We investigate the optimal configurations of n points on the unit sphere for a class of potential functions. In particular, we characterize these optimal configurations in terms of their approximation properties within frame theory. Furthermore, we consider similar optimal configurations in terms of random distributions of points on the sphere. In this probabilistic setting, we characterize these optimal distributions by means of special classes of probabilistic frames. Our work also indicates some connections between statistical shape analysis and frame theory.

Fusion frames enable signal decompositions into weighted linear subspace components. For positive integers p, we introduce p-fusion frames, a sharpening of the notion of fusion frames. Tight p-fusion frames are closely related to the classical notions of designs and cubature formulas in Grassmann spaces and are analyzed with methods from harmonic analysis in the Grassmannians. We define the p-fusion frame potential, derive bounds for its value, and discuss the connections to tight p-fusion frames.

We introduce probabilistic frames to study finite frames whose elements are chosen at random. While finite tight frames generalize orthonormal bases by allowing redundancy, independent, uniformly distributed points on the sphere approximately form a finite unit norm tight frame (FUNTF). In the present paper, we develop probabilistic versions of tight frames and FUNTFs to significantly weaken the requirements on the random choice of points to obtain an approximate finite tight frame. Namely, points can be chosen from any probabilistic tight frame, they do not have to be identically distributed, nor have unit norm. We also observe that classes of random matrices used in compressed sensing are induced by probabilistic tight frames

Abstract. Inspired by significant real-life applications, in particular, sparse phase retrieval and sparse pulsation frequency detection in Asteroseismology, we investigate a general framework for compressed sensing, where the measurements are quasi-linear. We formulate natural generalizations of the well-known Restricted Isometry Property (RIP) towards nonlinear measurements, which allow us to prove both unique identifiability of sparse signals as well as the convergence of recovery algorithms to compute them efficiently. We show that for certain randomized quasi-linear measurements, including Lipschitz perturbations of classical RIP matrices and phase retrieval from random projections, the proposed restricted isometry properties hold with high probability. We analyze a generalized Orthogonal Least Squares (OLS) under the assumption that magnitudes of signal entries to be recovered decay fast. Greed is good again, as we show that this algorithm performs efficiently in phase retrieval and Asteroseismology. For situations where the decay assumption on the signal does not necessarily hold, we propose two alternative algorithms, which are natural generalizations of the well-known iterative hard and soft-thresholding. While these algorithms are rarely successful for the mentioned applications, we show their strong recovery guarantees for quasi-linear measurements which are Lipschitz perturbations of RIP matrices.Key words. compressed sensing, restricted isometry property, greedy algorithm, quasi-linear, iterative thresholding AMS subject classifications. 94A20, 47J25, 15B521. Introduction. Compressed sensing addresses the problem of recovering nearly-sparse signals from vastly incomplete measurements [11,12,14,15,21]. By using the prior assumptions on the signal, the number of measurements can be well below the Shannon sampling rate and effective reconstruction algorithms are available. The standard compressed sensing approach deals with linear measurements. The success of signal recovery algorithms often relies on the so-called Restricted Isometry Property (RIP) [12,15,27,35,38,39], which is a near-identity spectral property of small submatrices of the measurement Gramian. The RIP condition is satisfied with high probability and nearly optimal number of measurements for a large class of random measurements [3,4,14,35,38], which explains the popularity of all sorts of random sensing approaches. The most effective recovery algorithms are based either on a greedy approach or on variational models, such as 1 -norm minimization, leading to suitable iterative thresholded gradient descent methods. In the literature of mathematical signal processing, greedy algorithms for sparse recovery originate from the so-called Matching Pursuit [33], although several predecessors were well-known in other communities. Among astronomers and asteroseismologists, for instance, Orthogonal Least Squares (OLS) [31] was already in use in the '60s for the detection of significant frequencies of star light-spectra (the so-called prewhitening) [...

Noncontact optical imaging of curved objects can result in strong artifacts due to the object's shape, leading to curvature biased intensity distributions. This artifact can mask variations due to the object's optical properties, and makes reconstruction of optical/physiological properties difficult. In this work we demonstrate a curvature correction method that removes this artifact and recovers the underlying data, without the necessity of measuring the object's shape. This method is applicable to many optical imaging modalities that suffer from shape-based intensity biases. By separating the spatially varying data (e.g., physiological changes) from the background signal (dc component), we show that the curvature can be extracted by either averaging or fitting the rows and columns of the images. Numerical simulations show that our method is equivalent to directly removing the curvature, when the object's shape is known, and accurately recovers the underlying data. Experiments on phantoms validate the numerical results and show that for a given image with 16.5% error due to curvature, the method reduces that error to 1.2%. Finally, diffuse multispectral images are acquired on forearms in vivo. We demonstrate the enhancement in image quality on intensity images, and consequently on reconstruction results of blood volume and oxygenation distributions.

In this article, we construct compactly supported multivariate pairs of dual wavelet frames, shortly called bi-frames, for an arbitrary dilation matrix. Our construction is based on the mixed oblique extension principle, and it provides bi-frames with few wavelets. In the examples, we obtain optimal bi-frames, i.e., primal and dual wavelets are constructed from a single fundamental refinable function, whose mask size is minimal w.r.t. sum rule order and smoothness. Moreover, the wavelets reach the maximal approximation order w.r.t. the underlying refinable function. For special dilation matrices, we derive very simple but optimal arbitrarily smooth bi-frames in arbitrary dimensions with only two primal and dual wavelets.

The decomposition of polynomial spaces on unions of Grassmannians G k 1 ,d ∪ . . . ∪ G kr ,d into irreducible orthogonally invariant subspaces and their reproducing kernels are investigated. We also generalize the concepts of cubature points and t-designs from single Grassmannians to unions. We derive their characterization as minimizers of a suitable energy potential to enable t-design constructions by numerical optimization. We also present new analytic families of t-designs for t = 1, 2, 3.2010 Mathematics Subject Classification. Primary 42C10, 65D32; Secondary 46E22, 33C45. 1 2 M. EHLER AND M. GRÄFirreducible components. In Section 6 we introduce cubatures and t-designs on unions of Grassmannians and derive a characterization as minimizers of an energy functional induced by a reproducing kernel. We compute some analytical minimizers in Section 7. Polynomials on single GrassmanniansThis section is dedicated to summarize some facts about single Grassmannians, see, for instance, [5,34]. The Grassmannian space of all k-dimensional linear subspaces of R d is naturally identified with the set of orthogonal projectors on R d of rank k denoted bysym is the set of symmetric matrices in R d×d . Each Grassmannian G k,d admits a unique orthogonally invariant probability measure σ k,d induced by the Haar (probability) measure σ O(d) on the orthogonal group O(d), i.e., for any Q ∈ G k,d and measurable function f , we observeThe space of complex-valued, square-integrable functions L 2 (G k,d ), endowed with the inner product (f, g) G k,d , decomposes into orthogonally invariant subspaces, the irreducible representation of O(d) associated to the partition 2λ = (2λ 1 , . . . , 2λ t ), cf. [5,34]. Note that two representations are equivalent if there is a linear isomorphism that commutes with the group action. A partition of t is an integer vector λ = (λ 1 , . . . , λ t ) with λ 1 ≥ . . . ≥ λ t ≥ 0, |λ| = t, where |λ| := t i=1 λ i , and the length ℓ(λ) is the number of nonzero parts of λ. Note that we add and suppress zero entries in λ without further notice, so that we can also compare partitions of different lengths. For partitions λ, λ ′ of integers t, t ′ , respectively, we denote λ ≤ λ ′ if and only if λ i ≤ λ ′ i , for all i = 1, . . . , ℓ(λ). The space of polynomials of degree at most t on G k,d is given bywhere C[X] t is the set of polynomials of degree at most t in d 2 many variables arranged as a matrix X ∈ C d×d , and f | G k,d denotes the restriction of f to G k,d . This polynomial space decomposes into Pol t (G k,d ) = |λ|≤t, ℓ(λ)≤min{k,d−k}

Abstract-We introduce Schroedinger Eigenmaps, a new semi-supervised manifold learning and recovery technique. This method is based on an implementation of graph Schroedinger operators with appropriately constructed barrier potentials as carriers of labeled information. We use our approach for the analysis of standard bio-medical datasets and new multispectral retinal images.Index Terms-Schroedinger Eigenmaps, Laplacian Eigenmaps, Schroedinger operator on a graph, barrier potential, dimension reduction, manifold learning.