In this paper, we study the Besov regularity of Levy white noises on the $d$-dimensional torus. Due to their rough sample paths, the white noises that we consider are defined as generalized stochastic fields. We, initially, obtain regularity results
for general Levy white noises. Then, we focus on two subclasses of noises: compound Poisson and symmetric-$alpha$-stable (including Gaussian), for which we make more precise statements. Before measuring regularity, we show that the question is well-posed; we prove that Besov spaces are in the cylindrical $sigma$-field of the space of generalized functions. These results pave the way to the characterization of the $n$-term wavelet approximation properties of stochastic processes.
We consider $L^1$-TV regularization of univariate signals with values on the real line or on the unit circle. While the real data space leads to a convex optimization problem, the problem is non-convex for circle-valued data. In this paper, we derive
exact algorithms for both data spaces. A key ingredient is the reduction of the infinite search spaces to a finite set of configurations, which can be scanned by the Viterbi algorithm. To reduce the computational complexity of the involved tabulations, we extend the technique of distance transforms to non-uniform grids and to the circular data space. In total, the proposed algorithms have complexity $mathscr{O}(KN)$ where $N$ is the length of the signal and $K$ is the number of different values in the data set. In particular, the complexity is $mathscr{O}(N)$ for quantized data. It is the first exact algorithm for TV regularization with circle-valued data, and it is competitive with the state-of-the-art methods for scalar data, assuming that the latter are quantized.
It is known that the Karhunen-Lo`{e}ve transform (KLT) of Gaussian first-order auto-regressive (AR(1)) processes results in sinusoidal basis functions. The same sinusoidal bases come out of the independent-component analysis (ICA) and actually corres
pond to processes with completely independent samples. In this paper, we relax the Gaussian hypothesis and study how orthogonal transforms decouple symmetric-alpha-stable (S$alpha$S) AR(1) processes. The Gaussian case is not sparse and corresponds to $alpha=2$, while $0<alpha<2$ yields processes with sparse linear-prediction error. In the presence of sparsity, we show that operator-like wavelet bases do outperform the sinusoidal ones. Also, we observe that, for processes with very sparse increments ($0<alphaleq 1$), the operator-like wavelet basis is indistinguishable from the ICA solution obtained through numerical optimization. We consider two criteria for independence. The first is the Kullback-Leibler divergence between the joint probability density function (pdf) of the original signal and the product of the marginals in the transformed domain. The second is a divergence between the joint pdf of the original signal and the product of the marginals in the transformed domain, which is based on Steins formula for the mean-square estimation error in additive Gaussian noise. Our framework then offers a unified view that encompasses the discrete cosine transform (known to be asymptotically optimal for $alpha=2$) and Haar-like wavelets (for which we achieve optimality for $0<alphaleq1$).
Here we present a method of constructing steerable wavelet frames in $L_2(mathbb{R}^d)$ that generalizes and unifies previous approaches, including Simoncellis pyramid and Riesz wavelets. The motivation for steerable wavelets is the need to more accu
rately account for the orientation of data. Such wavelets can be constructed by decomposing an isotropic mother wavelet into a finite collection of oriented mother wavelets. The key to this construction is that the angular decomposition is an isometry, whereby the new collection of wavelets maintains the frame bounds of the original one. The general method that we propose here is based on partitions of unity involving spherical harmonics. A fundamental aspect of this construction is that Fourier multipliers composed of spherical harmonics correspond to singular integrals in the spatial domain. Such transforms have been studied extensively in the field of harmonic analysis, and we take advantage of this wealth of knowledge to make the proposed construction practically feasible and computationally efficient.
The Riesz transform is a natural multi-dimensional extension of the Hilbert transform, and it has been the object of study for many years due to its nice mathematical properties. More recently, the Riesz transform and its variants have been used to c
onstruct complex wavelets and steerable wavelet frames in higher dimensions. The flip side of this approach, however, is that the Riesz transform of a wavelet often has slow decay. One can nevertheless overcome this problem by requiring the original wavelet to have sufficient smoothness, decay, and vanishing moments. In this paper, we derive necessary conditions in terms of these three properties that guarantee the decay of the Riesz transform and its variants, and as an application, we show how the decay of the popular Simoncelli wavelets can be improved by appropriately modifying their Fourier transforms. By applying the Riesz transform to these new wavelets, we obtain steerable frames with rapid decay.
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 fr
om 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.
