1,165 research outputs found
Multiscale Representations for Manifold-Valued Data
We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere , the special orthogonal group , the positive definite matrices , and the Grassmann manifolds . The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the and maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as , , , where the and maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper
Definability and stability of multiscale decompositions for manifold-valued data
We discuss multiscale representations of discrete manifold-valued data. As it
turns out that we cannot expect general manifold-analogues of biorthogonal
wavelets to possess perfect reconstruction, we focus our attention on those
constructions which are based on upscaling operators which are either
interpolating or midpoint-interpolating. For definable multiscale
decompositions we obtain a stability result
Non-equispaced B-spline wavelets
This paper has three main contributions. The first is the construction of
wavelet transforms from B-spline scaling functions defined on a grid of
non-equispaced knots. The new construction extends the equispaced,
biorthogonal, compactly supported Cohen-Daubechies-Feauveau wavelets. The new
construction is based on the factorisation of wavelet transforms into lifting
steps. The second and third contributions are new insights on how to use these
and other wavelets in statistical applications. The second contribution is
related to the bias of a wavelet representation. It is investigated how the
fine scaling coefficients should be derived from the observations. In the
context of equispaced data, it is common practice to simply take the
observations as fine scale coefficients. It is argued in this paper that this
is not acceptable for non-interpolating wavelets on non-equidistant data.
Finally, the third contribution is the study of the variance in a
non-orthogonal wavelet transform in a new framework, replacing the numerical
condition as a measure for non-orthogonality. By controlling the variances of
the reconstruction from the wavelet coefficients, the new framework allows us
to design wavelet transforms on irregular point sets with a focus on their use
for smoothing or other applications in statistics.Comment: 42 pages, 2 figure
A global approach to the refinement of manifold data
A refinement of manifold data is a computational process, which produces a
denser set of discrete data from a given one. Such refinements are closely
related to multiresolution representations of manifold data by pyramid
transforms, and approximation of manifold-valued functions by repeated
refinements schemes. Most refinement methods compute each refined element
separately, independently of the computations of the other elements. Here we
propose a global method which computes all the refined elements simultaneously,
using geodesic averages. We analyse repeated refinements schemes based on this
global approach, and derive conditions guaranteeing strong convergence.Comment: arXiv admin note: text overlap with arXiv:1407.836
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
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