Antonio Marquina

High-Order Total Variation-Based Image Restoration

The total variation (TV) denoising method is a PDE-based technique that preserves edges well but has the sometimes undesirable staircase effect, namely, the transformation of smooth regions ( ramps) into piecewise constant regions ( stairs). In this paper we present an improved model, constructed by adding a nonlinear fourth order diffusive term to the Euler--Lagrange equations of the variational TV model. Our technique substantially reduces the staircase effect, while preserving sharp jump discontinuities (edges). We show numerical evidence of the power of resolution of this novel model with respect to the TV model in some 1D and 2D numerical examples.

Explicit Algorithms for a New Time Dependent Model Based on Level Set Motion for Nonlinear Deblurring and Noise Removal

In this paper we formulate a time dependent model to approximate the solution to the nonlinear total variation optimization problem for deblurring and noise removal introduced by Rudin and Osher [ Total variation based image restoration with free local constraints, in Proceedings IEEE Internat. Conf. Imag. Proc., IEEE Press, Piscataway, NJ, (1994), pp. 31--35] and Rudin, Osher, and Fatemi [ Phys. D, 60 (1992), pp. 259--268], respectively. Our model is based on level set motion whose steady state is quickly reached by means of an explicit procedure based on Roes scheme [ J. Comput. Phys., 43 (1981), pp. 357--372], used in fluid dynamics. We show numerical evidence of the speed of resolution and stability of this simple explicit procedure in some representative 1D and 2D numerical examples.

Image Super-Resolution by TV-Regularization and Bregman Iteration

In this paper we formulate a new time dependent convolutional model for super-resolution based on a constrained variational model that uses the total variation of the signal as a regularizing functional. We propose an iterative refinement procedure based on Bregman iteration to improve spatial resolution. The model uses a dataset of low resolution images and incorporates a downsampling operator to relate the high resolution scale to the low resolution one. We present an algorithm for the model and we perform a series of numerical experiments to show evidence of the good behavior of the numerical scheme and quality of the results.

Recurrence relations for rational cubic methods II: the Chebyshev method

We continue the analysis of rational cubic methods, initiated in [7]. In this paper, we obtain a system of a priori error bounds for the Chebyshev method in Banach spaces through a local convergence theorem that provides sufficient conditions on the initial point in order to ensure the convergence of Chebyshev iterates. The error estimates are exact for second degree polynomials. We also discuss some applications.ZusammenfassungWir betrachten ein System von a priori Fehlerabschätzungen für die Konvergenz des Chebyshev-Verfahrens in, Banachräumen. Unsere Sätze geben hinreichende Bedingungen an, den Startwert, welche die Konvergenz der Chebyshev-Iteration sichern. Sie bestehen aus einem System rekursiver Beziehungen, ähnlich den Bedingungen von Kantorvich für das Newton-Verfahren.

Morphology and Dynamics of Relativistic Jets

We present a comprehensive analysis of the morphology and dynamics of relativistic pressure-matched axisymmetric jets. The numerical simulations have been carried out with a high-resolution shock-capturing hydrocode based on an approximate relativistic Riemann solver derived from the spectral decomposition of the Jacobian matrices of relativistic hydrodynamics. We discuss the dependence of the jet morphology on several parameters, paying special attention to the relativistic effects caused by high Lorentz factors and large internal energies of the beam flow. The parameter space of our analysis is spanned by the ratio of the beam and ambient medium rest mass density (η), the beam Mach number (Mb), the beam Lorentz factor (Wb), and the adiabatic index (γ) of the equation of state (assuming an ideal gas). Both the ultrarelativistic regime (Wb ≥ 20) and the hypersonic regime (relativistic Mach number greater than 100) have been studied. Our results show that the enhancement of the effective inertial mass of the beam due to relativistic effects (through the specific enthalpy and the Lorentz factor) makes relativistic jets significantly more stable than Newtonian jets. We find that relativistic jets propagate very efficiently through the ambient medium, at speeds that agree very well with those obtained from an estimate based on a one-dimensional momentum balance. The propagation efficiency of a relativistic jet is an increasing function of the beam flow velocity. Relativistic jets seem to give rise to two different morphologies, according to the relevance of relativistic effects. Hot beams (i.e., with internal energies comparable to the beam rest-mass energy) show little internal structure (as they are almost in pressure equilibrium with their surroundings) and relatively smooth cocoons forming lobes near the head of the jet. Highly supersonic models, in which the kinematic relativistic effects due to high beam flow Lorentz factors dominate, display extended cocoons that are overpressured with respect to the environment. The cocoon thickness decreases, and its mean pressure increases with increasing beam Lorentz factor.

Local piecewise hyperbolic reconstruction of numerical fluxes for nonlinear scalar conservation laws

This paper constructs a local third order accurate shock capturing method for hyperbolic scalar conservation laws, based on numerical fluxes with a total variation diminishing (TVD) Runge–Kutta evolution in time, using the idea recently introduced by C. W. Shu and S. J. Osher for essentially nonoscillatory (ENO) methods. The constructed method is an upwind conservative scheme that is local in the sense that numerical fluxes are reconstructed without using extrapolation from the data of the smoothest neighboring cell. To design the method, a new concept of local smoothing is introduced to prevent the increasing of total variation of the solution near discontinuities and to achieve third order accuracy. The method becomes third order accurate in smooth regions of the solution, except at local extrema where it may degenerate to

Blind deconvolution using TV regularization and Bregman iteration

O(h^{3/2} )

A flux-split algorithm applied to conservative models for multicomponent compressible flows

, thus giving better accuracy than TVD methods at local extrema. The main advantage of this method lies on the property that is more local than that of ENO and TVD upwind...

Approximate Osher-Solomon schemes for hyperbolic systems

In this paper we formulate a new time dependent model for blind deconvolution based on a constrained variational model that uses the sum of the total variation norms of the signal and the kernel as a regularizing functional. We incorporate mass conservation and the nonnegativity of the kernel and the signal as additional constraints. We apply the idea of Bregman iterative regularization, first used for image restoration by Osher and colleagues [S.J. Osher, M. Burger, D. Goldfarb, J.J. Xu, and W. Yin, An iterated regularization method for total variation based on image restoration, UCLA CAM Report, 04‐13, (2004)]. to recover finer scales. We also present an analytical study of the model discussing uniqueness of the solution, convergence to steady state and a priori parameter estimation. We present a simple algorithmic implementation of the model and we perform a series of numerical experiments to show evidence of the good behavior of the numerical scheme and quality of the results, improving on results obtained by Chan and Wang [T.F. Chan and C.K. Wong, Total variation blind deconvolution, IEEE Trans Image Process 7 (1998), 370–375].

MRI resolution enhancement using total variation regularization

In this paper we consider a conservative extension of the Euler equations for gas dynamics to describe a two-component compressible flow in Cartesian coordinates. It is well known that classical shock-capturing schemes applied to conservative models are oscillatory near the interface between the two gases. Several authors have addressed this problem proposing either a primitive consistent algorithm [J. Comput. Phys. 112 (1994) 31] or Lagrangian ingredients (Ghost Fluid Method by Fedkiw et al. [J. Comput. Phys. 152 (1999) 452] and [J. Comput. Phys. 169 (2001) 594]). We solve directly this conservative model by a flux-split algorithm, due to the first author (see [J. Comput. Phys. 125 (1996) 42]), together with a high-order (WENO5) flux reconstruction [J. Comput. Phys. 115 (1994) 200; 83 (1989) 32]. This algorithm seems to reduce the oscillations near the interfaces in a way that does not affect the physics of the experiments. We validate our algorithm with the numerical simulation of the interaction of a Mach 1.22 shock wave impinging a helium bubble in air, under the same conditions studied by Haas and Sturtevant [J. Fluid Mech. 181 (1987) 41] and successfully simulated by Quirk and Karni [J. Fluid Mech. 318 (1996) 129].