32,300 research outputs found
Geometric and Photometric Data Fusion in Non-Rigid Shape Analysis
In this paper, we explore the use of the diffusion geometry framework for the fusion of geometric and photometric information in local and global shape descriptors. Our construction is based on the definition of a diffusion process on the shape manifold embedded into a high-dimensional space where the embedding coordinates represent the photometric information. Experimental results show that such data fusion is useful in coping with different challenges of shape analysis where pure geometric and pure photometric methods fai
Improving embedding of graphs with missing data by soft manifolds
Embedding graphs in continous spaces is a key factor in designing and
developing algorithms for automatic information extraction to be applied in
diverse tasks (e.g., learning, inferring, predicting). The reliability of graph
embeddings directly depends on how much the geometry of the continuous space
matches the graph structure. Manifolds are mathematical structure that can
enable to incorporate in their topological spaces the graph characteristics,
and in particular nodes distances. State-of-the-art of manifold-based graph
embedding algorithms take advantage of the assumption that the projection on a
tangential space of each point in the manifold (corresponding to a node in the
graph) would locally resemble a Euclidean space. Although this condition helps
in achieving efficient analytical solutions to the embedding problem, it does
not represent an adequate set-up to work with modern real life graphs, that are
characterized by weighted connections across nodes often computed over sparse
datasets with missing records. In this work, we introduce a new class of
manifold, named soft manifold, that can solve this situation. In particular,
soft manifolds are mathematical structures with spherical symmetry where the
tangent spaces to each point are hypocycloids whose shape is defined according
to the velocity of information propagation across the data points. Using soft
manifolds for graph embedding, we can provide continuous spaces to pursue any
task in data analysis over complex datasets. Experimental results on
reconstruction tasks on synthetic and real datasets show how the proposed
approach enable more accurate and reliable characterization of graphs in
continuous spaces with respect to the state-of-the-art
On topological data analysis for structural dynamics: an introduction to persistent homology
Topological methods can provide a way of proposing new metrics and methods of
scrutinising data, that otherwise may be overlooked. In this work, a method of
quantifying the shape of data, via a topic called topological data analysis
will be introduced. The main tool within topological data analysis (TDA) is
persistent homology. Persistent homology is a method of quantifying the shape
of data over a range of length scales. The required background and a method of
computing persistent homology is briefly discussed in this work. Ideas from
topological data analysis are then used for nonlinear dynamics to analyse some
common attractors, by calculating their embedding dimension, and then to assess
their general topologies. A method will also be proposed, that uses topological
data analysis to determine the optimal delay for a time-delay embedding. TDA
will also be applied to a Z24 Bridge case study in structural health
monitoring, where it will be used to scrutinise different data partitions,
classified by the conditions at which the data were collected. A metric, from
topological data analysis, is used to compare data between the partitions. The
results presented demonstrate that the presence of damage alters the manifold
shape more significantly than the effects present from temperature
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
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