Approximation theory of wavelet frame based image restoration

It is shown that there exists a frame wavelet with fast decay in the time domain and compact support in the frequency domain generating a wavelet system whose canonical dual frame cannot be generated by an arbitrary number of generators. We show that there exist wavelet frame generated by two functions which have good dual wavelet frames, but for which the canonical dual wavelet frame does not consist of wavelets, according to scaling functions.

The wavelet frame systems have been widely investigated and applied for image restoration and many other image processing problems over the past decades, attributing to their good capability of sparsely approximating piece-wise smooth functions such as images. Most wavelet frame based models exploit the $l_1$ norm of frame coefficients for a sparsity constraint in the past. The authors in \cite{ZhangY2013, Dong2013} proposed an $l_0$ minimization model, where the $l_0$ norm of wavelet frame coefficients is penalized instead, and have demonstrated that significant improvements can be achieved compared to the commonly used $l_1$ minimization model. Very recently, the authors in \cite{Chen2015} proposed $l_0$-$l_2$ minimization model, where the nonlocal prior of frame coefficients is incorporated. This model proved to outperform the single $l_0$ minimization based model in terms of better recovered image quality. In this paper, we propose a truncated $l_0$-$l_2$ minimization model which combines sparsity, nonlocal and support prior of the frame coefficients. The extensive experiments have shown that the recovery results from the proposed regularization method performs better than existing state-of-the-art wavelet frame based methods, in terms of edge enhancement and texture preserving performance.

Wavelet analysis is a particular time-or space-scale representation of signals which has found a wide range of applications in physics, signal processing and applied Mathematics in the last few years. We show that there exist wavelet frame generated by two functions which have good dual wavelet frames, but for which the canonical dual wavelet frame does not consist of wavelets, according to scaling functions.

Exact travelling wave solutions of the dissipative coupled Korteweg-de Vries equation

This paper deals with a way of constructing reproducing kernel Hilbert spaces and their associated kernels from frame theory. After introducing briefly frame theory, we give mild conditions on frame elements for spanning a RKHS. Examples of different kernels are then given based on wavelet frame. Thus, issues of this way of building kernel for semiparametric learning are discussed and an application example on a toy problem is described.

We consider the backward error associated with a given approximate solution of a linear least squares problem. The backward error can be very expensive to compute, as it involves the minimal singular value of a certain matrix that depends on the problem data and the approximate solution. An estimate based on a regularized projection of the residual vector has been proposed in the literature and analyzed by several authors. Although numerical experiments in the literature suggest that it is a reliable estimate of the backward error for any given approximate least squares solution, to date no satisfactory explanation for this behavior had been found. We derive new bounds which confirm this experimental observation.

Wavelet frames for (not necessarily reducing) affine subspaces II: The structure of affine subspaces

In this paper, the Cauchy problem of the nonlinear Helmholtz-type equation is discussed. This problem is well known to be severely ill-posed. Compared with linear equations, nonlinear equations are more difficult to deal with because of their lack of linearity. In order to obtain the approximate solution of this nonlinear problem, a combination of the quasi-boundary value method and the quasi-reversibility method is proposed. Using the variable separation method, the approximate solution is equivalent to solving a class of integral equations. The well-posedness of the approximation problem is proved by the Banach fixed-point theorem. Convergence analysis and error estimation are discussed. Since the convergence of error estimates cannot be obtained by the traditional a-priori bound, we introduce a new a-priori bound to obtain the convergence of error estimates, and the Hölder-type error estimate is achieved. Finally, some numerical experiments are given to corroborate the qualitative analysis and show the regularization method works well.

In this paper, we propose a neuro-fuzzy finite element method with an iterative learning scheme, as an application of Gaussian radial basis function (RBF) networks, in which the finite elements are fuzzy subsets defined by Gaussian membership functions. For almost all problems in practical engineering where FEM is applied, analytical solutions are not obtained. Since the finite element concept is based on the discretization of a continuum, the subject of developing reliable a posteriori error estimates is central to the effective use of FEM. In the FEM some error estimators which are reasonably accurate and useful in the existing problem have been proposed. The approximate solutions of neuro-fuzzy FEM also need some tools for error estimation. Hence, we propose a method of error estimation based on the same iterative learning scheme. Illustrative examples of solving Laplace equations are shown with the computational results of approximate solutions and estimated errors by using the proposed methods. >

Wavelet frame based surface reconstruction from unorganized points

The method of least squares is used to construct approximate solutions to the boundary value problem rf=g0, B¡(f) = 0 for z'=l, ...,/c, on the interval [a, b], where t is an zzth order formal differential operator, g0it) is a given function in L? [a, b], and By, ..., Bk are linearly independent boundary values.Letting H"[a, b] denote the space of all functions/(z) in C"_1[a, b] with/"'""1' absolutely continuous on [a, bl and/"" in L2[a, b], a sequence of functions f,(Z) (z'=l, 2, ...) in H"[a, ¿>] is constructed satisfying the boundary conditions and a completeness condition.Assuming the boundary value problem has a solution, the approximate solutions fit ) = 2} = i a'iit(t) (z'=l,2, ...) are constructed; the coefficients a\ are determined uniquely from the system of equations 2 irS"riM = igo,rSt), 1= 1, ...,/, where (/, g) denotes the inner product in L2[a, b].The approximate solutions are shown to converge to a solution of the boundary value problem, and error estimates are established.functions f(t) which are square-integrable on [a, b].The usual inner product and norm in S are denoted by (fig) and ||/||, and we denote convergence in S by ft -*-f.For any operator L in S we denote its domain, range, and null space by 3>(L), 3i(L), and Jf(L), respectively.

This paper makes qualitative analysis to the bounded traveling wave solutions for a kind of nonlinear dispersive-dissipative equation, and considers its solving problem. The relation is investigated between behavior of its solution and the dissipation coefficient. Further, all approximate damped oscillatory solutions when dissipation coefficient is small are presented by utilizing the method of undetermined coefficients according to the theory of rotated vector field in planar dynamical systems. Finally, error estimate is given by establishing the integral equation which reflects the relation between approximate and exact damped oscillatory solutions applying the idea of homogenization principle.

As for an orthonormal basis, a frame allows each element in the underlying Hilbert space to be written as an unconditionally convergent infinite linear combination of the frame elements. The coefficients are called frame coefficients. Peter G. Casazza and Ole Christensen introduced some methods to approximate frame coefficients. In this article, we investigate some of these results for a continuous frame. As a consequence, approximation of the solution to a moment problem is also discussed. We also apply the results to wavelet frames and Weyl-Heisenberg frames.

Solutions to the equations of motion of the low-energy, effective field theory emerging out of compactified heterotic string theory are constructed by making use of the well known duality symmetries. Beginning with four-dimensional solutions of the Einstein massless scalar field theory in the canonical frame we first rewrite the corresponding solutions in the string frame. Thereafter, using the T- and S-duality symmetries of the low-energy string effective action we arrive at the corresponding uncharged, electrically charged and magnetically charged solutions. Brief comments on the construction of dual versions of the Kerr-Sen type using the dilatonic Kerr solution as the seed are also included. Thereafter, we verify the status of the energy conditions for the solutions in the string frame. Several of the metrics found here are shown to possess naked singularities although the energy conditions are obeyed. Dual solutions exhibit a duality in the conservation/violation of the null and averaged null energy conditions (NEC/ANEC), a fact demonstrated previously in the context of black holes (Kar S 1997 Phys. Rev. D 55 4872) and cosmologies (Kar S 1997 Phys. Lett. B 404 234). Additionally, those backgrounds which conserve the NEC/ANEC in spite of possessing naked singularities serve as counterexamples to cosmic censorship in the context of low-energy, effective string theory.

In a Banach space, for the approximate solution of the Cauchy problem for the evolution equation with an operator generating an analytic semigroup, a purely implicit three-level semidiscrete scheme that can be reduced to two-level schemes is considered. Using these schemes, an approximate solution to the original problem is constructed. Explicit bounds on the approximate solution error are proved using properties of semigroups under minimal assumptions about the smoothness of the data of the problem. An intermediate step in this proof is the derivation of an explicit estimate for the semidiscrete Crank–Nicolson scheme. To demonstrate the generality of the perturbation algorithm as applied to difference schemes, a four-level scheme that is also reduced to two-level schemes is considered.