8,362 research outputs found
Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs
Laplacian mixture models identify overlapping regions of influence in
unlabeled graph and network data in a scalable and computationally efficient
way, yielding useful low-dimensional representations. By combining Laplacian
eigenspace and finite mixture modeling methods, they provide probabilistic or
fuzzy dimensionality reductions or domain decompositions for a variety of input
data types, including mixture distributions, feature vectors, and graphs or
networks. Provable optimal recovery using the algorithm is analytically shown
for a nontrivial class of cluster graphs. Heuristic approximations for scalable
high-performance implementations are described and empirically tested.
Connections to PageRank and community detection in network analysis demonstrate
the wide applicability of this approach. The origins of fuzzy spectral methods,
beginning with generalized heat or diffusion equations in physics, are reviewed
and summarized. Comparisons to other dimensionality reduction and clustering
methods for challenging unsupervised machine learning problems are also
discussed.Comment: 13 figures, 35 reference
On the optimality of shape and data representation in the spectral domain
A proof of the optimality of the eigenfunctions of the Laplace-Beltrami
operator (LBO) in representing smooth functions on surfaces is provided and
adapted to the field of applied shape and data analysis. It is based on the
Courant-Fischer min-max principle adapted to our case. % The theorem we present
supports the new trend in geometry processing of treating geometric structures
by using their projection onto the leading eigenfunctions of the decomposition
of the LBO. Utilisation of this result can be used for constructing numerically
efficient algorithms to process shapes in their spectrum. We review a couple of
applications as possible practical usage cases of the proposed optimality
criteria. % We refer to a scale invariant metric, which is also invariant to
bending of the manifold. This novel pseudo-metric allows constructing an LBO by
which a scale invariant eigenspace on the surface is defined. We demonstrate
the efficiency of an intermediate metric, defined as an interpolation between
the scale invariant and the regular one, in representing geometric structures
while capturing both coarse and fine details. Next, we review a numerical
acceleration technique for classical scaling, a member of a family of
flattening methods known as multidimensional scaling (MDS). There, the
optimality is exploited to efficiently approximate all geodesic distances
between pairs of points on a given surface, and thereby match and compare
between almost isometric surfaces. Finally, we revisit the classical principal
component analysis (PCA) definition by coupling its variational form with a
Dirichlet energy on the data manifold. By pairing the PCA with the LBO we can
handle cases that go beyond the scope defined by the observation set that is
handled by regular PCA
On the Two-View Geometry of Unsynchronized Cameras
We present new methods for simultaneously estimating camera geometry and time
shift from video sequences from multiple unsynchronized cameras. Algorithms for
simultaneous computation of a fundamental matrix or a homography with unknown
time shift between images are developed. Our methods use minimal correspondence
sets (eight for fundamental matrix and four and a half for homography) and
therefore are suitable for robust estimation using RANSAC. Furthermore, we
present an iterative algorithm that extends the applicability on sequences
which are significantly unsynchronized, finding the correct time shift up to
several seconds. We evaluated the methods on synthetic and wide range of real
world datasets and the results show a broad applicability to the problem of
camera synchronization.Comment: 12 pages, 9 figures, Computer Vision and Pattern Recognition (CVPR)
201
Revisiting Complex Moments For 2D Shape Representation and Image Normalization
When comparing 2D shapes, a key issue is their normalization. Translation and
scale are easily taken care of by removing the mean and normalizing the energy.
However, defining and computing the orientation of a 2D shape is not so simple.
In fact, although for elongated shapes the principal axis can be used to define
one of two possible orientations, there is no such tool for general shapes. As
we show in the paper, previous approaches fail to compute the orientation of
even noiseless observations of simple shapes. We address this problem. In the
paper, we show how to uniquely define the orientation of an arbitrary 2D shape,
in terms of what we call its Principal Moments. We show that a small subset of
these moments suffice to represent the underlying 2D shape and propose a new
method to efficiently compute the shape orientation: Principal Moment Analysis.
Finally, we discuss how this method can further be applied to normalize
grey-level images. Besides the theoretical proof of correctness, we describe
experiments demonstrating robustness to noise and illustrating the method with
real images.Comment: 69 pages, 20 figure
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