Kendall, D.G., D. Bardon, T.K. Carne and H. Le
（1999年刊行，John Wiley & Sons，ISBN:0471968234）
【メモ】
近年の形態測定学（morphometrics）の数ある理論の中でもDavid Kendall が1970年代以降つくってきた「形状理論」（shape theory）は理論上の【骨格】に相当する部分を担ってきました。生物形態のサイズを除去したシェイプは Kendall の言う非線形の「形状空間」（shape space）に属するため、形態データに関する線形統計学を行なうためには、形状空間を適当な線形接空間に射影する必要があります。われわれがなじんでいるさまざまな線形数学の諸手法（分散分析とかアロメトリー解析）はあくまでも「近似的方法」にすぎません。では、「正確」な方法とは何か−それはある種の非線形数学−一般化された幾何学−を前提にした理論の構築を要求します。Kendall の形状空間論はそのための道具を提供していると私は理解しています。本書はこの Kendall 形状理論に関する、著者自身によるモノグラフです。
【目次】
Preface
Chapter 1: Shapes and shape spaces
1.1: Origins
1.2: Some preliminary observations
1.3: A matrix representation for the shape of a k-ad
1.4: 'Elementary' shape space Σ[k,1] and Σ[k,2]
1.5: The Fubini-Study metric on Σ[k,2]
1.6: The proof of Casson's theorem
※ 幾何学的形態から出発して Kendall 形状空間Σ[k,m]−ｍ次元のｋ標識点がなす空間−にいたる導出の過程を述べる。
Chapter 2: The global structure of shape spaces
2.1: The problem
2.2: When is a space familiar
2.3: CW complexes
2.4: A cellular decomposition of the unit sphare
2.5: The cellular decomposition of shape spaces
2.6: Inclusions and isometries
2.7: Simple connectivity and higher homotopy groups
2.8: The mapping cone decomposition
2.9: Homotopy type and Casson's theorem
Chapter 3: Computing the homology of cell complexes
3.1: The orientation of certain spaces
3.2: The orientation of spherical cells
3.3: The boundary of an oriented cell
3.4: The chain complex, homology and cohomology groups
3.5: Reduced homology
3.6: The homology exact sequence for shape spaces
3.7: Applications of the exact sequence
3.8: Topological invariants that distinguish between shape spaces
Chapter 4: A chain complex for shape spaces
4.1: The chain complex
4.2: The space of unoriented shapes
4.3: The boundary map in the chain complex
4.4: Decomposing the chain complex
4.5: Homology and cohomology of the spaces
4.6: Connectivity of shape spaces
4.7: Limits of shape spaces
Chapter 5: The homology groups of shape spaces
5.1: Spaces of shapes in 2-space
5.2: Spaces of shapes in 3-space
5.3: Spaces of shapes in 4-space
5.4: Spaces of unoriented shapes in 2-space
5.5: Spaces of unoriented shapes in 3-space
5.6: Spaces of unoriented shapes in 4-space
5.7: Decomposing the essential complexes
5.8: Closed formulae for the homology groups
5.9: Duality in shape spaces
※ ２〜５章では、形状空間の位相構造−とくに位相同型について−を議論する。
Chapter 6: Geodesics in shape spaces
6.1: The action of _SO_(m) on the pre-shape sphere
6.2: Viewing the induced Riemannian metric through horizontal geodesics
6.3: The singular points and the nesting principle
6.4: The distance between two shapes
6.5: The set of geodesics between two shapes
6.6: The non-uniqueness of minimal geodesics
6.7: The cut locus in shape spaces
6.8: The distances and projections to lower strata
Chapter 7: The Riemannian structure of shape spaces
7.1: The Riemannian metric
7.2: The metric re-expressed through natural local vector fields
7.3: The Riemannian curvature tensor
※ ６〜７章では、形状空間上の測地線距離がリーマン計量であること、そして Kendall 形状空間がリーマン多様体であることを示す。
Chapter 8: Induced shape-measures
8.1: Geometric preliminaries
8.2: The shape-measure on Σ[k,m]induced by k labelled iid isotropic Gaussian distributions on =R[m]=
8.3: Shape-measures on Σ[m+1,m] of Poisson-Delaunay tiles
8.4: Shape-measures on Σ[k,2] induced by k labelled iid non-isotropic Gaussian distributions on =R[m]=
8.5: Shape-measures on Σ[k,2] induced by complex normal distributions
8.6: The shape-measure on Σ[3,2] induced by three labelled iid uniform distributions in compact convex set
8.7: The shape-measure on Σ[3,2] induced by three labelled iid uniform distributions in a convex polygon I: the singular tesselation
8.8: The shape-measure on Σ[3,2] induced by three labelled iid uniform distributions in a convex polygon II: the explicit formula
Chapter 9: Mean shapes and the shape of the means
9.1: Concept of means in non-linear spaces
9.2: Metrics on shape space
9.3: Uniqueness of Frechet means of shape-measures
9.4: Frechet means and the shape of the means
9.5: Uniqueness of Frechet means of n given shapes
9.6: Procrustean mean shapes
※ ８〜９章では、Kendall 形状空間の上での非線形な確率論と統計学の構築を行なう。
Chapter 10: Visualising the higher dimensional shape spaces
10.1: The two-dimensional representation of Σ[4,3]
10.2: The cell-decomposition of _SO(3)_
10.3: The action of the group _D_
10.4: The geodesics of Σ[4,3]
10.5: Some distributions on Σ[4,3]
10.6: A diffusion on Σ[4,3]
※ 高次元形状空間の視覚化の試み。
Chapter 11: General shape spaces
11.1: Shape spaces for general manifolds
11.2: Size-and-shape spaces
11.3: Size-and-shape spaces for the plane
11.4: Spheres and hyperbolic spaces
11.5: Relative metrics
11.6: Elliptic functions
11.7: Shape spaces for triangles in the sphere and the hyperbolic plane
※ 変位・回転・スケーリングに対する不変量である shape よりも緩い不変量 size-and-shape（変位と回転のみに対する不変量）についての議論。
Appendix
A.1: Unary operators on groups
A.2: Binary operators on groups
A.3: The Universal Coefficient Theorems
A.4: Duality in manifolds
A.5: The splitting of exact sequences
※ 幾何学（位相幾何学・多様体・群論・ホモロジー／コホモロジー／ホモトピー論などなど）について補足。
Bibliography
Index
Kendall形状空間論についてはここ数年の間に２冊の解説書が出版されています。本書が出たことでさらにはずみがつくかな。それにしても「数学」だ！
- Small, C.G. 1996. The statistical theory of shape. Springer-Verlag, New York, x+227pp., ISBN:0387947299
- Dryden, I.L. and K.V. Mardia 1998. Statistical shape analysis. John Wiley & Sons, Chichester,xx+347pp., ISBN:0471958166
