14,461 research outputs found
Filterbank optimization with convex objectives and the optimality of principal component forms
This paper proposes a general framework for the optimization of orthonormal filterbanks (FBs) for given input statistics. This includes as special cases, many previous results on FB optimization for compression. It also solves problems that have not been considered thus far. FB optimization for coding gain maximization (for compression applications) has been well studied before. The optimum FB has been known to satisfy the principal component property, i.e., it minimizes the mean-square error caused by reconstruction after dropping the P weakest (lowest variance) subbands for any P. We point out a much stronger connection between this property and the optimality of the FB. The main result is that a principal component FB (PCFB) is optimum whenever the minimization objective is a concave function of the subband variances produced by the FB. This result has its grounding in majorization and convex function theory and, in particular, explains the optimality of PCFBs for compression. We use the result to show various other optimality properties of PCFBs, especially for noise-suppression applications. Suppose the FB input is a signal corrupted by additive white noise, the desired output is the pure signal, and the subbands of the FB are processed to minimize the output noise. If each subband processor is a zeroth-order Wiener filter for its input, we can show that the expected mean square value of the output noise is a concave function of the subband signal variances. Hence, a PCFB is optimum in the sense of minimizing this mean square error. The above-mentioned concavity of the error and, hence, PCFB optimality, continues to hold even with certain other subband processors such as subband hard thresholds and constant multipliers, although these are not of serious practical interest. We prove that certain extensions of this PCFB optimality result to cases where the input noise is colored, and the FB optimization is over a larger class that includes biorthogonal FBs. We also show that PCFBs do not exist for the classes of DFT and cosine-modulated FBs
Mutual Information and Minimum Mean-square Error in Gaussian Channels
This paper deals with arbitrarily distributed finite-power input signals
observed through an additive Gaussian noise channel. It shows a new formula
that connects the input-output mutual information and the minimum mean-square
error (MMSE) achievable by optimal estimation of the input given the output.
That is, the derivative of the mutual information (nats) with respect to the
signal-to-noise ratio (SNR) is equal to half the MMSE, regardless of the input
statistics. This relationship holds for both scalar and vector signals, as well
as for discrete-time and continuous-time noncausal MMSE estimation. This
fundamental information-theoretic result has an unexpected consequence in
continuous-time nonlinear estimation: For any input signal with finite power,
the causal filtering MMSE achieved at SNR is equal to the average value of the
noncausal smoothing MMSE achieved with a channel whose signal-to-noise ratio is
chosen uniformly distributed between 0 and SNR
Bayesian reconstruction of the cosmological large-scale structure: methodology, inverse algorithms and numerical optimization
We address the inverse problem of cosmic large-scale structure reconstruction
from a Bayesian perspective. For a linear data model, a number of known and
novel reconstruction schemes, which differ in terms of the underlying signal
prior, data likelihood, and numerical inverse extra-regularization schemes are
derived and classified. The Bayesian methodology presented in this paper tries
to unify and extend the following methods: Wiener-filtering, Tikhonov
regularization, Ridge regression, Maximum Entropy, and inverse regularization
techniques. The inverse techniques considered here are the asymptotic
regularization, the Jacobi, Steepest Descent, Newton-Raphson,
Landweber-Fridman, and both linear and non-linear Krylov methods based on
Fletcher-Reeves, Polak-Ribiere, and Hestenes-Stiefel Conjugate Gradients. The
structures of the up-to-date highest-performing algorithms are presented, based
on an operator scheme, which permits one to exploit the power of fast Fourier
transforms. Using such an implementation of the generalized Wiener-filter in
the novel ARGO-software package, the different numerical schemes are
benchmarked with 1-, 2-, and 3-dimensional problems including structured white
and Poissonian noise, data windowing and blurring effects. A novel numerical
Krylov scheme is shown to be superior in terms of performance and fidelity.
These fast inverse methods ultimately will enable the application of sampling
techniques to explore complex joint posterior distributions. We outline how the
space of the dark-matter density field, the peculiar velocity field, and the
power spectrum can jointly be investigated by a Gibbs-sampling process. Such a
method can be applied for the redshift distortions correction of the observed
galaxies and for time-reversal reconstructions of the initial density field.Comment: 40 pages, 11 figure
Foreground separation methods for satellite observations of the cosmic microwave background
A maximum entropy method (MEM) is presented for separating the emission due
to different foreground components from simulated satellite observations of the
cosmic microwave background radiation (CMBR). In particular, the method is
applied to simulated observations by the proposed Planck Surveyor satellite.
The simulations, performed by Bouchet and Gispert (1998), include emission from
the CMBR, the kinetic and thermal Sunyaev-Zel'dovich (SZ) effects from galaxy
clusters, as well as Galactic dust, free-free and synchrotron emission. We find
that the MEM technique performs well and produces faithful reconstructions of
the main input components. The method is also compared with traditional Wiener
filtering and is shown to produce consistently better results, particularly in
the recovery of the thermal SZ effect.Comment: 31 pages, 19 figures (bitmapped), accpeted for publication in MNRA
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