Michael Unser
On the Hilbert transform of wavelets.
Tue, 07/26/2011 - 15:01 — SomNambulAuthors: Kunal Narayan Chaudhury, Michael Unser
Subjects: Functional Analysis
AbstractA wavelet is a localized function having a prescribed number of vanishing
moments. In this correspondence, we provide precise arguments as to why the
Hilbert transform of a wavelet is again a wavelet. In particular, we provide
sharp estimates of the localization, vanishing moments, and smoothness of the
transformed wavelet. We work in the general setting of non-compactly supported
wavelets.Fast O(1) bilateral filtering using trigonometric range kernels.
Tue, 05/24/2011 - 15:01 — SomNambulAuthors: Kunal Narayan Chaudhury, Michael Unser, Daniel Sage
Subjects: Computer Vision and Pattern Recognition
AbstractIt is well-known that spatial averaging can be realized (in space or
frequency domain) using algorithms whose complexity does not depend on the size
or shape of the filter. These fast algorithms are generally referred to as
constant-time or O(1) algorithms in the image processing literature. Along with
the spatial filter, the edge-preserving bilateral filter [bilateralFilter]
involves an additional range kernel. This is used to restrict the averaging to
those neighborhood pixels whose intensity are similar or close to that of the
pixel of interest.Left-Inverses of Fractional Laplacian and Sparse Stochastic Processes.
Wed, 09/15/2010 - 15:01 — SomNambulAuthors: Michael Unser, Qiyu Sun
Subjects: Information Theory
AbstractThe fractional Laplacian $(-\triangle)^{\gamma/2}$ commutes with the primary
coordination transformations in the Euclidean space $\RR^d$: dilation,
translation and rotation, and has tight link to splines, fractals and stable
Levy processes. For $0<\gamma<d$, its inverse is the classical Riesz potential
$I_\gamma$ which is dilation-invariant and translation-invariant. In this work,
we investigate the functional properties (continuity, decay and invertibility)
of an extended class of differential operators that share those invariance
properties.Fast space-variant elliptical filtering using box splines.
Thu, 03/11/2010 - 16:01 — SomNambulAuthors: Kunal Narayan Chaudhury, Michael Unser, Arrate Munoz-Barrutia
Subjects: Computer Vision and Pattern Recognition
AbstractThe efficient realization of linear space-variant (non-convolution) filters
is a challenging computational problem in image processing. In this paper, we
demonstrate that it is possible to filter an image with a Gaussian-like
elliptic window of varying size, elongation and orientation using a fixed
number of computations per pixel. The associated algorithm, which is based on a
family of smooth compactly supported piecewise polynomials, the
radially-uniform box splines, is realized using pre-integration and local
finite-differences.Fast adaptive elliptical filtering using box splines.
Thu, 08/27/2009 - 13:23 — SomNambulAuthors: Kunal Narayan Chaudhury, Michael Unser, Arrate Munoz Barrutia
Subjects: Information Theory
AbstractWe demonstrate that it is possible to filter an image with an elliptic window
of varying size, elongation and orientation with a fixed computational cost per
pixel. Our method involves the application of a suitable global pre-integrator
followed by a pointwise-adaptive localization mesh. We present the basic theory
for the 1D case using a B-spline formalism and then appropriately extend it to
2D using radially-uniform box splines. The size and ellipticity of these
radially-uniform box splines is adaptively controlled. Moreover, they converge
to Gaussians as the order increases.Fast adaptive elliptical filtering using box splines.
Thu, 08/27/2009 - 13:23 — SomNambulAuthors: Kunal Narayan Chaudhury, Michael Unser, Arrate Munoz Barrutia
Subjects: Information Theory
AbstractWe demonstrate that it is possible to filter an image with an elliptic window
of varying size, elongation and orientation with a fixed computational cost per
pixel. Our method involves the application of a suitable global pre-integrator
followed by a pointwise-adaptive localization mesh. We present the basic theory
for the 1D case using a B-spline formalism and then appropriately extend it to
2D using radially-uniform box splines. The size and ellipticity of these
radially-uniform box splines is adaptively controlled. Moreover, they converge
to Gaussians as the order increases.Gabor wavelet analysis and the fractional Hilbert transform.
Thu, 08/27/2009 - 13:23 — SomNambulAuthors: Kunal Narayan Chaudhury, Michael Unser
Subjects: Information Theory
AbstractWe propose an amplitude-phase representation of the dual-tree complex wavelet
transform (DT-CWT) which provides an intuitive interpretation of the associated
complex wavelet coefficients.Gabor wavelet analysis and the fractional Hilbert transform.
Thu, 08/27/2009 - 13:23 — SomNambulAuthors: Kunal Narayan Chaudhury, Michael Unser
Subjects: Information Theory
AbstractWe propose an amplitude-phase representation of the dual-tree complex wavelet
transform (DT-CWT) which provides an intuitive interpretation of the associated
complex wavelet coefficients.Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms.
Tue, 08/25/2009 - 22:41 — SomNambulAuthors: Kunal Narayan Chaudhury, Michael Unser
Subjects: Information Theory
AbstractWe propose a novel method for constructing Hilbert transform (HT) pairs of
wavelet bases based on a fundamental approximation-theoretic characterization
of scaling functions--the B-spline factorization theorem. In particular,
starting from well-localized scaling functions, we construct HT pairs of
biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet
filters via a discrete form of the continuous HT filter.On the Shiftability of Dual-Tree Complex Wavelet Transforms.
Tue, 08/25/2009 - 22:41 — SomNambulAuthors: Kunal Narayan Chaudhury, Michael Unser
Subjects: Information Theory
AbstractThe dual-tree complex wavelet transform (DT-CWT) is known to exhibit better
shift-invariance than the conventional discrete wavelet transform. We propose
an amplitude-phase representation of the DT-CWT which, among other things,
offers a direct explanation for the improvement in the shift-invariance.
User login
Loading