1,228 research outputs found
Wavelets and their use
This review paper is intended to give a useful guide for those who want to
apply discrete wavelets in their practice. The notion of wavelets and their use
in practical computing and various applications are briefly described, but
rigorous proofs of mathematical statements are omitted, and the reader is just
referred to corresponding literature. The multiresolution analysis and fast
wavelet transform became a standard procedure for dealing with discrete
wavelets. The proper choice of a wavelet and use of nonstandard matrix
multiplication are often crucial for achievement of a goal. Analysis of various
functions with the help of wavelets allows to reveal fractal structures,
singularities etc. Wavelet transform of operator expressions helps solve some
equations. In practical applications one deals often with the discretized
functions, and the problem of stability of wavelet transform and corresponding
numerical algorithms becomes important. After discussing all these topics we
turn to practical applications of the wavelet machinery. They are so numerous
that we have to limit ourselves by some examples only. The authors would be
grateful for any comments which improve this review paper and move us closer to
the goal proclaimed in the first phrase of the abstract.Comment: 63 pages with 22 ps-figures, to be published in Physics-Uspekh
Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples
In this paper, we consider the problem of recovering a compactly supported
multivariate function from a collection of pointwise samples of its Fourier
transform taken nonuniformly. We do this by using the concept of weighted
Fourier frames. A seminal result of Beurling shows that sample points give rise
to a classical Fourier frame provided they are relatively separated and of
sufficient density. However, this result does not allow for arbitrary
clustering of sample points, as is often the case in practice. Whilst keeping
the density condition sharp and dimension independent, our first result removes
the separation condition and shows that density alone suffices. However, this
result does not lead to estimates for the frame bounds. A known result of
Groechenig provides explicit estimates, but only subject to a density condition
that deteriorates linearly with dimension. In our second result we improve
these bounds by reducing the dimension dependence. In particular, we provide
explicit frame bounds which are dimensionless for functions having compact
support contained in a sphere. Next, we demonstrate how our two main results
give new insight into a reconstruction algorithm---based on the existing
generalized sampling framework---that allows for stable and quasi-optimal
reconstruction in any particular basis from a finite collection of samples.
Finally, we construct sufficiently dense sampling schemes that are often used
in practice---jittered, radial and spiral sampling schemes---and provide
several examples illustrating the effectiveness of our approach when tested on
these schemes
Spread spectrum-based video watermarking algorithms for copyright protection
Merged with duplicate record 10026.1/2263 on 14.03.2017 by CS (TIS)Digital technologies know an unprecedented expansion in the last years. The consumer can
now benefit from hardware and software which was considered state-of-the-art several years
ago. The advantages offered by the digital technologies are major but the same digital
technology opens the door for unlimited piracy. Copying an analogue VCR tape was certainly
possible and relatively easy, in spite of various forms of protection, but due to the analogue
environment, the subsequent copies had an inherent loss in quality. This was a natural way of
limiting the multiple copying of a video material. With digital technology, this barrier
disappears, being possible to make as many copies as desired, without any loss in quality
whatsoever. Digital watermarking is one of the best available tools for fighting this threat.
The aim of the present work was to develop a digital watermarking system compliant with the
recommendations drawn by the EBU, for video broadcast monitoring. Since the watermark
can be inserted in either spatial domain or transform domain, this aspect was investigated and
led to the conclusion that wavelet transform is one of the best solutions available. Since
watermarking is not an easy task, especially considering the robustness under various attacks
several techniques were employed in order to increase the capacity/robustness of the system:
spread-spectrum and modulation techniques to cast the watermark, powerful error correction
to protect the mark, human visual models to insert a robust mark and to ensure its invisibility.
The combination of these methods led to a major improvement, but yet the system wasn't
robust to several important geometrical attacks. In order to achieve this last milestone, the
system uses two distinct watermarks: a spatial domain reference watermark and the main
watermark embedded in the wavelet domain. By using this reference watermark and techniques
specific to image registration, the system is able to determine the parameters of the attack and
revert it. Once the attack was reverted, the main watermark is recovered. The final result is a
high capacity, blind DWr-based video watermarking system, robust to a wide range of attacks.BBC Research & Developmen
The Construction of Nonseparable Wavelet Bi-Frames and Associated Approximation Schemes
Wavelet analysis and its fast algorithms are widely used in many fields of applied mathematics such as in signal and image processing. In the present thesis, we circumvent the restrictions of orthogonal and biorthogonal wavelet bases by constructing wavelet frames. They still allow for a stable decomposition, and so-called wavelet bi-frames provide a series expansion very similar to those of pairs of biorthogonal wavelet bases. Contrary to biorthogonal bases, primal and dual wavelets are no longer supposed to satisfy any geometrical conditions, and the frame setting allows for redundancy. This provides more flexibility in their construction. Finally, we construct families of optimal wavelet bi-frames in arbitrary dimensions with arbitrarily high smoothness. Then we verify that the n-term approximation can be described by Besov spaces and we apply the theoretical findings to image denoising
A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity
The richness of natural images makes the quest for optimal representations in
image processing and computer vision challenging. The latter observation has
not prevented the design of image representations, which trade off between
efficiency and complexity, while achieving accurate rendering of smooth regions
as well as reproducing faithful contours and textures. The most recent ones,
proposed in the past decade, share an hybrid heritage highlighting the
multiscale and oriented nature of edges and patterns in images. This paper
presents a panorama of the aforementioned literature on decompositions in
multiscale, multi-orientation bases or dictionaries. They typically exhibit
redundancy to improve sparsity in the transformed domain and sometimes its
invariance with respect to simple geometric deformations (translation,
rotation). Oriented multiscale dictionaries extend traditional wavelet
processing and may offer rotation invariance. Highly redundant dictionaries
require specific algorithms to simplify the search for an efficient (sparse)
representation. We also discuss the extension of multiscale geometric
decompositions to non-Euclidean domains such as the sphere or arbitrary meshed
surfaces. The etymology of panorama suggests an overview, based on a choice of
partially overlapping "pictures". We hope that this paper will contribute to
the appreciation and apprehension of a stream of current research directions in
image understanding.Comment: 65 pages, 33 figures, 303 reference
Multidimensional Wavelets and Computer Vision
This report deals with the construction and the mathematical analysis of multidimensional nonseparable wavelets and their efficient application in computer vision. In the first part, the fundamental principles and ideas of multidimensional wavelet filter design such as the question for the existence of good scaling matrices and sensible design criteria are presented and extended in various directions. Afterwards, the analytical properties of these wavelets are investigated in some detail. It will turn out that they are especially well-suited to represent (discretized) data as well as large classes of operators in a sparse form - a property that directly yields efficient numerical algorithms. The final part of this work is dedicated to the application of the developed methods to the typical computer vision problems of nonlinear image regularization and the computation of optical flow in image sequences. It is demonstrated how the wavelet framework leads to stable and reliable results for these problems of generally ill-posed nature. Furthermore, all the algorithms are of order O(n) leading to fast processing
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