61,120 research outputs found
Spectral Generalized Multi-Dimensional Scaling
Multidimensional scaling (MDS) is a family of methods that embed a given set
of points into a simple, usually flat, domain. The points are assumed to be
sampled from some metric space, and the mapping attempts to preserve the
distances between each pair of points in the set. Distances in the target space
can be computed analytically in this setting. Generalized MDS is an extension
that allows mapping one metric space into another, that is, multidimensional
scaling into target spaces in which distances are evaluated numerically rather
than analytically. Here, we propose an efficient approach for computing such
mappings between surfaces based on their natural spectral decomposition, where
the surfaces are treated as sampled metric-spaces. The resulting spectral-GMDS
procedure enables efficient embedding by implicitly incorporating smoothness of
the mapping into the problem, thereby substantially reducing the complexity
involved in its solution while practically overcoming its non-convex nature.
The method is compared to existing techniques that compute dense correspondence
between shapes. Numerical experiments of the proposed method demonstrate its
efficiency and accuracy compared to state-of-the-art approaches
On Nonrigid Shape Similarity and Correspondence
An important operation in geometry processing is finding the correspondences
between pairs of shapes. The Gromov-Hausdorff distance, a measure of
dissimilarity between metric spaces, has been found to be highly useful for
nonrigid shape comparison. Here, we explore the applicability of related shape
similarity measures to the problem of shape correspondence, adopting spectral
type distances. We propose to evaluate the spectral kernel distance, the
spectral embedding distance and the novel spectral quasi-conformal distance,
comparing the manifolds from different viewpoints. By matching the shapes in
the spectral domain, important attributes of surface structure are being
aligned. For the purpose of testing our ideas, we introduce a fully automatic
framework for finding intrinsic correspondence between two shapes. The proposed
method achieves state-of-the-art results on the Princeton isometric shape
matching protocol applied, as usual, to the TOSCA and SCAPE benchmarks
Similarity Search Over Graphs Using Localized Spectral Analysis
This paper provides a new similarity detection algorithm. Given an input set
of multi-dimensional data points, where each data point is assumed to be
multi-dimensional, and an additional reference data point for similarity
finding, the algorithm uses kernel method that embeds the data points into a
low dimensional manifold. Unlike other kernel methods, which consider the
entire data for the embedding, our method selects a specific set of kernel
eigenvectors. The eigenvectors are chosen to separate between the data points
and the reference data point so that similar data points can be easily
identified as being distinct from most of the members in the dataset.Comment: Published in SampTA 201
Robust Temporally Coherent Laplacian Protrusion Segmentation of 3D Articulated Bodies
In motion analysis and understanding it is important to be able to fit a
suitable model or structure to the temporal series of observed data, in order
to describe motion patterns in a compact way, and to discriminate between them.
In an unsupervised context, i.e., no prior model of the moving object(s) is
available, such a structure has to be learned from the data in a bottom-up
fashion. In recent times, volumetric approaches in which the motion is captured
from a number of cameras and a voxel-set representation of the body is built
from the camera views, have gained ground due to attractive features such as
inherent view-invariance and robustness to occlusions. Automatic, unsupervised
segmentation of moving bodies along entire sequences, in a temporally-coherent
and robust way, has the potential to provide a means of constructing a
bottom-up model of the moving body, and track motion cues that may be later
exploited for motion classification. Spectral methods such as locally linear
embedding (LLE) can be useful in this context, as they preserve "protrusions",
i.e., high-curvature regions of the 3D volume, of articulated shapes, while
improving their separation in a lower dimensional space, making them in this
way easier to cluster. In this paper we therefore propose a spectral approach
to unsupervised and temporally-coherent body-protrusion segmentation along time
sequences. Volumetric shapes are clustered in an embedding space, clusters are
propagated in time to ensure coherence, and merged or split to accommodate
changes in the body's topology. Experiments on both synthetic and real
sequences of dense voxel-set data are shown. This supports the ability of the
proposed method to cluster body-parts consistently over time in a totally
unsupervised fashion, its robustness to sampling density and shape quality, and
its potential for bottom-up model constructionComment: 31 pages, 26 figure
Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber
We study theoretically the generation of photon pairs by spontaneous
four-wave mixing (SFWM) in photonic crystal optical fiber. We show that it is
possible to engineer two-photon states with specific spectral correlation
(``entanglement'') properties suitable for quantum information processing
applications. We focus on the case exhibiting no spectral correlations in the
two-photon component of the state, which we call factorability, and which
allows heralding of single-photon pure-state wave packets without the need for
spectral post filtering. We show that spontaneous four wave mixing exhibits a
remarkable flexibility, permitting a wider class of two-photon states,
including ultra-broadband, highly-anticorrelated states.Comment: 17 pages, 7 figures, submitte
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
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