668 research outputs found
A Note on BIBO Stability
The statements on the BIBO stability of continuous-time convolution systems
found in engineering textbooks are often either too vague (because of lack of
hypotheses) or mathematically incorrect. What is more troubling is that they
usually exclude the identity operator. The purpose of this note is to clarify
the issue while presenting some fixes. In particular, we show that a linear
shift-invariant system is BIBO-stable in the -sense if and only if
its impulse response is included in the space of bounded Radon measures, which
is a superset of (Lebesgue's space of absolutely integrable
functions). As we restrict the scope of this characterization to the
convolution operators whose impulse response is a measurable function, we
recover the classical statement
Autofocus for digital Fresnel holograms by use of a Fresnelet-sparsity criterion
We propose a robust autofocus method for reconstructing digital Fresnel holograms. The numerical reconstruction
involves simulating the propagation of a complex wave front to the appropriate distance. Since the latter value is difficult to determine manually, it is desirable to rely on an automatic procedure for finding the optimal distance to achieve high-quality reconstructions. Our algorithm maximizes a sharpness metric related to the sparsity of the signal’s expansion in distance-dependent waveletlike Fresnelet bases. We show results from simulations and experimental situations that confirm its applicability
Left-Inverses of Fractional Laplacian and Sparse Stochastic Processes
The fractional Laplacian 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 , its inverse is the classical Riesz potential
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. In particular, we extend the definition of the classical Riesz
potential to any non-integer number larger than and
show that it is the unique left-inverse of the fractional Laplacian
which is dilation-invariant and
translation-invariant. We observe that, for any and
, there exists a Schwartz function such that is not -integrable. We then introduce the new unique left-inverse
of the fractional Laplacian with the
property that is dilation-invariant (but not
translation-invariant) and that is -integrable for any
Schwartz function . We finally apply that linear operator
with to solve the stochastic partial differential equation
with white Poisson noise as its driving term
.Comment: Advances in Computational Mathematics, accepte
On the Hilbert transform of wavelets
A 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. Our main result is that, in the presence of some minimal smoothness
and decay, the Hilbert transform of a wavelet is again as smooth and
oscillating as the original wavelet, whereas its localization is controlled by
the number of vanishing moments of the original wavelet. We motivate our
results using concrete examples.Comment: Appears in IEEE Transactions on Signal Processing, vol. 59, no. 4,
pp. 1890-1894, 201
Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms
We 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. As a concrete
application of this methodology, we identify HT pairs of spline wavelets of a
specific flavor, which are then combined to realize a family of complex
wavelets that resemble the optimally-localized Gabor function for sufficiently
large orders.
Analytic wavelets, derived from the complexification of HT wavelet pairs,
exhibit a one-sided spectrum. Based on the tensor-product of such analytic
wavelets, and, in effect, by appropriately combining four separable
biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for
constructing 2D directional-selective complex wavelets. In particular,
analogous to the HT correspondence between the components of the 1D
counterpart, we relate the real and imaginary components of these complex
wavelets using a multi-dimensional extension of the HT--the directional HT.
Next, we construct a family of complex spline wavelets that resemble the
directional Gabor functions proposed by Daugman. Finally, we present an
efficient FFT-based filterbank algorithm for implementing the associated
complex wavelet transform.Comment: 36 pages, 8 figure
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