1,013 research outputs found
L1 Control Theoretic Smoothing Splines
In this paper, we propose control theoretic smoothing splines with L1
optimality for reducing the number of parameters that describes the fitted
curve as well as removing outlier data. A control theoretic spline is a
smoothing spline that is generated as an output of a given linear dynamical
system. Conventional design requires exactly the same number of base functions
as given data, and the result is not robust against outliers. To solve these
problems, we propose to use L1 optimality, that is, we use the L1 norm for the
regularization term and/or the empirical risk term. The optimization is
described by a convex optimization, which can be efficiently solved via a
numerical optimization software. A numerical example shows the effectiveness of
the proposed method.Comment: Accepted for publication in IEEE Signal Processing Letters. 4 pages
(twocolumn), 5 figure
Sampled-data Lâ smoothing: fixed-size ARE solution with free hold function
The problem of estimating an analog signal from its noisy sampled measurements is studied in the Lâ (induced L2-norm) framework. The main emphasis is placed on relaxing causality requirements. Namely, it is assumed that l future measurements are available to the estimator, which corresponds to the fixed-lag smoothing formulation. A closed-form solution to the problem is derived. The solution has the complexity of O(l) and is based on two discrete algebraic Riccati equations, whose size does not depend on the smoothing lag l
Sampled-data L-infinity smoothing: fixed-size ARE solution with free hold function
The problem of estimating an analog signal from its noisy sampled measurements is studied in the L-infinity (induced L2-norm) framework. The main emphasis is placed on relaxing causality requirements. Namely, it is assumed that l future measurements are available to the estimator, which corresponds to the fixed-lag smoothing formulation. A closed-form solution to the problem is derived. The solution has the complexity of O(l) and is based on two discrete algebraic Riccati equations, whose size does not depend on the smoothing lag l
Optimising Spatial and Tonal Data for PDE-based Inpainting
Some recent methods for lossy signal and image compression store only a few
selected pixels and fill in the missing structures by inpainting with a partial
differential equation (PDE). Suitable operators include the Laplacian, the
biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The
quality of such approaches depends substantially on the selection of the data
that is kept. Optimising this data in the domain and codomain gives rise to
challenging mathematical problems that shall be addressed in our work.
In the 1D case, we prove results that provide insights into the difficulty of
this problem, and we give evidence that a splitting into spatial and tonal
(i.e. function value) optimisation does hardly deteriorate the results. In the
2D setting, we present generic algorithms that achieve a high reconstruction
quality even if the specified data is very sparse. To optimise the spatial
data, we use a probabilistic sparsification, followed by a nonlocal pixel
exchange that avoids getting trapped in bad local optima. After this spatial
optimisation we perform a tonal optimisation that modifies the function values
in order to reduce the global reconstruction error. For homogeneous diffusion
inpainting, this comes down to a least squares problem for which we prove that
it has a unique solution. We demonstrate that it can be found efficiently with
a gradient descent approach that is accelerated with fast explicit diffusion
(FED) cycles. Our framework allows to specify the desired density of the
inpainting mask a priori. Moreover, is more generic than other data
optimisation approaches for the sparse inpainting problem, since it can also be
extended to nonlinear inpainting operators such as EED. This is exploited to
achieve reconstructions with state-of-the-art quality.
We also give an extensive literature survey on PDE-based image compression
methods
Option Pricing in an Imperfect World
In a model with no given probability measure, we consider asset pricing in
the presence of frictions and other imperfections and characterize the property
of coherent pricing, a notion related to (but much weaker than) the no
arbitrage property. We show that prices are coherent if and only if the set of
pricing measures is non empty, i.e. if pricing by expectation is possible. We
then obtain a decomposition of coherent prices highlighting the role of
bubbles. eventually we show that under very weak conditions the coherent
pricing of options allows for a very clear representation from which it is
possible, as in the original work of Breeden and Litzenberger, to extract the
implied probability. Eventually we test this conclusion empirically via a new
non parametric approach.Comment: The paper has been withdrawn because in the newer version it was
split into two different papers, each of which have been uploaded into Arxi
Past, Present, and Future of Simultaneous Localization And Mapping: Towards the Robust-Perception Age
Simultaneous Localization and Mapping (SLAM)consists in the concurrent
construction of a model of the environment (the map), and the estimation of the
state of the robot moving within it. The SLAM community has made astonishing
progress over the last 30 years, enabling large-scale real-world applications,
and witnessing a steady transition of this technology to industry. We survey
the current state of SLAM. We start by presenting what is now the de-facto
standard formulation for SLAM. We then review related work, covering a broad
set of topics including robustness and scalability in long-term mapping, metric
and semantic representations for mapping, theoretical performance guarantees,
active SLAM and exploration, and other new frontiers. This paper simultaneously
serves as a position paper and tutorial to those who are users of SLAM. By
looking at the published research with a critical eye, we delineate open
challenges and new research issues, that still deserve careful scientific
investigation. The paper also contains the authors' take on two questions that
often animate discussions during robotics conferences: Do robots need SLAM? and
Is SLAM solved
- âŠ