A full second order variational model for multiscale texture analysis

We present a second order image decomposition model to perform denoising and texture extraction. We look for the decomposition f=u+v+w where u is a first order term, v a second order term and w the (0 order) remainder term. For highly textured images the model gives a two-scale texture decomposition: u can be viewed as a macro-texture (larger scale) whose oscillations are not too large and w is the micro-texture (very oscillating) that may contain noise. We perform mathematical analysis of the model and give numerical examples.     

A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising

The total variation model proposed by Rudin, Osher and Fatemi performs very well for removing noise while preserving edges. However, it favors a piecewise constant solution in BV space which often leads to the staircase effect, and small details such as textures are often filtered out with noise in the process of denoising. To preserve the textures and eliminate the staircase effect, we improve the total variation model in this paper. This is accomplished by the following steps: (1) we define a new space of functions of fractional-order bounded variation called the BV_α space by using the Grünwald–Letnikov definition of fractional-order derivative; (2) we model the structure of the image as a function belonging to the BV_α space, and the textures in different scales as functions belonging to different negative Sobolev spaces. Thus, we propose a class of fractional-order multi-scale variational models for image denoising. (3) We analyze some properties of the fraction-order total variation operator and its conjugate operator. By using these properties, we develop an alternation projection algorithm for the new model and propose an efficient condition of the convergence of the algorithm. The numerical results show that the fractional-order multi-scale variational model can improve the peak signal to noise ratio of image, preserve textures and eliminate the staircase effect efficiently in the process of denoising.     

Graduated adaptive image denoising: local compromise between total variation and isotropic diffusion

We introduce variants of the variational image denoising method proposed by Blomgren et al. (In: Numerical Analysis 1999 (Dundee), pp. 43–67. Chapman & Hall, Boca Raton, FL, 2000), which interpolates between total-variation denoising and isotropic diffusion denoising. We study how parameter choices affect results and allow tuning between TV denoising and isotropic diffusion for respecting texture on one spatial scale while denoising features assumed to be noise on finer spatial scales. Furthermore, we prove existence and (where appropriate) uniqueness of minimizers. We consider both L ² and L ¹ data fidelity terms.     

Curves in Banach spaces which allow a C 1,BV parametrization or a parametrization with finite convexity

We give a complete characterization of those f: [0, 1] → X (where X is a Banach space) which allow an equivalent C ^1,BV parametrization (i.e., a C ¹ parametrization whose derivative has bounded variation) or a parametrization with bounded convexity. Our results are new also for X = ? ⁿ . We present examples which show applicability of our characterizations. For example, we show that the C ^1,BV and C ² parametrization problems are equivalent for X = ? but are not equivalent for X = ? ⁿ .     

Convergence of a fixed point iteration method for the OSV model

In image processing, image denoising and texture extraction are important problems in which many new methods recently have been developed. One of the most important models is the OSV model [S. Osher, A. Solé, L. Vese, Image decomposition and restoration using total variation minimization and the H^-1 norm, Multiscale Model. Simul. A SIAM Interdisciplinary J. 1(3) (2003) 349-370] which is constructed by the total variation and H^-1 norm. This paper proves the existence of the minimizer of the functional from the OSV model and analyzes the convergence of an iterative method for solving the problems. Our iteration method is constructed by a fixed point iteration on the fourth order partial differential equation from the computation of the associated Euler-Lagrange equation, and the limit of our iterations satisfies the minimizer of the functional from the OSV model. In numerical experiments, we compare the numerical results of our works with those of the ROF model [L.I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992) 259-268].     

A characteristics-mixed covolume method for a convection-dominated transport problem

In this paper, we propose a characteristics-mixed covolume method for approximating the solution to a convection dominated transport problem. The method is a combination of characteristic approximation to handle the convection term in time and mixed covolume method spatial approximation to deal with the diffusion term. The velocity and press are approximated by the lowest order Raviart-Thomas mixed finite element space on rectangles. The projection of a mixed covolume element is introduced. We prove its first order optimal rate of convergence for the approximate velocities in the L² norm as well as for the approximate pressures in the L² norm.     

Separability of the space of continuous functions that are continuous in Λ-variation

The Shao-Sablin index of a Λ-sequence Λ=(λi) is defined by . The main result of the paper states that the Banach space CΛBV of continuous functions of bounded Λ-variation with the standard Λ-variation norm is separable if and only if SΛ<2. Also, ΛBV=ΛBVc if and only if SΛ<2, where ΛBVc denotes the space of functions continuous in Λ-variation. A number of corollaries is drawn, and one of them being that the Garsia-Sawyer class GS is a dense subset of the Banach space HBV of functions of bounded harmonic variation.     

Lifting of BV functions with values in S1

We show that for every $\text{u∈}\text{BV}\text{(Ω;S}{}^1\text{)}$, there exists a bounded variation function $\text{?∈}\text{BV}\text{(Ω;}\text{R}\text{)}$ such that u=e^i? a.e. on $\text{Ω}$ and |?|_BV?2|u|_BV. The constant 2 is optimal in dimension n>1. To cite this article: J. Dávila, R. Ignat, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Lacunary interpolation by quartic splines on uniform meshes

The interpolation of a discrete set of data on the interval [0, 1], representing the first and the second derivatives (except at 0) of a smooth function f is investigated via quartic C²-splines. Error bounds in the uniform norm for ∥s⁽ⁱ⁾ − f⁽ⁱ⁾∥, i=0(1)2, if f ∈ C^l[0, 1], l=3, 5 and ⁽³⁾ ∈ BV[0, 1], together with computational examples will also be presented.     

An Aumann–Shapley type operator in Pseudo-D-lattices

Let L be a \(\sigma \)-complete pseudo-D-lattice and let BV be the Banach space of all real-valued, vanishing at zero, functions of bounded variation on L endowed with the variation norm. We prove the existence of a continuous Aumann–Shapley type operator \(\phi \) on the closed subspace of BV spanned by powers of nonatomic \(\sigma \)-additive positive modular measures on L. Moreover we give an integral representation of \(\phi \) on a class of functions that correspond to measure games.     

Functions of bounded variation on “good” metric spaces

In this paper we give a natural definition of Banach space valued BV functions defined on complete metric spaces endowed with a doubling measure (for the sake of simplicity we will say doubling metric spaces) supporting a Poincaré inequality (see Definition 2.5 below). The definition is given starting from Lipschitz functions and taking closure with respect to a suitable convergence; more precisely, we define a total variation functional for every Lipschitz function; then we take the lower semicontinuous envelope with respect to the L¹ topology and define the BV space as the domain of finiteness of the envelope. The main problem of this definition is the proof that the total variation of any BV function is a measure; the techniques used to prove this fact are typical of Γ-convergence and relaxation. In Section 4 we define the sets of finite perimeter, obtaining a Coarea formula and an Isoperimetric inequality. In the last section of this paper we also compare our definition of BV functions with some definitions already existing in particular classes of doubling metric spaces, such as Weighted spaces, Ahlfors-regular spaces and Carnot–Carathéodory spaces.     

The ultimate estimate of the upper norm bound for the summation of operators

Let A and B be bounded linear operators acting on a Hilbert space H. It is shown that the triangular inequality serves as the ultimate estimate of the upper norm bound for the sum of two operators in the sense that

sup{∥U^*AU+V^*BV∥:U and V are unitaries}=min{∥A+μI∥+∥B-μI∥:μ∈C}.     

Well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge

For a supersonic Euler flow past a straight-sided wedge whose vertex angle is less than the extreme angle, there exists a shock-front emanating from the wedge vertex, and the shock-front is usually strong especially when the vertex angle of the wedge is large. In this paper, we establish the L¹ well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge whose boundary slope function has small total variation, when the total variation of the incoming flow is small. In this case, the Lipschitz wedge perturbs the flow, and the waves reflect after interacting with the strong shock-front and the wedge boundary. We first obtain the existence of solutions in BV when the incoming flow has small total variation by the wave front tracking method and then establish the L¹ stability of the solutions with respect to the incoming flows. In particular, we incorporate the nonlinear waves generated from the wedge boundary to develop a Lyapunov functional between two solutions containing strong shock-fronts, which is equivalent to the L¹ norm, and prove that the functional decreases in the flow direction. Then the L¹ stability is established, so is the uniqueness of the solutions by the wave front tracking method. Finally, the uniqueness of solutions in a broader class, the class of viscosity solutions, is also obtained.     

Image Restoration and Decomposition via Bounded Total Variation and Negative Hilbert-Sobolev Spaces

We propose a new class of models for image restoration and decomposition by functional minimization. Following ideas of Y. Meyer in a total variation minimization framework of L. Rudin, S. Osher, and E. Fatemi, our model decomposes a given (degraded or textured) image u ₀ into a sum u+v. Here u∈BV is a function of bounded variation (a cartoon component), while the noisy (or textured) component v is modeled by tempered distributions belonging to the negative Hilbert-Sobolev space H ^−s . The proposed models can be seen as generalizations of a model proposed by S. Osher, A. Solé, L. Vese and have been also motivated by D. Mumford and B. Gidas. We present existence, uniqueness and two characterizations of minimizers using duality and the notion of convex functions of measures with linear growth, following I. Ekeland and R. Temam, F. Demengel and R. Temam. We also give a numerical algorithm for solving the minimization problem, and we present numerical results of denoising, deblurring, and decompositions of both synthetic and real images.     

Combined ℓ<Subscript>2</Subscript> data and gradient fitting in conjunction with ℓ<Subscript>1</Subscript> regularization

We are interested in minimizing functionals with ℓ₂ data and gradient fitting term and ℓ₁ regularization term with higher order derivatives in a discrete setting. We examine the structure of the solution in 1D by reformulating the original problem into a contact problem which can be solved by dual optimization techniques. The solution turns out to be a ’smooth’ discrete polynomial spline whose knots coincide with the contact points while its counterpart in the contact problem is a discrete version of a spline with higher defect and contact points as knots. In 2D we modify Chambolle’s algorithm to solve the minimization problem with the ℓ₁ norm of interacting second order partial derivatives as regularization term. We show that the algorithm can be implemented efficiently by applying the fast cosine transform. We demonstrate by numerical denoising examples that the ℓ₂ gradient fitting term can be used to avoid both edge blurring and staircasing effects.      

A variational problem with lack of compactness for H(S;S) maps of prescribed degree

We consider, for maps in H^1/2(S¹;S¹), a family of (semi)norms equivalent to the standard one. We ask whether, for such a norm, there is some map in H^1/2(S¹;S¹) of prescribed topological degree equal to 1 and minimal norm. In general, the answer is no, due to concentration phenomena. The existence of a minimal map is sensitive to small perturbations of the norm. We derive a sufficient condition for the existence of minimal maps. In particular, we prove that, for every given norm, there are arbitrarily small perturbations of it for which the minimum is attained. In case there is no minimizer, we determine the asymptotic behavior of minimizing sequences. We prove that, for such minimizing sequences, the energy concentrates near a point of S¹. We describe this concentration in terms of bubbling-off of circles.     

Convergence analysis of the rectangular Morley element scheme for second order problem in arbitrary dimensions

We present the convergence analysis of the rectangular Morley element scheme utilised on the second order problem in arbitrary dimensions. Specifically, we prove that the convergence of the scheme is of O(h) order in energy norm and of O(h²) order in L² norm on general d-rectangular triangulations. Moreover, when the triangulation is uniform, the convergence rate can be of O(h²) order in energy norm, and the convergence rate in L² norm is still of O(h²) order, which cannot be improved. Numerical examples are presented to demonstrate our theoretical results.     

Approximate controllability and homogenization of a semilinear elliptic problem

The L²- and H¹-approximate controllability and homogenization of a semilinear elliptic boundary-value problem is studied in this paper. The principal term of the state equation has rapidly oscillating coefficients and the control region is locally distributed. The observation region is a subset of codimension 1 in the case of L²-approximate controllability or is locally distributed in the case of H¹-approximate controllability. By using the classical Fenchel-Rockafellar's duality theory, the existence of an approximate control of minimal norm is established by means of a fixed point argument. We consider its asymptotic behavior as the rapidly oscillating coefficients H-converge. We prove its convergence to an approximate control of minimal norm for the homogenized problem.     

On the best constants in Markov-type inequalities involving Laguerre norms with different weights

The paper concerns best constants in Markov-type inequalities between the norm of a higher derivative of a polynomial and the norm of the polynomial itself. The norm of the polynomial is taken in L ² on the half-line with the weight t ^α e ^?t and the derivative is measured in L ² on the half-line with the weight t ^β e ^?t . Under an additional assumption on the difference β ? α, we determine the leading term of the asymptotics of the constants as the degree of the polynomial goes to infinity.     

On the function S(T) in the theory of the Riemann zeta-function

The function S(T) is the error term in the formula for the number of zeros of the Riemann zeta-function above the real axis and up to height T in the complex plane. We assume the Riemann hypothesis, and examine how well S(T) can be approximated by a Dirichlet polynomial in the L² norm.