By r = ( y + Ut ) T B ( y + Ut ) y T setting the gradient of this expression to zero straightforwardly generates the solution (Bookstein 1997) � − 1 U T BU U T By � t = −

  66. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves TPS relaxation along curves y r = Vec ( Y r ) , B = diag ( B e , B e ) , B e depends only on some configuration X y r = y + Ut the task is now to minimize the form r By r = ( y + Ut ) T B ( y + Ut ) y T setting the gradient of this expression to zero straightforwardly generates the solution (Bookstein 1997) � − 1 U T BU U T By � t = −

  67. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves Data

  68. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves Data

  69. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves TPS relaxation along curves Let the curve defined by y j be interpolated by cubic spline or B-spline ˜ f (De Boor (1972) or Eilers & Marx (1996)), ) T ∈ ˜ y j = ( y ( 1 ) , y ( 2 ) f , j = 1 , 2 , . . . k j j ) T ∈ ˜ re-sampled points y i = ( y ( 1 ) , y ( 2 ) f , i = 1 , 2 , . . . M i i ( M = 500) and M = { y 1 , y 2 , ... y M } suppose that y ( s ) sj ) T ∈ ˜ = ( y ( 1 ) sj , y ( 2 ) f (the rows of Y s ) are j free to slid away from their old position y j along the curve ˜ f

  70. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves TPS relaxation along curves Let the curve defined by y j be interpolated by cubic spline or B-spline ˜ f (De Boor (1972) or Eilers & Marx (1996)), ) T ∈ ˜ y j = ( y ( 1 ) , y ( 2 ) f , j = 1 , 2 , . . . k j j ) T ∈ ˜ re-sampled points y i = ( y ( 1 ) , y ( 2 ) f , i = 1 , 2 , . . . M i i ( M = 500) and M = { y 1 , y 2 , ... y M } suppose that y ( s ) sj ) T ∈ ˜ = ( y ( 1 ) sj , y ( 2 ) f (the rows of Y s ) are j free to slid away from their old position y j along the curve ˜ f

  71. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves TPS relaxation along curves Let the curve defined by y j be interpolated by cubic spline or B-spline ˜ f (De Boor (1972) or Eilers & Marx (1996)), ) T ∈ ˜ y j = ( y ( 1 ) , y ( 2 ) f , j = 1 , 2 , . . . k j j ) T ∈ ˜ re-sampled points y i = ( y ( 1 ) , y ( 2 ) f , i = 1 , 2 , . . . M i i ( M = 500) and M = { y 1 , y 2 , ... y M } suppose that y ( s ) sj ) T ∈ ˜ = ( y ( 1 ) sj , y ( 2 ) f (the rows of Y s ) are j free to slid away from their old position y j along the curve ˜ f

  72. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves TPS relaxation along curves J ( y s ) = y T s By s , has to be minimized and y r is obtained as a minimizer of J ( y s ) given by y s J ( y s ) y r = arg min (3) the minimization starts with substitution of y 1 by y i ∈ M , ... and ends with substitution of y k by y i ∈ M , where y j , j = 1 , 2 , . . . k , are the rows of Y and i = 1 , 2 , . . . M : � � y ( r ) y s J ( y s ) = arg min , (4) j j , k + j where ( j , k + j ) th entry of y s is substituted by y ( s ) ∈ M for i j = 1 , 2 , . . . k ; i = 1 , 2 , . . . M , y r = Vec ( Y r ) and y ( r ) are the j rows of Y r

  73. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves TPS relaxation along curves J ( y s ) = y T s By s , has to be minimized and y r is obtained as a minimizer of J ( y s ) given by y s J ( y s ) y r = arg min (3) the minimization starts with substitution of y 1 by y i ∈ M , ... and ends with substitution of y k by y i ∈ M , where y j , j = 1 , 2 , . . . k , are the rows of Y and i = 1 , 2 , . . . M : � � y ( r ) y s J ( y s ) = arg min , (4) j j , k + j where ( j , k + j ) th entry of y s is substituted by y ( s ) ∈ M for i j = 1 , 2 , . . . k ; i = 1 , 2 , . . . M , y r = Vec ( Y r ) and y ( r ) are the j rows of Y r

  74. Introduction Cubic splines TPS for shape data Acknowledgement TPS relaxation along curves Data

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