introduction to shape manifolds - github pages · shape statistics: hypothesis testing testing...
TRANSCRIPT
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Introduction to Shape Manifolds
Geometry of Data
September 26, 2019
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Shape Statistics: Averages
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Shape Statistics: Averages
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Shape Statistics: Variability
Shape priors in segmentation
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Shape Statistics: Classification
http://sites.google.com/site/xiangbai/animaldataset
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Shape Statistics: Hypothesis Testing
Testing group differences
Cates, et al. IPMI 2007 and ISBI 2008
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Shape Application: Bird Identification
American Crow Common Raven
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Shape Application: Bird Identification
Glaucous Gull Iceland Gull
http://notendur.hi.is/yannk/specialities.htm
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Shape Application: Box Turtles
Male Female
http://www.bio.davidson.edu/people/midorcas/research/Contribute/boxturtle/boxinfo.htm
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Shape Statistics: Regression
Application: Healthy Brain Aging
35 37 39 41 43
45 47 49 51 53
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What is Shape?
Shape is the geometry of an object modulo position,orientation, and size.
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What is Shape?
Shape is the geometry of an object modulo position,orientation, and size.
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Geometry Representations
I Landmarks (key identifiable points)I Boundary models (points, curves, surfaces, level
sets)I Interior models (medial, solid mesh)I Transformation models (splines, diffeomorphisms)
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Landmarks
FromGalileo (1638) illustrating the differences in shapes
of the bones of small and large animals.
5
Landmark: point of correspondence on each object
that matches between and within populations.
Different types: anatomical (biological), mathematical,
pseudo, quasi
6
T2 mouse vertebra with six mathematical landmarks
(line junctions) and 54 pseudo-landmarks.
7
Bookstein (1991)
Type I landmarks (joins of tissues/bones)
Type II landmarks (local properties such as maximal
curvatures)
Type III landmarks (extremal points or constructed land-
marks)
Labelled or un-labelled configurations
8
From Dryden & Mardia, 1998
I A landmark is an identifiable point on an object thatcorresponds to matching points on similar objects.
I This may be chosen based on the application (e.g.,by anatomy) or mathematically (e.g., by curvature).
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Landmark CorrespondenceShape and Registration
Homology:
Corresponding
(homologous)
features in all
skull images.
Ch. G. Small, The Statistical Theory of Shape
From C. Small, The Statistical Theory of Shape
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More Geometry Representations
Dense BoundaryPoints
Continuous Boundary(Fourier, splines)
Medial Axis(solid interior)
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Transformation Models
From D’Arcy Thompson, On Growth and Form, 1917.
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Shape Spaces
A shape is a point in a high-dimensional, nonlinearmanifold, called a shape space.
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Shape Spaces
A shape is a point in a high-dimensional, nonlinearmanifold, called a shape space.
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Shape Spaces
A shape is a point in a high-dimensional, nonlinearmanifold, called a shape space.
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Shape Spaces
A shape is a point in a high-dimensional, nonlinearmanifold, called a shape space.
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Shape Spaces
x
y
d(x, y)
A metric space structure provides a comparisonbetween two shapes.
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Examples: Shape Spaces
Kendall’s Shape Space Space ofDiffeomorphisms
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Tangent Spaces
p
X
M
Infinitesimal change in shape:
p X
A tangent vector is the velocity of a curve on M.
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Tangent Spaces
p
X
M
Infinitesimal change in shape:
p X
A tangent vector is the velocity of a curve on M.
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The Exponential Map
pT M pExp (X)p
X
M
Notation: Expp(X)
I p: starting point on MI X: initial velocity at pI Output: endpoint of geodesic segment, starting at
p, with velocity X, with same length as ‖X‖
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The Log Map
pT M pExp (X)p
X
M
Notation: Logp(q)I Inverse of ExpI p, q: two points in MI Output: tangent vector at p, such that
Expp(Logp(q)) = qI Gives distance between points:
d(p, q) = ‖Logp(q)‖.
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Shape Equivalences
Two geometry representations, x1, x2, are equivalent ifthey are just a translation, rotation, scaling of each other:
x2 = λR · x1 + v,
where λ is a scaling, R is a rotation, and v is atranslation.
In notation: x1 ∼ x2
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Equivalence Classes
The relationship x1 ∼ x2 is an equivalencerelationship:I Reflexive: x1 ∼ x1
I Symmetric: x1 ∼ x2 implies x2 ∼ x1
I Transitive: x1 ∼ x2 and x2 ∼ x3 imply x1 ∼ x3
We call the set of all equivalent geometries to x theequivalence class of x:
[x] = {y : y ∼ x}
The set of all equivalence classes is our shape space.
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Equivalence Classes
The relationship x1 ∼ x2 is an equivalencerelationship:I Reflexive: x1 ∼ x1
I Symmetric: x1 ∼ x2 implies x2 ∼ x1
I Transitive: x1 ∼ x2 and x2 ∼ x3 imply x1 ∼ x3
We call the set of all equivalent geometries to x theequivalence class of x:
[x] = {y : y ∼ x}
The set of all equivalence classes is our shape space.
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Equivalence Classes
The relationship x1 ∼ x2 is an equivalencerelationship:I Reflexive: x1 ∼ x1
I Symmetric: x1 ∼ x2 implies x2 ∼ x1
I Transitive: x1 ∼ x2 and x2 ∼ x3 imply x1 ∼ x3
We call the set of all equivalent geometries to x theequivalence class of x:
[x] = {y : y ∼ x}
The set of all equivalence classes is our shape space.
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Kendall’s Shape Space
I Define object with k points.I Represent as a vector in R2k.I Remove translation, rotation, and
scale.I End up with complex projective
space, CPk−2.
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Quotient Spaces
What do we get when we “remove” scaling from R2?
x
Notation: [x] ∈ R2/R+
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Quotient Spaces
What do we get when we “remove” scaling from R2?
x
[x]
Notation: [x] ∈ R2/R+
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Quotient Spaces
What do we get when we “remove” scaling from R2?
x
[x]
Notation: [x] ∈ R2/R+
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Quotient Spaces
What do we get when we “remove” scaling from R2?
x
[x]
Notation: [x] ∈ R2/R+
![Page 37: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/37.jpg)
Quotient Spaces
What do we get when we “remove” scaling from R2?
x
[x]
Notation: [x] ∈ R2/R+
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Constructing Kendall’s Shape Space
I Consider planar landmarks to be points in thecomplex plane.
I An object is then a point (z1, z2, . . . , zk) ∈ Ck.I Removing translation leaves us with Ck−1.I How to remove scaling and rotation?
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Constructing Kendall’s Shape Space
I Consider planar landmarks to be points in thecomplex plane.
I An object is then a point (z1, z2, . . . , zk) ∈ Ck.
I Removing translation leaves us with Ck−1.I How to remove scaling and rotation?
![Page 40: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/40.jpg)
Constructing Kendall’s Shape Space
I Consider planar landmarks to be points in thecomplex plane.
I An object is then a point (z1, z2, . . . , zk) ∈ Ck.I Removing translation leaves us with Ck−1.
I How to remove scaling and rotation?
![Page 41: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/41.jpg)
Constructing Kendall’s Shape Space
I Consider planar landmarks to be points in thecomplex plane.
I An object is then a point (z1, z2, . . . , zk) ∈ Ck.I Removing translation leaves us with Ck−1.I How to remove scaling and rotation?
![Page 42: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/42.jpg)
Scaling and Rotation in the Complex PlaneIm
Re0
!
r
Recall a complex number can be writ-ten as z = reiφ, with modulus r andargument φ.
Complex Multiplication:
seiθ ∗ reiφ = (sr)ei(θ+φ)
Multiplication by a complex number seiθ is equivalent toscaling by s and rotation by θ.
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Scaling and Rotation in the Complex PlaneIm
Re0
!
r
Recall a complex number can be writ-ten as z = reiφ, with modulus r andargument φ.
Complex Multiplication:
seiθ ∗ reiφ = (sr)ei(θ+φ)
Multiplication by a complex number seiθ is equivalent toscaling by s and rotation by θ.
![Page 44: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/44.jpg)
Removing Scale and Rotation
Multiplying a centered point set, z = (z1, z2, . . . , zk−1),by a constant w ∈ C, just rotates and scales it.
Thus the shape of z is an equivalence class:
[z] = {(wz1,wz2, . . . ,wzk−1) : ∀w ∈ C}
This gives complex projective space CPk−2 – much likethe sphere comes from equivalence classes of scalarmultiplication in Rn.
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Removing Scale and Rotation
Multiplying a centered point set, z = (z1, z2, . . . , zk−1),by a constant w ∈ C, just rotates and scales it.
Thus the shape of z is an equivalence class:
[z] = {(wz1,wz2, . . . ,wzk−1) : ∀w ∈ C}
This gives complex projective space CPk−2 – much likethe sphere comes from equivalence classes of scalarmultiplication in Rn.
![Page 46: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/46.jpg)
Removing Scale and Rotation
Multiplying a centered point set, z = (z1, z2, . . . , zk−1),by a constant w ∈ C, just rotates and scales it.
Thus the shape of z is an equivalence class:
[z] = {(wz1,wz2, . . . ,wzk−1) : ∀w ∈ C}
This gives complex projective space CPk−2 – much likethe sphere comes from equivalence classes of scalarmultiplication in Rn.
![Page 47: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/47.jpg)
Alternative: Shape Matrices
Represent an object as a real d × k matrix.Preshape process:I Remove translation: subtract the row means from
each row (i.e., translate shape centroid to 0).I Remove scale: divide by the Frobenius norm.
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Orthogonal Procrustes Analysis
Problem:Find the rotation R∗ that minimizes distance betweentwo d × k matrices A, B:
R∗ = arg minR∈SO(d)
‖RA− B‖2
Solution:Let UΣVT be the SVD of BAT , then
R∗ = UVT
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Orthogonal Procrustes Analysis
Problem:Find the rotation R∗ that minimizes distance betweentwo d × k matrices A, B:
R∗ = arg minR∈SO(d)
‖RA− B‖2
Solution:Let UΣVT be the SVD of BAT , then
R∗ = UVT
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Geodesics in 2D Kendall Shape Space
Let A and B be 2× k shape matrices
1. Remove centroids from A and B
2. Project onto sphere: A← A/‖A‖, B← B/‖B‖3. Align rotation of B to A with OPA
4. Now a geodesic is simply that of the sphere, S2k−1
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Geodesics in 2D Kendall Shape Space
Let A and B be 2× k shape matrices
1. Remove centroids from A and B2. Project onto sphere: A← A/‖A‖, B← B/‖B‖
3. Align rotation of B to A with OPA
4. Now a geodesic is simply that of the sphere, S2k−1
![Page 52: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/52.jpg)
Geodesics in 2D Kendall Shape Space
Let A and B be 2× k shape matrices
1. Remove centroids from A and B2. Project onto sphere: A← A/‖A‖, B← B/‖B‖3. Align rotation of B to A with OPA
4. Now a geodesic is simply that of the sphere, S2k−1
![Page 53: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/53.jpg)
Geodesics in 2D Kendall Shape Space
Let A and B be 2× k shape matrices
1. Remove centroids from A and B2. Project onto sphere: A← A/‖A‖, B← B/‖B‖3. Align rotation of B to A with OPA
4. Now a geodesic is simply that of the sphere, S2k−1
![Page 54: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/54.jpg)
Intrinsic Means (Frechet)
The intrinsic mean of a collection of points x1, . . . , xN ina metric space M is
µ = arg minx∈M
N∑i=1
d(x, xi)2,
where d(·, ·) denotes distance in M.
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Gradient of the Geodesic Distance
The gradient of the Riemannian distance function is
gradxd(x, y)2 = −2 Logx(y).
So, gradient of the sum-of-squared distance function is
gradx
N∑i=1
d(x, xi)2 = −2
N∑i=1
Logx(xi).
![Page 56: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/56.jpg)
Computing Means
Gradient Descent Algorithm:
Input: x1, . . . , xN ∈ M
µ0 = x1
Repeat:
δµ = 1N
∑Ni=1 Logµk
(xi)
µk+1 = Expµk(δµ)
![Page 57: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/57.jpg)
Computing Means
Gradient Descent Algorithm:
Input: x1, . . . , xN ∈ M
µ0 = x1
Repeat:
δµ = 1N
∑Ni=1 Logµk
(xi)
µk+1 = Expµk(δµ)
![Page 58: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/58.jpg)
Computing Means
Gradient Descent Algorithm:
Input: x1, . . . , xN ∈ M
µ0 = x1
Repeat:
δµ = 1N
∑Ni=1 Logµk
(xi)
µk+1 = Expµk(δµ)
![Page 59: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/59.jpg)
Computing Means
Gradient Descent Algorithm:
Input: x1, . . . , xN ∈ M
µ0 = x1
Repeat:
δµ = 1N
∑Ni=1 Logµk
(xi)
µk+1 = Expµk(δµ)
![Page 60: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/60.jpg)
Computing Means
Gradient Descent Algorithm:
Input: x1, . . . , xN ∈ M
µ0 = x1
Repeat:
δµ = 1N
∑Ni=1 Logµk
(xi)
µk+1 = Expµk(δµ)
![Page 61: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/61.jpg)
Computing Means
Gradient Descent Algorithm:
Input: x1, . . . , xN ∈ M
µ0 = x1
Repeat:
δµ = 1N
∑Ni=1 Logµk
(xi)
µk+1 = Expµk(δµ)
![Page 62: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/62.jpg)
Computing Means
Gradient Descent Algorithm:
Input: x1, . . . , xN ∈ M
µ0 = x1
Repeat:
δµ = 1N
∑Ni=1 Logµk
(xi)
µk+1 = Expµk(δµ)
![Page 63: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/63.jpg)
Computing Means
Gradient Descent Algorithm:
Input: x1, . . . , xN ∈ M
µ0 = x1
Repeat:
δµ = 1N
∑Ni=1 Logµk
(xi)
µk+1 = Expµk(δµ)
![Page 64: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/64.jpg)
Computing Means
Gradient Descent Algorithm:
Input: x1, . . . , xN ∈ M
µ0 = x1
Repeat:
δµ = 1N
∑Ni=1 Logµk
(xi)
µk+1 = Expµk(δµ)
![Page 65: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/65.jpg)
Computing Means
Gradient Descent Algorithm:
Input: x1, . . . , xN ∈ M
µ0 = x1
Repeat:
δµ = 1N
∑Ni=1 Logµk
(xi)
µk+1 = Expµk(δµ)
![Page 66: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/66.jpg)
Example of Mean on Kendall Shape Space
Hand data from Tim Cootes
→
![Page 67: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/67.jpg)
Example of Mean on Kendall Shape Space
Hand data from Tim Cootes
→
![Page 68: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/68.jpg)
Principal Geodesic Analysis
Linear Statistics (PCA) Curved Statistics (PGA)
![Page 69: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/69.jpg)
Principal Geodesic Analysis
Linear Statistics (PCA) Curved Statistics (PGA)
![Page 70: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/70.jpg)
Principal Geodesic Analysis
Linear Statistics (PCA) Curved Statistics (PGA)
![Page 71: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/71.jpg)
Principal Geodesic Analysis
Linear Statistics (PCA) Curved Statistics (PGA)
![Page 72: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/72.jpg)
Principal Geodesic Analysis
Linear Statistics (PCA) Curved Statistics (PGA)
![Page 73: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/73.jpg)
Principal Geodesic Analysis
Linear Statistics (PCA) Curved Statistics (PGA)
![Page 74: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/74.jpg)
Principal Geodesic Analysis
Linear Statistics (PCA) Curved Statistics (PGA)
![Page 75: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/75.jpg)
PGA of Kidney
Mode 1 Mode 2 Mode 3
![Page 76: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/76.jpg)
PGA DefinitionFirst principal geodesic direction:
v1 = arg max‖v‖=1
N∑i=1
‖Logy(πH(yi))‖2,
where H = Expy(span({v}) ∩ U).
Remaining principal directions are defined recursively as
vk = arg max‖v‖=1
N∑i=1
‖Logy(πH(yi))‖2,
where H = Expy(span({v1, . . . , vk−1, v}) ∩ U).
![Page 77: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/77.jpg)
PGA DefinitionFirst principal geodesic direction:
v1 = arg max‖v‖=1
N∑i=1
‖Logy(πH(yi))‖2,
where H = Expy(span({v}) ∩ U).
Remaining principal directions are defined recursively as
vk = arg max‖v‖=1
N∑i=1
‖Logy(πH(yi))‖2,
where H = Expy(span({v1, . . . , vk−1, v}) ∩ U).
![Page 78: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/78.jpg)
Tangent Approximation to PGA
Input: Data y1, . . . , yN ∈ MOutput: Principal directions, vk ∈ TµM, variances,
λk ∈ Ry = Frechet mean of {yi}ui = Logµ(yi)
S = 1N−1
∑Ni=1 uiuT
i{vk, λk} = eigenvectors/eigenvalues of S.
![Page 79: Introduction to Shape Manifolds - GitHub Pages · Shape Statistics: Hypothesis Testing Testing group differences Cates, et al. IPMI 2007 and ISBI 2008](https://reader033.vdocument.in/reader033/viewer/2022053012/5f0fdc547e708231d4463f50/html5/thumbnails/79.jpg)
Where to Learn More
Books
I Dryden and Mardia, Statistical Shape Analysis, Wiley, 1998.
I Small, The Statistical Theory of Shape, Springer-Verlag,1996.
I Kendall, Barden and Carne, Shape and Shape Theory, Wiley,1999.
I Krim and Yezzi, Statistics and Analysis of Shapes,Birkhauser, 2006.