The Construction of Nonseparable Wavelet Bi-Frames and Associated Approximation Schemes

Wavelet analysis and its fast algorithms are widely used in many fields of applied mathematics such as in signal and image processing. In the present thesis, we circumvent the restrictions of orthogonal and biorthogonal wavelet bases by constructing wavelet frames. They still allow for a stable decomposition, and so-called wavelet bi-frames provide a series expansion very similar to those of pairs of biorthogonal wavelet bases. Contrary to biorthogonal bases, primal and dual wavelets are no longer supposed to satisfy any geometrical conditions, and the frame setting allows for redundancy. This provides more flexibility in their construction. Finally, we construct families of optimal wavelet bi-frames in arbitrary dimensions with arbitrarily high smoothness. Then we verify that the n-term approximation can be described by Besov spaces and we apply the theoretical findings to image denoising

Constructing pairs of dual bandlimited frame wavelets in L2(Rn)L^2(\mathbb{R}^n)

Given a real, expansive dilation matrix we prove that any bandlimited function $\psi \in L^2(\mathbb{R}^n)$, for which the dilations of its Fourier transform form a partition of unity, generates a wavelet frame for certain translation lattices. Moreover, there exists a dual wavelet frame generated by a finite linear combination of dilations of $\psi$ with explicitly given coefficients. The result allows a simple construction procedure for pairs of dual wavelet frames whose generators have compact support in the Fourier domain and desired time localization. The construction relies on a technical condition on $\psi$, and we exhibit a general class of function satisfying this condition.Comment: 21 pages, 6 figure

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

Generating Tight Wavelet Frames From Sums of Squares Representations

We construct multivariate tight wavelet frames in several settings by using the theory of sums of squares representations for nonnegative trigonometric polynomials. This is done by way of two extension principles which allow us to translate the problem of constructing these frames to one of designing collections of trigonometric polynomials satisfying certain orthogonality and normalization conditions. We consider first the setting of dyadic dilation, and assume that the lowpass masks are constructed by the coset sum method, which lifts a univariate lowpass mask to a nonseparable multivariate lowpass mask, with several properties of the input being preserved. The existence of the necessary sums of squares representations is proved utilizing the special structure of these lowpass masks. We extend this first construction to the setting of prime dilation, focusing on the case of interpolatory input masks. We prove lower bounds on the vanishing moments of the highpass masks in these two constructions, and new results about the properties of the prime coset sum method. In the first two settings, we use lowpass masks satisfying the sub-QMF condition, and apply the unitary extension principle to ensure that our filter banks result in tight wavelet frames. In the third setting, we use lowpass masks satisfying a generalization of this condition, which we dub the oblique sub-QMF condition. In fact, it turns out that for a fixed lowpass mask and vanishing moment recovery function, this condition is equivalent to the existence of highpass masks satisfying the oblique extension principle conditions. This allows us to construct multivariate tight wavelet frames for any lowpass mask satisfying the oblique sub-QMF condition, under some mild assumptions on the vanishing moment recovery function. To establish this equivalence, we first prove a new result on sums of squares representations for nonnegative multivariate trigonometric polynomials, which says that any such function may be written as a finite sum of squares of quotients of trigonometric polynomials. We will also prove a generalization of the sum of squares result for matrices rather than functions, in which we show that a matrix with trigonometric polynomial entries which is positive semidefinite for all evaluations has a representation as a sum of squares of commuting symmetric matrices with rational trigonometric polynomial entries. We suspect that these sums of squares results for trigonometric polynomials and matrices with such entries will be of interest far beyond the wavelet construction community

Bifurcation analysis of the Topp model

In this paper, we study the 3-dimensional Topp model for the dynamicsof diabetes. We show that for suitable parameter values an equilibrium of this modelbifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests thatnear this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arisethrough period doubling cascades of limit cycles.Keywords Dynamics of diabetes · Topp model · Reduced planar quartic Toppsystem · Singular point · Limit cycle · Hopf-saddle-node bifurcation · Perioddoubling bifurcation · Shilnikov homoclinic orbit · Chao