Families of multiresolution and wavelet spaces with optimal properties

Under suitable conditions, if the scaling functions ?₁ and ?₂ generate the multiresolutions V _(j)(?₁) and V _(j)(?₂), then their convolution ?₁*?₂also generates a multiresolution V _(j)(?₁*?₂) More over, if p is an appropriate convolution operator from l ₂ into itself and if ? is a scaling function generating the multiresolution V _(j)(?),then p*?is a scaling function generating the same multiresolution V _(j)(?)=V _(j)(p*?). Using these two properties, we group the scaling and wavelet functions into equivalent classes and consider various equivalent basis functions of the associated function spaces We use the n-fold convolution product to construct sequences of multiresolution and wavelet spaces V _(j)(?ⁿ) and W _(j)(?ⁿ) with increasing regularity. We discuss the link between multiresolution analysis and Shannon's sampling theory. We then show that the interpolating and orthogonal pre- and post-filters associated with the multiresolution sequence V ₍₀₎(?ⁿ)asymptotically converge to the ideal lowpass filter of Shannon. We also prove that the filters associated with the sequence of wavelet spaces W ₍₀₎(?ⁿ)convergeto the ideal bandpass filter. Finally, we construct the basic wavelet sequences ψ ^b _nand show that they tend to Gabor functions. Thisprovides wavelets that are nearly time-frequency optimal. The theory is illustrated with the example of polynomial splines.     

A Unifying Representer Theorem for Inverse Problems and Machine Learning

Foundations of Computational Mathematics - Regularization addresses the ill-posedness of the training problem in machine learning or the reconstruction of a signal from a limited number of...     

On the Continuity of Characteristic Functionals and Sparse Stochastic Modeling

The characteristic functional is the infinite-dimensional generalization of the Fourier transform for measures on function spaces. It characterizes the statistical law of the associated stochastic process in the same way as a characteristic function specifies the probability distribution of its corresponding random variable. Our goal in this work is to lay the foundations of the innovation model, a (possibly) non-Gaussian probabilistic model for sparse signals. This is achieved by using the characteristic functional to specify sparse stochastic processes that are defined as linear transformations of general continuous-domain white Lévy noises (also called innovation processes). We prove the existence of a broad class of sparse processes by using the Minlos–Bochner theorem. This requires a careful study of the regularity properties, especially the \(L^p\) -boundedness, of the characteristic functional of the innovations. We are especially interested in the functionals that are only defined for \(p<1\) since they appear to be associated with the sparser kind of processes. Finally, we apply our main theorem of existence to two specific subclasses of processes with specific invariance properties.     

Beyond Wiener’s Lemma: Nuclear Convolution Algebras and the Inversion of Digital Filters

Journal of Fourier Analysis and Applications - A convolution algebra is a topological vector space $${\mathcal {X}}$$ that is closed under the convolution operation. It is said to be inverse-closed...     

Approximation of Non-decaying Signals from Shift-Invariant Subspaces

In our recent work, the sampling and reconstruction of non-decaying signals, modeled as members of weighted-\(L_p\) spaces, were shown to be stable with an appropriate choice of the generating kernel for the shift-invariant reconstruction space. In this paper, we extend the Strang–Fix theory to show that, for d-dimensional signals whose derivatives up to order L are all in some weighted-\(L_p\) space, the weighted norm of the approximation error can be made to go down as \(O(h^L)\) when the sampling step h tends to 0. The sufficient condition for this decay rate is that the generating kernel belongs to a particular hybrid-norm space and satisfies the Strang–Fix conditions of order L. We show that the \(O(h^L)\) behavior of the error is attainable for both approximation schemes using projection (when the signal is prefiltered with the dual kernel) and interpolation (when a prefilter is unavailable). The requirement on the signal for the interpolation method, however, is slightly more stringent than that of the projection because we need to increase the smoothness of the signal by a margin of \(d/p+\varepsilon \), for arbitrary \(\varepsilon >0\). This extra amount of derivatives is used to make sure that the direct sampling is stable.

     

The distribution of particles characterized by size and free mobility within polydisperse populations of protein-polysaccharide conjugates, determined from two-dimensional agarose electropherograms

New approaches for the characterization of polydisperse particle populations are presented*. The investigated samples contain virus-sized protein-polysaccharide conjugates which had previously been prepared as immunogens against bacterial meningitis (Hib). The analysis is based on two-dimensional agarose electrophoresis (Serwer-type). This method, like the one of O'Farrell, achieves a separation according to size and charge. It relies on a different principle, however, and is applicable to nondenatured particles which are 100 to more than 1000 times larger in mass than regular uncrosslinked proteins. Data from stained gel patterns are evaluated by the computer program ELPHOFIT, which makes it possible to standardize the gel and to construct a nomogram which defines every position on the gel in terms of particle size and free mobility (related to surface net charge density). The output of ELPHOFIT, consisting of nomogram parameters, is transferred to the image processing program GELFIT. This software is used to evaluate the computer images obtained by digitizing the stained gel patterns: (i) The nomogram is electronically superimposed on the computer image. (ii) The gel pattern is transformed from a curvilinear to a rectangular coordinate system of particle size and free mobility. The center of gravity as well as density maxima are given in coordinates of particle size and free mobility. Ranges of grey levels can be accentuated by adding 16 pseudocolors. (iii) Using surface-stripping techniques, GELFIT provides an estimate for the number of major subpopulations within each preparation. (iv) Numerical values for the distribution of particle size and free mobility are determined. Using program IMAGE, the quantitative physical assessment of a given conjugate preparation is presented in the form of a computer-generated three-dimensional plot, the shape of which serves to identify and characterize the preparation visually. The data analysis based on digitized two-dimensional gel patterns is automated to an extent that a technician can perform routine evaluations. It uses the Macintosh II personal computer.     

Computerized methods for analyzing two-dimensional agarose gel electropherograms

Previous methods interpret zonal or polydisperse gel patterns of two-dimensional Serwer-type gels in terms of size and free mobility (surface net charge density). These two parameters have been determined for each component without quantitatively measuring the abundance of the components. The present study advances these previous methods by determining the relative concentration of each component by computer evaluation of densitometrically analyzed gel patterns. Suitable procedures and their underlying algorithms are presented. The mathematical routines are implemented in a user-friendly software package, called GelFit and designed for a Macintosh personal computer. The program input consists of digitized images of gel staining patterns exemplified by those obtained from electrophoresis of native subcellular-sized particles. The data are processed through the following steps: (i) Noise reduction and calibration. (ii) Geometrical transformation of the pattern onto a rectangular size/free mobility coordinate system using rationales of the extended Ogston model. (iii) Analysis of the transformed image to determine density maxima, density profiles along iso-free-mobility or iso-size lines, curve fitting of one-dimensional profiles or two-dimensional surfaces using Gaussian functions and curve stripping of surfaces to determine the possible number of particle populations.     

Generalized Poisson Summation Formulas for Continuous Functions of Polynomial Growth

The Poisson summation formula (PSF) describes the equivalence between the sampling of an analog signal and the periodization of its frequency spectrum. In engineering textbooks, the PSF is usually stated formally without explicit conditions on the signal for the formula to hold. By contrast, in the mathematics literature, the PSF is commonly stated and proven in the pointwise sense for various types of \(L_1\) signals. This \(L_1\) assumption is, however, too restrictive for many signal-processing tasks that demand the sampling of possibly growing signals. In this paper, we present two generalized versions of the PSF for d-dimensional signals of polynomial growth. In the first generalization, we show that the PSF holds in the space of tempered distributions for every continuous and polynomially growing signal. In the second generalization, the PSF holds in a particular negative-order Sobolev space if we further require that \(d/2+\varepsilon \) derivatives of the signal are bounded by some polynomial in the \(L_2\) sense.     

Operator-Like Wavelet Bases of L_{2}(\mathbb{R}^{d})

The connection between derivative operators and wavelets is well known. Here we generalize the concept by constructing multiresolution approximations and wavelet basis functions that act like Fourier multiplier operators. This construction follows from a stochastic model: signals are tempered distributions such that the application of a whitening (differential) operator results in a realization of a sparse white noise. Using wavelets constructed from these operators, the sparsity of the white noise can be inherited by the wavelet coefficients. In this paper, we specify such wavelets in full generality and determine their properties in terms of the underlying operator.