time dependent differential inclusions cocycle attractors and fuzzy differential equations traditional formulations of fuzzy differential equations do not reproduce the rich and varied behavior of crisp differential equations des a recent interpretation in terms of differential inclusions expressed level setwise overcomes this deficiency and opens up for profitable investigation such properties as stability attraction periodicity and the like this is especially important for investigating continuous systems which are uncertain or incompletely specified this paper studies attractors of fuzzy des in terms of cocycles and encompasses both the time dependent and autonomous cases

anatomically informed interpolation of fmri data on the cortical surface abstract analyzing functional magnetic resonance imaging fmri data restricted to the cortical surface is of particular interest for two reasons 1 to increase detection sensitivity using anatomical constraints and 2 to compare or use fmri results in the context of source localization from magneto electro encephalography meeg data which requires data to be projected on the same spatial support designing an optimal scheme to interpolate fmri raw data or resulting activation maps on the cortical surface relies on a trade off between choosing large enough interpolation kernels because of the distributed nature of the hemodynamic response and avoiding mixing data issued from different anatomical structures we propose an original method that automatically adjusts the level of such a trade off by defining interpolation kernels around each vertex of the cortical surface using a geodesic voronoi diagram this voronoi based interpolation method was evaluated using simulated fmri activation maps manually generated on an anatomical mri and compared with a more standard approach where interpolation kernels were defined as local spheres of radius r 3 or 5 mm several validation parameters were considered the spatial resolution of the simulated activation map the spatial resolution of the cortical mesh the level of anatomical functional data misregistration and the location of the vertices within the gray matter ribbon using an activation map at the spatial resolution of standard fmri data robustness to misregistration errors was observed for both methods whereas only the voronoi based approach was insensitive to the position of the vertices within the gray matter ribbon

cortical surface based analysis ii inflation flattening and a surface based coordinate system the surface of the human cerebral cortex is a highly folded sheet with the majority of its surface area buried within folds as such it is a difficult domain for computational as well as visualization purposes we have therefore designed a set of procedures for modifying the representation of the cortical surface to i inflate it so that activity buried inside sulci may be visualized ii cut and flatten an entire hemisphere and iii transform a hemisphere into a simple parameterizable surface such as a sphere for the purpose of establishing a surface based coordinate system r 1999 academic press

differential equations with fuzzy parameters via differential inclusions abstract we give a definition of solutions of ordinary differential equations in r n containing parameters which are described by changing in time fuzzy sets they are defined as fuzzy subsets of the space of absolutely continuous functions we introduce a hypograph metric in the space of fuzzy sets and prove a theorem on continuous dependence of fuzzy solutions on parameters and initial conditions with respect to that metric

characterization of compact subsets of fuzzy sets a characterization of compact subsets is presented for the metric space of normal fuzzy convex fuzzy sets on the base space rn the metric for which is the supremum over the hausdorff distances between corresponding level sets it is shown that a closed subset is compact if and only if it is uniformly support bounded and the corresponding set of support functions is equileftcontinuous in the membership grade variable

on the fuzzy initial value problem abstract the initial value problem x t f t x t x 0 x 0 with fuzzy initial value and with deterministic or fuzzy function f is considered two different approaches viz the extension principle and the use of extremal solutions of deterministic initial value problems are applied generalizations to fuzzy integral equations and fuzzy functional differential equations are indicated

geometry based demosaicking demosaicking is a particular case of interpolation problems where from a scalar image in which each pixel has either the red the green or the blue component we want to interpolate the full color image state of the art demosaicking algorithms perform interpolation along edges but these edges are estimated locally we propose a level set based geometric method to estimate image edges inspired by the image in painting literature this method has a time complexity of o s where s is the number of pixels in the image and compares favorably with the state of the art algorithms both visually and in most relevant image quality measures

geometric atlas modeling the cortex as an organized surface recent atlases of the cortical surface are based on a modelization of the cerebral cortex as a topological sphere this captures effectively its organization as a regular bidimensional sheet of layers parallel to the surface and with perpendicular cortical columns yet while in the vertical direction cortices are almost the same throughout phylia in the sense of its surface the cerebral cortex is one of the most variable and distinctive parts of the nervous system indeed gyri and sulci appear to have a crucial organizing role in an architectonic connectional and functional sense this organization is not explicitly captured by the surface model of the cortex we propose a geometric model of the cortical anatomy based on flat representations of principal sulci obtained from surface reconstructions of mri data and on neuroanatomical and theoretical considerations concerning the folding patterns of the cortex the cortex is modeled by a sphere where primary sulci are included as axes the arrangement of the axes is a simplification of the arrangement of principal sulci observed in flat stereographic representations of the whole cortical surface the position of secondary and tertiary sulci is then defined by a field of orientations parallel and orthogonal to the axes we consider the use of the geometric model as a synthetic reference cortex for addressing reconstructions of cortical surfaces we present a method which establishes a bijection between the geometric model and a cortical surface reconstruction by using the axes of the model as boundary conditions for a set of partial differential equations solved over both surfaces using the geometric model as atlas provides a natural parameterization of the cortical surface that unlike angular coordinates allows for a localization based on the surface distance to its main organizing landmarks and folding patterns

the cauchy problem for continuous fuzzy differential equations abstract we prove a version of the classical peano s theorem for the initial value problem for a fuzzy differential equation in the metric space of normal fuzzy convex sets with distance given by the maximum of the hausdorff distances between level sets

anatomically informed basis functions abstract this paper introduces the general framework concepts and procedures of anatomically informed basis functions aibf a new method for the analysis of functional magnetic resonance imaging fmri data in contradistinction to existing voxel based univariate or multivariate methods the approach described here can incorporate various forms of prior anatomical knowledge to specify sophisticated spatiotemporal models for fmri time series in particular we focus on anatomical prior knowledge based on reconstructed gray matter surfaces and assumptions about the location and spatial smoothness of the blood oxygenation level dependent bold effect after reconstruction of the grey matter surface from an individual s high resolution t1 weighted mri we specify a set of anatomically informed basis functions fit the model parameters for a single time point using a regularized solution and finally make inferences about the estimated parameters over time significant effects induced by the experimental paradigm can then be visualized in the native voxel space or on the reconstructed folded inflated or flattened cortical surface as an example we apply the approach to a fmri study finger opposition task and compare the results to those of a voxel based analysis as implemented in the statistical parametric mapping package spm99 additionally we show using simulated data that the approach offers several desirable features particularly in terms of superresolution and localization

total variation projection with first order schemes this article proposes a new algorithm to compute the projection on the set of images whose total variation is bounded by a constant the projection is computed through a dual formulation that is solved by first order non smooth optimization methods this yields an iterative algorithm that applies iterative soft thresholding to the dual vector field and for which we establish convergence rate on the primal iterates this projection algorithm can then be used as a building block in a variety of applications such as solving inverse problems under a total variation constraint or for texture synthesis numerical results are reported to illustrate the usefulness and potential applicability of our tv projection algorithm on various examples including denoising texture synthesis inpainting deconvolution and tomography problems we also show that our projection algorithm competes favorably with state of the art tv projection methods in terms of convergence speed

fuzzy quasilinear spaces and applications we introduce the concept of fuzzy quasilinear space and fuzzy quasilinear operator moreover we state some properties and give results which extend to the fuzzy context some results of linear functional analysis

existence and uniqueness of solution for fuzzy random differential equations with non lipschitz coefficients in this paper a class of fuzzy random differential equations with non lipschitz coefficients is studied the existence and uniqueness of solutions for fuzzy random differential equations with non lipschitz coefficients is first proved then the dependence of fuzzy random differential equations on initial values is discussed finally the non confluence property of the solution for fuzzy random differential equation is investigated our main tool is the gronwall lemma

notes on fuzzy complex analysis several fuzzy complex analysis problems are discussed pointing out some errors in literature and some weaknesses of introduced concepts a modified approach to fuzzy complex analysis is proposed and its impact is discussed

caratheodory solutions of fuzzy differential equations in this paper the existence of solutions for fuzzy differential equations with right hand side satisfying a caratheodory condition on a class of fuzzy sets is established

interpolation of missing data in image sequences this paper presents a number of model based interpolation schemes tailored to the problem of interpolating missing regions in image sequences these missing regions may be of arbitrary size and of random but known location this problem occurs regularly with archived film material the film is abraded or obscured in patches giving rise to bright and dark flashes known as dirt and sparkle in the motion picture industry both 3 d autoregressive models and 3 d markov random fields are considered in the formulation of the different reconstruction processes the models act along motion directions estimated using a multiresolution block matching scheme it is possible to address this sort of impulsive noise suppression problem with median filters and comparisons with earlier work using multilevel median filters are performed these comparisons demonstrate the higher reconstruction fidelity of the new interpolators

variational approach for the fusion of exposure bracketed pairs when taking pictures of a dark scene with artificial lighting ambient light is not sufficient for most cameras to obtain both accurate color and detail information the exposure bracketing feature usually available in many camera models enables the user to obtain a series of pictures taken in rapid succession with different exposure times the implicit idea is that the user picks the best image from this set but in many cases none of these images is good enough in general good brightness and color information are retained from longer exposure settings whereas sharp details are obtained from shorter ones in this paper we propose a variational method for automatically combining an exposure bracketed pair of images within a single picture that reflects the desired properties of each one we introduce an energy functional consisting of two terms one measuring the difference in edge information with the short exposure image and the other measuring the local color difference with a warped version of the long exposure image this method is able to handle camera and subject motion as well as noise and the results compare favorably with the state of the art

the dirichlet problem for the total variation flow suppose that is an open bounded domain with a lipschitz boundary the purpose of this chapter is to study the dirichlet problem left begin gathered frac partial u partial t div left frac du left du right right in q left 0 infty right times omega hfill u left t x right phi left x right on s left 0 infty right times partial omega hfill u left 0 x right u 0 left x right in x in omega hfill end gathered right 5 1 where u0 l1 and l1 this evolution equation is related to the gradient descent method used to solve the problem begin gathered minimize int omega left du right int omega f udx int partial omega left u phi right d mathcal h n 1 hfill u in bv left omega right hfill end gathered 2 where f l1 l existence for this variational problem was proved in 118 theorem 1 4

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

on the validity of the peano theorem for fuzzy differential equations abstract kaleva 6 attempted to derive the validity of the classical peano theorem over a locally compact subset of en d based on this attempt kloeden 8 concluded that the peano theorem is also valid over en dp this paper challenges both attempts by counterexamples

generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations the usual concept of differentiability of fuzzy number valued functions has the following shortcoming if c is a fuzzy number and g a b r is an usual real valued function differentiable on x 0 a b with g x 0 0 then f x c g x is not differentiable on x 0 in this paper we introduce and study generalized concepts of differentiability of any order n n which solves this shortcoming newton leibnitz type formula is obtained and existence of the solutions of fuzzy differential equations involving generalized differentiability is studied also some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form c g x are given

perceptual color correction through variational techniques in this paper we present a discussion about perceptual based color correction of digital images in the framework of variational techniques we propose a novel image functional whose minimization produces a perceptually inspired color enhanced version of the original the variational formulation permits a more flexible local control of contrast adjustment and attachment to data we show that a numerical implementation of the gradient descent technique applied to this energy functional coincides with the equation of automatic color enhancement ace a particular perceptual based model of color enhancement moreover we prove that a numerical approximation of the euler lagrange equation reduces the computational complexity of ace from o n2 to o nlogn where n is the total number of pixels in the image

image editing with spatiograms transfer histogram equalization is a well known method for image contrast enhancement nevertheless as histograms do not include any information on the spatial repartition of colors their application to local image editing problems remains limited to cope with this lack of spatial information spatiograms have been recently proposed for tracking purposes a spatiogram is an image descriptor that combines a histogram with the mean and the variance of the position of each color in this paper we address the problem of local retouching of images by proposing a variational method for spatiogram transfer more precisely a reference spatiogram is used to modify the color value of a given region of interest of the processed image experiments on shadow removal and inpainting demonstrate the strength of the proposed approach

stability and periodicity in fuzzy differential equations formulations of fuzzy differential equations des in terms of the hukuhara derivative do not reproduce the rich and varied behavior of crisp des another interpretation in terms of differential inclusions expressed level setwise overcomes much of this deficiency and opens up for profitable investigation such properties as stability attraction periodicity and the like this is especially important for investigating continuous systems which are uncertain or incompletely specified this paper studies studies lyapunov stability of fuzzy des and the periodicity of the fuzzy solution set for both the time dependent and autonomous cases

highly accurate optic flow computation with theoretically justified warping in this paper we suggest a variational model for optic flow computation based on non linearised and higher order constancy assumptions besides the common grey value constancy assumption also gradient constancy as well as the constancy of the hessian and the laplacian are proposed since the model strictly refrains from a linearisation of these assumptions it is also capable to deal with large displacements for the minimisation of the rather complex energy functional we present an efficient numerical scheme employing two nested fixed point iterations following a coarse to fine strategy it turns out that there is a theoretical foundation of so called warping techniques hitherto justified only on an experimental basis since our algorithm consists of the integration of various concepts ranging from different constancy assumptions to numerical implementation issues a detailed account of the effect of each of these concepts is included in the experimental section the superior performance of the proposed method shows up by significantly smaller estimation errors when compared to previous techniques further experiments also confirm excellent robustness under noise and insensitivity to parameter variations

towards the theory of fuzzy differential equations comparative analysis of two ideologically distinct approaches in the theory of fuzzy differential equations is given domains of applications advantages and shortcomings of those approaches are described new definitions of solutions to fuzzy differential equations are proposed

orientation matching minimization for image denoising and inpainting in this paper we propose an orientation matching functional minimization for image denoising and image inpainting following the two step tv stokes algorithm rahman et al in scale space and variational methods in computer vision pp 473 482 springer heidelberg 2007 tai et al in image processing based on partial differential equations pp 3 22 springer heidelberg 2006 bertalmio et al in proc conf comp vision pattern rec pp 355 362 2001 a regularized tangential vector field with zero divergence condition is first obtained then a novel approach to reconstruct the image is proposed instead of finding an image that fits the regularized normal direction from the first step we propose to minimize an orientation matching cost measuring the alignment between the image gradient and the regularized normal direction this functional yields a new nonlinear partial differential equation pde for reconstructing denoised and inpainted images the equation has an adaptive diffusivity depending on the orientation of the regularized normal vector field providing reconstructed images which have sharp edges and smooth regions the additive operator splitting aos scheme is used for discretizing euler lagrange equations we present the results of various numerical experiments that illustrate the improvements obtained with the new functional

numerical solutions of fuzzy differential equations abstract numerical algorithms for solving fuzzy ordinary differential equations fode are considered a scheme based on the classical euler method is discussed in detail and this is followed by a complete error analysis the algorithm is illustrated by solving several linear and nonlinear fuzzy cauchy problems

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

filling in by joint interpolation of vector fields and gray levels a variational approach for filling in regions of missing data in digital images is introduced the approach is based on joint interpolation of the image gray levels and gradient isophotes directions smoothly extending in an automatic fashion the isophote lines into the holes of missing data this interpolation is computed by solving the variational problem via its gradient descent flow which leads to a set of coupled second order partial differential equations one for the gray levels and one for the gradient orientations the process underlying this approach can be considered as an interpretation of the gestaltist s principle of good continuation no limitations are imposed on the topology of the holes and all regions of missing data can be simultaneously processed even if they are surrounded by completely different structures applications of this technique include the restoration of old photographs and removal of superimposed text like dates subtitles or publicity examples of these applications are given we conclude the paper with a number of theoretical results on the proposed variational approach and its corresponding gradient descent flow

existence and uniqueness of cauchy problem for fuzzy differential equations under dissipative conditions the existence theorem of peano for the fuzzy differential equation x t f t x t x t 0 x 0 does not hold in general except in the special case where the fuzzy number space e n d is finite dimensional 1 or f is assumed to be continuous and bounded 2 in this paper the dissipative type conditions which guarantee the existence theorem for peano are specified based on the existence theorem of approximate solutions to above cauchy problem

on the structure of solutions for fuzzy initial value problem in this paper by means of banach fixed point theorem we establish the existence and uniqueness of solutions for set differential equations the continuous dependence of solutions on initial values and the structural stability of solutions on the basis of these results we define level sets of solutions to fuzzy differential equations by the solutions of set differential equations and then we obtain the corresponding results for the large solutions of the fuzzy initial value problem fivp finally discuss the relationship between small solutions and large solutions of fivp

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

multichannel blind deconvolution of spatially misaligned images existing multichannel blind restoration techniques assume perfect spatial alignment of channels correct estimation of blur size and are prone to noise we developed an alternating minimization scheme based on a maximum a posteriori estimation with a priori distribution of blurs derived from the multichannel framework and a priori distribution of original images defined by the variational integral this stochastic approach enables us to recover the blurs and the original image from channels severely corrupted by noise we observe that the exact knowledge of the blur size is not necessary and we prove that translation misregistration up to a certain extent can be automatically removed in the restoration process

generalized hukuhara differentiability of interval valued functions and interval differential equations abstract in the present paper we use a recently proposed generalization of the hukuhara difference for compact convex sets to introduce and study a generalization of the hukuhara differentiability to the case of interval valued functions we also consider several possible definitions for the derivative of an interval valued function recently proposed in the literature and we study connections between them and their properties using these concepts we study interval differential equations local existence and uniqueness of two solutions is obtained together with characterizations of the solutions of an interval differential equation by ode systems several examples are also shown

fuzzy laplace transform based on the henstock integral and its applications in discontinuous fuzzy systems abstract the existing results on the fuzzy laplace transform and their applications were based on zaheh s decomposition theorem and were formally characterized by the integrals of real valued functions directly that is the existence of the fuzzy laplace transform in essence has not been solved in this article the fuzzy laplace transform is incorporated into the framework of the henstock integral and proposed by use of fuzzy henstock integrals on infinite intervals in addition as a theoretical basis the existence and the basic properties of the fuzzy laplace transform are investigated the convolution of fuzzy valued functions and real valued functions is defined and the convolution theorem of the fuzzy laplace transform is given finally discontinuous fuzzy initial value problems and two kinds of fuzzy volterra integral equations are discussed with use of the fuzzy laplace transform presented in this article

uncertain systems and fuzzy differential equations the study of fuzzy differential equations is initiated and some typical results are given in order to understand the intricacies involved in incorporating fuzziness in the terms of differential equations

numerical solutions of fuzzy differential and integral equations using the embedding method numerical procedures for solving fuzzy differential equations fdes and fuzzy integral equations fies with arbitrary kernels have been investigated sufficient conditions for convergence of the proposed algorithms are given and their applicability is illustrated with examples this work and its conclusions may narrow the gap between the theoretical research on fdes and fies and the practical applications already existing in the design of various fuzzy dynamical systems

nth order fuzzy linear differential equations in this paper a numerical method for solving nth order linear differential equations with fuzzy initial conditions is considered the idea is based on the collocation method the existence theorem of the fuzzy solution is considered this method is illustrated by solving several examples

a peano theorem for fuzzy differential equations with evolving membership grade abstract the peano theorem on the existence without possible uniqueness of solutions has been a perplexing problem in the theory of fuzzy differential equations the difficulty appears to be due to the standard use of the supremum metric defined by the supremum over the hausdorff metric between the level sets of the fuzzy sets another may have been the classical formulation of fuzzy differential equations in terms of the hukuhara derivative of the level sets here a peano theorem is established for fuzzy differential equations formulated in a recent paper by the authors by combining hullermeier s suggestion of defining fuzzy differential equations at each level set via differential inclusions with aubin s morphological equations which allow non local set evolution a major difference from previous publications is the use of the endograph metric end essentially the hausdorff metric between the endographs in r n 0 1 of fuzzy sets instead of the supremum metric another is that the membership grades of the fuzzy sets are also allowed to evolve under the fuzzy differential equations the result applies for a very general class of fuzzy sets without additional assumptions of fuzzy convexity compact supports or even normality

embedding of level continuous fuzzy sets on banach spaces abstract in this paper we present an extension of the minkowski embedding theorem showing the existence of an isometric embedding between the class f c x of compact convex and level continuous fuzzy sets on a real separable banach space x and c 0 1 b x the banach space of real continuous functions defined on the cartesian product between 0 1 and the unit ball b x in the dual space x also by using this embedding we give some applications to the characterization of relatively compact subsets of f c x in particular an ascoli arzela type theorem is proved and applied to solving the cauchy problem x t f t x t x t0 x0 on f c x

detection of missing data in image sequences bright and dark flashes are typical artifacts in degraded motion picture material the distortion is referred to as dirt and sparkle in the motion picture industry this is caused either by dirt becoming attached to the frames of the film or by the film material being abraded the visual result is random patches of the frames having grey level values totally unrelated to the initial information at those sites to restore the film without causing distortion to areas of the frames that are not affected the locations of the blotches must be identified heuristic and model based methods for the detection of these missing data regions are presented in this paper and their action on simulated and real sequences is compared

remarks on peano like theorems for fuzzy differential equations the validity of the classical peano theorem is noted for differential equations on the metric space en dp of normal fuzzy convex sets in rn where dp is the p th mean of the hausdorff distances between corresponding level sets this contrasts with differential equations on the space en d where d is the max metric which is not locally compact so more than just continuity is required to ensure the existence of solutions some additional properties differing from that given by kaleva are mentioned

projection of fmri data onto the cortical surface using anatomically informed convolution kernels as surface based data analysis offer an attractive approach for intersubject matching and comparison the projection of voxel based 3d volumes onto the cortical surface is an essential problem we present here a method that aims at producing representations of functional brain data on the cortical surface from functional mri volumes such representations are for instance required for subsequent cortical based functional analysis we propose a projection technique based on the definition around each node of the gray white matter interface mesh of convolution kernels whose shape and distribution rely on the geometry of the local anatomy for one anatomy a set of convolution kernels is computed that can be used to project any functional data registered with this anatomy therefore resulting in anatomically informed projections of data onto the cortical surface this kernel based approach offers better sensitivity specificity than other classical methods and robustness to misregistration errors influences of mesh and volumes spatial resolutions were also estimated for various projection techniques using simulated functional maps

a variational model for histogram transfer of color images in this paper we propose a variational formulation for histogram transfer of two or more color images we study an energy functional composed by three terms one tends to approach the cumulative histograms of the transformed images the other two tend to maintain the colors and geometry of the original images by minimizing this energy we obtain an algorithm that balances equalization and the conservation of features of the original images as a result they evolve while approaching an intermediate histogram between them this intermediate histogram does not need to be specified in advance but it is a natural result of the model finally we provide experiments showing that the proposed method compares well with the state of the art

existence of fixed points of fuzzy mappings via theory of fuzzy differential equations recently the study of fuzzy differential equations has gained importance due to its application in this paper we prove the existence of a unique solution for fuzzy initial value problems further employing stability theory and contraction mapping principle we obtain fixed points of fuzzy nonlinear mappings