In the past few decades, mathematics based approaches have been widely adopted in various image restoration problems; the partial differential equation (PDE) based approach (e.g., the total variation model [L. ], etc.) and wavelet frame based approach are some successful examples. These approaches were developed through different paths and generally provided understanding from different angles of the same problem. As shown in numerical simulations, implementations of the wavelet frame based approach and the PDE based approach quite often end up solving a similar numerical problem with similar numerical behaviors, even though different approaches have advantages in different applications. Since wavelet frame based and PDE based approaches have all been modeling the same types of problems with success, it is natural to ask whether the wavelet frame based approach is fundamentally connected with the PDE based approach when we trace them all the way back to their roots. A fundamental connection of a wavelet frame based approach with a total variation model and its generalizations was established in [J. Cai, B. Dong, S. Osher, and Z. Shen, J. Amer. Math. Soc., 25 (2012), pp. 1033-1089]. This connection gives the wavelet frame based approach a geometric explanation and, at the same time, it equips a PDE based approach with a time frequency analysis. Cai et al. showed that a special type of wavelet frame model using generic wavelet frame systems can be regarded as an approximation of a generic variational model (with the total variation model as a special case) in the discrete setting. A systematic convergence analysis, as the resolution of the image goes to infinity, which is the key step in linking the two approaches, is also given in Cai et al. Motivated by Cai et al. and [Q. Jiang, Appl. Numer. Math., 62 (2012), pp. 51-66], this paper establishes a fundamental connection between the wavelet frame based approach and nonlinear evolution PDEs, provides interpretations and analytical studies of such connections, and proposes new algorithms for image restoration based on the new understandings. Together with the results in [J. Cai et al., J. Amer. Math. Soc., 25 (2012), pp. 1033-1089], we now have a better picture of how the wavelet frame based approach can be used to interpret the general PDE based approach (e.g., the variational models or nonlinear evolution PDEs) and can be used as a new and useful tool in numerical analysis to discretize and solve various variational and PDE models. To be more precise, we shall establish the following: (1) The connections between wavelet frame shrinkage and nonlinear evolution PDEs provide new and inspiring interpretations of both approaches that enable us to derive new PDE models and (better) wavelet frame shrinkage algorithms for image restoration.(2) A generic nonlinear evolution PDE (of parabolic or hyperbolic type) can be approximated by wavelet frame shrinkage with properly chosen wavelet frame systems and carefully designed shrinkage functions. (3) The main idea of this work i...

The purpose of this paper is to investigate spectral properties of the transition operator associated to a multivariate vector refinement equation and their applications to the study of smoothness of the corresponding refinable vector of functions.Let, where M is an expansive integer matrix. We assume that M is isotropic, i.e., M is similar to a diagonal matrix diag(σ 1 , . . . , σ s ) withThe smoothness of Φ is measured by the critical exponentwhereWe assume that the mask a is finitely supported, i.e., the set suppa := {α ∈ Z Z s : a(α) = 0} is finite. Note that each a(α) is an r × r complex matrix. Let A := α∈Z Z s a(α)/d, where d := | det M |. We assume that spec (A) (the spectrum of A) is {η 1 , η 2 . . . . , η r }, where η 1 = 1 and η j = 1 for j = 2, . . . , r., where ⊗ denotes the (right) Kronecker product. Suppose the highest degree of polynomials reproduced by Φ is k − 1. LetThe main result of this paper asserts that if Φ is stable, then λ(Φ) = − log d ρ k s/2, where. This result is obtained through an extensive use of linear algebra and matrix theory. Three examples are provided to illustrate the general theory. All these examples have background of practical applications.

The synchrosqueezing transform, a kind of reassignment method, aims to sharpen the timefrequency representation and to separate the components of a multicomponent non-stationary signal. In this paper, we consider the short-time Fourier transform (STFT) with a time-varying parameter, called the adaptive STFT. Based on the local approximation of linear frequency modulation mode, we analyze the well-separated condition of non-stationary multicomponent signals using the adaptive STFT with the Gaussian window function. We propose the STFT-based synchrosqueezing transform (FSST) with a time-varying parameter, named the adaptive FSST, to enhance the time-frequency concentration and resolution of a multicomponent signal, and to separate its components more accurately. In addition, we also propose the 2nd-order adaptive FSST to further improve the adaptive FSST for the non-stationary signals with fast-varying frequencies. Furthermore, we present a localized optimization algorithm based on our well-separated condition to estimate the time-varying parameter adaptively and automatically. Simulation results on synthetic signals and the bat echolocation signal are provided to demonstrate the effectiveness and robustness of the proposed method.where A k (t), φ k (t) > 0, has been a very active research area over the past few years. Note that the number of component K may change with time t, but it should be constant for long enough time intervals. The representation of x(t) in (1) with A k (t) and φ k (t) varying slowly or more slowly than φ k (t) is called an adaptive harmonic model (AHM) representation of x(t), where A k (t) are called the instantaneous amplitudes and φ k (t) the instantaneous frequencies (IFs). To decompose x(t) as an AHM representation (1) is important to extract information, such as the underlying dynamics, hidden in x(t). Time-frequency (TF) analysis is widely used in engineering fields such as communication, radar and sonar as a powerful tool for analyzing time-varying non-stationary signals [1]. Time-frequency analysis is especially useful for signals containing many oscillatory components with slowly timevarying amplitudes and instantaneous frequencies. The short-time Fourier transform (STFT), the continuous wavelet transform (CWT) and the Wigner-Ville distribution are the most typical TF analysis, see details in [1]-[6]. Other TF distributions of Cohen's class include the exponential distribution [7], a smoothed pseudo Wigner distribution [8] and the complex-lag distribution [9].In addition, the TF signal analysis and synthesis using the eigenvalue decomposition method has been studied [10,11]. In particular, an eigenvalue decomposition-based approach which enables the separation of non-stationary components with overlapped supports in the TF plane has been proposed in [12].Recently a number of new TF analysis methods such as the Hilbert spectrum analysis with empirical mode decomposition (EMD) [13], the reassignment method [14] and synchrosqueezed wavelet transform (SST) [15] have also been proposed ...

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The objective of this paper is to introduce a direct approach for generating local averaging rules for both the √ 3 and 1-to-4 vector subdivision schemes for computer-aided design of smooth surfaces. Our innovation is to directly construct refinable bivariate spline function vectors with minimum supports and highest approximation orders on the six-directional mesh, and to compute their refinement masks which give rise to the matrix-valued coefficient stencils for the surface subdivision schemes. Both the C 1-quadratic and C 2-cubic spaces are studied in some detail. In particular, we show that our C 2-cubic refinement mask for the 1-to-4 subdivision can be slightly modified to yield an adaptive version of Loop's surface subdivision scheme.

Abstract. The notion of K-balancing was introduced a few years ago as a condition for the construction of orthonormal scaling function vectors and multi-wavelets to ensure the property of preservation/annihilation of scalarvalued discrete polynomial data of order K (or degree K − 1), when decomposed by the corresponding matrix-valued low-pass/high-pass filters. While this condition is indeed precise, to the best of our knowledge only the proof for K = 1 is known. In addition, the formulation of the K-balancing condition for K ≥ 2 is so prohibitively difficult to satisfy that only a very few examples for K = 2 and vector dimension 2 have been constructed in the open literature. The objective of this paper is to derive various characterizations of the Kbalancing condition that include the polynomial preservation property as well as equivalent formulations that facilitate the development of methods for the construction purpose. These results are established in the general multivariate and bi-orthogonal settings for any K ≥ 1.

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