compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

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

artistic edge and corner enhancing smoothing two important visual properties of paintings and painting like images are the absence of texture details and the increased sharpness of edges as compared to photographic images painting like artistic effects can be achieved from photographic images by filters that smooth out texture details while preserving or enhancing edges and corners however not all edge preserving smoothers are suitable for this purpose we present a simple nonlinear local operator that generalizes both the well known kuwahara filter and the more general class of filters known in the literature as ldquocriterion and value filter structure rdquo this class of operators suffers from intrinsic theoretical limitations which give rise to a dramatic instability in presence of noise especially on shadowed areas such limitations are discussed in the paper and overcome by the proposed operator a large variety of experimental results shows that the output of the proposed operator is visually similar to a painting comparisons with existing techniques on a large set of natural images highlight conditions on which traditional edge preserving smoothers fail whereas our approach produces good results in particular unlike many other well established approaches the proposed operator is robust to degradations of the input image such as blurring and noise contamination

edge preserving filtering of images with low photon counts edge preserving filters such as local m smoothers or bilateral filtering are usually designed for gaussian noise this paper investigates how these filters can be adapted in order to efficiently deal with poissonian noise in addition the issue of photometry invariance is addressed by changing the way filter coefficients are normalized the proposed normalization is additive instead of being multiplicative and leads to a strong connection with anisotropic diffusion experiments show that ensuring the photometry invariance leads to comparable denoising performances in terms of the root mean square error computed on the signal

a new statistical approach to detecting significant activation in functional mri there are many ways to detect activation patterns in a time series of observations at a single voxel in a functional magnetic resonance imaging study the critical problem is to estimate the statistical significance which depends on the estimation of both the magnitude of the response to the stimulus and the serial dependence of the time series and especially on the assumptions made in that estimation we show that for experimental designs with periodic stimuli only a few aspects of the serial dependence are important and these can be estimated reliably via nonparametric estimation of the spectral density of the time series whereas existing techniques are biased by their assumptions the linear model with stationary serially dependent errors can be analyzed entirely in frequency domain and doing so provides many insights in particular we introduce a technique to detect periodic activations and show that it has a distribution theory that enables us to assign significance levels down to 1 in 100 000 levels which are needed when a whole brain image is under consideration nonparametric spectral density estimation is shown to be self calibrating and accurate when compared to several other time domain approaches the technique is especially resistant to high frequency artefacts that we have found in some datasets and we demonstrate that time domain approaches may be sufficiently susceptible to these effects to give misleading results the method is easily generalized to handle event related designs we found it necessary to consider the trends in the time series carefully and use nonlinear filters to remove the trends and robust techniques to remove spikes using this in connection with our techniques allows us to detect activations in clumps of a few even one voxel in periodic designs yet produce essentially no false positive detections at any voxels in null datasets

scaling theorems for zero crossings we characterize some properties of the zero crossings of the laplacian of signals in particular images filtered with linear filters as a function of the scale of the filter extending recent work by witkin 16 we prove that in any dimension the only filter that does not create generic zero crossings as the scale increases is the gaussian this result can be generalized to apply to level crossings of any linear differential operator it applies in particular to ridges and ravines in the image intensity in the case of the second derivative along the gradient there is no filter that avoids creation of zero crossings unless the filtering is performed after the derivative is applied

on feller processes with sample paths in besov spaces under mild regularity assumptions on its domain the infinitesimal generator of a feller process is known to be a pseudo differential operator we give a simple condition on the symbol of the generator in order to characterize the smoothness of the sample paths of real valued feller processes in terms of besov spaces b s pq bbb r our result extends previous papers on the paths of gaussian symmetric alpha stable 6 20 and levy processes 11

discriminative analysis of skull morphology in adolescent idiopathic scoliosis patients comparative study with normal controls statistical morphometric analysis is used in this paper to assess the skull morphological difference in adolescent idiopathic scoliosis ais patients as compared with normal controls the skull morphology analysis starts from a 3 d segmentation of the skull in mri data using the adaptive thresholding and mathematical morphological operations the skull vault is then separated from the whole skull surface using a demarcation plane determined by three anthropometrically meaningful landmarks since the segmented skull surface is not sufficiently smooth which will introduce errors in the subsequent shape description we devise a surface smoothing method to improve the quality of the skull surface this mesh smoothing task is formulated as a quadratic optimization problem in which the objective function is designed to be a linear combination of the shape term and the smoothing term thus it overcomes the shrinkage problem in the traditional laplacian smoothing a spherically uniform sampling on the smoothed skull surface is performed afterwards the scale normalized distances from the skull vault centroid to the sample points are defined as features on which the recursive feature elimination strategy is applied to select the features that contribute most to the classification of ais patients and normal controls using the support vector machine the results of this study quantify the skull vault surface changes and provide new evidence of bone formation abnormality in ais patients

nonlinear multiscale analysis of three dimensional echocardiographic sequences introduces a new model for multiscale analysis of space time echocardiographic sequences the proposed nonlinear partial differential equation representing the multiscale analysis filters the sequence while keeping the space time coherent structures it combines the ideas of regularized perona malik anisotropic diffusion and the galilean invariant movie multiscale analysis of alvarez et al arch rat mech anal vol 123 p 200 57 1993 a numerical method for solving the proposed partial differential equation is suggested and its stability is shown computational results on synthesized and real sequences are provided a qualitative and quantitative evaluation of the accuracy of the method is presented

on the equivalence of soft wavelet shrinkage total variation diffusion total variation regularization and sides soft wavelet shrinkage total variation tv diffusion tv regularization and a dynamical system called sides are four useful techniques for discontinuity preserving denoising of signals and images in this paper we investigate under which circumstances these methods are equivalent in the one dimensional case first we prove that haar wavelet shrinkage on a single scale is equivalent to a single step of space discrete tv diffusion or regularization of two pixel pairs in the translationally invariant case we show that applying cycle spinning to haar wavelet shrinkage on a single scale can be regarded as an absolutely stable explicit discretization of tv diffusion we prove that space discrete tv diffusion and tv regularization are identical and that they are also equivalent to the sides system when a specific force function is chosen afterwards we show that wavelet shrinkage on multiple scales can be regarded as a single step diffusion filtering or regularization of the laplacian pyramid of the signal we analyze possibilities to avoid gibbs like artifacts for multiscale haar wavelet shrinkage by scaling the thresholds finally we present experiments where hybrid methods are designed that combine the advantages of wavelets and pde variational approaches these methods are based on iterated shift invariant wavelet shrinkage at multiple scales with scaled thresholds

shearlet based total variation diffusion for denoising we propose a shearlet formulation of the total variation tv method for denoising images shearlets have been mathematically proven to represent distributed discontinuities such as edges better than traditional wavelets and are a suitable tool for edge characterization common approaches in combining wavelet like representations such as curvelets with tv or diffusion methods aim at reducing gibbs type artifacts after obtaining a nearly optimal estimate we show that it is possible to obtain much better estimates from a shearlet representation by constraining the residual coefficients using a projected adaptive total variation scheme in the shearlet domain we also analyze the performance of a shearlet based diffusion method numerical examples demonstrate that these schemes are highly effective at denoising complex images and outperform a related method based on the use of the curvelet transform furthermore the shearlet tv scheme requires far fewer iterations than similar competitors

support vector clustering we present a novel clustering method using the approach of support vector machines data points are mapped by means of a gaussian kernel to a high dimensional feature space where we search for the minimal enclosing sphere this sphere when mapped back to data space can separate into several components each enclosing a separate cluster of points we present a simple algorithm for identifying these clusters the width of the gaussian kernel controls the scale at which the data is probed while the soft margin constant helps coping with outliers and overlapping clusters the structure of a dataset is explored by varying the two parameters maintaining a minimal number of support vectors to assure smooth cluster boundaries we demonstrate the performance of our algorithm on several datasets

growth and holder conditions for the sample paths of feller processes let a d a be the infinitesimal generator of a feller semigroup such that cc n d a and a cc n is a pseudo differential operator with symbol p x satisfying p c 1 2 and imp x c0rep x we show that the associated feller process xt t 0 on n is a semimartingale even a homogeneous diffusion with jumps in the sense of 21 and characterize the limiting behaviour of its trajectories as t 0 and to this end we introduce various indices e g x 0 lim x y 2 p y 0 or x 0 liminf x y 2 e 1 p y e 0 and obtain a s x that lim t 0t 1 s t xs x 0 or according to x or x similar statements hold for the limit inferior and superior and also for t our results extend the constant coefficient i e levy case considered by w pruitt 27

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

bounded self weights estimation method for non local means image denoising using minimax estimators a non local means nlm filter is a weighted average of a large number of non local pixels with various image intensity values the nlm filters have been shown to have powerful denoising performance excellent detail preservation by averaging many noisy pixels and using appropriate values for the weights respectively the nlm weights between two different pixels are determined based on the similarities between two patches that surround these pixels and a smoothing parameter another important factor that influences the denoising performance is the self weight values for the same pixel the recently introduced local james stein type center pixel weight estimation method ljs outperforms other existing methods when determining the contribution of the center pixels in the nlm filter however the ljs method may result in excessively large self weight estimates since no upper bound is assumed and the method uses a relatively large local area for estimating the self weights which may lead to a strong bias in this paper we investigated these issues in the ljs method and then propose a novel local self weight estimation methods using direct bounds lmm db and reparametrization lmm rp based on the baranchik s minimax estimator both the lmm db and lmm rp methods were evaluated using a wide range of natural images and a clinical mri image together with the various levels of additive gaussian noise our proposed parameter selection methods yielded an improved bias variance trade off a higher peak signal to noise psnr ratio and fewer visual artifacts when compared with the results of the classical nlm and ljs methods our proposed methods also provide a heuristic way to select a suitable global smoothing parameters that can yield psnr values that are close to the optimal values

robust estimation approach for blind denoising this work develops a new robust statistical framework for blind image denoising robust statistics addresses the problem of estimation when the idealized assumptions about a system are occasionally violated the contaminating noise in an image is considered as a violation of the assumption of spatial coherence of the image intensities and is treated as an outlier random variable a denoised image is estimated by fitting a spatially coherent stationary image model to the available noisy data using a robust estimator based regression method within an optimal size adaptive window the robust formulation aims at eliminating the noise outliers while preserving the edge structures in the restored image several examples demonstrating the effectiveness of this robust denoising technique are reported and a comparison with other standard denoising filters is presented

path regularity for feller semigroups via gaussian kernel estimates and generalizations to arbitrary semigroups on c0 given 1 1 we present a dyadic growth condition open image in new window on the finite dimensional distributions of operator semigroups on c0 e which for 0 and feller semigroups assures that the corresponding feller process has paths in local holder spaces and in weighted besov spaces of order we show that for operator semigroups satisfying gaussian kernel estimates of order m 1 condition open image in new window holds for all open image in new window and even for all open image in new window in the case of feller semigroups such gaussian kernel estimates are typical for feller semigroups on fractals of walk dimension m and for semigroups generated by elliptic operators on d of order m d

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

singularity theory and phantom edges in scale space the process of detecting edges in a one dimensional signal by finding the zeros of the second derivative of the signal can be interpreted as the process of detecting the critical points of a general class of contrast functions that are applied to the signal it is shown that the second derivative of the contrast function at the critical point is related to the classification of the associated edge as being phantom or authentic the contrast of authentic edges decreases with filter scale while the contrast of phantom edges are shown to increase with scale as the filter scale increases an authentic edge must either turn into a phantom edge or join with a phantom edge and vanish the points in the scale space at which these events occur are seen to be singular points of the contrast function using ideas from singularity or catastrophy theory the scale map contours near these singular points are found to be either vertical or parabolic

stochastic differential equations and geometric flows in previous years curve evolution applied to a single contour or to the level sets of an image via partial differential equations has emerged as an important tool in image processing and computer vision curve evolution techniques have been utilized in problems such as image smoothing segmentation and shape analysis we give a local stochastic interpretation of the basic curve smoothing equation the so called geometric heat equation and show that this evolution amounts to a tangential diffusion movement of the particles along the contour moreover assuming that a priori information about the shapes of objects in an image is known we present modifications of the geometric heat equation designed to preserve certain features in these shapes while removing noise we also show how these new flows may be applied to smooth noisy curves without destroying their larger scale features in contrast to the original geometric heat flow which tends to circularize any closed curve

geometrically induced force interaction for three dimensional deformable models in this paper we propose a novel 3 d deformable model that is based upon a geometrically induced external force field which can be conveniently generalized to arbitrary dimensions this external force field is based upon hypothesized interactions between the relative geometries of the deformable model and the object boundary characterized by image gradient the evolution of the deformable model is solved using the level set method so that topological changes are handled automatically the relative geometrical configurations between the deformable model and the object boundaries contribute to a dynamic vector force field that changes accordingly as the deformable model evolves the geometrically induced dynamic interaction force has been shown to greatly improve the deformable model performance in acquiring complex geometries and highly concave boundaries and it gives the deformable model a high invariancy in initialization configurations the voxel interactions across the whole image domain provide a global view of the object boundary representation giving the external force a long attraction range the bidirectionality of the external force field allows the new deformable model to deal with arbitrary cross boundary initializations and facilitates the handling of weak edges and broken boundaries in addition we show that by enhancing the geometrical interaction field with a nonlocal edge preserving algorithm the new deformable model can effectively overcome image noise we provide a comparative study on the segmentation of various geometries with different topologies from both synthetic and real images and show that the proposed method achieves significant improvements against existing image gradient techniques

variational problems and partial differential equations on implicit surfaces a novel framework for solving variational problems and partial differential equations for scalar and vector valued data defined on surfaces is introduced in this paper the key idea is to implicitly represent the surface as the level set of a higher dimensional function and to solve the surface equations in a fixed cartesian coordinate system using this new embedding function the equations are then both intrinsic to the surface and defined in the embedding space this approach thereby eliminates the need for performing complicated and inaccurate computations on triangulated surfaces as is commonly done in the literature we describe the framework and present examples in computer graphics and image processing applications including texture synthesis flow field visualization and image and vector field intrinsic regularization for data defined on 3d surfaces

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

ellipsoidal support vector clustering for functional mri analysis as an exploratory approach the clustering of fmri time series has proved its effectiveness in analyzing the functional mri especially in the detection of activated regions due to the arbitrary distribution of fmri time series in the temporal domain imposing simple assumption on the data structure usually could be misleading and limit the detector s performance therefore a true data driven clustering algorithm that adapts to the data structure is preferred and only high level control over the clustering procedure is desired support vector clustering svc is a suitable one in some extent because of its advantages such as no cluster shape restriction no need to explicitly specify the number of clusters and the mechanism in outlier elimination in this work we propose an extension of the svc to step further toward a data sensitive detector this approach is named as ellipsoidal support vector clustering esvc to be robust to noise the clustering is performed on features extracted from the fmri time series via fourier transform experimental results on simulated and real data sets demonstrate the effectiveness of incorporating data structure in clustering fmri time series

a global sparse gradient based coupled system for image denoising abstract we consider a coupled system for image denoising which consists of an anisotropic diffusion equation and a global sparse gradient model gsg the global sparse gradient model is used to estimate the edge map from the noisy image accurately and robustly the anisotropic diffusion process is guided by the edge map furthermore we prove the existence and uniqueness of the solutions of the coupled system with dirichlet initial boundary value problem finally comparing with the edge indicators based on local methods experimental results demonstrate that our proposed method has the state of the art performance among the pde based anisotropic diffusion approaches both in objective measurement and visual evaluation

plancherel type estimates and sharp spectral multipliers abstract we study general spectral multiplier theorems for self adjoint positive definite operators on l2 x where x is any open subset of a space of homogeneous type we show that the sharp hormander type spectral multiplier theorems follow from the appropriate estimates of the l2 norm of the kernel of spectral multipliers and the gaussian bounds for the corresponding heat kernel the sharp hormander type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the riesz means summability which we also study here we discuss several examples which include sharp spectral multiplier theorems for a class of scattering operators on r3 and new spectral multiplier theorems for the laguerre and hermite expansions

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

interpreting line drawings of curved objects in this thesis we study the problem of interpreting line drawings of scenes composed or opaque regular solid objects bounded by piecewise smooth surfaces with no markings or texture on them it is assumed that the line drawing has been formed by orthographic projection of such a scene under general viewpoint that the line drawing is error free and that there are no lines due to shadows or specularities our definition implicitly excludes laminae wires and the apices of cones a major component of the interpretation of line drawings is line labelling by line labelling we mean a classification of each image curve as corresponding to either a depth or orientation discontinuity in the scene and b further subclassification of each kind of discontinuity for a depth discontinuity we determine whether it is a limb a locus of points on the surface where the line of sight is tangent to the surface or an occluding edge a tangent plane discontinuity of the surface for an orientation discontinuity we determine whether it corresponds to a convex or concave edge this thesis presents the first mathematically rigorous scheme for labelling line drawings of the class of scenes described by analysing the projection of the neighborhoods of different kinds of points on a piecewise smooth surface we are able to catalog all local labelling possibilities for the different types of junctions in a line drawing an algorithm is developed which utilizes this catalog to determine all legal labellings of the line drawing a local minimum complexity rule at each vertex select those labellings which correspond to the minimum number of faces meeting at the vertex is used in order to prune highly counter intuitive interpretations the labelling scheme was implemented and tested on a number of line drawings the labellings obtained are few and by and large in accordance with human interpretations

on a class of forward backward parabolic equations formation of singularities abstract we study the formation of singularities for the problem u t u x x e u t x x in 0 t u e u t 0 in 0 t u u 0 0 in 0 where and t are positive constants a bounded interval u 0 a nonnegative radon measure on a nonmonotone and nonnegative function with 0 0 and an increasing bounded function we show that if u 0 is a bounded or continuous function singularities may appear spontaneously the class of singularities which can arise in finite time is remarkably large and includes infinitely many dirac masses and singular continuous measures

on clustering fmri time series analysis of fmri time series is often performed by extracting one or more parameters for the individual voxels methods based e g on various statistical tests are then used to yield parameters corresponding to probability of activation or activation strength however these methods do not indicate whether sets of voxels are activated in a similar way or in different ways typically delays between two activated signals are not identified in this article we use clustering methods to detect similarities in activation between voxels we employ a novel metric that measures the similarity between the activation stimulus and the fmri signal we present two different clustering algorithms and use them to identify regions of similar activations in an fmri experiment involving a visual stimulus

the growth of random walks and levy processes let x i be a sequence of independent identically distributed non degenerate random variables taking values in mathbb r d and s n sum n i 1 x i m n max 1 leqq i leqq n s i define for x 0 g x p x 1 x k x x 2 e x 1 2 1 x 1 leq x m x x 1 e x 1 1 x 1 leq x and h x g x k x m x then if beta sup alpha lim sup x alpha h x 0 delta sup alpha lim inf x alpha h x 0 it is proved that n 1 alpha m n rightarrow 0 for alpha delta while the lim inf is 0 and the lim sup is infty for beta alpha delta some alternative characterizations of the indices beta delta are obtained as well as the analogous results for levy processes