Pep Mulet

A Nonlinear Primal-Dual Method for Total Variation-Based Image Restoration

We present a new method for solving total variation (TV) minimization problems in image restoration. The main idea is to remove some of the singularity caused by the nondifferentiability of the quantity

High-Order Total Variation-Based Image Restoration

|\nabla u|

Total variation image restoration: numerical methods and extensions

in the definition of the TV-norm before we apply a linearization technique such as Newtons method. This is accomplished by introducing an additional variable for the flux quantity appearing in the gradient of the objective function, which can be interpreted as the normal vector to the level sets of the image u. Our method can be viewed as a primal-dual method as proposed by Conn and Overton [ A Primal-Dual Interior Point Method for Minimizing a Sum of Euclidean Norms, preprint, 1994] and Andersen [Ph. D. thesis, Odense University, Denmark, 1995] for the minimization of a sum of Euclidean norms. In addition to possessing local quadratic convergence, experimental results show that the new method seems to be globally convergent.

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

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.

Faster minimization of linear wirelength for global placement

We describe some numerical techniques for the total variation image restoration method, namely a primal-dual linearization for the Euler-Lagrange equations and some preconditioning issues. We also highlight extension of this technique to color images, blind deconvolution and the staircasing effect.

Extensions to Total Variation denoising

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].

Analysis of WENO Schemes for Full and Global Accuracy

A linear wirelength objective more effectively cap- tures timing, congestion, and other global placement considera- tions than a squared wirelength objective. The GORDIAN-L cell placement tool (19) minimizes linear wirelength by first approx- imating the linear wirelength objective by a modified squared wirelength objective, then executing the following loop—1) min- imize the current objective to yield some approximate solution and 2) use the resulting solution to construct a more accurate objective—until the solution converges. This paper shows how to apply a generalization (5), (6) of a 1937 algorithm due to Weiszfeld (22) to placement with a linear wirelength objective and that the main GORDIAN-L loop is actually a special case of this algorithm. We then propose applying a regularization parameter to the generalized Weiszfeld algorithm to control the tradeoff between convergence and solution accuracy; the GORDIAN-L iteration is equivalent to setting this regularization parameter to zero. We also apply novel numerical methods, such as the primal- Newton and primal-dual Newton iterations, to optimize the linear wirelength objective. Finally, we show both theoretically and empirically that the primal-dual Newton iteration stably attains quadratic convergence, while the generalized Weiszfeld iteration is linear convergent. Hence, primal-dual Newton is a superior choice for implementing a placer such as GORDIAN-L, or for any linear wirelength optimization.

Adaptive interpolation of images

The total variation denoising method, proposed by Rudin, Osher and Fatermi, 92, is a PDE-based algorithm for edge-preserving noise removal. The images resulting from its application are usually piecewise constant, possibly with a staircase effect at smooth transitions and may contain significantly less fine details than the original non-degraded image. In this paper we present some extensions to this technique that aim to improve the above drawbacks, through redefining the total variation functional or the noise constraints.

Hyperbolicity Analysis of Polydisperse Sedimentation Models via a Secular Equation for the Flux Jacobian

Liu, Osher, and Chan introduced weighted essentially nonoscillatory (WENO) reconstructions in [X.-D. Liu, S. Osher, and T. Chan, J. Comput. Phys., 115 (1994), pp. 200-212] to improve the order of accuracy of essentially nonoscillatory (ENO) reconstructions [A. Harten et al., J. Comput. Phys., 71 (1987), pp. 231-303]. In [G.-S. Jiang and C.-W. Shu, J. Comput. Phys., 126 (1996), pp. 202-228], the authors proposed smoothness indicators to obtain a WENO fifth order reconstruction from third order ENO reconstructions. With these smoothness indicators, Balsara and Shu [J. Comput. Phys., 160 (2000), pp. 405-452] and, later, [G. A. Gerolymos, D. Senechal, and I. Vallet, J. Comput. Phys., 228 (2009), pp. 8481-8524] obtained

Characteristic-Based Schemes for Multi-Class Lighthill-Whitham-Richards Traffic Models

(2r-1)