scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

a generalized formula for integrable classes of surfaces in lie algebras we discuss relations between the approach of fokas and gelfand to immersions on lie algebras and the theory of soliton surfaces of sym we show that many results concerning immersions on lie algebras can be reduced to or interpreted within the soliton surfaces approach we present also some new results including a generalization of the fokas gelfand formula for integrable classes of surfaces in lie algebras and in particular in pseudo euclidean n dim spaces the generalized formula is used to formulate a method of constructing integrable classes of surfaces as an example we discuss the class of linear weingarten surfaces defined by the linear relationship between gaussian and mean curvatures we construct explicitly a one parameter family of linear weingarten surfaces parallel equidistant to a given pseudospherical surface

some remarks about perona malik equation abstract we study the cauchy problem for the one dimensional perona malik equation see 1 1 below with periodic boundary conditions it was proved see siam j appl math 57 5 1997 1328 1342 that in general case no global weak solution exists in spite of good numerical results obtained by several authors see ieee trans pattern anal machine intell 12 1990 629 639 siam j numer anal 29 3 1992 845 866 29 1 1992 182 193 the present paper deals with the existence of solutions in a weaker sense the outline of the paper following some suggestions contained in su alcuni problemi instabili legati alla teoria della visione manuscript is based on the study of a discretized form of the problem involving ordinary differential equations in a suitable banach space jump discontinuities of the initial datum are allowed we prove that the piecewise linear interpolations of the corresponding solutions obey the maximum principle while their essential total variation is decreasing in time we furthermore establish an a priori bound for their time derivative and obtain the convergence of a subsequence to an ultraweak solution see definitions at the end of section 1

harmonic maps backlund darboux transformations and line solution analogues harmonic maps from or one connected domain into and u m are treated the gbdt version of the b cklund darboux transformation is applied to the case of the harmonic maps and a new and simple algebraic procedure to construct new harmonic maps from the initial ones is given using some methods from system theory a new general formula on the gbdt transformations of the sym tafel immersions is derived a new class of the unitary harmonic maps with asymptotics along one line essentially different from the asymptotics in all other directions similar in certain ways to line solutions is obtained explicitly and studied

iterative parameter choice and multigrid methods for anisotropic diffusion denoising anisotropic diffusion methods are well known for giving good qualitative results for image denoising this paper gives a review of the anisotropic diffusion methodology and its application to image denoising we propose a fixed point iteration using a multigrid solver to solve a regularized anisotropic diffusion equation which is not only well posed but also has a nontrivial steady state solution a new regularization parameter choice method brent ncp combining brent s method and the normalized cumulative periodogram information of the misfit is also introduced we test our algorithm on several common test images with different noise levels the experimental results demonstrate the effectiveness of the anisotropic diffusion with a multigrid approach and the broad applicability of the brent ncp parameter choice algorithm

a discrete model for an ill posed nonlinear parabolic pde we study a finite difference discretization of an ill posed nonlinear parabolic partial differential equation the pde is the one dimensional version of a simplified two dimensional model for the formation of shear bands via anti plane shear of a granular medium for the discretized initial value problem we derive analytically and observed numerically a two stage evolution leading to a steady state i an initial growth of grid scale instabilities and ii coarsening dynamics elaborating the second phase at any fixed time the solution has a piecewise linear profile with a finite number of shear bands in this coarsening phase one shear band after another collapses until a steady state with just one jump discontinuity is achieved the amplitude of this steady state shear band is derived analytically but due to the ill posedness of the underlying problem its position exhibits sensitive dependence analyzing data from the simulations we observe that the number of shear bands at time t decays like t 1 3 from this scaling law we show that the time scale of the coarsening phase in the evolution of this model for granular media critically depends on the discreteness of the model our analysis also has implications to related ill posed nonlinear pdes for the one dimensional perona malik equation in image processing and to models for clustering instabilities in granular materials 2001 elsevier science b v all rights reserved

the perona malik paradox the perona malik equation is a formally ill posed parabolic equation for which simple discretizations are nevertheless numerically found to be stable after discussing the background of this parad

interior peak solutions for an elliptic system of fitzhugh nagumo type abstract we consider the dirichlet problem for an elliptic system of fitzhugh nagumo type we prove that the problem has a solution with a sharp peak inside the domain

a new approach to the darboux backlund transformation versus the standard dressing method we present a new approach to the construction of the darboux matrix this is a generalization of a recently formulated method based on the assumption that the square of the darboux matrix vanishes for some values of the spectral parameter we consider the multisoliton case the reduction problem and the discrete case the relationships between our approach the zakharov shabat dressing method and the neugebauer meinel method are discussed in detail

existence and continuation of solutions for a nonlinear neumann problem abstract in this article we study the existence continuation and bifurcation from infinity of nonconstant solutions for a nonlinear neumann problem we apply the leray schauder degree and the degree for s o 2 equivariant gradient operators defined by the second author in s rybicki s o 2 degree for orthogonal maps and its applications to bifurcation theory nonlinear anal tma 23 1 1994 83 102

multipeak solutions for the neumann problem of an elliptic system of fitzhugh nagumo type for systems of elliptic equations of fitzhugh nagumo type on bounded domains and with small diffusion in one equation we construct solutions with multiple sharp peaks close to each other and close to but not on the boundary this is a striking contrast to results for scalar equations for some symmetric domains we also construct similar multipeak solutions except that here the peaks are not close to each other

an analysis of the perona malik scheme the perona malik scheme is a numerical procedure for de noising digital pictures in this work we investigate how it evolves piecewise smooth initial data in one space dimension by scaling a natural parameter that appears in the scheme in an appropriate way with respect to the grid size we obtain a meaningful continuum limit the resulting evolution is unusual it involves solving a system of heat equations coupled to each other through nonlinear boundary conditions that become singular at special times leading to gradient blow up for the solutions however the scheme suggests a natural continuation beyond each of the singular times that involves a change in the p d e system we give a convergence proof that applies under some technical restrictions on any bounded interval of time which might include singular times furthermore the proposed continuum limit has some of the essential features observed in applications of the numerical scheme

constrained restoration and the recovery of discontinuities the linear image restoration problem is to recover an original brightness distribution x sup 0 given the blurred and noisy observations y kx sup 0 b where k and b represent the point spread function and measurement error respectively this problem is typical of ill conditioned inverse problems that frequently arise in low level computer vision a conventional method to stabilize the problem is to introduce a priori constraints on x sup 0 and design a cost functional h x over images x which is a weighted average of the prior constraints regularization term and posterior constraints data term the reconstruction is then the image x which minimizes h a prominent weakness in this approach especially with quadratic type stabilizers is the difficulty in recovering discontinuities the authors therefore examine prior smoothness constraints of a different form which permit the recovery of discontinuities without introducing auxiliary variables for marking the location of jumps and suspending the constraints in their vicinity in this sense discontinuities are addressed implicitly rather than explicitly

an algebraic method to construct the darboux matrix we present an effective procedure to construct the 1 soliton darboux matrix our approach based on the zakharov shabat mikhailov s dressing method is especially useful in the case of non canonical normalization and for non isospectral linear problems the construction is divided into two steps first we represent a given linear problem as a system of some algebraic constraints on two matrices in this context we introduce and discuss invariants of the darboux matrix second we derive the darboux matrix demanding that it preserves the algebraic constraints in particular we consider in details the restrictions imposed by various reduction groups on the form of the darboux matrix

algebraic construction of the darboux matrix revisited we present algebraic construction of darboux matrices for 1 1 dimensional integrable systems of nonlinear partial differential equations with a special stress on the nonisospectral case we discuss different approaches to the darboux backlund transformation based on different dependences of the darboux matrix polynomial sum of partial fractions or the transfer matrix form we derive symmetric n soliton formulae in the general case the matrix spectral parameter and dressing actions in loop groups are also discussed we describe reductions to twisted loop groups unitary reductions the matrix lax pair for the kdv equation and reductions of chiral models harmonic maps to su n and to grassmann spaces we show that in the kdv case the nilpotent darboux matrix generates the binary darboux transformation the paper is intended as a review of known results usually presented in a novel context but some new results are included as well e g general compact formulae for n soliton surfaces and linear and bilinear constraints on the nonisospectral lax pair matrices which are preserved by darboux transformations

stability properties of the perona malik scheme the perona malik scheme is a numerical technique for denoising digital images without blurring object boundaries edges in general solutions generated by this scheme do not satisfy a comparison principle we identify conditions under which two solutions initially ordered remain ordered and we state restricted comparison principles these allow us to study stability properties of the scheme we also explore what these results say in the limit as the discretization size goes to 0

group interpretation of the spectral parameter in the case of nonhomogeneous nonlinear schrodinger system a one parameter group inserting the spectral parameter into the nonparametric lax pair of the nonhomogeneous nonlinear schrodinger system is presented and discussed being a nonlocal extension of a lie point transformation this group is probably not contained in any known class of symmetries

multipeak solutions for an elliptic system of fitzhugh nagumo type we consider the dirichlet problem for an elliptic system of fitzhugh nagumo type we prove that the problem has solutions with arbitrary many sharp peaks near the boundary but not on the boundary

young measure solutions and instability of the one dimensional perona malik equation we apply a variational approach to the one dimensional version of the widely used perona malik equation in image processing we rephrase the problem into the one related to the quasiconvex hull of a graph in the space of 2 x 2 matrices m 2x2 we then use the solutions of some heat equations as the centre of the mass for the young measure valued solutions to construct the approximate solutions by using simple laminates the approximate solutions can be viewed as solutions of a perturbation problem by w 1 w p or w 1 w infinity functions the sequences of the approximate solutions generates young measure valued solutions our results also show that the solutions of the one dimensional perona malik equation are unstable under small w 1 w infinity perturbations

nonlinear regularized reaction diffusion filters for denoising of images with textures denoising is always a challenging problem in natural imaging and geophysical data processing in this paper we consider the denoising of texture images using a nonlinear reaction diffusion equation and directional wavelet frames in our model a curvelet shrinkage is used for regularization of the diffusion process to preserve important features in the diffusion smoothing and a wave atom shrinkage is used as the reaction in order to preserve and enhance interesting oriented textures we derive a digital reaction diffusion filter that lives on graphs and show convergence of the corresponding iteration process experimental results and comparisons show very good performance of the proposed model for texture preserving denoising

gradient estimates for the perona malik equation we consider the cauchy problem for the perona malik equation u t mathrm div left frac nabla u 1 nabla u 2 right in a bounded open set omega subseteq mathbb r n with neumann boundary conditions

an effective method to compute n fold darboux matrix and n soliton surfaces fairly simple formulas for an n fold darboux matrix and an n soliton addition to an arbitrary soliton surface embedded in e3 are proven they depend directly on the background wave function and are evidently symmetric with respect to n added solitons as an example an explicit expression for the interaction of n solitons on a single vortex filament is presented moreover there is given a simplified version of the n soliton formulas of neugebauer and meinel

a new nonlocal nonlinear diffusion of image processing abstract a novel nonlocal nonlinear diffusion is analyzed which has proven useful as a denoising tool in image processing the equation can be viewed as a new paradigm for the regularization of the well known perona malik equation the regularization is implemented via nonlinearity intensity reduction through fractional derivatives well posedness in the weak setting is established global existence and convergence to the average holds in the purely diffusive limit whereas an interesting dynamic behavior is engendered by the presence of nontrivial equilibria as the intensity of the nonlinearity is increased and comes close to the one of perona malik

semi implicit complementary volume scheme for solving level set like equations in image processing and curve evolution we introduce linear semi implicit complementary volume numerical scheme for solving level set like nonlinear degenerate diffusion equations arising in image processing and curve evolution problems we study discretization of image selective smoothing equation of mean curvature flow type given by alvarez lions and morel 3 solution of the level set equation of osher and sethian 26 30 is also included in the study we prove l infty and w 1 1 estimates for the proposed scheme and give existence of its generalized solution in every discrete time scale step efficiency of the scheme is given by its linearity and stability preconditioned iterative solvers are used for computing arising linear systems we present computational results related to image processing and plane curve evolution