26,346 research outputs found
Functional quantization and metric entropy for Riemann-Liouville processes
We derive a high-resolution formula for the -quantization errors of
Riemann-Liouville processes and the sharp Kolmogorov entropy asymptotics for
related Sobolev balls. We describe a quantization procedure which leads to
asymptotically optimal functional quantizers. Regular variation of the
eigenvalues of the covariance operator plays a crucial role
High-resolution product quantization for Gaussian processes under sup-norm distortion
We derive high-resolution upper bounds for optimal product quantization of
pathwise contionuous Gaussian processes respective to the supremum norm on
[0,T]^d. Moreover, we describe a product quantization design which attains this
bound. This is achieved under very general assumptions on random series
expansions of the process. It turns out that product quantization is
asymptotically only slightly worse than optimal functional quantization. The
results are applied e.g. to fractional Brownian sheets and the
Ornstein-Uhlenbeck process.Comment: Version publi\'ee dans la revue Bernoulli, 13(3), 653-67
Asymptotically optimal quantization schemes for Gaussian processes
We describe quantization designs which lead to asymptotically and order
optimal functional quantizers. Regular variation of the eigenvalues of the
covariance operator plays a crucial role to achieve these rates. For the
development of a constructive quantization scheme we rely on the knowledge of
the eigenvectors of the covariance operator in order to transform the problem
into a finite dimensional quantization problem of normal distributions.
Furthermore we derive a high-resolution formula for the -quantization
errors of Riemann-Liouville processes.Comment: 2
Functional quantization
Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (p. 119-121).Data is rarely obtained for its own sake; oftentimes, it is a function of the data that we care about. Traditional data compression and quantization techniques, designed to recreate or approximate the data itself, gloss over this point. Are performance gains possible if source coding accounts for the user's function? How about when the encoders cannot themselves compute the function? We introduce the notion of functional quantization and use the tools of high-resolution analysis to get to the bottom of this question. Specifically, we consider real-valued raw data Xn/1 and scalar quantization of each component Xi of this data. First, under the constraints of fixed-rate quantization and variable-rate quantization, we obtain asymptotically optimal quantizer point densities and bit allocations. Introducing the notions of functional typicality and functional entropy, we then obtain asymptotically optimal block quantization schemes for each component. Next, we address the issue of non-monotonic functions by developing a model for high-resolution non-regular quantization. When these results are applied to several examples we observe striking improvements in performance.Finally, we answer three questions by means of the functional quantization framework: (1) Is there any benefit to allowing encoders to communicate with one another? (2) If transform coding is to be performed, how does a functional distortion measure influence the optimal transform? (3) What is the rate loss associated with a suboptimal quantizer design? In the process, we demonstrate how functional quantization can be a useful and intuitive alternative to more general information-theoretic techniques.by Vinith Misra.M.Eng
Distributed Functional Scalar Quantization Simplified
Distributed functional scalar quantization (DFSQ) theory provides optimality
conditions and predicts performance of data acquisition systems in which a
computation on acquired data is desired. We address two limitations of previous
works: prohibitively expensive decoder design and a restriction to sources with
bounded distributions. We rigorously show that a much simpler decoder has
equivalent asymptotic performance as the conditional expectation estimator
previously explored, thus reducing decoder design complexity. The simpler
decoder has the feature of decoupled communication and computation blocks.
Moreover, we extend the DFSQ framework with the simpler decoder to acquire
sources with infinite-support distributions such as Gaussian or exponential
distributions. Finally, through simulation results we demonstrate that
performance at moderate coding rates is well predicted by the asymptotic
analysis, and we give new insight on the rate of convergence
Multiresolution vector quantization
Multiresolution source codes are data compression algorithms yielding embedded source descriptions. The decoder of a multiresolution code can build a source reproduction by decoding the embedded bit stream in part or in whole. All decoding procedures start at the beginning of the binary source description and decode some fraction of that string. Decoding a small portion of the binary string gives a low-resolution reproduction; decoding more yields a higher resolution reproduction; and so on. Multiresolution vector quantizers are block multiresolution source codes. This paper introduces algorithms for designing fixed- and variable-rate multiresolution vector quantizers. Experiments on synthetic data demonstrate performance close to the theoretical performance limit. Experiments on natural images demonstrate performance improvements of up to 8 dB over tree-structured vector quantizers. Some of the lessons learned through multiresolution vector quantizer design lend insight into the design of more sophisticated multiresolution codes
Scale-Dependent Functions, Stochastic Quantization and Renormalization
We consider a possibility to unify the methods of regularization, such as the
renormalization group method, stochastic quantization etc., by the extension of
the standard field theory of the square-integrable functions to the theory of functions that depend on coordinate
and resolution . In the simplest case such field theory turns out to be a
theory of fields defined on the affine group ,
, which consists of dilations and translation of
Euclidean space. The fields are constructed using the
continuous wavelet transform. The parameters of the theory can explicitly
depend on the resolution . The proper choice of the scale dependence
makes such theory free of divergences by construction.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Loop Quantum Mechanics and the Fractal Structure of Quantum Spacetime
We discuss the relation between string quantization based on the Schild path
integral and the Nambu-Goto path integral. The equivalence between the two
approaches at the classical level is extended to the quantum level by a
saddle--point evaluation of the corresponding path integrals. A possible
relationship between M-Theory and the quantum mechanics of string loops is
pointed out. Then, within the framework of ``loop quantum mechanics'', we
confront the difficult question as to what exactly gives rise to the structure
of spacetime. We argue that the large scale properties of the string condensate
are responsible for the effective Riemannian geometry of classical spacetime.
On the other hand, near the Planck scale the condensate ``evaporates'', and
what is left behind is a ``vacuum'' characterized by an effective fractal
geometry.Comment: 19pag. ReVTeX, 1fig. Invited paper to appear in the special issue of
{\it Chaos, Solitons and Fractals} on ``Super strings, M,F,S,...Theory''
(M.S. El Naschie and C.Castro, ed
Low-cost, high-resolution, fault-robust position and speed estimation for PMSM drives operating in safety-critical systems
In this paper it is shown how to obtain a low-cost, high-resolution and fault-robust position sensing system for permanent magnet synchronous motor drives operating in safety-critical systems, by combining high-frequency signal injection with binary Hall-effect sensors. It is shown that the position error signal obtained via high-frequency signal injection can be merged easily into the quantization-harmonic-decoupling vector tracking observer used to process the Hall-effect sensor signals. The resulting algorithm provides accurate, high-resolution estimates of speed and position throughout the entire speed range; compared to state-of-the-art drives using Hall-effect sensors alone, the low speed performance is greatly improved in healthy conditions and also following position sensor faults. It is envisaged that such a sensing system can be successfully used in applications requiring IEC 61508 SIL 3 or ISO 26262 ASIL D compliance, due to its extremely high mean time to failure and to the very fast recovery of the drive following Hall-effect sensor faults at low speeds. Extensive simulation and experimental results are provided on a 3.7 kW permanent magnet drive
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