linearized multidimensional earth mover s distance gradient flows this paper presents the first framework capable of performing active contour segmentation using earth mover s distance emd to measure dissimilarity between multidimensional feature distributions emd is the best known and understood cross bin histogram distance measure and as such it allows for meaningful comparisons between distributions unlike bin to bin measures that only account for discrepancies on a bin to bin basis because emd is obtained with linear programming techniques its differential structure with respect to variations in bin weights as the active contour evolves is expressed through sensitivity analysis euler lagrange equations are then derived from the computed sensitivity at every iteration to produce gradient descent flows we validate our approach with color image segmentation in comparison with state of the art bhattacharyya bin to bin and 1d emd cross bin active contours some unique advantages of cross bin comparison are highlighted in our segmentation results better perceptual value and increased robustness with respect to the initialization

on the length and area regularization for multiphase level set segmentation in this paper we introduce novel regularization techniques for level set segmentation that target specifically the problem of multiphase segmentation when the multiphase model is used to obtain a partitioning of the image in more than two regions a new set of issues arise with respect to the single phase case in terms of regularization strategies for example if smoothing or shrinking each contour individually could be a good model in the single phase case this is not necessarily true in the multiphase scenario in this paper we address these issues designing enhanced length and area regularization terms whose minimization yields evolution equations in which each level set function involved in the multiphase segmentation can sense the presence of the other level set functions and evolve accordingly in other words the coupling of the level set function which before was limited to the data term i e the proper segmentation driving force is extended in a mathematically principled way to the regularization terms as well the resulting regularization technique is more suitable to eliminate spurious regions and other kind of artifacts an extensive experimental evaluation supports the model we introduce in this paper showing improved segmentation performance with respect to traditional regularization techniques

similarity and dissimilarity in correlations of genomic dna we analyze auto correlations of human chromosomes 1 22 and rice chromosomes 1 12 for seven binary mapping rules and find that the correlation patterns are different for different rules but almost identical for all of the chromosomes despite their varying lengths and gc contents we propose a simple stochastic process for modeling these correlations and we find that the proposed process can reproduce quantitatively and qualitatively the correlation patterns found in the genomes of human and rice

coupling image restoration and segmentation a generalized linear model bregman perspective we introduce a new class of data fitting energies that couple image segmentation with image restoration these functionals model the image intensity using the statistical framework of generalized linear models by duality we establish an information theoretic interpretation using bregman divergences we demonstrate how this formulation couples in a principled way image restoration tasks such as denoising deblurring deconvolution and inpainting with segmentation we present an alternating minimization algorithm to solve the resulting composite photometric geometric inverse problem we use fisher scoring to solve the photometric problem and to provide asymptotic uncertainty estimates we derive the shape gradient of our data fitting energy and investigate convex relaxation for the geometric problem we introduce a new alternating split bregman strategy to solve the resulting convex problem and present experiments and comparisons on both synthetic and real world images

active contours without edges we propose a new model for active contours to detect objects in a given image based on techniques of curve evolution mumford shah 1989 functional for segmentation and level sets our model can detect objects whose boundaries are not necessarily defined by the gradient we minimize an energy which can be seen as a particular case of the minimal partition problem in the level set formulation the problem becomes a mean curvature flow like evolving the active contour which will stop on the desired boundary however the stopping term does not depend on the gradient of the image as in the classical active contour models but is instead related to a particular segmentation of the image we give a numerical algorithm using finite differences finally we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable also the initial curve can be anywhere in the image and interior contours are automatically detected

a review of statistical approaches to level set segmentation integrating color texture motion and shape since their introduction as a means of front propagation and their first application to edge based segmentation in the early 90 s level set methods have become increasingly popular as a general framework for image segmentation in this paper we present a survey of a specific class of region based level set segmentation methods and clarify how they can all be derived from a common statistical framework region based segmentation schemes aim at partitioning the image domain by progressively fitting statistical models to the intensity color texture or motion in each of a set of regions in contrast to edge based schemes such as the classical snakes region based methods tend to be less sensitive to noise for typical images the respective cost functionals tend to have less local minima which makes them particularly well suited for local optimization methods such as the level set method we detail a general statistical formulation for level set segmentation subsequently we clarify how the integration of various low level criteria leads to a set of cost functionals we point out relations between the different segmentation schemes in experimental results we demonstrate how the level set function is driven to partition the image plane into domains of coherent color texture dynamic texture or motion moreover the bayesian formulation allows to introduce prior shape knowledge into the level set method we briefly review a number of advances in this domain

geodesic active regions and level set methods for supervised texture segmentation this paper presents a novel variational framework to deal with frame partition problems in computer vision this framework exploits boundary and region based segmentation modules under a curve based optimization objective function the task of supervised texture segmentation is considered to demonstrate the potentials of the proposed framework the textured feature space is generated by filtering the given textured images using isotropic and anisotropic filters and analyzing their responses as multi component conditional probability density functions the texture segmentation is obtained by unifying region and boundary based information as an improved geodesic active contour model the defined objective function is minimized using a gradient descent method where a level set approach is used to implement the obtained pde according to this pde the curve propagation towards the final solution is guided by boundary and region based segmentation forces and is constrained by a regularity force the level set implementation is performed using a fast front propagation algorithm where topological changes are naturally handled the performance of our method is demonstrated on a variety of synthetic and real textured frames

non logarithmic jensen shannon divergence abstract the jensen shannon divergence is a symmetrized and smoothed version of the kullback leibler divergence recently it has been widely applied to the analysis and characterization of symbolic sequences in this paper we investigate a generalization of the jensen shannon divergence this generalization is done in the framework of the non extensive tsallis statistics we study its basic properties and we investigate its applicability as a tool for segmentating symbolic sequences

on local region models and a statistical interpretation of the piecewise smooth mumford shah functional the mumford shah functional is a general and quite popular variational model for image segmentation in particular it provides the possibility to represent regions by smooth approximations in this paper we derive a statistical interpretation of the full piecewise smooth mumford shah functional by relating it to recent works on local region statistics moreover we show that this statistical interpretation comes along with several implications firstly one can derive extended versions of the mumford shah functional including more general distribution models secondly it leads to faster implementations finally thanks to the analytical expression of the smooth approximation via gaussian convolution the coordinate descent can be replaced by a true gradient descent

computation of discontinuous optical flow by domain decomposition and shape optimization a well known method for the reconstruction of motion fields from noisy image data is to determine flow fields by the minimization of a quadratic functional the first approach of this class has been proposed by horn and schunck 1981 a drawback of such approaches is that an explicit representation of the discontinuities of the motion field is lacking and that in general the resulting flow fields approximate the motion fields only badly at the corresponding locations in the image plane in this article we discuss the possibility to improve the results by hypothesizing the qualitative structure of the motion field in terms of certain parameters we decompose the image plane into disjoint sets restrict the domain of definition of the functionals to these sets and use the hypotheses to deform and to move the boundaries of the sets within the image plane we discuss the range of applicability of this new technique and illustrate the algorithm by numerical examples this article is a revised and extended version of schnorr 1990

stable distributions in stochastic fragmentation we investigate a class of stochastic fragmentation processes involving stable and unstable fragments we solve analytically for the fragment length density and find that a generic algebraic divergence characterizes its small size tail furthermore the entire range of acceptable values of decay exponent consistent with length conservation can be realized we show that the stochastic fragmentation process is non self averaging as moments exhibit significant sample to sample fluctuations additionally we find that the distributions of the moments and of extremal characteristics possess an infinite set of progressively weaker singularities

quasi linear transformations and discrete tilings tilings of the discrete plane z 2 generated by quasi linear transformations qlt have been introduced by nehlig p nehlig applications quasi affines pavages par images reciproques theoretical computer science 156 1995 1 38 we studied these tilings and gave some results such as periodicity and the number of neighbours of each of them m a jacob da col applications quasi affines et pavages du plan discret theoretical computer science 259 2001 245 269 also available in english http dpt info u strasbg fr jacob articles paving pdf the aim of this paper is to go on with this study in the discrete n dimensional space z 2 we give a lower and an upper bound to the number of distinct tiles we also give an algorithm to determine the points of a given tile this algorithm will induce another algorithm to determine the number of distinct tiles associated to a qlt

a geometric model for active contours in image processing we propose a new model for active contours based on a geometric partial differential equation our model is intrinsec stable satisfies the maximum principle and permits a rigorous mathematical analysis it enables us to extract smooth shapes we cannot retrieve angles and it can be adapted to find several contours simultaneously moreover as a consequence of the stability we can design robust algorithms which can be engineed with no parameters in applications numerical experiments are presented

spatiotemporal motion boundary detection and motion boundary velocity estimation for tracking moving objects with a moving camera a level sets pdes approach with concurrent camera motion compensation the purpose of this study is to investigate a method of tracking moving objects with a moving camera this method estimates simultaneously the motion induced by camera movement the problem is formulated as a bayesian motion based partitioning problem in the spatiotemporal domain of the image sequence an energy functional is derived from the bayesian formulation the euler lagrange descent equations determine simultaneously an estimate of the image motion field induced by camera motion and an estimate of the spatiotemporal motion boundary surface the euler lagrange equation corresponding to the surface is expressed as a level set partial differential equation for topology independence and numerically stable implementation the method can be initialized simply and can track multiple objects with nonsimultaneous motions velocities on motion boundaries can be estimated from geometrical properties of the motion boundary several examples of experimental verification are given using synthetic and real image sequences

multi reference shape priors for active contours in this paper we present a new way of constraining the evolution of an active contour with respect to a set of fixed reference shapes this approach is based on a description of shapes by the legendre moments computed from their characteristic function this provides a region based representation that can handle arbitrary shape topologies moreover exploiting the properties of moments it is possible to include intrinsic affine invariance in the descriptor which solves the issue of shape alignment without increasing the number of d o f of the initial problem and allows introducing geometric shape variabilities our new shape prior is based on a distance in terms of descriptors between the evolving curve and the reference shapes minimizing the corresponding shape energy leads to a geometric flow that does not rely on any particular representation of the contour and can be implemented with any contour evolution algorithm we introduce our prior into a two class segmentation functional showing its benefits on segmentation results in presence of severe occlusions and clutter examples illustrate the ability of the model to deal with large affine deformation and to take into account a set of reference shapes of different topologies

a generic framework for tracking using particle filter with dynamic shape prior tracking deforming objects involves estimating the global motion of the object and its local deformations as functions of time tracking algorithms using kalman filters or particle filters pfs have been proposed for tracking such objects but these have limitations due to the lack of dynamic shape information in this paper we propose a novel method based on employing a locally linear embedding in order to incorporate dynamic shape information into the particle filtering framework for tracking highly deformable objects in the presence of noise and clutter the pf also models image statistics such as mean and variance of the given data which can be useful in obtaining proper separation of object and background

a level set model for image classification we present a supervised classification model based on a variational approach this model is devoted to find an optimal partition composed of homogeneous classes with regular interfaces the originality of the proposed approach concerns the definition of a partition by the use of level sets each set of regions and boundaries associated to a class is defined by a unique level set function we use as many level sets as different classes and all these level sets are moving together thanks to forces which interact in order to get an optimal partition we show how these forces can be defined through the minimization of a unique fonctional the coupled partial differential equations pde related to the minimization of the functional are considered through a dynamical scheme given an initial interface set zero level set the different terms of the pde s are governing the motion of interfaces such that at convergence we get an optimal partition as defined above each interface is guided by internal forces regularity of the interface and external ones data term no vacuum no regions overlapping several experiments were conducted on both synthetic and real images

compositional segmentation and long range fractal correlations in dna sequences a segmentation algorithm based on the jensen shannon entropic divergence is used to decompose long range correlated dna sequences into statistically significant compositionally homogeneous patches by adequately setting the significance level for segmenting the sequence the underlying power law distribution of patch lengths can be revealed some of the identified dna domains were uncorrelated but most of them continued to display long range correlations even after several steps of recursive segmentation thus indicating a complex multi length scaled structure for the sequence on the other hand by separately shuffling each segment or by randomly rearranging the order in which the different segments occur in the sequence shuffled sequences preserving the original statistical distribution of patch lengths were generated both types of random sequences displayed the same correlation scaling exponents as the original dna sequence thus demonstrating that neither the internal structure of patches nor the order in which these are arranged in the sequence is critical therefore long range correlations in nucleotide sequences seem to rely only on the power law distribution of patch lengths textcopyright 1996 the american physical society

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

exploiting information geometry to improve the convergence of nonparametric active contours this paper presents a fast converging riemannian steepest descent method for nonparametric statistical active contour models with application to image segmentation unlike other fast algorithms the proposed method is general and can be applied to any statistical active contour model from the exponential family which comprises most of the models considered in the literature this is achieved by first identifying the intrinsic statistical manifold associated with this class of active contours and then constructing a steepest descent on that manifold a key contribution of this paper is to derive a general and tractable closed form analytic expression for the manifold s riemannian metric tensor which allows computing discrete gradient flows efficiently the proposed methodology is demonstrated empirically and compared with other state of the art approaches on several standard test images a phantom positron emission tomography scan and a b mode echography of in vivo human dermis

variational region based segmentation using multiple texture statistics this paper investigates variational region level criterion for supervised and unsupervised texture based image segmentation the focus is given to the demonstration of the effectiveness and robustness of this region based formulation compared to most common variational approaches the main contributions of this global criterion are twofold first the proposed methods circumvent a major problem related to classical texture based segmentation approaches existing methods even if they use different and various texture features are mainly stated as the optimization of a criterion evaluating punctual pixel likelihoods or similarity measure computed within a local neighborhood these approaches require sufficient dissimilarity between the considered texture features an additional limitation is the choice of the neighborhood size and shape these two parameters and especially the neighborhood size significantly influence the classification performances the neighborhood must be large enough to capture texture structures and small enough to guarantee segmentation accuracy these parameters are often set experimentally these limitations are mitigated with the proposed variational methods stated at the region level it resorts to an energy criterion defined on image where regions are characterized by nonparametric distributions of their responses to a set of filters in the supervised case the segmentation algorithm consists in the minimization of a similarity measure between region level statistics and texture prototypes and a boundary based functional that imposes smoothness and regularity on region boundaries in the unsupervised case the data driven term involves the maximization of the dissimilarity between regions the proposed similarity measure is generic and permits optimally fusing various types of texture features it is defined as a weighted sum of kullback leibler divergences between feature distributions the optimization of the proposed variational criteria is carried out using a level set formulation the effectiveness and the robustness of this formulation at region level compared to classical active contour methods are evaluated for various brodatz and natural images

local histogram based segmentation using the wasserstein distance we propose and analyze a nonparametric region based active contour model for segmenting cluttered scenes the proposed model is unsupervised and assumes pixel intensity is independently identically distributed our proposed energy functional consists of a geometric regularization term that penalizes the length of the partition boundaries and a region based image term that uses histograms of pixel intensity to distinguish different regions more specifically the region data encourages segmentation so that local histograms within each region are approximately homogeneous an advantage of using local histograms in the data term is that histogram differentiation is not required to solve the energy minimization problem we use wasserstein distance with exponent 1 to determine the dissimilarity between two histograms the wasserstein distance is a metric and is able to faithfully measure the distance between two histograms compared to many pointwise distances moreover it is insensitive to oscillations and therefore our model is robust to noise a fast global minimization method based on chan et al in siam j appl math 66 5 1632 1648 2006 bresson et al in j math imaging vis 28 2 151 167 2007 is employed to solve the proposed model the advantages of using this method are two fold first the computational time is less than that of the method by gradient descent of the associated euler lagrange equation chan et al in proc of ssvm pp 697 708 2007 second it is able to find a global minimizer finally we propose a variant of our model that is able to properly segment a cluttered scene with local illumination changes

a mumford shah level set approach for the inversion and segmentation of x ray tomography data a level set based approach for the determination of a piecewise constant density function from data of its radon transform is presented simultaneously a segmentation of the reconstructed density is obtained the segmenting contour and the corresponding density are found as minimizers of a mumford shah like functional over the set of admissible contours and for a fixed contour over the space of piecewise constant densities which may be discontinuous across the contour shape sensitivity analysis is used to find a descent direction for the cost functional which leads to an update formula for the contour in the level set framework the descent direction can be chosen with respect to different metrics the use of an l 2 type and an h 1 type metric is proposed and the corresponding steepest descent flow equations are derived a heuristic approach for the insertion of additional components of the density is presented the method is tested for several data sets including synthetic as well as real world data it is shown that the method works especially well for large data noise 10 noise the choice of the h 1 metric for the determination of the descent direction is found to have positive effect on the number of level set steps necessary for finding the optimal contours and densities

an alternative description of power law correlations in dna sequences abstract we analyze the coding sequence for the homo sapiens via a model which naturally embraces power law correlations plc among the bases in dna sequences of living organisms this model is based on a principle of universal optimization which is the core of all statistical arguments being associated with the power law distribution function of the length of dna measured in base pairs bp this distribution provides a plc parameter introduced through a nonadditive framework in which such parameter measures the plc in the dna sequence the results show that the short range correlations src always present in coding dna sequences are appropriately captured through the power law distribution adequately describing the cumulative length distribution of dna bases in contrast with the case of the traditional exponential statistical model we use an empirical cumulative distribution function and the database of proteins compiled by the ensembl project to show that the power law distribution provides the best description of the data a bayesian analysis of the data further confirms this result

a novel level set method for image segmentation by incorporating local statistical analysis and global similarity measurement this paper presents a novel level set method for complex image segmentation where the local statistical analysis and global similarity measurement are both incorporated into the construction of energy functional the intensity statistical analysis is performed on local circular regions centered in each pixel so that the local energy term is constructed in a piecewise constant way meanwhile the bhattacharyya coefficient is utilized to measure the similarity between probability distribution functions for intensities inside and outside the evolving contour the global energy term can be formulated by minimizing the bhattacharyya coefficient to avoid the time consuming re initialization step the penalty energy term associated with a new double well potential is constructed to maintain the signed distance property of level set function the experiments and comparisons with four popular models on synthetic and real images have demonstrated that our method is efficient and robust for segmenting noisy images images with intensity inhomogeneity texture images and multiphase images the intensity statistical analysis is performed on local circular regions the global energy term is formulated by minimizing the bhattacharyya coefficient the penalty energy term associated with a new double well potential is proposed the proposed method is efficient and robust for segmenting complex images

conformation dependent evolution of copolymer sequences a toy model of molecular evolution of sequences in copolymers is proposed and implemented using a molecular dynamics based algorithm the model involves coupling of conformation dependent and sequence dependent properties it is shown that this model allows the realization of two main possibilities ascending and descending branches of evolution in terms of information content of a sequence depending on the interaction parameters shaping the conformation of a polymer globule the problem of adequate description of information complexity of copolymer sequences is studied it is shown that shannon s entropy or compressibility of a sequence gives preference to random sequences and therefore cannot be applied for this purpose on the other hand the jensen shannon divergence measure turns out to give the description of information complexity which corresponds to our intuitive expectations in particular this characteristic can adequately describe two branches of evolution mentioned above exhibiting a singularity on the boundary of these regimes

multifractal detrended cross correlation analysis of genome sequences using chaos game representation abstract we characterized the multifractal nature and power law cross correlation between any pair of genome sequence through an integrative approach combining 2d multifractal detrended cross correlation analysis and chaos game representation in this paper we have analyzed genomes of some prokaryotes and calculated fractal spectra h q and f from our analysis we observed existence of multifractal nature and power law cross correlation behavior between any pair of genome sequences cluster analysis was performed on the calculated scaling exponents to identify the class affiliation and the same is represented as a dendrogram we suggest this approach may find applications in next generation sequence analysis big data analytics etc

variational b spline level set a linear filtering approach for fast deformable model evolution in the field of image segmentation most level set based active contour approaches take advantage of a discrete representation of the associated implicit function we present in this paper a different formulation where the implicit function is modeled as a continuous parametric function expressed on a b spline basis starting from the active contour energy functional we show that this formulation allows us to compute the solution as a restriction of the variational problem on the space spanned by the b splines as a consequence the minimization of the functional is directly obtained in terms of the b spline coefficients we also show that each step of this minimization may be expressed through a convolution operation because the b spline functions are separable this convolution may in turn be performed as a sequence of simple 1 d convolutions which yields an efficient algorithm as a further consequence each step of the level set evolution may be interpreted as a filtering operation with a b spline kernel such filtering induces an intrinsic smoothing in the algorithm which can be controlled explicitly via the degree and the scale of the chosen b spline kernel we illustrate the behavior of this approach on simulated as well as experimental images from various fields

from inpainting to active contours background subtraction is an elementary method for detection of foreground objects and their segmentations obviously it requires an observation image as well as a background one in this work we attempt to remove the last requirement by reconstructing the background from the observation image and a guess on the location of the object to be segmented via variational inpainting method a numerical evaluation of this reconstruction provides a disocclusion measure and the correct foreground segmentation region is expected to maximize this measure this formulation is in fact an optimal control problem where controls are shapes regions and states are the corresponding inpaintings optimization of the disocclusion measure leads formally to a coupled contour evolution equation an inpainting equation the state equation as well as a linear pde depending on the inpainting the adjoint state equation the contour evolution is implemented in the framework of level sets finally the proposed method is validated on various examples we focus among others in the segmentation of calcified plaques observed in radiographs from human lumbar aortic regions

region based strategies for active contour models the variational method has been introduced by kass et al 1987 in the field of object contour modeling as an alternative to the more traditional edge detection edge thinning edge sorting sequence since the method is based on a pre processing of the image to yield an edge map it shares the limitations of the edge detectors it uses in this paper we propose a modified variational scheme for contour modeling which uses no edge detection step but local computations instead only around contour neighborhoods as well as an anticipating strategy that enhances the modeling activity of deformable contour curves many of the concepts used were originally introduced to study the local structure of discontinuity in a theoretical and formal statement by leclerc zucker 1987 but never in a practical situation such as this one the first part of the paper introduces a region based energy criterion for active contours and gives an examination of its implications as compared to the gradient edge map energy of snakes then a simplified optimization scheme is presented accounting for internal and external energy in separate steps this leads to a complete treatment which is described in the last sections of the paper 4 and 5 the optimization technique used here is mostly heuristic and is thus presented without a formal proof but is believed to fill a gap between snakes and other useful image representations such as split and merge regions or mixed line labels image fields

stabilization of parametric active contours using a tangential redistribution term depending on implementation active contours have been classified as geometric or parametric active contours parametric contours irrespective of representation are known to suffer from the problem of irregular bunching and spacing out of curve points during the curve evolution in a spline based implementation of active contours this leads to occasional formation of loops locally and subsequently the curve blows up due to instabilities in this paper we analyze the reason for this problem and propose a solution to alleviate the same we propose an ordinary differential equation ode for controlling the curve parametrization during evolution by including a tangential force we show that the solution of the proposed ode is bounded we demonstrate the effectiveness of the proposed method for segmentation and tracking tasks on closed as well as open contours

dream 2 s deformable regions driven by an eulerian accurate minimization method for image and video segmentation this paper deals with image and video segmentation using active contours we propose a general form for the energy functional related to region based active contours we compute the associated evolution equation using shape derivation tools and accounting for the evolving region based terms then we apply this general framework to compute the evolution equation from functionals that include various statistical measures of homogeneity for the region to be segmented experimental results show that the determinant of the covariance matrix appears to be a very relevant tool for segmentation of homogeneous color regions as an example it has been successfully applied to face segmentation in real video sequences

joint multiregion segmentation and parametric estimation of image motion by basis function representation and level set evolution the purpose of this study is to investigate a variational method for joint segmentation and parametric estimation of image motion by basis function representation of motion and level set evolution the functional contains three terms one term is of classic regularization to bias the solution toward a segmentation with smooth boundaries a second term biases the solution toward a segmentation with boundaries which coincide with motion discontinuities following a description of motion discontinuities by a function of the image spatio temporal variations the third term refers to region information and measures conformity of the parametric representation of the motion of each region of segmentation to the image spatio temporal variations the components of motion in each region of segmentation are represented as functions in a space generated by a set of basis functions the coefficients of the motion components considered combinations of the basis functions are the parameters of representation the necessary conditions for a minimum of the functional which are derived taking into consideration the dependence of the motion parameters on segmentation lead to an algorithm which condenses to concurrent curve evolution implemented via level sets and estimation of the parameters by least squares within each region of segmentation the algorithm and its implementation are verified on synthetic and real images using a basis of cosine transforms

space time segmentation using level set active contours applied to myocardial gated spect this paper presents a new variational method for the segmentation of a moving object against a still background over a sequence of two dimensional or three dimensional 3 d image frames the method is illustrated in application to myocardial gated single photon emission computed tomography spect data and incorporates a level set framework to handle topological changes while providing closed boundaries the key innovation is the introduction of a geometrical constraint into the derivation of the euler lagrange equations such that the segmentation of each individual frame can be interpreted as a closed boundary of an object an isolevel of a set of hyper surfaces while integrating information over the entire sequence this results in the definition of an evolution velocity normal to the object boundary applying this method to 3 d myocardial gated spect sequences the left ventricle endocardial and epicardial limits can be computed in each frame this space time segmentation method was tested on simulated and clinical 3 d myocardial gated spect sequences and the corresponding ejection fractions were computed

a comprehensive segmentation analysis of crude oil market based on time irreversibility abstract in this paper we perform a comprehensive entropic segmentation analysis of crude oil future prices from 1983 to 2014 which used the jensen shannon divergence as the statistical distance between segments and analyze the results from original series s and series begin at 1986 marked as s to find common segments which have same boundaries then we apply time irreversibility analysis of each segment to divide all segments into two groups according to their asymmetry degree based on the temporal distribution of the common segments and high asymmetry segments we figure out that these two types of segments appear alternately and do not overlap basically in daily group while the common portions are also high asymmetry segments in weekly group in addition the temporal distribution of the common segments is fairly close to the time of crises wars or other events because the hit from severe events to oil price makes these common segments quite different from their adjacent segments the common segments can be confirmed in daily group series or weekly group series due to the large divergence between common segments and their neighbors while the identification of high asymmetry segments is helpful to know the segments which are not affected badly by the events and can recover to steady states automatically finally we rearrange the segments by merging the connected common segments or high asymmetry segments into a segment and conjoin the connected segments which are neither common nor high asymmetric

high dimensional statistical measure for region of interest tracking this paper deals with region of interest roi tracking in video sequences the goal is to determine in successive frames the region which best matches in terms of a similarity measure a roi defined in a reference frame some tracking methods define similarity measures which efficiently combine several visual features into a probability density function pdf representation thus building a discriminative model of the roi this approach implies dealing with pdfs with domains of definition of high dimension to overcome this obstacle a standard solution is to assume independence between the different features in order to bring out low dimension marginal laws and or to make some parametric assumptions on the pdfs at the cost of generality we discard these assumptions by proposing to compute the kullback leibler divergence between high dimensional pdfs using the k th nearest neighbor framework in consequence the divergence is expressed directly from the samples i e without explicit estimation of the underlying pdfs as an application we defined 5 7 and 13 dimensional feature vectors containing color information including pixel based gradient based and patch based and spatial layout the proposed procedure performs tracking allowing for translation and scaling of the roi experiments show its efficiency on a movie excerpt and standard test sequences selected for the specific conditions they exhibit partial occlusions variations of luminance noise and complex motion

bayesian analysis of isochores statistical identification of isochore structure the variation in large scale gc composition proportion of dna bases that are g or c as opposed to a or t of mammalian genomes is a necessary requirement for understanding both the evolution of base composition and the many genomic features such as mutation and recombination rates which covary with base composition we have developed a bayesian method for isochore analysis that we demonstrate to be more accurate than the commonly used binary segmentation approach implemented within the program isofinder the method accounts for both fine scale and large scale structure we adapt direct simulation methods to allow for iid samples from the posterior distribution of our model and provide an accurate approximation to this that can analyze data from a chromosome in a matter of seconds we apply our method to human chromosome 1 the resulting estimate of how gc content varies across this region is shown to be a better predictor of local recombination rates t

a statistical overlap prior for variational image segmentation this study investigates variational image segmentation with an original data term referred to as statistical overlap prior which measures the conformity of overlap between the nonparametric distributions of image data within the segmentation regions to a learned statistical description this leads to image segmentation and distribution tracking algorithms that relax the assumption of minimal overlap and as such are more widely applicable than existing algorithms we propose to minimize active curve functionals containing the proposed overlap prior compute the corresponding euler lagrange curve evolution equations and give an interpretation of how the overlap prior controls such evolution we model the overlap measured via the bhattacharyya coefficient with a gaussian prior whose parameters are estimated from a set of relevant training images quantitative and comparative performance evaluations of the proposed algorithms over several experiments demonstrate the positive effects of the overlap prior in regard to segmentation accuracy and convergence speed

long range correlation properties of coding and noncoding dna sequences genbank analysis an open question in computational molecular biology is whether long range correlations are present in both coding and noncoding dna or only in the latter to answer this question we consider all 33301 coding and all 29453 noncoding eukaryotic sequences each of length larger than 512 base pairs bp in the present release of the genbank to dtermine whether there is any statistically significant distinction in their long range correlation properties standard fast fourier transform fft analysis indicates that coding sequences have practically no correlations in the range from 10 bp to 100 bp spectral exponent beta 0 00 0 04 where the uncertainty is two standard deviations in contrast for noncoding sequences the average value of the spectral exponent beta is positive 0 16 0 05 which unambiguously shows the presence of long range correlations we also separately analyze the 874 coding and the 1157 noncoding sequences that have more than 4096 bp and find a larger region of power law behavior we calculate the probability that these two data sets coding and noncoding were drawn from the same distribution and we find that it is less than 10 10 we obtain independent confirmation of these findings using the method of detrended fluctuation analysis dfa which is designed to treat sequences with statistical heterogeneity such as dna s known mosaic structure patchiness arising from the nonstationarity of nucleotide concentration the near perfect agreement between the two independent analysis methods fft and dfa increases the confidence in the reliability of our conclusion

efficient energies and algorithms for parametric snakes parametric active contour models are one of the preferred approaches for image segmentation because of their computational efficiency and simplicity however they have a few drawbacks which limit their performance in this paper we identify some of these problems and propose efficient solutions to get around them the widely used gradient magnitude based energy is parameter dependent its use will negatively affect the parametrization of the curve and consequently its stiffness hence we introduce a new edge based energy that is independent of the parameterization it is also more robust since it takes into account the gradient direction as well we express this energy term as a surface integral thus unifying it naturally with the region based schemes the unified framework enables the user to tune the image energy to the application at hand we show that parametric snakes can guarantee low curvature curves but only if they are described in the curvilinear abscissa since normal curve evolution do not ensure constant arc length we propose a new internal energy term that will force this configuration the curve evolution can sometimes give rise to closed loops in the contour which will adversely interfere with the optimization algorithm we propose a curve evolution scheme that prevents this condition

effects of coarse graining on the scaling behavior of long range correlated and anti correlated signals abstract we investigate how various coarse graining signal quantization methods affect the scaling properties of long range power law correlated and anti correlated signals quantified by the detrended fluctuation analysis specifically for coarse graining in the magnitude of a signal we consider i the floor ii the symmetry and iii the centro symmetry coarse graining methods we find that for anti correlated signals coarse graining in the magnitude leads to a crossover to random behavior at large scales and that with increasing the width of the coarse graining partition interval this crossover moves to intermediate and small scales in contrast the scaling of positively correlated signals is less affected by the coarse graining with no observable changes when 1 while for 1 a crossover appears at small scales and moves to intermediate and large scales with increasing for very rough coarse graining 3 based on the floor and symmetry methods the position of the crossover stabilizes in contrast to the centro symmetry method where the crossover continuously moves across scales and leads to a random behavior at all scales thus indicating a much stronger effect of the centro symmetry compared to the floor and the symmetry method for coarse graining in time where data points are averaged in non overlapping time windows we find that the scaling for both anti correlated and positively correlated signals is practically preserved the results of our simulations are useful for the correct interpretation of the correlation and scaling properties of symbolic sequences

distribution matching with the bhattacharyya similarity a bound optimization framework we present efficient graph cut algorithms for three problems 1 finding a region in an image so that the histogram or distribution of an image feature within the region most closely matches a given model 2 co segmentation of image pairs and 3 interactive image segmentation with a user provided bounding box each algorithm seeks the optimum of a global cost function based on the bhattacharyya measure a convenient alternative to other matching measures such as the kullback leibler divergence our functionals are not directly amenable to graph cut optimization as they contain non linear functions of fractional terms which make the ensuing optimization problems challenging we first derive a family of parametric bounds of the bhattacharyya measure by introducing an auxiliary labeling then we show that these bounds are auxiliary functions of the bhattacharyya measure a result which allows us to solve each problem efficiently via graph cuts we show that the proposed optimization procedures converge within very few graph cut iterations comprehensive and various experiments including quantitative and comparative evaluations over two databases demonstrate the advantages of the proposed algorithms over related works in regard to optimality computational load accuracy and flexibility

image segmentation using active contours calculus of variations or shape gradients we consider the problem of segmenting an image through the minimization of an energy criterion involving region and boundary functionals we show that one can go from one class to the otherby solving poisson s orhelmholtz s equation with well chosen boundar y conditions using this equivalence we study the case of a large class of region functionals by standard methods of the calculus of variations and derive the corresponding euler lagrange equations we revisit this problem using the notion of a shape derivative and show that the same equations can be elegantly derived without going through the unnatural step of converting the region integrals into boundary integrals we also define a larger class of region functionals based on the estimation and comparison to a prototype of the probability density distribution of image features and show how the shape derivative tool allows us to easily compute the corresponding gateaux derivatives and euler lagrange equations finally we apply this new functional to the problem of regions segmentation in sequences of color images we briefly describe our numerical scheme and show some experimental results

statistical region snake based segmentation adapted to different physical noise models algorithms for object segmentation are crucial in many image processing applications during past years active contour models snakes have been widely used for finding the contours of objects this segmentation strategy is classically edge based in the sense that the snake is driven to fit the maximum of an edge map of the scene we propose a region snake approach and we determine fast algorithms for the segmentation of an object in an image the algorithms developed in a maximum likelihood approach are based on the calculation of the statistics of the inner and the outer regions defined by the snake it has thus been possible to develop optimal algorithms adapted to the random fields which describe the gray levels in the input image if we assume that their probability density function family are known we demonstrate that this approach is still efficient when no boundary s edge exists in the image we also show that one can obtain fast algorithms by transforming the summations over a region for the calculation of the statistics into summations along the boundary of the region finally we will provide numerical simulation results for different physical situations in order to illustrate the efficiency of this approach

region competition unifying snakes region growing and bayes mdl for multiband image segmentation we present a novel statistical and variational approach to image segmentation based on a new algorithm named region competition this algorithm is derived by minimizing a generalized bayes minimum description length mdl criterion using the variational principle the algorithm is guaranteed to converge to a local minimum and combines aspects of snakes balloons and region growing the classic snakes balloons and region growing algorithms can be directly derived from our approach we provide theoretical analysis of region competition including accuracy of boundary location criteria for initial conditions and the relationship to edge detection using filters it is straightforward to generalize the algorithm to multiband segmentation and we demonstrate it on gray level images color images and texture images the novel color model allows us to eliminate intensity gradients and shadows thereby obtaining segmentation based on the albedos of objects it also helps detect highlight regions

a variational framework for multiregion pairwise similarity based image segmentation variational cost functions that are based on pairwise similarity between pixels can be minimized within level set framework resulting in a binary image segmentation in this paper we extend such cost functions and address multi region image segmentation problem by employing a multi phase level set framework for multi modal images cost functions become more complicated and relatively difficult to minimize we extend our previous work proposed for background foreground separation to the segmentation of images in more than two regions we also demonstrate an efficient implementation of the curve evolution which reduces the computational time significantly finally we validate the proposed method on the berkeley segmentation data set by comparing its performance with other segmentation techniques

on scene segmentation and histograms based curve evolution we consider curve evolution based on comparing distributions of features and its applications for scene segmentation in the first part we promote using cross bin metrics such as the earth mover s distance emd instead of standard bin wise metrics as the bhattacharyya or kullback leibler metrics to derive flow equations for minimizing functionals involving the emd we employ a tractable expression for calculating emd between one dimensional distributions we then apply the derived flows to various examples of single image segmentation and to scene analysis using video data in the latter we consider the problem of segmenting a scene to spatial regions in which different activities occur we use a nonparametric local representation of the regions by considering multiple one dimensional histograms of normalized spatiotemporal derivatives we then obtain semisupervised segmentation of regions using the flows derived in the first part of the paper our results are demonstrated on challenging surveillance scenes and compare favorably with state of the art results using parametric representations by dynamic systems or mixtures of them

local region descriptors for active contours evolution edge based and region based active contours are frequently used in image segmentation while edges characterize small neighborhoods of pixels region descriptors characterize entire image regions that may have overlapping probability densities in this paper we propose to characterize image regions locally by defining local region descriptors lrds these are essentially feature statistics from pixels located within windows centered on the evolving contour and they may reduce the overlap between distributions lrds are used to define general form energies based on level sets in general a particular energy is associated with an active contour by means of the logarithm of the probability density of features conditioned on the region in order to reduce the number of local minima of such energies we introduce two novel functions for constructing the energy functional which are both based on the assumption that local densities are approximately gaussian the first uses a similarity measure between features of pixels that involves confidence intervals the second employs a local markov random field mrf model by minimizing the associated energies we obtain active contours that can segment objects that have largely overlapping global probability densities our experiments show that the proposed method can accurately segment natural large images in very short time when using a fast level set implementation

metamorphs deformable shape and appearance models this paper presents a new deformable modeling strategy that is aimed at integrating shape and appearance in a unified space if we think of traditional deformable models as active contours or evolving curve fronts the new deformable shape and appearance models that we propose are deforming disks or volumes each model not only has boundary shape but also interior appearance the model shape is implicitly embedded in a higher dimensional space of distance transforms and is thus represented by a distance map image this way both the shape and the appearance of the model are defined in the pixel space a common deformation scheme that is the free form deformations ffds parameterizes warping deformations of the volumetric space in which the model is embedded hence simultaneously deforming both model boundary and interior when applied to segmentation a metamorphs model can be initialized by covering a seed region far from the object boundary and then the model efficiently evolves and converges to an optimal solution the model dynamics are derived in a unified variational framework that consists of edge based and region based energy terms both of which are differentiable with respect to the common set of ffd parameters as the model deforms its interior appearance statistics are adaptively learned and then toward the next step deformation the model examines not only edge information but also its exterior region statistics to ensure that it only expands to new territory with consistent appearance statistics the metamorphs formulation also allows natural merging and competition of multiple models we demonstrate the robustness of metamorphs by using both natural and medical images that have high noise levels intensity inhomogeneity and complex texture

effect of trends on detrended fluctuation analysis scaling behavior we find that crossovers result from the competition between the scaling of the noise and the apparent scaling of the trend we study how the characteristics of these crossovers depend on i the slope of the linear trend ii the amplitude and period of the periodic trend iii the amplitude and power of the power law trend and iv the length as well as the correlation properties of the noise surprisingly we find that the crossovers in the scaling of noisy signals with trends also follow scaling laws i e long range power law dependence of the position of the crossover on the parameters of the trends we show that the dfa result of noise with a trend can be exactly determined by the superposition of the separate results of the dfa on the noise and on the trend assuming that the noise and the trend are not correlated if this superposition rule is not followed this is an indication that the noise and the superposed trend are not independent so that removing the trend could lead to changes in the correlation properties of the noise in addition we show how to use dfa appropriately to minimize the effects of trends how to recognize if a crossover indicates indeed a transition from one type to a different type of underlying correlation or if the crossover is due to a trend without any transition in the dynamical properties of the noise

a nonparametric statistical method for image segmentation using information theory and curve evolution in this paper we present a new information theoretic approach to image segmentation we cast the segmentation problem as the maximization of the mutual information between the region labels and the image pixel intensities subject to a constraint on the total length of the region boundaries we assume that the probability densities associated with the image pixel intensities within each region are completely unknown a priori and we formulate the problem based on nonparametric density estimates due to the nonparametric structure our method does not require the image regions to have a particular type of probability distribution and does not require the extraction and use of a particular statistic we solve the information theoretic optimization problem by deriving the associated gradient flows and applying curve evolution techniques we use level set methods to implement the resulting evolution the experimental results based on both synthetic and real images demonstrate that the proposed technique can solve a variety of challenging image segmentation problems furthermore our method which does not require any training performs as good as methods based on training

active contours driven by the salient edge energy model in this brief we present a new indicator i e salient edge energy for guiding a given contour robustly and precisely toward the object boundary specifically we define the salient edge energy by exploiting the higher order statistics on the diffusion space and incorporate it into a variational level set formulation with the local region based segmentation energy for solving the problem of curve evolution in contrast to most previous methods the proposed salient edge energy allows the curve to find only significant local minima relevant to the object boundary even in the noisy and cluttered background moreover the segmentation performance derived from our new energy is less sensitive to the size of local windows compared with other recently developed methods owing to the ability of our energy function to suppress diverse clutters the proposed method has been tested on various images and experimental results show that the salient edge energy effectively drives the active contour both qualitatively and quantitatively compared to various state of the art methods

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

motion and appearance nonparametric joint entropy for video segmentation this paper deals with video segmentation based on motion and spatial information classically the motion term is based on a motion compensation error mce between two consecutive frames defining a motion based energy as the integral of a function of the mce over the object domain implicitly results in making an assumption on the mce distribution gaussian for the square function and more generally parametric distributions for functions used in robust estimation however these assumptions are not necessarily appropriate instead we propose to define the energy as a function of an estimation of the mce distribution this function was chosen to be a continuous version of the ahmad lin entropy approximation the purpose being to be more robust to outliers inherently present in the mce since a motion only constraint can fail with homogeneous objects the motion based energy is enriched with spatial information using a joint entropy formulation the resulting energy is minimized iteratively using active contours this approach provides a general framework which consists in defining a statistical energy as a function of a multivariate distribution independently of the features associated with the object of interest the link between the energy and the features observed or computed on the video sequence is then made through a nonparametric kernel based distribution estimation it allows for example to keep the same energy definition while using different features or different assumptions on the features

extended aic model based on high order moments and its application in the financial market abstract in this paper an extended method of traditional akaike information criteria aic is proposed to detect the volatility of time series by combining it with higher order moments such as skewness and kurtosis since measures considering higher order moments are powerful in many aspects the properties of asymmetry and flatness can be observed furthermore in order to reduce the effect of noise and other incoherent features we combine the extended aic algorithm with multiscale wavelet analysis in which the newly extended aic algorithm is applied to wavelet coefficients at several scales and the time series are reconstructed by wavelet transform after that we create aic planes to derive the relationship among aic values using variance skewness and kurtosis respectively when we test this technique on the financial market the aim is to analyze the trend and volatility of the closing price of stock indices and classify them and we also adapt multiscale analysis to measure complexity of time series over a range of scales empirical results show that the singularity of time series in stock market can be detected via extended aic algorithm

long range correlation measures for quantifying patchiness deviations from uniform power law scaling in genomic dna abstract we introduce and develop new techniques to quantify dna patchiness and characteristics of their mosaic structure these techniques which involve calculating two functions l and l measure correlation exponents at length scale l and detect distinct characteristic patch sizes embedded in scale invariant patch size distributions using these methods it is possible to address a number of issues relating to the mosaic structure of genomic dna we find several distinct characteristic patch sizes in yeast human and prokaryotic sequences we also find that the distinct patch sizes in all 16 yeast chromosomes are similar

quantitative linguistic study of dna sequences abstract a new family of compound poisson distribution functions from quantitative linguistics is used to study the linguistic features of dna sequences that go beyond the zipf s law the relative frequency distribution of n tuples and the compositional segmentation study can be fit reasonably well using this new family of distribution functions on the other hand the absolute values of the relative frequency come out naturally from the linguistic model without ambiguity it is suggesting that dna sequences have features that resemble natural language and it may be modeled by linguistic methodology

affine invariant geometric shape priors for region based active contours we present a new way of constraining the evolution of a region based active contour with respect to a reference shape minimizing a shape prior defined as a distance between shape descriptors based on the legendre moments of the characteristic function leads to a geometric flow that can be used with benefits in a two class segmentation application the shape model includes intrinsic invariance with regard to pose and affine deformations

3 d carotid multi region mri segmentation by globally optimal evolution of coupled surfaces in this paper we propose a novel global optimization based 3 d multi region segmentation algorithm for t1 weighted black blood carotid magnetic resonance mr images the proposed algorithm partitions a 3 d carotid mr image into three regions wall lumen and background the algorithm performs such partitioning by simultaneously evolving two coupled 3 d surfaces of carotid artery adventitia boundary ab and lumen intima boundary lib while preserving their anatomical inter surface consistency such that the lib is always located within the ab in particular we show that the proposed algorithm results in a fully time implicit scheme that propagates the two linearly ordered surfaces of the ab and lib to their globally optimal positions during each discrete time frame by convex relaxation in this regard we introduce the continuous max flow model and prove its duality equivalence to the convex relaxed optimization problem with respect to each evolution step we then propose a fully parallelized continuous max flow based algorithm which can be readily implemented on a gpu to achieve high computational efficiency extensive experiments with four users using 12 3t mr and 26 1 5t mr images demonstrate that the proposed algorithm yields high accuracy and low operator variability in computing vessel wall volume in addition we show the algorithm outperforms previous methods in terms of high computational efficiency and robustness with fewer user interactions

divergence measures based on the shannon entropy a novel class of information theoretic divergence measures based on the shannon entropy is introduced unlike the well known kullback divergences the new measures do not require the condition of absolute continuity to be satisfied by the probability distributions involved more importantly their close relationship with the variational distance and the probability of misclassification error are established in terms of bounds these bounds are crucial in many applications of divergence measures the measures are also well characterized by the properties of nonnegativity finiteness semiboundedness and boundedness

phase transition in a random fragmentation problem with applications to computer science we study a fragmentation problem where an initial object of sizex is broken into m random pieces provided x x 0 where x0 is an atomic cut off subsequently the fragmentation process continues for each of those daughter pieces whose sizes are bigger than x0 t heprocess stops when all the fragments have sizes smaller than x0 w es ho wt ha tt he fluctuation of the total number of splitting events characterized by the variance generically undergoes a nontrivial phase transition as one tunes the branching numberm through a critical value m mc for m m c they are anomalously large and non gaussian we apply this general result to analyse two different search algorithms in computer science

a global minimization hybrid active contour model with applications to oil spill images active contour model is popularly and widely used in the field of image segmentation which is based on superior theoretical properties and efficient numerical methods nevertheless one of the prominent disadvantages of this kind of model is the existence of local minima in its functional energy in this paper we propose a novel global minimization hybrid active contour model this model effectively integrates the edge information the local region information and the global region information of the image which is relatively sufficient to extract the object boundaries furthermore we introduce an efficient and fast numerical approach to globally minimize the proposed model which is through a dual formulation of the minimization problem and easy to implement the proposed model is robust enough to the initial condition and does not need to initialize the contour in a distance function and re initialize it periodically during the evolution process specially we applied the proposed model to segment oil spill images in which there usually exist the noise blurry boundaries and intensity inhomogeneity compared with the state of the art models experiment results demonstrate the performance and effectiveness of the proposed model with applications to synthetical and real images especially for oil spill images

optical flow 3d segmentation and interpretation a variational method with active curve evolution and level sets this study investigates a variational active curve evolution method for dense three dimensional 3d segmentation and interpretation of optical flow in an image sequence of a scene containing moving rigid objects viewed by a possibly moving camera this method jointly performs 3d motion segmentation 3d interpretation recovery of 3d structure and motion and optical flow estimation the objective functional contains two data terms for each segmentation region one based on the motion only equation which relates the essential parameters of 3d rigid body motion to optical flow and the other on the horn and schunck optical flow constraint it also contains two regularization terms for each region one for optical flow the other for the region boundary the necessary conditions for a minimum of the functional result in concurrent 3d motion segmentation by active curve evolution via level sets and linear estimation of each region essential parameters and optical flow subsequently the screw of 3d motion and regularized relative depth are recovered analytically for each region from the estimated essential parameters and optical flow examples are provided which verify the method and its implementation