1,707 research outputs found
Uniform Gaussian bounds for subelliptic heat kernels and an application to the total variation flow of graphs over Carnot groups
In this paper we study heat kernels associated to a Carnot group , endowed
with a family of collapsing left-invariant Riemannian metrics \sigma_\e which
converge in the Gromov-Hausdorff sense to a sub-Riemannian structure on as
\e\to 0. The main new contribution are Gaussian-type bounds on the heat
kernel for the \sigma_\e metrics which are stable as \e\to 0 and extend the
previous time-independent estimates in \cite{CiMa-F}. As an application we
study well posedness of the total variation flow of graph surfaces over a
bounded domain in (G,\s_\e). We establish interior and boundary gradient
estimates, and develop a Schauder theory which are stable as \e\to 0. As a
consequence we obtain long time existence of smooth solutions of the
sub-Riemannian flow (\e=0), which in turn yield sub-Riemannian minimal
surfaces as .Comment: We have corrected a few typos and added a few more details to the
proof of the Gaussian estimate
Highly corrupted image inpainting through hypoelliptic diffusion
We present a new image inpainting algorithm, the Averaging and Hypoelliptic
Evolution (AHE) algorithm, inspired by the one presented in [SIAM J. Imaging
Sci., vol. 7, no. 2, pp. 669--695, 2014] and based upon a semi-discrete
variation of the Citti-Petitot-Sarti model of the primary visual cortex V1. The
AHE algorithm is based on a suitable combination of sub-Riemannian hypoelliptic
diffusion and ad-hoc local averaging techniques. In particular, we focus on
reconstructing highly corrupted images (i.e. where more than the 80% of the
image is missing), for which we obtain reconstructions comparable with the
state-of-the-art.Comment: 15 pages, 10 figure
An Infinitesimal Probabilistic Model for Principal Component Analysis of Manifold Valued Data
We provide a probabilistic and infinitesimal view of how the principal
component analysis procedure (PCA) can be generalized to analysis of nonlinear
manifold valued data. Starting with the probabilistic PCA interpretation of the
Euclidean PCA procedure, we show how PCA can be generalized to manifolds in an
intrinsic way that does not resort to linearization of the data space. The
underlying probability model is constructed by mapping a Euclidean stochastic
process to the manifold using stochastic development of Euclidean
semimartingales. The construction uses a connection and bundles of covariant
tensors to allow global transport of principal eigenvectors, and the model is
thereby an example of how principal fiber bundles can be used to handle the
lack of global coordinate system and orientations that characterizes manifold
valued statistics. We show how curvature implies non-integrability of the
equivalent of Euclidean principal subspaces, and how the stochastic flows
provide an alternative to explicit construction of such subspaces. We describe
estimation procedures for inference of parameters and prediction of principal
components, and we give examples of properties of the model on embedded
surfaces
Principal Sub-manifolds
We revisit the problem of finding principal components to the multivariate
datasets, that lie on an embedded nonlinear Riemannian manifold within the
higher-dimensional space. Our aim is to extend the geometric interpretation of
PCA, while being able to capture the non-geodesic form of variation in the
data. We introduce the concept of a principal sub-manifold, a manifold passing
through the center of the data, and at any point on the manifold, it moves in
the direction of the highest curvature in the space spanned by eigenvectors of
the local tangent space PCA. Compared to the recent work in the case where the
sub-manifold is of dimension one (Panaretos, Pham and Yao 2014)--essentially a
curve lying on the manifold attempting to capture the one-dimensional
variation--the current setting is much more general. The principal sub-manifold
is therefore an extension of the principal flow, accommodating to capture the
higher dimensional variation in the data. We show the principal sub-manifold
yields the usual principal components in Euclidean space. By means of examples,
we illustrate how to find, use and interpret principal sub-manifold with an
extension of using it in shape analysis
Active Contour Models for Manifold Valued Image Segmentation
Image segmentation is the process of partitioning a image into different
regions or groups based on some characteristics like color, texture, motion or
shape etc. Active contours is a popular variational method for object
segmentation in images, in which the user initializes a contour which evolves
in order to optimize an objective function designed such that the desired
object boundary is the optimal solution. Recently, imaging modalities that
produce Manifold valued images have come up, for example, DT-MRI images, vector
fields. The traditional active contour model does not work on such images. In
this paper, we generalize the active contour model to work on Manifold valued
images. As expected, our algorithm detects regions with similar Manifold values
in the image. Our algorithm also produces expected results on usual gray-scale
images, since these are nothing but trivial examples of Manifold valued images.
As another application of our general active contour model, we perform texture
segmentation on gray-scale images by first creating an appropriate Manifold
valued image. We demonstrate segmentation results for manifold valued images
and texture images
Differential geometric regularization for supervised learning of classifiers
We study the problem of supervised learning for both binary and multiclass classification from a unified geometric perspective. In particular, we propose a geometric regularization technique to find the submanifold corresponding to an estimator of the class probability P(y|\vec x). The regularization term measures the volume of this submanifold, based on the intuition that overfitting produces rapid local oscillations and hence large volume of the estimator. This technique can be applied to regularize any classification function that satisfies two requirements: firstly, an estimator of the class probability can be obtained; secondly, first and second derivatives of the class probability estimator can be calculated. In experiments, we apply our regularization technique to standard loss functions for classification, our RBF-based implementation compares favorably to widely used regularization methods for both binary and multiclass classification.http://proceedings.mlr.press/v48/baia16.pdfPublished versio
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