Abstract—Suppose is an unknown vector in (a digital image or signal); we plan to measure general linear functionals of and then reconstruct. If is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements can be dramatically smaller than the size. Thus, certain natural classes of images with pixels need only = ( 1 4 log 5 2 ()) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual pixel samples. More specifically, suppose has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)—so the coefficients belong to an ball for 0 1. The most important coefficients in that expansion allow reconstruction with 2 error ( 1 2 1

this paper we explore the relationship between orthonormal ridgelets and true ridge functions r(x 1 cos # + x 2 sin #). We derive a formula giving the ridgelet coe#cients of a ridge function in terms of the 1-D wavelet coe#cients of the ridge profile r(t), and we study the properties of the linear approximation operator which `kills' coe#cients at high angular scale or high ridge scale. We also show that partial orthonormal ridgelet expansions can give e#cient nonlinear approximations to pure ridge functions. In e#ect, the rearranged weighted ridgelet coe#cients of a ridge function decay at essentially the same rate as the rearranged weighted 1-D wavelet coe#cients of the 1-D ridge profile r(t). This shows that simple thresholding in the ridgelet basis is, for certain purposes, equally as good as ideal nonlinear ridge approximation. Key Words and Phrases. Wavelets. Ridge function. Ridgelet. Radon transform. Best m-term

The key vehicle of the recent development of a topological theory of regular variation based on topological dynamics [BOst13], and embracing its classical univariate counterpart (cf. [BGT]) as well as fragmentary multivariate (mostly Euclidean) theories (eg [MeSh], [Res], [Ya]), are groups with a right-invariant metric carrying ‡ows. Following the vector paradigm, they are best seen as normed groups. That concept only occasionally appears explicitly in the literature despite its frequent disguised presence, and despite a respectable lineage traceable back to the Pettis closed-graph theorem, to the Birkho¤-Kakutani metrization theorem and further back still to Banach’s Théorie des opérations linéaires. We collect together known salient features and develop their theory including Steinhaus theory uni…ed by the Category Embedding Theorem [BOst11], the associated themes of subadditivity and convexity, and a topological duality inherent to topological

We introduce new anisotropic wavelet decompositions associated with the smoothness #, # =(#1 ,...,# d ), #1 ,...,# d > 0 of multivariate functions as measured in anisotropic Besov spaces B # . We give the rate of nonlinear approximation of functions f # B # by these wavelets. Finally, we prove that, among a general class of anisotropic wavelet decompositions of a function f # B # , the anisotropic wavelet decomposition associated with # gives the optimal rate of compression of the wavelet decomposition of f . Chapter 1

Given the wavelet expansion of a function v, a non-linear adaptive approximation of v is obtained by neglecting those coefficients whose size drops below a certain threshold. We propose several ways to define the threshold: all are based on the characterization of the local regularity of v (in a Sobolev or Besov scale) in terms of summability of properly defined subsets of its coefficients. A-priori estimates of the approximation error are derived. For the Haar system, the asymptotic behavior of both the approximation error and the number of survived coefficients is thoroughly investigated for a class of functions having Holder-type singularities. Corresponding author: Claudio Canuto Dipartimento di Matematica, Politecnico di Torino Corso Duca degli Abruzzi, 24 I-10129 TORINO Italy phone: +39 11 564 7543 fax: +39 11 564 7599 email: ccanuto@polito.it 1 Introduction Wavelets have proven a powerful tool in signal processing and related topics (see [11], [16], [30]). For instance, the i...