987 research outputs found
Parametric Regression on the Grassmannian
We address the problem of fitting parametric curves on the Grassmann manifold
for the purpose of intrinsic parametric regression. As customary in the
literature, we start from the energy minimization formulation of linear
least-squares in Euclidean spaces and generalize this concept to general
nonflat Riemannian manifolds, following an optimal-control point of view. We
then specialize this idea to the Grassmann manifold and demonstrate that it
yields a simple, extensible and easy-to-implement solution to the parametric
regression problem. In fact, it allows us to extend the basic geodesic model to
(1) a time-warped variant and (2) cubic splines. We demonstrate the utility of
the proposed solution on different vision problems, such as shape regression as
a function of age, traffic-speed estimation and crowd-counting from
surveillance video clips. Most notably, these problems can be conveniently
solved within the same framework without any specifically-tailored steps along
the processing pipeline.Comment: 14 pages, 11 figure
A Geometrical Study of Matching Pursuit Parametrization
This paper studies the effect of discretizing the parametrization of a
dictionary used for Matching Pursuit decompositions of signals. Our approach
relies on viewing the continuously parametrized dictionary as an embedded
manifold in the signal space on which the tools of differential (Riemannian)
geometry can be applied. The main contribution of this paper is twofold. First,
we prove that if a discrete dictionary reaches a minimal density criterion,
then the corresponding discrete MP (dMP) is equivalent in terms of convergence
to a weakened hypothetical continuous MP. Interestingly, the corresponding
weakness factor depends on a density measure of the discrete dictionary.
Second, we show that the insertion of a simple geometric gradient ascent
optimization on the atom dMP selection maintains the previous comparison but
with a weakness factor at least two times closer to unity than without
optimization. Finally, we present numerical experiments confirming our
theoretical predictions for decomposition of signals and images on regular
discretizations of dictionary parametrizations.Comment: 26 pages, 8 figure
Efficient probabilistic and geometric anatomical mapping using particle mesh approximation on GPUs
pre-printDeformable image registration in the presence of considerable contrast differences and large size and shape changes presents significant research challenges. First, it requires a robust registration framework that does not depend on intensity measurements and can handle large nonlinear shape variations. Second, it involves the expensive computation of nonlinear deformations with high degrees of freedom. Often it takes a significant amount of computation time and thus becomes infeasible for practical purposes. In this paper, we present a solution based on two key ideas: a new registration method that generates a mapping between anatomies represented as a multicompartment model of class posterior images and geometries and an implementation of the algorithm using particle mesh approximation on Graphical Processing Units (GPUs) to fulfill the computational requirements. We show results on the registrations of neonatal to 2-year old infant MRIs. Quantitative validation demonstrates that our proposed method generates registrations that better maintain the consistency of anatomical structures over time and provides transformations that better preserve structures undergoing large deformations than transformations obtained by standard intensity-only registration. We also achieve the speedup of three orders of magnitudes compared to a CPU reference implementation, making it possible to use the technique in time-critical applications
Shape description and matching using integral invariants on eccentricity transformed images
Matching occluded and noisy shapes is a problem frequently encountered in medical image analysis and more generally in computer vision. To keep track of changes inside the breast, for example, it is important for a computer aided detection system to establish correspondences between regions of interest. Shape transformations, computed both with integral invariants (II) and with geodesic distance, yield signatures that are invariant to isometric deformations, such as bending and articulations. Integral invariants describe the boundaries of planar shapes. However, they provide no information about where a particular feature lies on the boundary with regard to the overall shape structure. Conversely, eccentricity transforms (Ecc) can match shapes by signatures of geodesic distance histograms based on information from inside the shape; but they ignore the boundary information. We describe a method that combines the boundary signature of a shape obtained from II and structural information from the Ecc to yield results that improve on them separately
Point-wise Map Recovery and Refinement from Functional Correspondence
Since their introduction in the shape analysis community, functional maps
have met with considerable success due to their ability to compactly represent
dense correspondences between deformable shapes, with applications ranging from
shape matching and image segmentation, to exploration of large shape
collections. Despite the numerous advantages of such representation, however,
the problem of converting a given functional map back to a point-to-point map
has received a surprisingly limited interest. In this paper we analyze the
general problem of point-wise map recovery from arbitrary functional maps. In
doing so, we rule out many of the assumptions required by the currently
established approach -- most notably, the limiting requirement of the input
shapes being nearly-isometric. We devise an efficient recovery process based on
a simple probabilistic model. Experiments confirm that this approach achieves
remarkable accuracy improvements in very challenging cases
A Revisit of Shape Editing Techniques: from the Geometric to the Neural Viewpoint
3D shape editing is widely used in a range of applications such as movie
production, computer games and computer aided design. It is also a popular
research topic in computer graphics and computer vision. In past decades,
researchers have developed a series of editing methods to make the editing
process faster, more robust, and more reliable. Traditionally, the deformed
shape is determined by the optimal transformation and weights for an energy
term. With increasing availability of 3D shapes on the Internet, data-driven
methods were proposed to improve the editing results. More recently as the deep
neural networks became popular, many deep learning based editing methods have
been developed in this field, which is naturally data-driven. We mainly survey
recent research works from the geometric viewpoint to those emerging neural
deformation techniques and categorize them into organic shape editing methods
and man-made model editing methods. Both traditional methods and recent neural
network based methods are reviewed
Efficient Probabilistic and Geometric Anatomical Mapping Using Particle Mesh Approximation on GPUs
Deformable image registration in the presence of considerable contrast differences and
large size and shape changes presents significant research challenges. First, it requires a
robust registration framework that does not depend on intensity measurements and can
handle large nonlinear shape variations. Second, it involves the expensive computation of
nonlinear deformations with high degrees of freedom. Often it takes a significant amount
of computation time and thus becomes infeasible for practical purposes. In this paper, we
present a solution based on two key ideas: a new registration method that generates a mapping
between anatomies represented as a multicompartment model of class posterior images
and geometries and an implementation of the algorithm using particle mesh approximation
on Graphical Processing Units (GPUs) to fulfill the computational requirements. We show
results on the registrations of neonatal to 2-year old infant MRIs. Quantitative
validation demonstrates that our proposed method generates registrations that better maintain
the consistency of anatomical structures over time and provides transformations that
better preserve structures undergoing large deformations than transformations obtained by
standard intensity-only registration. We also achieve the speedup of three orders of magnitudes
compared to a CPU reference implementation, making it possible to use the technique
in time-critical applications
Local Multidimensional Scaling for Nonlinear Dimension Reduction, Graph Drawing and Proximity Analysis
In the past decade there has been a resurgence of interest in nonlinear dimension reduction. Among new proposals are “Local Linear Embedding,” “Isomap,” and Kernel Principal Components Analysis which all construct global low-dimensional embeddings from local affine or metric information. We introduce a competing method called “Local Multidimensional Scaling” (LMDS). Like LLE, Isomap, and KPCA, LMDS constructs its global embedding from local information, but it uses instead a combination of MDS and “force-directed” graph drawing. We apply the force paradigm to create localized versions of MDS stress functions with a tuning parameter to adjust the strength of nonlocal repulsive forces.
We solve the problem of tuning parameter selection with a meta-criterion that measures how well the sets of K-nearest neighbors agree between the data and the embedding. Tuned LMDS seems to be able to outperform MDS, PCA, LLE, Isomap, and KPCA, as illustrated with two well-known image datasets. The meta-criterion can also be used in a pointwise version as a diagnostic tool for measuring the local adequacy of embeddings and thereby detect local problems in dimension reductions
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