Patrick Guidotti

Confetti: object-space point blending and splatting

We present Confetti, a novel point-based rendering approach based on object-space point interpolation of densely sampled surfaces. We introduce the concept of a transformation-invariant covariance matrix of a set of points which can efficiently be used to determine splat sizes in a multiresolution point hierarchy. We also analyze continuous point interpolation in object-space and we define a new class of parameterized blending kernels as well as a normalization procedure to achieve smooth blending. Furthermore, we present a hardware accelerated rendering algorithm based on texture mapping and /spl alpha/-blending as well as programmable vertex and pixel-shaders.

Two new nonlinear nonlocal diffusions for noise reduction

Two new nonlocal nonlinear diffusion models for noise reduction are proposed, analyzed and implemented. They are both a close relative of the celebrated Perona-Malik equation. In a way, they can be viewed as a new regularization paradigm for Perona-Malik. They do preserve and enhance the most cherished features of Perona-Malik while delivering well-posed equations which admit a stable natural discretization. Unlike other regularizations, however, certain piecewise smooth functions are (meta)stable equilibria and, as a consequence, their dynamical behavior and that of their discrete implementations can be fully understood and do not lead to any “paradox”. The presence of nontrivial equilibria also explains why blurring is kept in check. One of the models has been proved to be well-posed. Numerical experiments are presented that illustrate the main features of the new models and that provide insight into their interesting dynamical behavior as well as demonstrate their effectiveness as a denoising tool.

Two Enhanced Fourth Order Diffusion Models for Image Denoising

This paper presents two new higher order diffusion models for removing noise from images. The models employ fractional derivatives and are modifications of an existing fourth order partial differential equation (PDE) model which was developed by You and Kaveh as a generalization of the well-known second order Perona-Malik equation. The modifications serve to cure the ill-posedness of the You-Kaveh model without sacrificing performance. Also proposed in this paper is a simple smoothing technique which can be used in numerical experiments to improve denoising and reduce processing time. Numerical experiments are shown for comparison.

Analysis of Wave Propagation in 1D Inhomogeneous Media

In this paper, we consider the one-dimensional inhomogeneous wave equation with particular focus on its spectral asymptotic properties and its numerical resolution. In the first part of the paper, we analyze the asymptotic nodal point distribution of high-frequency eigenfunctions, which, in turn, gives further information about the asymptotic behavior of eigenvalues and eigenfunctions. We then turn to the behavior of eigenfunctions in the high- and low-frequency limit. In the latter case, we derive a homogenization limit, whereas in the first we show that a sort of self-homogenization occurs at high frequencies. We also remark on the structure of the solution operator and its relation to desired properties of any numerical approximation. We subsequently shift our focus to the latter and present a Galerkin scheme based on a spectral integral representation of the propagator in combination with Gaussian quadrature in the spectral variable with a frequency-dependent measure. The proposed scheme yields accurate resolution of both high- and low-frequency components of the solution and as a result proves to be more accurate than available schemes at large time steps for both smooth and nonsmooth speeds of propagation.

Image Restoration with a New Class of Forward-Backward-Forward Diffusion Equations of Perona--Malik Type with Applications to Satellite Image Enhancement

A new class of anisotropic diffusion models is proposed for image processing which can be viewed either as a novel kind of regularization of the classical Perona--Malik model or, as advocated by the authors, as a new independent model. The models are diffusive in nature and are characterized by the presence of both forward and backward regimes. In contrast to the Perona--Malik model, in the proposed model the backward regime is confined to a bounded region, and gradients are only allowed to grow up to a large but tunable size, thus effectively preventing indiscriminate singularity formation, i.e., staircasing. Extensive numerical experiments demonstrate that the method is a viable denoising/deblurring tool. The method is significantly faster than competing state-of-the-art methods and appears to be particularly effective for simultaneous denoising and deblurring. An application to satellite image enhancement is also presented.

A new well-posed nonlinear nonlocal diffusion

A modification of the Perona-Malik equation is proposed for which the local nonlinear diffusion term is replaced by a nonlocal term of slightly reduced “strength”. The new equation is globally well-posed (in spaces of classical regularity) and possesses desirable properties from the perspective of image processing. It admits characteristic functions as (formally linearly “stable”) stationary solutions and can therefore be reliably employed for denoising keeping blurring in check. Its numerical implementation is stable, enhances some of the features of Perona-Malik, and avoids problems known to affect the latter.

Transient instability in Case II diffusion

A well-known model of one-dimensional Case II diffusion is reformulated in two dimensions. This 2-D model is used to study the stability of 1-D planar Case II diffusion to small spatial perturbations. An asymptotic solution based on the assumption of small perturbations and a small driving force is developed. This analysis reveals that while 1-D planar diffusion is indeed asymptotically stable to small spatial perturbations, it may exhibit a transient instability. That is, although any small perturbation is damped out over sufficiently long times, the amplitude of any perturbation initially grows with time.

Object-space point blending and splatting

We present a novel point-based rendering approach based on object-space point interpolation. We introduce the concept of a transformation-invariant covariance matrix of a set of points to efficiently determine splat sizes in a multiresolution hierarchy. We analyze continuous point interpolation in object-space, and define a new class of parametrized blending kernels to achieve smooth blending. Furthermore, we present a hardware accelerated rendering algorithm based on α-texture mapping and α-blending.

A 2-D free boundary problem with onset of a phase and singular elliptic boundary value problems

Abstract. Numerous models of industrial processes, such as diffusion in glassy polymers or solidification phenomena, lead to general one phase free boundary value problems with phase onset.The classical well-posedness of a fast diffusion approximation to the concerned free boundary value problems is proved. The analysis is performed via a singular change of variables leading to a singular system in a fixed domain. An existence and regularity theory for classical solutions is developed for the relevant underlying class of singular elliptic boundary value problems and is then used to prove the well-posedness for the models considered in which these are coupled to Hamilton-Jacobi or to parabolic evolution equations.

Eigenvalue Characterization and Computation for the Laplacian on General 2-D Domains

In this paper, we address the problem of determining and efficiently computing an approximation to the eigenvalues of the negative Laplacian − ▵ on a general domain Ω ⊂ ℝ2 subject to homogeneous Dirichlet or Neumann boundary conditions. The basic idea is to look for eigenfunctions as the superposition of generalized eigenfunctions of the corresponding free space operator, in the spirit of the classical method of particular solutions (MPS). The main novelties of the proposed approach are the possibility of targeting each eigenvalue independently without the need for extensive scanning of the positive real axis and the use of small matrices. This is made possible by iterative inclusion of more basis functions in the expansions and a projection idea that transforms the minimization problem associated with MPS and its variants into a relatively simple zero-finding problem, even for expansions with very few basis functions.