shape analysis in r · shape analysis in r gm library in the light of recent methodological...
TRANSCRIPT
![Page 1: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/1.jpg)
Outline
Shape Analysis in RGM library in the light of recent methodological
developments
Stanislav [email protected]
Department of Applied Mathematics and Statistics, Comenius University,Bratislava, Slovakia
Neurospin, Institut d’Imagerie BioMedicale Commissariat a l’Energie Atomique, Gifsur Yvette, France
5th R useR Conference, Rennes, France, July 8-10, 2009
![Page 2: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/2.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Outline
1 IntroductionNotation and problems
2 Cubic splinesExample 1 – shape dataNCS for bivariate data
3 TPS for shape dataTPS for shape dataTPS relaxation along curves
4 Acknowledgement
![Page 3: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/3.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Outline
1 IntroductionNotation and problems
2 Cubic splinesExample 1 – shape dataNCS for bivariate data
3 TPS for shape dataTPS for shape dataTPS relaxation along curves
4 Acknowledgement
![Page 4: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/4.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
Ian Dryden’s R-package — shapes
Statistical shape analysis
Version: 1.1-3
http://www.maths.nott.ac.uk/ ild/shapes
Generalized Procrustes Analysis (GPA), Relative WarpAnalysis (RWA), statistical inference
Thin-plate spline grids, 3D visualization via librariesscatterplot3d and rgl
![Page 5: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/5.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
Ian Dryden’s R-package — shapes
Statistical shape analysis
Version: 1.1-3
http://www.maths.nott.ac.uk/ ild/shapes
Generalized Procrustes Analysis (GPA), Relative WarpAnalysis (RWA), statistical inference
Thin-plate spline grids, 3D visualization via librariesscatterplot3d and rgl
![Page 6: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/6.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
Ian Dryden’s R-package — shapes
Statistical shape analysis
Version: 1.1-3
http://www.maths.nott.ac.uk/ ild/shapes
Generalized Procrustes Analysis (GPA), Relative WarpAnalysis (RWA), statistical inference
Thin-plate spline grids, 3D visualization via librariesscatterplot3d and rgl
![Page 7: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/7.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
Ian Dryden’s R-package — shapes
Statistical shape analysis
Version: 1.1-3
http://www.maths.nott.ac.uk/ ild/shapes
Generalized Procrustes Analysis (GPA), Relative WarpAnalysis (RWA), statistical inference
Thin-plate spline grids, 3D visualization via librariesscatterplot3d and rgl
![Page 8: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/8.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
Ian Dryden’s R-package — shapes
Statistical shape analysis
Version: 1.1-3
http://www.maths.nott.ac.uk/ ild/shapes
Generalized Procrustes Analysis (GPA), Relative WarpAnalysis (RWA), statistical inference
Thin-plate spline grids, 3D visualization via librariesscatterplot3d and rgl
![Page 9: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/9.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
New R-package — GMM
Statistical shape analysis
upcoming in autumn 2009
http://www.defm.fmph.uniba.sk/ katina/katina.htm
sliding of semilandmarks on open and closed curvesand surfaces, missing value estimation, affine andnon-affine component, unwarping, Multivariate MultipleLinear Regression Model of shape on size, Relative WarpAnalysis, shape-space PCA, form-space PCA ,size-adjusted PCA, 2-block PLS (two shape blocks, oneshape block and one block of external variables), analysisof asymmetry, statistical inference
GMM toolbox (Hull/York Medical School, University ofVienna)
![Page 10: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/10.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
New R-package — GMM
Statistical shape analysis
upcoming in autumn 2009
http://www.defm.fmph.uniba.sk/ katina/katina.htm
sliding of semilandmarks on open and closed curvesand surfaces, missing value estimation, affine andnon-affine component, unwarping, Multivariate MultipleLinear Regression Model of shape on size, Relative WarpAnalysis, shape-space PCA, form-space PCA ,size-adjusted PCA, 2-block PLS (two shape blocks, oneshape block and one block of external variables), analysisof asymmetry, statistical inference
GMM toolbox (Hull/York Medical School, University ofVienna)
![Page 11: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/11.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
New R-package — GMM
Statistical shape analysis
upcoming in autumn 2009
http://www.defm.fmph.uniba.sk/ katina/katina.htm
sliding of semilandmarks on open and closed curvesand surfaces, missing value estimation, affine andnon-affine component, unwarping, Multivariate MultipleLinear Regression Model of shape on size, Relative WarpAnalysis, shape-space PCA, form-space PCA ,size-adjusted PCA, 2-block PLS (two shape blocks, oneshape block and one block of external variables), analysisof asymmetry, statistical inference
GMM toolbox (Hull/York Medical School, University ofVienna)
![Page 12: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/12.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
New R-package — GMM
Statistical shape analysis
upcoming in autumn 2009
http://www.defm.fmph.uniba.sk/ katina/katina.htm
sliding of semilandmarks on open and closed curvesand surfaces, missing value estimation, affine andnon-affine component, unwarping, Multivariate MultipleLinear Regression Model of shape on size, Relative WarpAnalysis, shape-space PCA, form-space PCA ,size-adjusted PCA, 2-block PLS (two shape blocks, oneshape block and one block of external variables), analysisof asymmetry, statistical inference
GMM toolbox (Hull/York Medical School, University ofVienna)
![Page 13: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/13.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
New R-package — GMM
Statistical shape analysis
upcoming in autumn 2009
http://www.defm.fmph.uniba.sk/ katina/katina.htm
sliding of semilandmarks on open and closed curvesand surfaces, missing value estimation, affine andnon-affine component, unwarping, Multivariate MultipleLinear Regression Model of shape on size, Relative WarpAnalysis, shape-space PCA, form-space PCA ,size-adjusted PCA, 2-block PLS (two shape blocks, oneshape block and one block of external variables), analysisof asymmetry, statistical inference
GMM toolbox (Hull/York Medical School, University ofVienna)
![Page 14: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/14.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
Introduction
xj ∈ R, k-vector x
yj ∈ R, k-vector y
x j =(
x (1)j , x (2)
j
)T∈ R
2, k × 2 matrix X
y j =(
y (1)j , y (2)
j
)T∈ R
2, k × 2 matrix Y
j = 1, 2, . . . k
![Page 15: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/15.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
Introduction
xj ∈ R, k-vector x
yj ∈ R, k-vector y
x j =(
x (1)j , x (2)
j
)T∈ R
2, k × 2 matrix X
y j =(
y (1)j , y (2)
j
)T∈ R
2, k × 2 matrix Y
j = 1, 2, . . . k
![Page 16: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/16.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
Introduction
xj ∈ R, k-vector x
yj ∈ R, k-vector y
x j =(
x (1)j , x (2)
j
)T∈ R
2, k × 2 matrix X
y j =(
y (1)j , y (2)
j
)T∈ R
2, k × 2 matrix Y
j = 1, 2, . . . k
![Page 17: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/17.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
Introduction
xj ∈ R, k-vector x
yj ∈ R, k-vector y
x j =(
x (1)j , x (2)
j
)T∈ R
2, k × 2 matrix X
y j =(
y (1)j , y (2)
j
)T∈ R
2, k × 2 matrix Y
j = 1, 2, . . . k
![Page 18: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/18.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
Introduction
xj ∈ R, k-vector x
yj ∈ R, k-vector y
x j =(
x (1)j , x (2)
j
)T∈ R
2, k × 2 matrix X
y j =(
y (1)j , y (2)
j
)T∈ R
2, k × 2 matrix Y
j = 1, 2, . . . k
![Page 19: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/19.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
Introduction
natural cubic splines
thin-plate splines
![Page 20: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/20.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
Introduction
natural cubic splines
thin-plate splines
![Page 21: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/21.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
Introduction
f : R → R
f : R2 → R
2
![Page 22: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/22.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
Introduction
f : R → R
f : R2 → R
2
![Page 23: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/23.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
Introduction
yj = f(
xj)
+ εj
y j = f(
x j)
+ εj
![Page 24: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/24.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Notation and problems
Introduction
yj = f(
xj)
+ εj
y j = f(
x j)
+ εj
![Page 25: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/25.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Outline
1 IntroductionNotation and problems
2 Cubic splinesExample 1 – shape dataNCS for bivariate data
3 TPS for shape dataTPS for shape dataTPS relaxation along curves
4 Acknowledgement
![Page 26: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/26.jpg)
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 shapechanges of modern humans and chimpanzees (Pantroglodytes) from fetal life to the complete eruption of thedeciduous dentition. The Anatomical Record (accepted)
computed tomographies (CT) of 151 modern humans(78 females and 73 males) of mixed ethnicity, living inFrance, from birth to adulthood. [Pellegrin Hospital(Bordeaux), Necker Hospital (Paris) and Clinique Pasteur(Toulouse)]
![Page 27: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/27.jpg)
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 shapechanges of modern humans and chimpanzees (Pantroglodytes) from fetal life to the complete eruption of thedeciduous dentition. The Anatomical Record (accepted)
computed tomographies (CT) of 151 modern humans(78 females and 73 males) of mixed ethnicity, living inFrance, from birth to adulthood. [Pellegrin Hospital(Bordeaux), Necker Hospital (Paris) and Clinique Pasteur(Toulouse)]
![Page 28: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/28.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Example 1 – shape data
Data
each mandibular surface was reconstructed from theCT-scans via the software package Amira (MercuryComputer Systems, Chelmsford, MA)
open-source software Edgewarp3D (Bookstein & Green2002), a 3D-template of 415 landmarks andsemilandmarks was created to measure the mandibularsurface and was warped onto each mandible
![Page 29: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/29.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Example 1 – shape data
Data
each mandibular surface was reconstructed from theCT-scans via the software package Amira (MercuryComputer Systems, Chelmsford, MA)
open-source software Edgewarp3D (Bookstein & Green2002), a 3D-template of 415 landmarks andsemilandmarks was created to measure the mandibularsurface and was warped onto each mandible
![Page 30: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/30.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Example 1 – shape data
Data
![Page 31: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/31.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Interpolation model
Consider a NCS given by
f (x) = c + ax +k
∑
j=1
wjφj (x) , j = 1, 2, . . . k ,
where
xj are the knots, φj (x) = φ(
x−xj)
= 112
∣
∣x − xj∣
∣
3 with theconstraints
∑kj=1 wj =
∑kj=1 wjxj = 0, f ′′ and f ′′′ are both
zero outside the interval [x1, xk ]
function φ (x) = 112 |x |3 is a continuous function known as a
radial (nodal) basis function (Jackson 1989)
![Page 32: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/32.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Interpolation model
Consider a NCS given by
f (x) = c + ax +k
∑
j=1
wjφj (x) , j = 1, 2, . . . k ,
where
xj are the knots, φj (x) = φ(
x−xj)
= 112
∣
∣x − xj∣
∣
3 with theconstraints
∑kj=1 wj =
∑kj=1 wjxj = 0, f ′′ and f ′′′ are both
zero outside the interval [x1, xk ]
function φ (x) = 112 |x |3 is a continuous function known as a
radial (nodal) basis function (Jackson 1989)
![Page 33: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/33.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Interpolation model
Consider a NCS given by
f (x) = c + ax +k
∑
j=1
wjφj (x) , j = 1, 2, . . . k ,
where
xj are the knots, φj (x) = φ(
x−xj)
= 112
∣
∣x − xj∣
∣
3 with theconstraints
∑kj=1 wj =
∑kj=1 wjxj = 0, f ′′ and f ′′′ are both
zero outside the interval [x1, xk ]
function φ (x) = 112 |x |3 is a continuous function known as a
radial (nodal) basis function (Jackson 1989)
![Page 34: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/34.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Interpolation model
Let (S)ij = φj(xi) = φ(xi − xj) = 112
∣
∣xi − xj∣
∣
3,w = (w1, . . . wk )T
constraint (1k , x)T w = 0
NCS interpolation to the data(
xj , yj)
y00
=
S 1k x1T
k 0 0xT 0 0
wca
, (1)
where xk×1 = (x1, . . . xk )T and yk×1=(y1, y2, . . . yk )T
![Page 35: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/35.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Interpolation model
Let (S)ij = φj(xi) = φ(xi − xj) = 112
∣
∣xi − xj∣
∣
3,w = (w1, . . . wk )T
constraint (1k , x)T w = 0
NCS interpolation to the data(
xj , yj)
y00
=
S 1k x1T
k 0 0xT 0 0
wca
, (1)
where xk×1 = (x1, . . . xk )T and yk×1=(y1, y2, . . . yk )T
![Page 36: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/36.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Interpolation model
Let (S)ij = φj(xi) = φ(xi − xj) = 112
∣
∣xi − xj∣
∣
3,w = (w1, . . . wk )T
constraint (1k , x)T w = 0
NCS interpolation to the data(
xj , yj)
y00
=
S 1k x1T
k 0 0xT 0 0
wca
, (1)
where xk×1 = (x1, . . . xk )T and yk×1=(y1, y2, . . . yk )T
![Page 37: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/37.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Interpolation model
Let (S)ij = φj(xi) = φ(xi − xj) = 112
∣
∣xi − xj∣
∣
3,w = (w1, . . . wk )T
constraint (1k , x)T w = 0
NCS interpolation to the data(
xj , yj)
y00
=
S 1k x1T
k 0 0xT 0 0
wca
, (1)
where xk×1 = (x1, . . . xk )T and yk×1=(y1, y2, . . . yk )T
![Page 38: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/38.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Interpolation model
Let matrix L be defined as
L =
S 1k x1T
k 0 0xT 0 0
inverse of L is equal to
L−1=
(
L11k×k L12
k×2L21
2×k L222×2
)
![Page 39: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/39.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Interpolation model
Let matrix L be defined as
L =
S 1k x1T
k 0 0xT 0 0
inverse of L is equal to
L−1=
(
L11k×k L12
k×2L21
2×k L222×2
)
![Page 40: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/40.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Interpolation model
bending energy matrix – k × k matrix Be = L11
constrains of this matrix 1Tk Be = 0, xT Be = 0, so the rank
of the Be is k − 2
w = Bey
(c, a)T = L21y
J (f ) = wT Sw = yT Bey
![Page 41: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/41.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Interpolation model
bending energy matrix – k × k matrix Be = L11
constrains of this matrix 1Tk Be = 0, xT Be = 0, so the rank
of the Be is k − 2
w = Bey
(c, a)T = L21y
J (f ) = wT Sw = yT Bey
![Page 42: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/42.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Interpolation model
bending energy matrix – k × k matrix Be = L11
constrains of this matrix 1Tk Be = 0, xT Be = 0, so the rank
of the Be is k − 2
w = Bey
(c, a)T = L21y
J (f ) = wT Sw = yT Bey
![Page 43: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/43.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Interpolation model
bending energy matrix – k × k matrix Be = L11
constrains of this matrix 1Tk Be = 0, xT Be = 0, so the rank
of the Be is k − 2
w = Bey
(c, a)T = L21y
J (f ) = wT Sw = yT Bey
![Page 44: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/44.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Interpolation model
bending energy matrix – k × k matrix Be = L11
constrains of this matrix 1Tk Be = 0, xT Be = 0, so the rank
of the Be is k − 2
w = Bey
(c, a)T = L21y
J (f ) = wT Sw = yT Bey
![Page 45: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/45.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Data pre-processing
SVD of Xc = ΓΛΓT =
∑2j=1 λjγjγ
Tj , Xc = X − 1kxT (Mardia
et al. 2000) [principal component analysis ]
the 1th principal component of X is equal to z1 = Xcγ1,where γ1 is the 1th column of Γ, and z1j , j = 1, 2, ...k areprincipal component scores of j th landmark (z1j is j thelement of k-vector z1)
re-ordering of the rows of X is done based on the ranks ofz1j in z1
![Page 46: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/46.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Data pre-processing
SVD of Xc = ΓΛΓT =
∑2j=1 λjγjγ
Tj , Xc = X − 1kxT (Mardia
et al. 2000) [principal component analysis ]
the 1th principal component of X is equal to z1 = Xcγ1,where γ1 is the 1th column of Γ, and z1j , j = 1, 2, ...k areprincipal component scores of j th landmark (z1j is j thelement of k-vector z1)
re-ordering of the rows of X is done based on the ranks ofz1j in z1
![Page 47: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/47.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Data pre-processing
SVD of Xc = ΓΛΓT =
∑2j=1 λjγjγ
Tj , Xc = X − 1kxT (Mardia
et al. 2000) [principal component analysis ]
the 1th principal component of X is equal to z1 = Xcγ1,where γ1 is the 1th column of Γ, and z1j , j = 1, 2, ...k areprincipal component scores of j th landmark (z1j is j thelement of k-vector z1)
re-ordering of the rows of X is done based on the ranks ofz1j in z1
![Page 48: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/48.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Data pre-processing
SVD of Ddc (Gower 1966) [principal coordinate analysis ]
D1 is k × k matrix of squared interlandmark Euklideandistances, D2 = −1
2D1 and
Ddc = D2 −1k
1k1Tk D2 −
1k
D21k1Tk +
1k2 1k1T
k D21k1Tk
doubly centered (both row- and column-centered)
![Page 49: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/49.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Data pre-processing
SVD of Ddc (Gower 1966) [principal coordinate analysis ]
D1 is k × k matrix of squared interlandmark Euklideandistances, D2 = −1
2D1 and
Ddc = D2 −1k
1k1Tk D2 −
1k
D21k1Tk +
1k2 1k1T
k D21k1Tk
doubly centered (both row- and column-centered)
![Page 50: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/50.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Data pre-processing
SVD of Ddc (Gower 1966) [principal coordinate analysis ]
D1 is k × k matrix of squared interlandmark Euklideandistances, D2 = −1
2D1 and
Ddc = D2 −1k
1k1Tk D2 −
1k
D21k1Tk +
1k2 1k1T
k D21k1Tk
doubly centered (both row- and column-centered)
![Page 51: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/51.jpg)
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 =
∑ji=2 d (i)
ch ,j = 2, 3, ...k
d (j)cch = dj , j = 1, 2, ...k , dcch = (d1, d2, ...dk )T , d1 = 0
NCS of x on dcch
NCS of y on dcch
![Page 52: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/52.jpg)
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 =
∑ji=2 d (i)
ch ,j = 2, 3, ...k
d (j)cch = dj , j = 1, 2, ...k , dcch = (d1, d2, ...dk )T , d1 = 0
NCS of x on dcch
NCS of y on dcch
![Page 53: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/53.jpg)
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 =
∑ji=2 d (i)
ch ,j = 2, 3, ...k
d (j)cch = dj , j = 1, 2, ...k , dcch = (d1, d2, ...dk )T , d1 = 0
NCS of x on dcch
NCS of y on dcch
![Page 54: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/54.jpg)
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 =
∑ji=2 d (i)
ch ,j = 2, 3, ...k
d (j)cch = dj , j = 1, 2, ...k , dcch = (d1, d2, ...dk )T , d1 = 0
NCS of x on dcch
NCS of y on dcch
![Page 55: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/55.jpg)
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 =
∑ji=2 d (i)
ch ,j = 2, 3, ...k
d (j)cch = dj , j = 1, 2, ...k , dcch = (d1, d2, ...dk )T , d1 = 0
NCS of x on dcch
NCS of y on dcch
![Page 56: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/56.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Data
For the purpose of re-sampling
21 digitized semilandmarks on the symphisisXP,2 = (xP,21, xP,22), dcch,2 (subject No.2)
NCS of y = xP,21 on x = dcch,2
NCS of y = xP,22 on x = dcch,2
![Page 57: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/57.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Data
For the purpose of re-sampling
21 digitized semilandmarks on the symphisisXP,2 = (xP,21, xP,22), dcch,2 (subject No.2)
NCS of y = xP,21 on x = dcch,2
NCS of y = xP,22 on x = dcch,2
![Page 58: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/58.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Data
For the purpose of re-sampling
21 digitized semilandmarks on the symphisisXP,2 = (xP,21, xP,22), dcch,2 (subject No.2)
NCS of y = xP,21 on x = dcch,2
NCS of y = xP,22 on x = dcch,2
![Page 59: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/59.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
NCS for bivariate data
Data
−0.3 −0.2 −0.1 0.0 0.1 0.2
−0.
2−
0.1
0.0
0.1
0.2
subject Nr.1Procrustes shape coordinates, symphisis
−0.3 −0.2 −0.1 0.0 0.1 0.2
−0.
2−
0.1
0.0
0.1
0.2
subject Nr.1Procrustes shape coordinates, symphisis
−0.3 −0.2 −0.1 0.0 0.1 0.2
−0.
2−
0.1
0.0
0.1
0.2
subject Nr.1Procrustes shape coordinates, symphisis
![Page 60: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/60.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Outline
1 IntroductionNotation and problems
2 Cubic splinesExample 1 – shape dataNCS for bivariate data
3 TPS for shape dataTPS for shape dataTPS relaxation along curves
4 Acknowledgement
![Page 61: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/61.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS for shape data
Penalized LRM
Penalized linear regression model (LRM)
y j = f(
x j)
+ εj , j = 1, 2, . . . k ,
where x j , y j ∈ R2, f = (f1, f2) ∈ D(2) (the class of
twice-differentiable, absolutely continuous functions f withsquare integrable second derivative (Wahba 1990)),fm:R2 → R, m = 1, 2
penalized sum of squares
Spen (f) =k
∑
j=1
∥
∥y j − f(x j)∥
∥
2+ λJ (f)
![Page 62: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/62.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS for shape data
Penalized LRM
Penalized linear regression model (LRM)
y j = f(
x j)
+ εj , j = 1, 2, . . . k ,
where x j , y j ∈ R2, f = (f1, f2) ∈ D(2) (the class of
twice-differentiable, absolutely continuous functions f withsquare integrable second derivative (Wahba 1990)),fm:R2 → R, m = 1, 2
penalized sum of squares
Spen (f) =k
∑
j=1
∥
∥y j − f(x j)∥
∥
2+ λJ (f)
![Page 63: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/63.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS for shape data
Penalized LRM
Penalized linear regression model (LRM)
y j = f(
x j)
+ εj , j = 1, 2, . . . k ,
where x j , y j ∈ R2, f = (f1, f2) ∈ D(2) (the class of
twice-differentiable, absolutely continuous functions f withsquare integrable second derivative (Wahba 1990)),fm:R2 → R, m = 1, 2
penalized sum of squares
Spen (f) =k
∑
j=1
∥
∥y j − f(x j)∥
∥
2+ λJ (f)
![Page 64: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/64.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS for shape data
Penalized LRM
penalty
J (f) =2
∑
m=1
∫ ∫
R2
∑
i,j
(
∂2fm∂x (i)∂x (j)
)2
dx (1)dx (2)
penalized least square estimator f is defined to be theminimizer of the functional Spen (f) over the class D(2) of fs,where
f = arg minf∈D(2)
Spen (f)
![Page 65: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/65.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS for shape data
Penalized LRM
penalty
J (f) =2
∑
m=1
∫ ∫
R2
∑
i,j
(
∂2fm∂x (i)∂x (j)
)2
dx (1)dx (2)
penalized least square estimator f is defined to be theminimizer of the functional Spen (f) over the class D(2) of fs,where
f = arg minf∈D(2)
Spen (f)
![Page 66: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/66.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS for shape data
Interpolation model
Consider a TPS given by
fm (x) = cm + aTmx+
k∑
j=1
wjmφj (x)
f (x) = c + AT x + WT s (x) ,
where
c = (c1, c2)T , A =(a1, a2), wm = (w1m, w2m, . . . wkm)T ,
m = 1, 2, W = (w1, w2), s (x)k×1 = [φ1 (x) , . . . φk (x)]T
function φ (x) = ‖x‖22 log
(
‖x‖22
)
is a continuous function
known as a radial (nodal) basis function (Jackson 1989)
![Page 67: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/67.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS for shape data
Interpolation model
Consider a TPS given by
fm (x) = cm + aTmx+
k∑
j=1
wjmφj (x)
f (x) = c + AT x + WT s (x) ,
where
c = (c1, c2)T , A =(a1, a2), wm = (w1m, w2m, . . . wkm)T ,
m = 1, 2, W = (w1, w2), s (x)k×1 = [φ1 (x) , . . . φk (x)]T
function φ (x) = ‖x‖22 log
(
‖x‖22
)
is a continuous function
known as a radial (nodal) basis function (Jackson 1989)
![Page 68: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/68.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS for shape data
Interpolation model
Consider a TPS given by
fm (x) = cm + aTmx+
k∑
j=1
wjmφj (x)
f (x) = c + AT x + WT s (x) ,
where
c = (c1, c2)T , A =(a1, a2), wm = (w1m, w2m, . . . wkm)T ,
m = 1, 2, W = (w1, w2), s (x)k×1 = [φ1 (x) , . . . φk (x)]T
function φ (x) = ‖x‖22 log
(
‖x‖22
)
is a continuous function
known as a radial (nodal) basis function (Jackson 1989)
![Page 69: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/69.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS for shape data
Interpolation model
(S)ij = φj (x i) = φ(
x i−x j)
, i , j = 1, 2, ...k , ∀ ‖x‖2 > 0
constraint(
1k...X
)T
W = 0
TPS interpolation to the data(
x j , y j)
Y00
=
S 1k X1T
k 0 0XT 0 0
WcT
A
, (2)
where Yk×2 = (y1, . . . yk )T and Xk×2 = (x1, . . . xk )T
![Page 70: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/70.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS for shape data
Interpolation model
(S)ij = φj (x i) = φ(
x i−x j)
, i , j = 1, 2, ...k , ∀ ‖x‖2 > 0
constraint(
1k...X
)T
W = 0
TPS interpolation to the data(
x j , y j)
Y00
=
S 1k X1T
k 0 0XT 0 0
WcT
A
, (2)
where Yk×2 = (y1, . . . yk )T and Xk×2 = (x1, . . . xk )T
![Page 71: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/71.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS for shape data
Interpolation model
(S)ij = φj (x i) = φ(
x i−x j)
, i , j = 1, 2, ...k , ∀ ‖x‖2 > 0
constraint(
1k...X
)T
W = 0
TPS interpolation to the data(
x j , y j)
Y00
=
S 1k X1T
k 0 0XT 0 0
WcT
A
, (2)
where Yk×2 = (y1, . . . yk )T and Xk×2 = (x1, . . . xk )T
![Page 72: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/72.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS for shape data
Interpolation model
(S)ij = φj (x i) = φ(
x i−x j)
, i , j = 1, 2, ...k , ∀ ‖x‖2 > 0
constraint(
1k...X
)T
W = 0
TPS interpolation to the data(
x j , y j)
Y00
=
S 1k X1T
k 0 0XT 0 0
WcT
A
, (2)
where Yk×2 = (y1, . . . yk )T and Xk×2 = (x1, . . . xk )T
![Page 73: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/73.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS for shape data
Interpolation model
Let matrix L be defined as
L =
S 1k X1T
k 0 0XT 0 0
inverse of L is equal to
L−1=
(
L11k×k L12
k×3L21
3×k L223×3
)
![Page 74: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/74.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS for shape data
Interpolation model
Let matrix L be defined as
L =
S 1k X1T
k 0 0XT 0 0
inverse of L is equal to
L−1=
(
L11k×k L12
k×3L21
3×k L223×3
)
![Page 75: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/75.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS for shape data
Interpolation model
bending energy matrix – k × k matrix Be = L11
constrains of this matrix 1Tk Be = 0, XT Be = 0, so the rank
of the Be is k − 2
W = BeY(
c, AT)T
= L21Y
J (f) = tr(WT SW) = tr(YT BeY)
![Page 76: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/76.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS for shape data
Interpolation model
bending energy matrix – k × k matrix Be = L11
constrains of this matrix 1Tk Be = 0, XT Be = 0, so the rank
of the Be is k − 2
W = BeY(
c, AT)T
= L21Y
J (f) = tr(WT SW) = tr(YT BeY)
![Page 77: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/77.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS for shape data
Interpolation model
bending energy matrix – k × k matrix Be = L11
constrains of this matrix 1Tk Be = 0, XT Be = 0, so the rank
of the Be is k − 2
W = BeY(
c, AT)T
= L21Y
J (f) = tr(WT SW) = tr(YT BeY)
![Page 78: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/78.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS for shape data
Interpolation model
bending energy matrix – k × k matrix Be = L11
constrains of this matrix 1Tk Be = 0, XT Be = 0, so the rank
of the Be is k − 2
W = BeY(
c, AT)T
= L21Y
J (f) = tr(WT SW) = tr(YT BeY)
![Page 79: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/79.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS for shape data
Interpolation model
bending energy matrix – k × k matrix Be = L11
constrains of this matrix 1Tk Be = 0, XT Be = 0, so the rank
of the Be is k − 2
W = BeY(
c, AT)T
= L21Y
J (f) = tr(WT SW) = tr(YT BeY)
![Page 80: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/80.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS relaxation along curves
Data
−1 0 1 2 3 4 5
−1
01
23
45
−6 −4 −2 0 2 4
−4
−2
02
46
−1 0 1 2 3 4 5
−1
01
23
45
−6 −4 −2 0 2 4
−4
−2
02
46
−1 0 1 2 3 4 5
−1
01
23
45
−6 −4 −2 0 2 4
−4
−2
02
46
![Page 81: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/81.jpg)
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 subjectNo.2
its Procrustes shape coordinates Y = XP,2 were relaxedonto Procrustes shape coordinates X = XP,1 of subjectNo.1, seeking the configuration Yr
![Page 82: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/82.jpg)
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 subjectNo.2
its Procrustes shape coordinates Y = XP,2 were relaxedonto Procrustes shape coordinates X = XP,1 of subjectNo.1, seeking the configuration Yr
![Page 83: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/83.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS relaxation along curves
Data
![Page 84: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/84.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS relaxation along curves
TPS relaxation along curves
Let Yk×2 = (y1, . . . yk )T be configuration matrix with the
rows y j=(
y (1)j , y (2)
j
)T
y(r)j is free to slid away from their old position y j along the
tangent directions u j =(
u(1)j , u(2)
j
)Twith ‖u‖2 = 1
new position y (r)j = y j + tju j
tangent directions u j =y j+1−y j−1
‖y j+1−y j−1‖2
U is a matrix of 2k rows and k columns in which the (j , j)thentry is u(1)
j and (k + j , j)th entry is u(2)j , otherwise zeros
![Page 85: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/85.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS relaxation along curves
TPS relaxation along curves
Let Yk×2 = (y1, . . . yk )T be configuration matrix with the
rows y j=(
y (1)j , y (2)
j
)T
y(r)j is free to slid away from their old position y j along the
tangent directions u j =(
u(1)j , u(2)
j
)Twith ‖u‖2 = 1
new position y (r)j = y j + tju j
tangent directions u j =y j+1−y j−1
‖y j+1−y j−1‖2
U is a matrix of 2k rows and k columns in which the (j , j)thentry is u(1)
j and (k + j , j)th entry is u(2)j , otherwise zeros
![Page 86: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/86.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS relaxation along curves
TPS relaxation along curves
Let Yk×2 = (y1, . . . yk )T be configuration matrix with the
rows y j=(
y (1)j , y (2)
j
)T
y(r)j is free to slid away from their old position y j along the
tangent directions u j =(
u(1)j , u(2)
j
)Twith ‖u‖2 = 1
new position y (r)j = y j + tju j
tangent directions u j =y j+1−y j−1
‖y j+1−y j−1‖2
U is a matrix of 2k rows and k columns in which the (j , j)thentry is u(1)
j and (k + j , j)th entry is u(2)j , otherwise zeros
![Page 87: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/87.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS relaxation along curves
TPS relaxation along curves
Let Yk×2 = (y1, . . . yk )T be configuration matrix with the
rows y j=(
y (1)j , y (2)
j
)T
y(r)j is free to slid away from their old position y j along the
tangent directions u j =(
u(1)j , u(2)
j
)Twith ‖u‖2 = 1
new position y (r)j = y j + tju j
tangent directions u j =y j+1−y j−1
‖y j+1−y j−1‖2
U is a matrix of 2k rows and k columns in which the (j , j)thentry is u(1)
j and (k + j , j)th entry is u(2)j , otherwise zeros
![Page 88: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/88.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS relaxation along curves
TPS relaxation along curves
Let Yk×2 = (y1, . . . yk )T be configuration matrix with the
rows y j=(
y (1)j , y (2)
j
)T
y(r)j is free to slid away from their old position y j along the
tangent directions u j =(
u(1)j , u(2)
j
)Twith ‖u‖2 = 1
new position y (r)j = y j + tju j
tangent directions u j =y j+1−y j−1
‖y j+1−y j−1‖2
U is a matrix of 2k rows and k columns in which the (j , j)thentry is u(1)
j and (k + j , j)th entry is u(2)j , otherwise zeros
![Page 89: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/89.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS relaxation along curves
TPS relaxation along curves
yr = Vec(Yr ), B = diag(Be, Be), Be depends only on someconfiguration X
yr = y + Ut
the task is now to minimize the form
yTr By r = (y + Ut)T B (y + Ut)
setting the gradient of this expression to zerostraightforwardly generates the solution (Bookstein 1997)
t = −(
UT BU)−1
UT By
![Page 90: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/90.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS relaxation along curves
TPS relaxation along curves
yr = Vec(Yr ), B = diag(Be, Be), Be depends only on someconfiguration X
yr = y + Ut
the task is now to minimize the form
yTr By r = (y + Ut)T B (y + Ut)
setting the gradient of this expression to zerostraightforwardly generates the solution (Bookstein 1997)
t = −(
UT BU)−1
UT By
![Page 91: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/91.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS relaxation along curves
TPS relaxation along curves
yr = Vec(Yr ), B = diag(Be, Be), Be depends only on someconfiguration X
yr = y + Ut
the task is now to minimize the form
yTr By r = (y + Ut)T B (y + Ut)
setting the gradient of this expression to zerostraightforwardly generates the solution (Bookstein 1997)
t = −(
UT BU)−1
UT By
![Page 92: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/92.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS relaxation along curves
TPS relaxation along curves
yr = Vec(Yr ), B = diag(Be, Be), Be depends only on someconfiguration X
yr = y + Ut
the task is now to minimize the form
yTr By r = (y + Ut)T B (y + Ut)
setting the gradient of this expression to zerostraightforwardly generates the solution (Bookstein 1997)
t = −(
UT BU)−1
UT By
![Page 93: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/93.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS relaxation along curves
Data
![Page 94: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/94.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS relaxation along curves
Data
![Page 95: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/95.jpg)
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 splineor B-spline f (De Boor (1972) or Eilers & Marx (1996)),y j = (y (1)
j , y (2)j )T ∈ f , j = 1, 2, . . . k
re-sampled points y i = (y (1)i , y (2)
i )T ∈ f , i = 1, 2, . . . M(M = 500) and M = {y1, y2, ...yM}
suppose that y(s)j = (y (1)
sj , y (2)sj )T ∈ f (the rows of Ys) are
free to slid away from their old position y j along the curve f
![Page 96: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/96.jpg)
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 splineor B-spline f (De Boor (1972) or Eilers & Marx (1996)),y j = (y (1)
j , y (2)j )T ∈ f , j = 1, 2, . . . k
re-sampled points y i = (y (1)i , y (2)
i )T ∈ f , i = 1, 2, . . . M(M = 500) and M = {y1, y2, ...yM}
suppose that y(s)j = (y (1)
sj , y (2)sj )T ∈ f (the rows of Ys) are
free to slid away from their old position y j along the curve f
![Page 97: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/97.jpg)
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 splineor B-spline f (De Boor (1972) or Eilers & Marx (1996)),y j = (y (1)
j , y (2)j )T ∈ f , j = 1, 2, . . . k
re-sampled points y i = (y (1)i , y (2)
i )T ∈ f , i = 1, 2, . . . M(M = 500) and M = {y1, y2, ...yM}
suppose that y(s)j = (y (1)
sj , y (2)sj )T ∈ f (the rows of Ys) are
free to slid away from their old position y j along the curve f
![Page 98: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/98.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS relaxation along curves
TPS relaxation along curves
J(ys) = yTs By s, has to be minimized and yr is obtained as
a minimizer of J(ys) given by
yr = arg minys
J(ys) (3)
the minimization starts with substitution of y1 by y i ∈ M, ...and ends with substitution of yk by y i ∈ M, wherey j , j = 1, 2, . . . k , are the rows of Y and i = 1, 2, . . . M:
y(r)j =
(
arg minys
J (ys)
)
j,k+j, (4)
where (j , k + j)th entry of ys is substituted by y(s)i ∈ M for
j = 1, 2, . . . k ; i = 1, 2, . . . M, yr = Vec(Yr ) and y(r)j are the
rows of Yr
![Page 99: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/99.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS relaxation along curves
TPS relaxation along curves
J(ys) = yTs By s, has to be minimized and yr is obtained as
a minimizer of J(ys) given by
yr = arg minys
J(ys) (3)
the minimization starts with substitution of y1 by y i ∈ M, ...and ends with substitution of yk by y i ∈ M, wherey j , j = 1, 2, . . . k , are the rows of Y and i = 1, 2, . . . M:
y(r)j =
(
arg minys
J (ys)
)
j,k+j, (4)
where (j , k + j)th entry of ys is substituted by y(s)i ∈ M for
j = 1, 2, . . . k ; i = 1, 2, . . . M, yr = Vec(Yr ) and y(r)j are the
rows of Yr
![Page 100: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/100.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS relaxation along curves
Data
![Page 101: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/101.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS relaxation along curves
Data
0 100 200 300 400 500
05
1015
2025
landmark 4 resampled 500 timesposition on the curve
TB
E
0 100 200 300 400 500
05
1015
2025
landmark 7 resampled 500 timesposition on the curve
TB
E
0 100 200 300 400 500
05
1015
2025
landmark 15 resampled 500 timesposition on the curve
TB
E
![Page 102: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/102.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS relaxation along curves
Results of form-space PCA
PC1 minus
PC1 plus
PC2 up
PC2 down
![Page 103: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/103.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
TPS relaxation along curves
Results of form-space PCA
![Page 104: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/104.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Outline
1 IntroductionNotation and problems
2 Cubic splinesExample 1 – shape dataNCS for bivariate data
3 TPS for shape dataTPS for shape dataTPS relaxation along curves
4 Acknowledgement
![Page 105: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/105.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Acknowledgement
Supported by MRTN-CT-2005-019564 (EVAN) to GerhardWeber , 1/3023/06 (VEGA) to Franti sek Stulajter ,1/0077/09 (VEGA) to Andrej P azman
For data acquisition and pre-processing I thank MichaelCoquerelle
Fred L Bookstein – University of Vienna, Vienna, Austriaand University of Washington, Seattle, US
Jean-Francois Mangin – Neurospin, Institut d’ImagerieBioMedicale Commissariat a l’Energie Atomique, Gif surYvette, France
Paul O’Higgins – Hull/York Medical School, University ofYork, York, UK
![Page 106: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/106.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Acknowledgement
Supported by MRTN-CT-2005-019564 (EVAN) to GerhardWeber , 1/3023/06 (VEGA) to Franti sek Stulajter ,1/0077/09 (VEGA) to Andrej P azman
For data acquisition and pre-processing I thank MichaelCoquerelle
Fred L Bookstein – University of Vienna, Vienna, Austriaand University of Washington, Seattle, US
Jean-Francois Mangin – Neurospin, Institut d’ImagerieBioMedicale Commissariat a l’Energie Atomique, Gif surYvette, France
Paul O’Higgins – Hull/York Medical School, University ofYork, York, UK
![Page 107: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/107.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Acknowledgement
Supported by MRTN-CT-2005-019564 (EVAN) to GerhardWeber , 1/3023/06 (VEGA) to Franti sek Stulajter ,1/0077/09 (VEGA) to Andrej P azman
For data acquisition and pre-processing I thank MichaelCoquerelle
Fred L Bookstein – University of Vienna, Vienna, Austriaand University of Washington, Seattle, US
Jean-Francois Mangin – Neurospin, Institut d’ImagerieBioMedicale Commissariat a l’Energie Atomique, Gif surYvette, France
Paul O’Higgins – Hull/York Medical School, University ofYork, York, UK
![Page 108: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/108.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Acknowledgement
Supported by MRTN-CT-2005-019564 (EVAN) to GerhardWeber , 1/3023/06 (VEGA) to Franti sek Stulajter ,1/0077/09 (VEGA) to Andrej P azman
For data acquisition and pre-processing I thank MichaelCoquerelle
Fred L Bookstein – University of Vienna, Vienna, Austriaand University of Washington, Seattle, US
Jean-Francois Mangin – Neurospin, Institut d’ImagerieBioMedicale Commissariat a l’Energie Atomique, Gif surYvette, France
Paul O’Higgins – Hull/York Medical School, University ofYork, York, UK
![Page 109: Shape Analysis in R · Shape Analysis in R GM library in the light of recent methodological developments Stanislav Katina stanislav.katina@gmail.com Department of Applied Mathematics](https://reader030.vdocuments.site/reader030/viewer/2022040500/5e1be2f5a4e6e30733252bcc/html5/thumbnails/109.jpg)
Introduction Cubic splines TPS for shape data Acknowledgement
Acknowledgement
Supported by MRTN-CT-2005-019564 (EVAN) to GerhardWeber , 1/3023/06 (VEGA) to Franti sek Stulajter ,1/0077/09 (VEGA) to Andrej P azman
For data acquisition and pre-processing I thank MichaelCoquerelle
Fred L Bookstein – University of Vienna, Vienna, Austriaand University of Washington, Seattle, US
Jean-Francois Mangin – Neurospin, Institut d’ImagerieBioMedicale Commissariat a l’Energie Atomique, Gif surYvette, France
Paul O’Higgins – Hull/York Medical School, University ofYork, York, UK