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## ‣ Denosing Using Wavelets and Projections onto the L1-Ball

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/06/2014
Português

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Both wavelet denoising and denosing methods using the concept of sparsity are
based on soft-thresholding. In sparsity based denoising methods, it is assumed
that the original signal is sparse in some transform domains such as the
wavelet domain and the wavelet subsignals of the noisy signal are projected
onto L1-balls to reduce noise. In this lecture note, it is shown that the size
of the L1-ball or equivalently the soft threshold value can be determined using
linear algebra. The key step is an orthogonal projection onto the epigraph set
of the L1-norm cost function.; Comment: Submitted to Signal Processing Magazine

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## ‣ Bivariate Daubechies Scaling Functions (Wavelets)

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/03/2001
Português

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Using the Daubechies conditions of compact support, orthogonal, and
regularity, we were able to derive bivariate scaling functions with which to
reproduce linear functions (planes). We describe how to create all possible
masks of refinement coefficients that satisfy those conditions.; Comment: 10 pages

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## ‣ Geophysical modelling with Colombeau functions: Microlocal properties and Zygmund regularity

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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In global seismology Earth's properties of fractal nature occur.
Zygmund classes appear as the most appropriate and systematic way to measure
this local fractality. For the purpose of seismic wave propagation, we model
the Earth's properties as Colombeau generalized functions. In one spatial
dimension, we have a precise characterization of Zygmund regularity in
Colombeau algebras. This is made possible via a relation between mollifiers and
wavelets.; Comment: corrected Definition 13; updated Remark 5

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## ‣ Linear independence of compactly supported separable shearlet systems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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This paper examines linear independence of shearlet systems. This property
has already been studied for wavelets and other systems such as, for instance,
for Gabor systems. In fact, for Gabor systems this problem is commonly known as
the HRT conjecture. In this paper we present a proof of linear independence of
compactly supported separable shearlet systems. For this, we employ a sampling
strategy to utilize the structure of an implicitly given underlying oversampled
wavelet system as well as the shape of the supports of the shearlet elements.

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## ‣ Sparse Principal Components Analysis

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/01/2009
Português

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Principal components analysis (PCA) is a classical method for the reduction
of dimensionality of data in the form of n observations (or cases) of a vector
with p variables. For a simple model of factor analysis type, it is proved that
ordinary PCA can produce a consistent (for n large) estimate of the principal
factor if and only if p(n) is asymptotically of smaller order than n.
There may be a basis in which typical signals have sparse representations:
most co-ordinates have small signal energies. If such a basis (e.g. wavelets)
is used to represent the signals, then the variation in many coordinates is
likely to be small. Consequently, we study a simple "sparse PCA" algorithm:
select a subset of coordinates of largest variance, estimate eigenvectors from
PCA on the selected subset, threshold and reexpress in the original basis. We
illustrate the algorithm on some exercise ECG data, and prove that in a single
factor model, under an appropriate sparsity assumption, it yields consistent
estimates of the principal factor.; Comment: This manuscript was written in late 2003; a much revised version is
to appear, with discussion and later references, in the Journal of the
American Statistical Association in 2009. The JASA revision cites this
archive version for the proof of inconsistency of PCA

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## ‣ On the Translation Invariance of Wavelet Subspaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/09/1999
Português

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An examination of the translation invariance of $V_0$ under dyadic rationals
is presented, generating a new equivalence relation on the collection of
wavelets. The equivalence classes under this relation are completely
characterized in terms of the support of the Fourier transform of the wavelet.
Using operator interpolation, it is shown that several equivalence classes are
non-empty.; Comment: 11 pages, AMS-Latex

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## ‣ Construction of bivariate symmetric orthonormal wavelets with short support

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 29/09/2011
Português

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In this paper, we give a parameterization of the class of bivariate symmetric
orthonormal scaling functions with filter size $6\times 6$ using the standard
dilation matrix 2I. In addition, we give two families of refinable functions
which are not orthonormal but have associated tight frames. Finally, we show
that the class of bivariate symmetric scaling functions with filter size
$8\times 8$ can not have two or more vanishing moments.; Comment: 33 pages

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## ‣ Separable representations, KMS states, and wavelets for higher-rank graphs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Let $\Lambda$ be a strongly connected, finite higher-rank graph. In this
paper, we construct representations of $C^*(\Lambda)$ on certain separable
Hilbert spaces of the form $L^2(X,\mu)$, by introducing the notion of a
$\Lambda$-semibranching function system (a generalization of the semibranching
function systems studied by Marcolli and Paolucci). In particular, when
$\Lambda$ is aperiodic, we obtain a faithful representation of $C^*(\Lambda)$
on $L^2(\Lambda^\infty, M)$, where $M$ is the Perron-Frobenius probability
measure on the infinite path space $\Lambda^\infty$ recently studied by an
Huef, Laca, Raeburn, and Sims. We also show how a $\Lambda$-semibranching
function system gives rise to KMS states for $C^*(\Lambda)$. For the
higher-rank graphs of Robertson and Steger, we also obtain a representation of
$C^*(\Lambda)$ on $L^2(X, \mu)$, where $X$ is a fractal subspace of $[0,1]$ by
embedding $\Lambda^{\infty}$ into $[0,1]$ as a fractal subset $X$ of $[0,1]$.
In this latter case we additionally show that there exists a KMS state for
$C^*(\Lambda)$ whose inverse temperature is equal to the Hausdorff dimension of
$X$. Finally, we construct a wavelet system for $L^2(\Lambda^\infty, M)$ by
generalizing the work of Marcolli and Paolucci from graphs to higher-rank
graphs.; Comment: Modified hypotheses in Theorem 3.5

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## ‣ Needlet algorithms for estimation in inverse problems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 02/05/2007
Português

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We provide a new algorithm for the treatment of inverse problems which
combines the traditional SVD inversion with an appropriate thresholding
technique in a well chosen new basis. Our goal is to devise an inversion
procedure which has the advantages of localization and multiscale analysis of
wavelet representations without losing the stability and computability of the
SVD decompositions. To this end we utilize the construction of localized frames
(termed "needlets") built upon the SVD bases. We consider two different
situations: the "wavelet" scenario, where the needlets are assumed to behave
similarly to true wavelets, and the "Jacobi-type" scenario, where we assume
that the properties of the frame truly depend on the SVD basis at hand (hence
on the operator). To illustrate each situation, we apply the estimation
algorithm respectively to the deconvolution problem and to the Wicksell
problem. In the latter case, where the SVD basis is a Jacobi polynomial basis,
we show that our scheme is capable of achieving rates of convergence which are
optimal in the $L_2$ case, we obtain interesting rates of convergence for other
$L_p$ norms which are new (to the best of our knowledge) in the literature, and
we also give a simulation study showing that the NEED-D estimator outperforms
other standard algorithms in almost all situations.; Comment: Published at http://dx.doi.org/10.1214/07-EJS014 in the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org)

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## ‣ Continuous characterizations of Besov-Lizorkin-Triebel spaces and new interpretations as coorbits

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We give characterizations for homogeneous and inhomogeneous
Besov-Lizorkin-Triebel spaces in terms of continuous local means for the full
range of parameters. In particular, we prove characterizations in terms of
Lusin functions and spaces involving the Peetre maximal function to apply the
classical coorbit space theory due to Feichtinger and Gr\"ochenig. This results
in atomic decompositions and wavelet bases for homogeneous spaces. In
particular we give sufficient conditions for suitable wavelets in terms of
moment, decay and smoothness conditions.

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## ‣ A note on reassignment for wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 29/09/2015
Português

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The reassignment method for the wavelet transform is investigated.
Particularly good results are obtained if the wavelet is an extremal for the
uncertainty relation of the affine group.

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## ‣ Oracle inequalities and minimax rates for non-local means and related adaptive kernel-based methods

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Statistics Theory#Computer Science - Computer Vision and Pattern Recognition#Computer Science - Information Theory

This paper describes a novel theoretical characterization of the performance
of non-local means (NLM) for noise removal. NLM has proven effective in a
variety of empirical studies, but little is understood fundamentally about how
it performs relative to classical methods based on wavelets or how various
parameters (e.g., patch size) should be chosen. For cartoon images and images
which may contain thin features and regular textures, the error decay rates of
NLM are derived and compared with those of linear filtering, oracle estimators,
variable-bandwidth kernel methods, Yaroslavsky's filter and wavelet
thresholding estimators. The trade-off between global and local search for
matching patches is examined, and the bias reduction associated with the local
polynomial regression version of NLM is analyzed. The theoretical results are
validated via simulations for 2D images corrupted by additive white Gaussian
noise.; Comment: 49 pages, 15 figures

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## ‣ Mysteries around the graph Laplacian eigenvalue 4

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We describe our current understanding on the phase transition phenomenon of
the graph Laplacian eigenvectors constructed on a certain type of unweighted
trees, which we previously observed through our numerical experiments. The
eigenvalue distribution for such a tree is a smooth bell-shaped curve starting
from the eigenvalue 0 up to 4. Then, at the eigenvalue 4, there is a sudden
jump. Interestingly, the eigenvectors corresponding to the eigenvalues below 4
are semi-global oscillations (like Fourier modes) over the entire tree or one
of the branches; on the other hand, those corresponding to the eigenvalues
above 4 are much more localized and concentrated (like wavelets) around
junctions/branching vertices. For a special class of trees called starlike
trees, we obtain a complete understanding of such phase transition phenomenon.
For a general graph, we prove the number of the eigenvalues larger than 4 is
bounded from above by the number of vertices whose degrees is strictly higher
than 2. Moreover, we also prove that if a graph contains a branching path, then
the magnitudes of the components of any eigenvector corresponding to the
eigenvalue greater than 4 decay exponentially from the branching vertex toward
the leaf of that branch.; Comment: 22 pages

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## ‣ A note on compressed sensing of structured sparse wavelet coefficients from subsampled Fourier measurements

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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This note complements the paper "The quest for optimal sampling:
Computationally efficient, structure-exploiting measurements for compressed
sensing" [2]. Its purpose is to present a proof of a result stated therein
concerning the recovery via compressed sensing of a signal that has structured
sparsity in a Haar wavelet basis when sampled using a multilevel-subsampled
discrete Fourier transform. In doing so, it provides a simple exposition of the
proof in the case of Haar wavelets and discrete Fourier samples of more general
result recently provided in the paper "Breaking the coherence barrier: A new
theory for compressed sensing" [1].; Comment: 8 pages, companion paper

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## ‣ Estimating the joint distribution of independent categorical variables via model selection

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/06/2009
Português

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Assume one observes independent categorical variables or, equivalently, one
observes the corresponding multinomial variables. Estimating the distribution
of the observed sequence amounts to estimating the expectation of the
multinomial sequence. A new estimator for this mean is proposed that is
nonparametric, non-asymptotic and implementable even for large sequences. It is
a penalized least-squares estimator based on wavelets, with a penalization term
inspired by papers of Birg\'{e} and Massart. The estimator is proved to satisfy
an oracle inequality and to be adaptive in the minimax sense over a class of
Besov bodies. The method is embedded in a general framework which allows us to
recover also an existing method for segmentation. Beyond theoretical results, a
simulation study is reported and an application on real data is provided.; Comment: Published in at http://dx.doi.org/10.3150/08-BEJ155 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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## ‣ The jump set under geometric regularisation. Part 1: Basic technique and first-order denoising

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Functional Analysis#Computer Science - Computer Vision and Pattern Recognition#26B30, 49Q20, 65J20

Let $u \in \mbox{BV}(\Omega)$ solve the total variation denoising problem
with $L^2$-squared fidelity and data $f$. Caselles et al. [Multiscale Model.
Simul. 6 (2008), 879--894] have shown the containment $\mathcal{H}^{m-1}(J_u
\setminus J_f)=0$ of the jump set $J_u$ of $u$ in that of $f$. Their proof
unfortunately depends heavily on the co-area formula, as do many results in
this area, and as such is not directly extensible to higher-order,
curvature-based, and other advanced geometric regularisers, such as total
generalised variation (TGV) and Euler's elastica. These have received increased
attention in recent times due to their better practical regularisation
properties compared to conventional total variation or wavelets. We prove
analogous jump set containment properties for a general class of regularisers.
We do this with novel Lipschitz transformation techniques, and do not require
the co-area formula. In the present Part 1 we demonstrate the general technique
on first-order regularisers, while in Part 2 we will extend it to higher-order
regularisers. In particular, we concentrate in this part on TV and, as a
novelty, Huber-regularised TV. We also demonstrate that the technique would
apply to non-convex TV models as well as the Perona-Malik anisotropic
diffusion...

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## ‣ On construction of multivariate symmetric MRA-based wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/01/2012
Português

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For an arbitrary matrix dilation, any integer n and any integer/semi-integer
c, we describe all masks that are symmetric with respect to the point c and
have sum rule of order n. For each such mask, we give explicit formulas for
wavelet functions that are point symmetric/antisymmetric and generate
frame-like wavelet system providing approximation order n. For any matrix
dilations (which are appropriate for axial symmetry group on R^2 in some
natural sense) and given integer n, axial symmetric/antisymmetric frame-like
wavelet systems providing approximation order n are constructed. Also, for
several matrix dilations the explicit construction of highly symmetric
frame-like wavelet systems providing approximation order n is presented.

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## ‣ A geometric approach to the cascade approximation operator for wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/12/1999
Português

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#Mathematics - Functional Analysis#46L60, 47D25, 42A16, 43A65 (Primary) 46L45, 42A65, 41A15 (Secondary)

This paper is devoted to an approximation problem for operators in Hilbert
space, that appears when one tries to study geometrically the cascade algorithm
in wavelet theory. Let $ H $ be a Hilbert space, and let $ \pi $ be a
representation of $ L^\infty(T) $ on $ H $. Let $ R $ be a positive operator in
$ L^\infty(T) $ such that $ R(1)=1 $, where $ 1 $ denotes the constant function
$ 1 $. We study operators $ M $ on $ H $ (bounded, but non-contractive) such
that $ \pi(f)M=M\pi(f(z^2)) $ and $ M^* \pi(f)M=\pi(R^* f) $, $ f \in L^\infty
(T) $, where the $ * $ refers to Hilbert space adjoint. We give a complete
orthogonal expansion of $ H $ which reduces $ \pi $ such that $ M $ acts as a
shift on one part, and the residual part is $ H^{(\infty)}=\bigcap_n[M^n H] $,
where $ [M^n H] $ is the closure of the range of $ M^n $. The shift part is
present, we show, if and only if $ \ker(M^*) \neq \{0\} $. We apply the
operator-theoretic results to the refinement operator (or cascade algorithm)
from wavelet theory. Using the representation $ \pi $, we show that, for this
wavelet operator $ M $, the components in the decomposition are unitarily, and
canonically, equivalent to spaces $ L^2(E_n) \subset L^2(R) $, where $ E_n
\subset R $, $ n=0,1...

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## ‣ Convex and Network Flow Optimization for Structured Sparsity

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We consider a class of learning problems regularized by a structured
sparsity-inducing norm defined as the sum of l_2- or l_infinity-norms over
groups of variables. Whereas much effort has been put in developing fast
optimization techniques when the groups are disjoint or embedded in a
hierarchy, we address here the case of general overlapping groups. To this end,
we present two different strategies: On the one hand, we show that the proximal
operator associated with a sum of l_infinity-norms can be computed exactly in
polynomial time by solving a quadratic min-cost flow problem, allowing the use
of accelerated proximal gradient methods. On the other hand, we use proximal
splitting techniques, and address an equivalent formulation with
non-overlapping groups, but in higher dimension and with additional
constraints. We propose efficient and scalable algorithms exploiting these two
strategies, which are significantly faster than alternative approaches. We
illustrate these methods with several problems such as CUR matrix
factorization, multi-task learning of tree-structured dictionaries, background
subtraction in video sequences, image denoising with wavelets, and topographic
dictionary learning of natural image patches.; Comment: to appear in the Journal of Machine Learning Research (JMLR)

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## ‣ Projective multi-resolution analyses arising from direct limits of Hilbert modules

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/01/2007
Português

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The authors have recently shown how direct limits of Hilbert spaces can be
used to construct multi-resolution analyses and wavelets in $L^2(\R)$. Here
they investigate similar constructions in the context of Hilbert modules over
$C^*$-algebras. For modules over $C(\T^n)$, the results shed light on work of
Packer and Rieffel on projective multi-resolution analyses for specific Hilbert
$C(\T^n)$-modules of functions on $\R^n$. There are also new applications to
modules over $C(C)$ when $C$ is the infinite path space of a directed graph.

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