On the Topology and Geometry of Spaces of Affine Shapes

Groisser, David; Tagare, Hemant D.

doi:10.1007/s10851-009-0143-4

Cited by 18 publications

(39 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Any open neighborhood of the trivial shape is actually already the full space a k d . This phenomenon occurs in similarity and affine shape space as well [3,4,5].…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…Finally, G 4 1 consists of configurations with no landmark coincidences (only type (a)), hence g 4 1 is homeomorphic to the circle with three points removed.…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…Example 4.4 In the case d " 1 and k " 4, t 4 1 consists of shapes with projective subspace constraints pI, jq with j " 1 and |I| " 1 ă 2 " 1¨4 2 , i.e. the shapes in general position (only type (a) from Section 3).…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…The intersection of all open neighborhoods to a point p P M is a useful tool towards understanding the separation properties of a space M . This set was introduced as the blur Blppq of p in M by Groisser and Tagare in their discussion of affine shape space [4]. We will call a point p P M unblurry if Blppq " tpu, and blurry in the case that its blur is a strict superset of tpu.…”

Section: The Manifold Of the Freementioning

confidence: 99%

“…For affine or similarity shapes, the topology of the shape space is well understood and there are natural choices for a Riemannian metric. Similarity shape space is a CW complex after removing the trivial shape [3], while affine shape space has a naturally ordered stratification with each stratum being diffeomorphic to a Grassmannian [4,5]. In both cases, the topological subspace of shapes with trivial isotropy group, i.e.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Manifolds of Projective Shapes

Hotz¹,

Kelma²,

Kent³

2016

Preprint

View full text Add to dashboard Cite

The projective shape of a configuration of k points or "landmarks" in RP d consists of the information that is invariant under projective transformations and hence is reconstructable from uncalibrated camera views. Mathematically, the space of projective shapes for these k landmarks can be described as the quotient space of k copies of RP d modulo the action of the projective linear group PGLpdq. Using homogeneous coordinates, such configurations can be described as real k ˆpd `1q-dimensional matrices given up to left-multiplication of non-singular diagonal matrices, while the group PGLpdq acts as GLpd `1q from the right. The main purpose of this paper is to give a detailed examination of the topology of projective shape space, and, using matrix notation, it is shown how to derive subsets that are in a certain sense maximal, differentiable Hausdorff manifolds which can be provided with a Riemannian metric. A special subclass of the projective shapes consists of the Tyler regular shapes, for which geometrically motivated pre-shapes can be defined, thus allowing for the construction of a natural Riemannian metric.

show abstract

“…Any open neighborhood of the trivial shape is actually already the full space a k d . This phenomenon occurs in similarity and affine shape space as well [3,4,5].…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…Finally, G 4 1 consists of configurations with no landmark coincidences (only type (a)), hence g 4 1 is homeomorphic to the circle with three points removed.…”

Section: Preliminaries and Notationmentioning

confidence: 99%

Section: Preliminaries and Notationmentioning

confidence: 99%

Section: The Manifold Of the Freementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations