on a fractional reaction diffusion system applied to image decomposition and restoration abstract in this paper we propose a new model using a fractional reaction diffusion system for image restoration and image decomposition which is motivated by work of s osher a sole and l vese s our model decomposes the degraded image into cartoon component belonging to a fractional sobolev space and textured component belonging to a negative hilbert space we are interested in the existence and uniqueness of weak solutions of the system obtained from the model to overcome the difficulties of the singularity in the system we use a regularization method we also give a numerical algorithm for solving the system and present numerical experimental results to show the efficiency of our model

interacting growth walk on a honeycomb lattice grows indefinitely in this paper we consider igw on a honeycomb lattice we take contact energy see text as 1 we show that igw at 1 t 0 is identical to interacting self avoiding walk isaw at ln4 kbt 1 ln4 0 7213 also igw at 0 t 1 corresponds to isaw at ln2 kbt 1 ln2 1 4427 for other temperatures we need to introduce a statistical weight factor to a walk of the igw ensemble to make correspondence with the isaw ensemble we shall be concerned with a linear homo polymer chain modelled by a lattice self avoiding walk saw let z denote the coordination number of the lattice an ensemble of saw is generated by the following simple non reversing blind ant algorithm a blind ant starts at a site say origin the ant steps into one of the z nearest neighbour sites with a probability p 1 z since the ant never reverses its step in the second and subsequent steps it moves to one of the z 1 nearest neighbour sites with a probability z 1 1 if the ant finds that the site has been visited earlier then the walk is terminated and we start all over again sample loss due to violation of self avoidance is called the problem of attrition because of which growing of large number of long polymer chains becomes difficult thus a walk havingn step is generated with a probability given by p saw c 1 1 z 1 n 1

laplacian and diffusional growth a unified theoretical description for symmetrical and parity broken patterns abstract nowadays laplacian growth patterns such as the saffman taylor instability are well understood mainly because of exact results derived by conformal mapping techniques this is not the case for crystal growth processes except for the free needle crystal normally it is thought that the crystal growth process can be understood by an analogy with the equivalent steady laplacian process the results that follow from this analogy however are in contradiction with numerical and experimental results even at very low velocities we present here a way to mimic the diffusion instabilities in a simple way which allow to describe the formation and the dynamics of new patterns we show that the problem of crystal growth in a channel is closely related to the saffman taylor problem in a sector by using this analogy we obtain the family of zero surface tension solutions for small peclet numbers including an asymmetric degree of freedom we study analytically the selection problem and confirm that parity broken solutions originate from the symmetrical solutions through a bifurcation in agreement with recent numerical investigations we obtain scaling laws for the parity broken solutions for a wide range of parameters we extend these results to the infinite geometry and show that parity broken double fingers can exist even in the range of small undercooling with fully isotropic surface tension the scaling law for the selected velocity of double fingering structures is very different from the scaling law of a free dendrite with anisotropy

selection of saffman taylor fingers in the sector geometry we study numerically and analytically the selection of saffman taylor fingers in cells having the shape of a disk sector for divergent fingers we find results which are qualitatively new compared to the standard linear geometry instead of a discrete set of solutions all converging to a relative width of 1 2 for zero surface tension we find that the fingers disappear below some surface tension this happens by the merging of neighboring branches of the discrete set of solutions this behavior is in quantitative agreement with experimental results

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

mean field diffusion limited aggregation a density model for viscous fingering phenomena we explore a universal density formalism to describe nonequilibrium growth processes specifically the immiscible viscous fingering in hele shaw cells usually referred to as the saffman taylor problem for that we develop an alternative approach to the viscous fingering phenomena whose basic concepts have been recently published in a rapid communication phys rev e 63 045305 r 2001 this approach uses the diffusion limited aggregation dla paradigm as a core we introduce a mean field dla generalization in stochastic and deterministic formulations the stochastic model a quasicontinuum dla simulates monte carlo patterns which demonstrate a striking resemblance to natural hele shaw fingers and for steady state growth regimes follow precisely the saffman taylor analytical solutions in channel and sector configurations the relevant deterministic theory a complete set of differential equations for a time development of density fields is derived from that stochastic model as a principal conclusion we prove an asymptotic equivalency of both the stochastic and deterministic mean field dla formulations to the classic saffman taylor hydrodynamics in terms of an interface evolution

dimension estimation of discrete time fractional brownian motion with applications to image texture classification fractional brownian motion fbm is a suitable description model for a large number of natural shapes and phenomena in applications it is imperative to estimate the fractal dimension from sampled data namely discrete time fbm dfbm to this aim the increment of dfbm referred to as discrete time fractional gaussian noise dfgn is invoked as an auxiliary tool the regular part of dfgn is first filtered out via levinson s algorithm the power spectral density of the regular process is found to satisfy a power law that its exponent can be well fitted by a quadratic function of fractal dimension a new method is then proposed to estimate the fractal dimension of dfbm from the given data set the computational complexity and statistical properties are investigated moreover the proposed algorithm is robust with respect to amplitude scaling and shifting as well as time shifting on the data finally the effectiveness of this estimator is demonstrated via a classification problem of natural texture images

estimation of 1 f noise several models have emerged for describing 1 f sup spl gamma noise processes based on these various techniques for estimating the properties of such processes have been developed this paper provides theoretical analysis of a new wavelet based approach which has the advantages of having low computational complexity and being able to handle the case where the 1 f sup spl gamma noise might be embedded in a further white noise process however the analysis conducted here shows that these advantages are balanced by the fact that the wavelet based scheme is only consistent for spectral exponents spl gamma in the range spl gamma spl isin 0 1 this is in contradiction to the results suggested in previous empirical studies when spl gamma spl isin 0 1 this paper also establishes that wavelet based maximum likelihood methods are asymptotically gaussian and efficient finally the asymptotic rate of mean square convergence of the parameter estimates is established and is shown to slow as spl gamma approaches one combined with a survey of non wavelet based methods these new results give a perspective on the various tradeoffs to be considered when modeling and estimating 1 f sup spl gamma noise processes

fast cartoon texture image filters can images be decomposed into the sum of a geometric part and a textural part in a theoretical breakthrough y meyer oscillating patterns in image processing and nonlinear evolution equations providence ri american mathematical society 2001 proposed variational models that force the geometric part into the space of functions with bounded variation and the textural part into a space of oscillatory distributions meyer s models are simple minimization problems extending the famous total variation model however their numerical solution has proved challenging it is the object of a literature rich in variants and numerical attempts this paper starts with the linear model which reduces to a low pass high pass filter pair a simple conversion of the linear filter pair into a nonlinear filter pair involving the total variation is introduced this new proposed nonlinear filter pair retains both the essential features of meyer s models and the simplicity and rapidity of the linear model it depends upon only one transparent parameter the texture scale measured in pixel mesh comparative experiments show a better and faster separation of cartoon from texture one application is illustrated edge detection

a fractional laplace equation regularity of solutions and finite element approximations this paper deals with the integral version of the dirichlet homogeneous fractional laplace equation for this problem weighted and fractional sobolev a priori estimates are provided in terms of the holder regularity of the data by relying on these results optimal order of convergence for the standard linear finite element method is proved for quasi uniform as well as graded meshes some numerical examples are given showing results in agreement with the theoretical predictions

pruned enriched rosenbluth method simulations of θ polymers of chain length up to 1 000 000 we present an algorithm for simulating flexible chain polymers it combines the rosenbluth rosenbluth method with recursive enrichment although it can be applied also in more general situations it is most efficient for three dimensional theta polymers on the simple cubic lattice there it allows high statistics simulations of chains of length up to n 10 sup 6 for storage reasons this is feasable only for polymers in a finite volume for free theta polymers in infinite volume we present very high statistics runs with n 10000 these simulations fully agree with previous simulations made by hegger and grassberger j chem phys bold 102 6681 1995 with a similar but less efficient algorithm showing that logarithmic corrections to mean field behavior are much stronger than predicted by field theory but the finite volume simulations show that the density inside a collapsed globule scales with the distance from the theta point as predicted by mean field theory in contrast to claims in the work mentioned above in addition to the simple cubic lattice we also studied two versions of the bond fluctuation model but with much shorter chains finally we show that our method can be applied also to off lattice models and illustrate this with simulations of amore model studied in detail by freire ital et al macromolecules bold 19 452 1986 and later work copyright ital 1997 ital the american physical society less

noise removal using fourth order partial differential equation with applications to medical magnetic resonance images in space and time we introduce a new method for image smoothing based on a fourth order pde model the method is tested on a broad range of real medical magnetic resonance images both in space and time as well as on nonmedical synthesized test images our algorithm demonstrates good noise suppression without destruction of important anatomical or functional detail even at poor signal to noise ratio we have also compared our method with related pde models

collapse of two dimensional linear polymers we study the collapse transition of a two dimensional very long polymer the model we consider is a lattice model where the chain is represented by a self avoiding walk with nearest neighbor attraction by using the transfer matrix technique we calculate exactly the thermal and geometrical properties of the polymer on strips of finite width we then use finite size scaling to determine the values of the tricritical point exponentsv1 0 55 0 01vu 1 15 0 15 1 v1 1 80 0 05 we compare these results to the other values already proposed in the literature

fourth order partial differential equations for noise removal a class of fourth order partial differential equations pdes are proposed to optimize the trade off between noise removal and edge preservation the time evolution of these pdes seeks to minimize a cost functional which is an increasing function of the absolute value of the laplacian of the image intensity function since the laplacian of an image at a pixel is zero if the image is planar in its neighborhood these pdes attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image piecewise planar images look more natural than step images which anisotropic diffusion second order pdes uses to approximate an observed image so the proposed pdes are able to avoid the blocky effects widely seen in images processed by anisotropic diffusion while achieving the degree of noise removal and edge preservation comparable to anisotropic diffusion although both approaches seem to be comparable in removing speckles in the observed images speckles are more visible in images processed by the proposed pdes because piecewise planar images are less likely to mask speckles than step images and anisotropic diffusion tends to generate multiple false edges speckles can be easily removed by simple algorithms such as the one presented in this paper

a new interpretation and improvement of the nonlinear anisotropic diffusion for image enhancement the purpose of the article is to give an analysis of the anisotropic diffusion ad and propose adaptive nonlinear filtering based on a judicious choice of the conductance function cf and the edgeness threshold a new undesirable effect which we call the pinhole effect may result when ad is introduced for the first time a robust solution to this effect is proposed and evaluated through experimental data the evolution of the diffused signal is analyzed through a physical model using the optical flow technique oft the overall strategy is evaluated through experimental results obtained on synthetic and actual images

adaptive smoothing respecting feature directions accurate extraction of the image feature directions is essential to image smoothing and other image processing tasks we show that the gradient based feature direction extraction method can be very erroneous the gradient is too local and it cannot detect oscillations we have developed two new methods the hessian method an approach using higher order differentiations and the gabor 1946 method an approach using space frequency analysis

fractional order anisotropic diffusion for image denoising this paper introduces a new class of fractional order anisotropic diffusion equations for noise removal these equations are euler lagrange equations of a cost functional which is an increasing function of the absolute value of the fractional derivative of the image intensity function so the proposed equations can be seen as generalizations of second order and fourth order anisotropic diffusion equations we use the discrete fourier transform to implement the numerical algorithm and give an iterative scheme in the frequency domain it is one important aspect of the algorithm that it considers the input image as a periodic image to overcome this problem we use a folded algorithm by extending the image symmetrically about its borders finally we list various numerical results on denoising real images experiments show that the proposed fractional order anisotropic diffusion equations yield good visual effects and better signal to noise ratio

noise removal with gauss curvature driven diffusion in this paper we propose the use of the gauss curvature in a gauss curvature driven diffusion equation for noise removal the proposed scheme uses the gauss curvature as the conductance term and controls the amount of diffusion the main advantage of the scheme is that it preserves important structures such as straight edges curvy edges ramps corners small scaled features etc

adaptive smoothing via contextual and local discontinuities a novel adaptive smoothing approach is proposed for noise removal and feature preservation where two distinct measures are simultaneously adopted to detect discontinuities in an image inhomogeneity underlying an image is employed as a multiscale measure to detect contextual discontinuities for feature preservation and control of the smoothing speed while local spatial gradient is used for detection of variable local discontinuities during smoothing unlike previous adaptive smoothing approaches two discontinuity measures are combined in our algorithm for synergy in preserving nontrivial features which leads to a constrained anisotropic diffusion process that inhomogeneity offers intrinsic constraints for selective smoothing thanks to the use of intrinsic constraints our smoothing scheme is insensitive to termination times and the resultant images in a wide range of iterations are applicable to achieve nearly identical results for various early vision tasks our algorithm is formally analyzed and related to anisotropic diffusion comparative results indicate that our algorithm yields favorable smoothing results and its application in extraction of hydrographic objects demonstrates its usefulness as a tool for early vision

the finite ridgelet transform for image representation the ridgelet transform was introduced as a sparse expansion for functions on continuous spaces that are smooth away from discontinuities along lines we propose an orthonormal version of the ridgelet transform for discrete and finite size images our construction uses the finite radon transform frat as a building block to overcome the periodization effect of a finite transform we introduce a novel ordering of the frat coefficients we also analyze the frat as a frame operator and derive the exact frame bounds the resulting finite ridgelet transform frit is invertible nonredundant and computed via fast algorithms furthermore this construction leads to a family of directional and orthonormal bases for images numerical results show that the frit is more effective than the wavelet transform in approximating and denoising images with straight edges

collapse transition and crossover scaling for self avoiding walks on the diamond lattice a monte carlo study of self avoiding walks on the diamond lattice is presented this model incorporates some of the steric effects and short range stiffness of real alkanes and for a nearest neighbour attractive interaction epsilon is found to have a collapse transition at kb theta epsilon 2 25 or 0 05 the behaviour of the chains in the vicinity of this theta temperature is analysed with the help of recent crossover scaling theories it is shown that for finite chain length n there is a rather broad theta region where the chains behaviour is quasi ideal the width of this region wtheta behaves as omega 0 varies as n 12 consistent with the blob picture the peak of the specific heat occurs at the boundary between the theta region and the region of collapsed chains the authors also give rough estimates for the scaling functions describing the crossovers of the end to end distance and structure factor of the chains

fractional splines and wavelets we extend schoenberg s family of polynomial splines with uniform knots to all fractional degrees alpha 1 these splines which involve linear combinations of the one sided power functions x alpha max 0 x alpha are alpha holder continuous for alpha 0 we construct the corresponding b splines by taking fractional finite differences and provide an explicit characterization in both time and frequency domains we show that these functions satisfy most of the properties of the traditional b splines including the convolution property and a generalized fractional differentiation rule that involves finite differences only we characterize the decay of the b splines that are not compactly supported for nonintegral alpha s their most astonishing feature in reference to the strang fix theory is that they have a fractional order of approximation alpha 1 while they reproduce the polynomials of degree lceil alpha rceil for alpha frac 1 2 they satisfy all the requirements for a multiresolution analysis of ll 2 riesz bounds two scale relation and may therefore be used to build new families of wavelet bases with a continuously varying order parameter our construction also yields symmetrized fractional b splines which provide the connection with duchon s general theory of radial m s splines including thin plate splines in particular we show that the symmetric version of our splines can be obtained as the solution of a variational problem involving the norm of a fractional derivative

numerical approximations for a nonlocal evolution equation in this paper we study numerical approximations of continuous solutions to the nonlocal p laplacian type diffusion equation u t t x int omega j x y u t y u t x p 2 u t y u t x dy first we find that a semidiscretization in space of this problem gives rise to an ode system whose solutions converge uniformly to the continuous one as the mesh size goes to zero moreover the semidiscrete approximation shares some properties of the continuous problem it preserves the total mass and the solution converges to the mean value of the initial condition as t goes to infinity next we also discretize the time variable and present a totally discrete method which also enjoys the above mentioned properties in addition we investigate the limit as p goes to infinity in these approximations and obtain a discrete model for the evolution of a sandpile finally we present some numerical experiments that illustrate our results

the dirichlet problem for the fractional p laplacian evolution equation abstract we consider a model of fractional diffusion involving a natural nonlocal version of the p laplacian operator we study the dirichlet problem posed in a bounded domain of r n with zero data outside of for which the existence and uniqueness of strong nonnegative solutions is proved and a number of quantitative properties are established a main objective is proving the existence of a special separate variable solution u x t t 1 p 2 f x called the friendly giant which produces a universal upper bound and explains the large time behaviour of all nontrivial nonnegative solutions in a sharp way moreover the spatial profile f of this solution solves an interesting nonlocal elliptic problem we also prove everywhere positivity of nonnegative solutions with any nontrivial data a property that separates this equation from the standard p laplacian equation

flatigw an inverse algorithm to compute the density of states of lattice self avoiding walks we show that the density of states dos for lattice self avoiding walks can be estimated by using an inverse algorithm called flatigw whose step growth rules are dynamically adjusted by requiring the energy histogram to be locally flat here the attractive energy associated with a configuration is taken to be proportional to the number of non bonded nearest neighbor pairs contacts the energy histogram is able to explicitly direct the growth of a walk because the step growth rule of the interacting growth walk narasimhan et al 2003 5 samples the available nearest neighbor sites according to the number of contacts they would make we have obtained the complex fisher zeros corresponding to the dos estimated for square lattice walks of various lengths and located the temperature by extrapolating the finite size values of the real zeros to their asymptotic value 1 49 reasonably close to the known value 1 50 barkema et al 1998 14

a new iterated two band diffusion equation theory and its application we propose an iterated two band filtering method to solve the selective image smoothing problem we prove that a discrete computation step in an iterated nonlinear diffusion based filtering algorithm is equivalent to a sequence of operations including decomposition regularization and then reconstruction in the proposed two band filtering scheme to correctly separate the high frequency components from the low frequency ones in the decomposition process we adopt a dyadic wavelet based approximation scheme in the regularization process we use a diffusivity function as a guide to retain useful data and suppress noises finally the signal of the next stage which is a smoother version of the signal at the previous stage can be computed by reconstructing the decomposed low frequency component and the regularized high frequency component based on the proposed scheme the smoothing operation can be applied to the correct targets experimental results show that our new approach is really efficient in noise removing

noise removal using smoothed normals and surface fitting in this work we use partial differential equation techniques to remove noise from digital images the removal is done in two steps we first use a total variation filter to smooth the normal vectors of the level curves of a noise image after this we try to find a surface to fit the smoothed normal vectors for each of these two stages the problem is reduced to a nonlinear partial differential equation finite difference schemes are used to solve these equations a broad range of numerical examples are given in the paper

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