884 research outputs found
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
Measuring the galaxy power spectrum with multiresolution decomposition -- II. diagonal and off-diagonal power spectra of the LCRS galaxies
The power spectrum estimator based on the discrete wavelet transform (DWT)
for 3-dimensional samples has been studied. The DWT estimator for
multi-dimensional samples provides two types of spectra with respect to
diagonal and off-diagonal modes, which are very flexible to deal with
configuration-related problems in the power spectrum detection. With simulation
samples and mock catalogues of the Las Campanas redshift survey (LCRS), we show
(1) the slice-like geometry of the LCRS doesn't affect the off-diagonal power
spectrum with ``slice-like'' mode; (2) the Poisson sampling with the LCRS
selection function doesn't cause more than 1- error in the DWT power
spectrum; and (3) the powers of peculiar velocity fluctuations, which cause the
redshift distortion, are approximately scale-independent. These results insure
that the uncertainties of the power spectrum measurement are under control. The
scatter of the DWT power spectra of the six strips of the LCRS survey is found
to be rather small. It is less than 1- of the cosmic variance of mock
samples in the wavenumber range h Mpc. To fit the detected
LCRS diagonal DWT power spectrum with CDM models, we find that the best-fitting
redshift distortion parameter is about the same as that obtained from
the Fourier power spectrum. The velocity dispersions for SCDM and
CDM models are also consistent with other detections with
the LCRS. A systematic difference between the best-fitting parameters of
diagonal and off-diagonal power spectra has been significantly measured. This
indicates that the off-diagonal power spectra are capable of providing
information about the power spectrum of galaxy velocity field.Comment: AAS LaTeX file, 41 pages, 10 figures included, accepted for
publication in Ap
Vector extension of monogenic wavelets for geometric representation of color images
14 pagesInternational audienceMonogenic wavelets offer a geometric representation of grayscale images through an AM/FM model allowing invariance of coefficients to translations and rotations. The underlying concept of local phase includes a fine contour analysis into a coherent unified framework. Starting from a link with structure tensors, we propose a non-trivial extension of the monogenic framework to vector-valued signals to carry out a non marginal color monogenic wavelet transform. We also give a practical study of this new wavelet transform in the contexts of sparse representations and invariant analysis, which helps to understand the physical interpretation of coefficients and validates the interest of our theoretical construction
A Representation Theorem for Singular Integral Operators on Spaces of Homogeneous Type
Let (X,d,\mu) be a space of homogeneous type and E a UMD Banach space. Under
the assumption mu({x})=0 for all x in X, we prove a representation theorem for
singular integral operators on (X,d,mu) as a series of simple shifts and
rearrangements plus two paraproducts. This gives a T(1) Theorem in this
setting
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