Multidimensional scaling of simplex shapes

Le, Huiling; Small, Christopher G.

doi:10.1016/s0031-3203(99)00023-0

Cited by 36 publications

(29 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We note that in the case of a hyperbolic model for simplex shape spaces, rather than Kendall's shape space model, which we treat here, much work has been done in finding algorithms converging to the intrinsic Fréchet mean; see [14], [16], and [18]. In [12] planar circular shapes were modelled in an infinite-dimensional Riemannian shape space and shape variation along geodesics was studied.…”

Section: Introductionmentioning

confidence: 99%

Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces

Huckemann

Ziezold

2006

Adv. Appl. Probab.

View full text Add to dashboard Cite

Classical principal component analysis on manifolds, for example on Kendall's shape spaces, is carried out in the tangent space of a Euclidean mean equipped with a Euclidean metric. We propose a method of principal component analysis for Riemannian manifolds based on geodesics of the intrinsic metric, and provide a numerical implementation in the case of spheres. This method allows us, for example, to compare principal component geodesics of different data samples. In order to determine principal component geodesics, we show that in general, owing to curvature, the principal component geodesics do not pass through the intrinsic mean. As a consequence, means other than the intrinsic mean are considered, allowing for several choices of definition of geodesic variance. In conclusion we apply our method to the space of planar triangular shapes and compare our findings with those of standard Euclidean principal component analysis.

show abstract

Section: Introductionmentioning

confidence: 99%

Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces

Huckemann

Ziezold

2006

Adv. Appl. Probab.

View full text Add to dashboard Cite

show abstract

“…Alignment is concerned with the recovery of the geometric transformation that minimises an error functional. If the transformation is a simple isometry, then one of the most elegant and effective methods is to use Procrutes alignment [22,23]. This involves co-centring and scaling the point-sets so that they have the same variance.…”

Section: Related Literaturementioning

confidence: 99%

A Riemannian approach to graph embedding

Robles-Kelly

Hancock

2007

Pattern Recognition

View full text Add to dashboard Cite

In this paper, we make use of the relationship between the Laplace-Beltrami operator and the graph Laplacian, for the purposes of embedding a graph onto a Riemannian manifold. To embark on this study, we review some of the basics of Riemannian geometry and explain the relationship between the Laplace-Beltrami operator and the graph Laplacian. Using the properties of Jacobi fields, we show how to compute an edge-weight matrix in which the elements reflect the sectional curvatures associated with the geodesic paths on the manifold between nodes. For the particular case of a constant sectional curvature surface, we use the Kruskal coordinates to compute edge weights that are proportional to the geodesic distance between points. We use the resulting edge-weight matrix to embed the nodes of the graph onto a Riemannian manifold. To do this, we develop a method that can be used to perform double centring on the Laplacian matrix computed from the edge-weights. The embedding coordinates are given by the eigenvectors of the centred Laplacian. With the set of embedding coordinates at hand, a number of graph manipulation tasks can be performed. In this paper, we are primarily interested in graph-matching. We recast the graph-matching problem as that of aligning pairs of manifolds subject to a geometric transformation. We show that this transformation is Pro-crustean in nature. We illustrate the utility of the method on image matching using the COIL database.

show abstract

“…So the induced metric on T + (n) is also right-invariant, in the sense that, for any Y 0 in GL(n), dg(π(Y 1 ), π(Y 2 )) = dg(π(Y 1 Y 0 ), π(Y 2 Y 0 )), where dg is the geodesic distance on T + (n). For π(Y ) ∈ ST + (n), Le and Small [7] have shown that…”

Section: Gl(n)/o(n)mentioning

confidence: 99%

“…Here we extend the shape-space developed by Small [7] from the domain of Cartesian landmark points to a field of surface normals. The model commences from a distance matrix between longvectors representing fields of surface normals.…”

Section: Introductionmentioning

confidence: 99%

Facial Shape Spaces from Surface Normals and Geodesic Distance

Ceolin

Smith

Hancock

2007

9th Biennial Conference of the Australian Pattern Recognition Society on Digital Image Computing Techniques and Applications (D

View full text Add to dashboard Cite

The analysis of shape-variations due to changes in facial expression and gender difference is a key problem in face recognition. In this paper, we draw on ideas from the field of statistical shape analysis to construct shape-spaces that span facial expressions and gender, and use the resulting shape-model to perform face recognition under varying expression and gender. Our novel contribution is to show how to construct shape-spaces over fields of surface normals rather than Cartesian landmark points. According to this model face needle-maps (or fields of surface normals) are points in a high-dimensional manifold referred to as a shape-space. The similarity between faces and gender difference is measured using a number of alternative geodesic, Euclidean and cosine distance between points on the manifold. In a recognition experiment we compare the perfomance distance with Euclidean, cosine and geodesic distance associated with the shape manifold. Here we explore if the distances used to distinguish gender and recognise the same for under different expressions.

show abstract

Multidimensional scaling of simplex shapes

Cited by 36 publications

References 0 publications

Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces

Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces

A Riemannian approach to graph embedding

Facial Shape Spaces from Surface Normals and Geodesic Distance

Contact Info

Product

Resources

About