Matrix-valued kernels for shape deformation analysis

@inproceedings{Micheli2013MatrixvaluedKF,
  title={Matrix-valued kernels for shape deformation analysis},
  author={Mario Micheli and Joan Alexis Glaun{\`e}s},
  year={2013}
}
The main purpose of this paper is providing a systematic study and classification of non-scalar kernels for Reproducing Kernel Hilbert Spaces (RKHS), to be used in the analysis of deformation in shape spaces endowed with metrics induced by the action of groups of diffeomorphisms. After providing an introduction to matrix-valued kernels and their relevant differential properties, we explore extensively those, that we call TRI kernels, that induce a metric on the corresponding Hilbert spaces of… 
Interpolation with uncoupled separable matrix-valued kernels.
TLDR
This paper uses matrix-valued reproducing kernels, which can be related to Reproducing kernel Hilbert spaces of vectorial functions, to consider the problem of approximating vector-valued functions over a domain $\Omega$.
Large-scale operator-valued kernel regression
TLDR
This thesis proposes and study scalable methods to perform regression with Operator-Valued Kernels, and develops a general framework devoted to the approximation of shift-invariant MErcer kernels on Locally Compact Abelian groups.
Metrics, Quantization and Registration in Varifold Spaces
TLDR
This work proposes a mathematical model for diffeomorphic registration of varifolds under a specific group action which is formulated in the framework of optimal control theory and addresses the problem of optimal finite approximations for those metrics and shows a $\Gamma-convergence property for the corresponding registration functionals.
Sub-Riemannian Methods in Shape Analysis
TLDR
This chapter provides a review of the methods that have been recently introduced in this context to study shapes, with a special focus on shape spaces defined as homogeneous spaces under the action of diffeomorphisms.
Kernel Metrics on Normal Cycles and Application to Curve Matching
TLDR
A new dissimilarity measure for shape registration is introduced using the notion of normal cycles, a concept from geometric measure theory which allows us to generalize curvature for nonsmooth subsets of the Euclidean space.
Random Fourier Features For Operator-Valued Kernels
TLDR
A general principle for Operator-valued Random Fourier Feature construction relies on a generalization of Bochner's theorem for translation-invariant operator-valued Mercer kernels and proves the uniform convergence of the kernel approximation for bounded and unbounded operator random Fourier features.
Weight metamorphosis of varifolds and the LDDMM-Fisher-Rao metric
This paper introduces and studies a metamorphosis framework for geometric measures known as varifolds, which extends the diffeomorphic registration model for objects such as curves, surfaces and
Stochastic Filtering on Shape Manifolds
This thesis addresses the problem of learning the dynamics of deforming objects in image time series. In many biomedical imaging and computer vision applications it is important to satisfy certain
Matrix valued positive definite kernels related to the generalized Aitken's integral for Gaussians
We introduce a method to construct general multivariate positive definite kernels on a nonempty set X that employs a prescribed bounded completely monotone function and special multivariate functions
...
1
2
3
4
...

References

SHOWING 1-10 OF 81 REFERENCES
On Learning Vector-Valued Functions
TLDR
This letter provides a study of learning in a Hilbert space of vector-valued functions and derives the form of the minimal norm interpolant to a finite set of data and applies it to study some regularization functionals that are important in learning theory.
An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach
Isotropic Energies, Filters and Splines for Vector Field Regularization
TLDR
This paper proposes a class of separable isotropic filters generalizing Gaussian filtering to vector fields, which enables fast smoothing in the spatial domain and solves the problem of approximating a dense and a sparse displacement field at the same time.
Sectional Curvature in Terms of the Cometric, with Applications to the Riemannian Manifolds of Landmarks
TLDR
This paper fully explore the case of geodesics on which only two points have nonzero momenta and compute the sectional curvatures of 2-planes spanned by the tangents to such geodesic, and gives insight into the geometry of the full manifolds of landmarks.
Probabilities and Statistics on Riemannian Manifolds : A Geometric approach
Measurements of geometric primitives are often noisy in real applications and we need to use statistics either to reduce the uncertainty (estimation), to compare measurements, or to test hypotheses.
Riemannian Geometries on Spaces of Plane Curves
We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of maps from the circle to the plane modulo the group of diffeomorphisms of the circle,
Large Deformation Diffeomorphic Metric Curve Mapping
TLDR
A discretized version of the matching criterion for curves is presented, in which discrete sequences of points along the curve are represented by vector-valued functionals, which gives a convenient and practical way to define a matching functional for curves.
Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds
Given a finite-dimensional manifold , the group of diffeomorphisms diffeomorphism of  which decrease suitably rapidly to the identity, acts on the manifold of submanifolds of  of diffeomorphism-type
...
1
2
3
4
5
...