1,068 research outputs found
Extrinsic local regression on manifold-valued data
We propose an extrinsic regression framework for modeling data with manifold
valued responses and Euclidean predictors. Regression with manifold responses
has wide applications in shape analysis, neuroscience, medical imaging and many
other areas. Our approach embeds the manifold where the responses lie onto a
higher dimensional Euclidean space, obtains a local regression estimate in that
space, and then projects this estimate back onto the image of the manifold.
Outside the regression setting both intrinsic and extrinsic approaches have
been proposed for modeling i.i.d manifold-valued data. However, to our
knowledge our work is the first to take an extrinsic approach to the regression
problem. The proposed extrinsic regression framework is general,
computationally efficient and theoretically appealing. Asymptotic distributions
and convergence rates of the extrinsic regression estimates are derived and a
large class of examples are considered indicating the wide applicability of our
approach
Optimal projection of observations in a Bayesian setting
Optimal dimensionality reduction methods are proposed for the Bayesian
inference of a Gaussian linear model with additive noise in presence of
overabundant data. Three different optimal projections of the observations are
proposed based on information theory: the projection that minimizes the
Kullback-Leibler divergence between the posterior distributions of the original
and the projected models, the one that minimizes the expected Kullback-Leibler
divergence between the same distributions, and the one that maximizes the
mutual information between the parameter of interest and the projected
observations. The first two optimization problems are formulated as the
determination of an optimal subspace and therefore the solution is computed
using Riemannian optimization algorithms on the Grassmann manifold. Regarding
the maximization of the mutual information, it is shown that there exists an
optimal subspace that minimizes the entropy of the posterior distribution of
the reduced model; a basis of the subspace can be computed as the solution to a
generalized eigenvalue problem; an a priori error estimate on the mutual
information is available for this particular solution; and that the
dimensionality of the subspace to exactly conserve the mutual information
between the input and the output of the models is less than the number of
parameters to be inferred. Numerical applications to linear and nonlinear
models are used to assess the efficiency of the proposed approaches, and to
highlight their advantages compared to standard approaches based on the
principal component analysis of the observations
Probabilistic Surfel Fusion for Dense LiDAR Mapping
With the recent development of high-end LiDARs, more and more systems are
able to continuously map the environment while moving and producing spatially
redundant information. However, none of the previous approaches were able to
effectively exploit this redundancy in a dense LiDAR mapping problem. In this
paper, we present a new approach for dense LiDAR mapping using probabilistic
surfel fusion. The proposed system is capable of reconstructing a high-quality
dense surface element (surfel) map from spatially redundant multiple views.
This is achieved by a proposed probabilistic surfel fusion along with a
geometry considered data association. The proposed surfel data association
method considers surface resolution as well as high measurement uncertainty
along its beam direction which enables the mapping system to be able to control
surface resolution without introducing spatial digitization. The proposed
fusion method successfully suppresses the map noise level by considering
measurement noise caused by laser beam incident angle and depth distance in a
Bayesian filtering framework. Experimental results with simulated and real data
for the dense surfel mapping prove the ability of the proposed method to
accurately find the canonical form of the environment without further
post-processing.Comment: Accepted in Multiview Relationships in 3D Data 2017 (IEEE
International Conference on Computer Vision Workshops
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