6,634 research outputs found
Sparse Coding on Symmetric Positive Definite Manifolds using Bregman Divergences
This paper introduces sparse coding and dictionary learning for Symmetric
Positive Definite (SPD) matrices, which are often used in machine learning,
computer vision and related areas. Unlike traditional sparse coding schemes
that work in vector spaces, in this paper we discuss how SPD matrices can be
described by sparse combination of dictionary atoms, where the atoms are also
SPD matrices. We propose to seek sparse coding by embedding the space of SPD
matrices into Hilbert spaces through two types of Bregman matrix divergences.
This not only leads to an efficient way of performing sparse coding, but also
an online and iterative scheme for dictionary learning. We apply the proposed
methods to several computer vision tasks where images are represented by region
covariance matrices. Our proposed algorithms outperform state-of-the-art
methods on a wide range of classification tasks, including face recognition,
action recognition, material classification and texture categorization
Sobolev gradients and image interpolation
We present here a new image inpainting algorithm based on the Sobolev
gradient method in conjunction with the Navier-Stokes model. The original model
of Bertalmio et al is reformulated as a variational principle based on the
minimization of a well chosen functional by a steepest descent method. This
provides an alternative of the direct solving of a high-order partial
differential equation and, consequently, allows to avoid complicated numerical
schemes (min-mod limiters or anisotropic diffusion). We theoretically analyze
our algorithm in an infinite dimensional setting using an evolution equation
and obtain global existence and uniqueness results as well as the existence of
an -limit. Using a finite difference implementation, we demonstrate
using various examples that the Sobolev gradient flow, due to its smoothing and
preconditioning properties, is an effective tool for use in the image
inpainting problem
Indirect Image Registration with Large Diffeomorphic Deformations
The paper adapts the large deformation diffeomorphic metric mapping framework
for image registration to the indirect setting where a template is registered
against a target that is given through indirect noisy observations. The
registration uses diffeomorphisms that transform the template through a (group)
action. These diffeomorphisms are generated by solving a flow equation that is
defined by a velocity field with certain regularity. The theoretical analysis
includes a proof that indirect image registration has solutions (existence)
that are stable and that converge as the data error tends so zero, so it
becomes a well-defined regularization method. The paper concludes with examples
of indirect image registration in 2D tomography with very sparse and/or highly
noisy data.Comment: 43 pages, 4 figures, 1 table; revise
Coupling of quantum angular momenta: an insight into analogic/discrete and local/global models of computation
In the past few years there has been a tumultuous activity aimed at
introducing novel conceptual schemes for quantum computing. The approach
proposed in (Marzuoli A and Rasetti M 2002, 2005a) relies on the (re)coupling
theory of SU(2) angular momenta and can be viewed as a generalization to
arbitrary values of the spin variables of the usual quantum-circuit model based
on `qubits' and Boolean gates. Computational states belong to
finite-dimensional Hilbert spaces labelled by both discrete and continuous
parameters, and unitary gates may depend on quantum numbers ranging over finite
sets of values as well as continuous (angular) variables. Such a framework is
an ideal playground to discuss discrete (digital) and analogic computational
processes, together with their relationships occuring when a consistent
semiclassical limit takes place on discrete quantum gates. When working with
purely discrete unitary gates, the simulator is naturally modelled as families
of quantum finite states--machines which in turn represent discrete versions of
topological quantum computation models. We argue that our model embodies a sort
of unifying paradigm for computing inspired by Nature and, even more
ambitiously, a universal setting in which suitably encoded quantum symbolic
manipulations of combinatorial, topological and algebraic problems might find
their `natural' computational reference model.Comment: 17 pages, 1 figure; Workshop `Natural processes and models of
computation' Bologna (Italy) June 16-18 2005; to appear in Natural Computin
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