5,218 research outputs found
A recursive scheme for computing autocorrelation functions of decimated complex wavelet subbands
This paper deals with the problem of the exact computation of the autocorrelation function of a real or complex discrete wavelet subband of a signal, when the autocorrelation function (or Power Spectral Density, PSD) of the signal in the time domain (or spatial domain) is either known or estimated using a separate technique. The solution to this problem allows us to couple time domain noise estimation techniques to wavelet domain denoising algorithms, which is crucial for the development of blind wavelet-based denoising techniques. Specifically, we investigate the Dual-Tree complex wavelet transform (DT-CWT), which has a good directional selectivity in 2-D and 3-D, is approximately shift-invariant, and yields better denoising results than a discrete wavelet transform (DWT). The proposed scheme gives an analytical relationship between the PSD of the input signal/image and the PSD of each individual real/complex wavelet subband which is very useful for future developments. We also show that a more general technique, that relies on Monte-Carlo simulations, requires a large number of input samples for a reliable estimate, while the proposed technique does not suffer from this problem
On The Continuous Steering of the Scale of Tight Wavelet Frames
In analogy with steerable wavelets, we present a general construction of
adaptable tight wavelet frames, with an emphasis on scaling operations. In
particular, the derived wavelets can be "dilated" by a procedure comparable to
the operation of steering steerable wavelets. The fundamental aspects of the
construction are the same: an admissible collection of Fourier multipliers is
used to extend a tight wavelet frame, and the "scale" of the wavelets is
adapted by scaling the multipliers. As an application, the proposed wavelets
can be used to improve the frequency localization. Importantly, the localized
frequency bands specified by this construction can be scaled efficiently using
matrix multiplication
Wavelet/shearlet hybridized neural networks for biomedical image restoration
Recently, new programming paradigms have emerged that combine parallelism and numerical computations with algorithmic differentiation. This approach allows for the hybridization of neural network techniques for inverse imaging problems with more traditional methods such as wavelet-based sparsity modelling techniques. The benefits are twofold: on the one hand traditional methods with well-known properties can be integrated in neural networks, either as separate layers or tightly integrated in the network, on the other hand, parameters in traditional methods can be trained end-to-end from datasets in a neural network "fashion" (e.g., using Adagrad or Adam optimizers). In this paper, we explore these hybrid neural networks in the context of shearlet-based regularization for the purpose of biomedical image restoration. Due to the reduced number of parameters, this approach seems a promising strategy especially when dealing with small training data sets
A generalized, parametric PR-QMF/wavelet transform design approach for multiresolution signal decomposition
This dissertation aims to emphasize the interrelations and the linkages of the theories of discrete-time filter banks and wavelet transforms. It is shown that the Binomial-QMF banks are identical to the interscale coefficients or filters of the compactly supported orthonormal wavelet transform bases proposed by Daubechies.
A generalized, parametric, smooth 2-band PR-QMF design approach based on Bernstein polynomial approximation is developed. It is found that the most regular compact support orthonormal wavelet filters, coiflet filters are only the special cases of the proposed filter bank design technique.
A new objective performance measure called Non-aliasing Energy Ratio(NER) is developed. Its merits are proven with the comparative performance studies of the well known orthonormal signal decomposition techniques.
This dissertation also addresses the optimal 2-band PR-QMF design problem. The variables of practical significance in image processing and coding are included in the optimization problem. The upper performance bounds of 2-band PR-QMF and their corresponding filter coefficients are derived.
It is objectively shown that there are superior filter bank solutions available over the standard block transform, DCT. It is expected that the theoretical contributions of this dissertation will find its applications particularly in Visual Signal Processing and Coding
Spherical 3D Isotropic Wavelets
Future cosmological surveys will provide 3D large scale structure maps with
large sky coverage, for which a 3D Spherical Fourier-Bessel (SFB) analysis in
spherical coordinates is natural. Wavelets are particularly well-suited to the
analysis and denoising of cosmological data, but a spherical 3D isotropic
wavelet transform does not currently exist to analyse spherical 3D data. The
aim of this paper is to present a new formalism for a spherical 3D isotropic
wavelet, i.e. one based on the SFB decomposition of a 3D field and accompany
the formalism with a public code to perform wavelet transforms. We describe a
new 3D isotropic spherical wavelet decomposition based on the undecimated
wavelet transform (UWT) described in Starck et al. 2006. We also present a new
fast Discrete Spherical Fourier-Bessel Transform (DSFBT) based on both a
discrete Bessel Transform and the HEALPIX angular pixelisation scheme. We test
the 3D wavelet transform and as a toy-application, apply a denoising algorithm
in wavelet space to the Virgo large box cosmological simulations and find we
can successfully remove noise without much loss to the large scale structure.
We have described a new spherical 3D isotropic wavelet transform, ideally
suited to analyse and denoise future 3D spherical cosmological surveys, which
uses a novel Discrete Spherical Fourier-Bessel Transform. We illustrate its
potential use for denoising using a toy model. All the algorithms presented in
this paper are available for download as a public code called MRS3D at
http://jstarck.free.fr/mrs3d.htmlComment: 9 pages + appendices. Public code can be downloaded at
http://jstarck.free.fr/mrs3d.html Corrected typos and updated references.
Accepted for publication in Astronomy and Astrophysic
Spherical 3D Isotropic Wavelets
Future cosmological surveys will provide 3D large scale structure maps with
large sky coverage, for which a 3D Spherical Fourier-Bessel (SFB) analysis in
spherical coordinates is natural. Wavelets are particularly well-suited to the
analysis and denoising of cosmological data, but a spherical 3D isotropic
wavelet transform does not currently exist to analyse spherical 3D data. The
aim of this paper is to present a new formalism for a spherical 3D isotropic
wavelet, i.e. one based on the SFB decomposition of a 3D field and accompany
the formalism with a public code to perform wavelet transforms. We describe a
new 3D isotropic spherical wavelet decomposition based on the undecimated
wavelet transform (UWT) described in Starck et al. 2006. We also present a new
fast Discrete Spherical Fourier-Bessel Transform (DSFBT) based on both a
discrete Bessel Transform and the HEALPIX angular pixelisation scheme. We test
the 3D wavelet transform and as a toy-application, apply a denoising algorithm
in wavelet space to the Virgo large box cosmological simulations and find we
can successfully remove noise without much loss to the large scale structure.
We have described a new spherical 3D isotropic wavelet transform, ideally
suited to analyse and denoise future 3D spherical cosmological surveys, which
uses a novel Discrete Spherical Fourier-Bessel Transform. We illustrate its
potential use for denoising using a toy model. All the algorithms presented in
this paper are available for download as a public code called MRS3D at
http://jstarck.free.fr/mrs3d.htmlComment: 9 pages + appendices. Public code can be downloaded at
http://jstarck.free.fr/mrs3d.html Corrected typos and updated references.
Accepted for publication in Astronomy and Astrophysic
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