hyperbolicity of the complex of free factors abstract we develop the geometry of folding paths in outer space and as an application prove that the complex of free factors of a free group of finite rank is hyperbolic

root properties and convergence rates of median filters median filters are a special class of ranked order filters used for smoothing signals repeated application of the filter on a quantized signal of finite length ultimately results in a sequence termed a root signal which is invariant to additional passes of the median filter in this paper the theory is developed both for determining the cardinality of the root signal space of arbitrary window width filters applied to signals with any number of quantization levels and for counting or estimating the number of passes required to produce a root for binary signals

reconstruction in a generalized model for translation invariant systems abstract we consider translation invariant ti operators on the set of maps from an abelian group g to called lg fuzzy sets where 0 is a complete lattice ordered group by defining minkowski and morphological operations on and considering order preserving operators we prove a reconstruction theorem this theorem which is called the strong reconstruction theorem srt is similar to the convolution theorem in the theory of linear and shift invariant systems and states that for an order preserving ti operator y one can explicitly compute y a for any a from a specific subset of called the base of y the introduced framework is a general model for the theory of translation invariant systems and srt shows the consistency of it for the special cases when g r z srt implies the results of maragos and schafer

image representation by a new optimal non uniform morphological sampling abstract this paper presents a new non uniform subsampling strategy based on mathematical morphology this sampling scheme is determinist and iterative at each iteration a sample position is determined in order to minimize the overall absolute reconstruction error we show that satisfying this criterion is equivalent to maximizing the volume of reconstructed image the sample selection strategy is repeated until reaching either a previously determined number of samples or a maximal admissible error the correspondent reconstruction procedure is also described the algorithm is first presented using only one structuring element then extended to the case of a family of structuring elements experiments on the choice of structuring elements in the family allow us to choose the optimal description set finally we present an application of this novel image representation method on image compression a comparison of results with jpeg standard shows the good performance of our approach especially for high compression ratios

microstatistics in signal decomposition and the optimal filtering problem the author introduces and analyzes a large class of nonlinear filters which are based on signal decomposition and where the estimation in the decomposed space uses linear combinations of either the observation vector the sorted observation vector or in general a nonlinear transformation of the observation vector thus nonlinear filter response characteristics are achieved but with the machinery of linear systems theory available for their optimization and design it is shown that linear filters are a subclass of microstatistic filters and that the optimal linear filter solution is suboptimal in the decomposed signal space the filtering problem reduces to a set of filters operating on the decomposed signals the output is a weighted sum of the decomposed filtered signals the optimal interconnection structure between decomposed signals has complexity o m 1 where m is the cardinality of the decomposition the formulation is given for a radix 1 decomposition and generalized for a radix q decomposition computer simulations illustrate the performance

color image coding using morphological pyramid decomposition presents a new algorithm that utilizes mathematical morphology for pyramidal coding of color images the authors obtain lossy color image compression by using block truncation coding at the pyramid levels to attain reduced data rates the pyramid approach is attractive due to low computational complexity simple parallel implementation and the ability to produce acceptable color images at moderate data rates in many applications the progressive transmission capability of the algorithm is very useful the authors show experimental results for color images at data rates of 1 89 bits pixel

the effect of morphological filters on texture boundary localization we extend the theoretical results obtained by stevenson et al 1987 to two distributions and provide a quantitative comparison of 1d morphological filter edge localization with two classical types of kernel based smoothing filters the mean and the close relative to morphological filters the median filter implications in the context of statistical texture segmentation are briefly discussed

theoretical foundations of spatially variant mathematical morphology part i binary images we develop a general theory of spatially variant sv mathematical morphology for binary images in the euclidean space the basic sv morphological operators that is sv erosion sv dilation sv opening and sv closing are defined we demonstrate the ubiquity of sv morphological operators by providing an sv kernel representation of increasing operators the latter representation is a generalization of matheron s representation theorem of increasing and translation invariant operators the sv kernel representation is redundant in the sense that a smaller subset of the sv kernel is sufficient for the representation of increasing operators we provide sufficient conditions for the existence of the basis representation in terms of upper semicontinuity in the hit or miss topology the latter basis representation is a generalization of maragos basis representation for increasing and translation invariant operators moreover we investigate the upper semicontinuity property of the basic sv morphological operators several examples are used to demonstrate that the theory of spatially variant mathematical morphology provides a general framework for the unification of various morphological schemes based on spatially variant geometrical structuring elements for example circular affine and motion morphology simulation results illustrate the theory of the proposed spatially variant morphological framework and show its potential power in various image processing applications

recursive binary dilation and erosion using digital line structuring elements in arbitrary orientations performing morphological operations such as dilation and erosion of binary images using very long line structuring elements is computationally expensive when performed brute force following definitions we present two pass algorithms that run at constant time for obtaining binary dilations and erosions with all possible length line structuring elements simultaneously the algorithms run at constant time for any orientation of the line structuring element another contribution of this paper is the use of the concept of orientation error between a continuous line and its discrete counterpart the orientation error is used in determining the minimum length of the basic digital line structuring element used in obtaining what we call dilation and erosion transforms the transforms are then thresholded by the length of the desired structuring element to obtain the dilation and erosion results the algorithms require only one maximum operation for erosion transform and only one minimum operation for dilation transform and one thresholding step and one translation step per result pixel we tested the algorithms on sun sparc station 10 on a set of 240 spl times 250 salt and pepper noise images with probability of a pixel being a 1 pixel set to 0 25 for orientations of the normals of the structuring elements in the range spl pi 2 3 spl pi 2 and lengths in pixels in the range 5 145 we achieved a speed up of about 50 and for special orientations spl theta spl isin spl pi 2 3 spl pi 4 spl pi 5 spl pi 4 3 spl pi 2 a speed up of about 100 when the structuring elements had lengths of 145 pixels over the brute force methods in these experiments we compared the results of our dilation algorithm with those of the algorithm discussed by soille et al see ieee trans pattern anal machine intell vol 18 p 562 67 1996 and showed that for binary dilation and erosion since it is just the dilation of the background with the reflected structuring element our algorithm performed better and achieved a speed up of about four when dilation or erosion transform alone is obtained

btc image coding using median filter roots in this paper we source encode the truncated block used in block truncation coding it is shown that the truncated block is well approximated by wide sense markoff statistics a signal having these characteristics has a high probability of belonging to the root signal set of median filters because the root signal space is much smaller than the binary space it takes fewer bits to specify the truncated block in the root signal space obtaining in this manner rate compression using two dimensional filtering we can reduce the standard btc rate of 1 63 bits pel to 1 31 bits pel using one dimensional filtering along with a trellis encoder rates close to 1 1 bits pel are obtained with this fixed length coding method

evolution equations for continuous scale morphological filtering multiscale signal analysis has emerged as a useful framework for many computer vision and signal processing tasks morphological filters can be used to develop nonlinear multiscale operations that have certain advantages over linear multiscale approaches in that they preserve important signal features such as edges the authors discuss several nonlinear partial differential equations that model the scale evolution associated with continuous space multiscale morphological erosions dilations openings and closings these equations relate the rate of change of the multiscale signal ensemble as scale increases to a nonlinear operator acting on the space of signals the nonlinear operator is characterized by the shape and dimensionality of the structuring element used by the morphological operators generally taking the form of a nonlinear function of certain partial differential operators

a theoretical analysis of the properties of median filters necessary and sufficient conditions for a signal to be invariant under a specific form of median filtering are derived these conditions state that a signal must be locally monotone to pass through a median filter unchanged it is proven that the form of successive median filtering of a signal i e the filtered output is itself again filtered eventually reduces the original signal to an invariant signal called a root signal for a signal of length l samples a maximum of frac 1 2 l 2 repeated filterings produces a root signal

rank order operators and the mean absolute error criterion the author shows that among all randomizing stack filters whose output decisions are based on the sum of the bits in the window on each level of the threshold decomposition architecture rank order operators are optimal under an error criterion which is the sum of the absolute error incurred on each level of the threshold decomposition architecture he also shows that if the filtering class being optimized obeys a deterministic stacking property then the traditional multilevel mean absolute error criterion reduces to the sum of the absolute error incurred to each level this leads to an efficient algorithm for choosing the rank order filter which minimizes the multilevel mean absolute error criterion

morphological filters part i their set theoretic analysis and relations to linear shift invariant filters this paper examines the set theoretic interpretation of morphological filters in the framework of mathematical morphology and introduces the representation of classical linear filters in terms of morphological correlations which involve supremum infimum operations and additions binary signals are classified as sets and multilevel signals as functions two set theoretic representations of signals are reviewed filters are classified as set processing sp or function processing fp conditions are provided for certain fp filters that pass binary signals to commute with signal thresholding because then they can be analyzed and implemented as sp filters the basic morphological operations of set erosion dilation opening and closing are related to minkowski set operations and are used to construct fp morphological filters emphasis is then given to analytically and geometrically quantifying the similarities and differences between morphological filtering of signals by sets and functions the latter case allows the definition of morphological convolutions and correlations toward this goal various properties of fp morphological filters are also examined linear shift invariant filters due to their translation invariance are uniquely characterized by their kernel which is a special collection of input signals increasing linear filters are represented as the supremum of erosions by their kernel functions if the filters are also discrete and have a finite extent impulse response they can be represented as the supremum of erosions only by their minimal with respect to a signal ordering kernel functions stable linear filters can be represented as the sum of at most two weighted suprema of erosions these results demonstrate the power of mathematical morphology as a unifying approach to both linear and nonlinear signal shaping strategies

threshold superposition in morphological image analysis systems it is shown that four composite morphological systems namely morphological edge detection peak valley extraction skeletonization and shape size distributions obey a weak linear superposition called threshold linear superposition the output image signal or measurement from each system is shown to be the sum of outputs due to input binary images that result from thresholding the input gray level image at all levels these results are generalized to a vector space formulation e g to any finite linear combination of simple morphological systems thus many such systems processing gray level images are reduced to corresponding binary image processing systems which are easier to analyze and implement

the classification and associative memory capability of stack filters the notion of the on set of a positive boolean function is used to classify stack filters into three different types called decreasing increasing and mixed the associative memory capability of each of these three types of stack filters is then investigated the associative memory of a stack filter is defined to be the set of root signals of that filter in a class of stack filters in which each filter s root set contains a desired set of patterns those filters whose root sets have the smallest cardinality are said to be minimal among all filters in that class for that set of patterns some learning schemes are proposed to find minimal decreasing and increasing stack filters it is also shown that for any specified set of patterns there is always a mixed stack filter which is minimal when one considers all stack filters which preserve those patterns in this sense mixed stack filters are always at least as good as decreasing or increasing stack filters

invariant norm quantifying nonlinear content of hamiltonian systems given a hamiltonian system one can represent it using a symplectic map this symplectic map is specified by a set of homogeneous polynomials which are uniquely determined by the hamiltonian in this paper we construct an invariant norm in the space of homogeneous polynomials of a given degree this norm is a function of parameters characterizing the original hamiltonian system such a norm has several potential applications

scale space using mathematical morphology in this paper we prove that the scale space of a one dimensional gray scale signal based on morphological filterings satisfies causality no new feature points are created as scale gets larger for this we refine the standard definition of zero crossing so as to allow signals with certain singularity and use them to define feature points this new definition of zero crossing agrees with the standard one in the case of functions with second order derivative in particular the scale space based on the gaussian kernel g does not need this concept because a filtered signal g f is always infinitely differentiable using this generalized concept of zero crossing we show that a morphological filtering based on opening and hence also closing by duality satisfies causality we note that some previous works have mistakes which are corrected in this paper our causality results do not apply to more general two dimensional gray scale images causality results on alternating sequential filter obtained as byproduct are also included

nonlinear scale space filtering and multiresolution system we derive and demonstrate a nonlinear scale space filter and its application in generating a nonlinear multiresolution system for each datum in a signal a neighborhood of weighted data is used for clustering the cluster center becomes the filter output the filter is governed by a single scale parameter that dictates the spatial extent of nearby data used for clustering this together with the local characteristic of the signal determines the scale parameter in the output space which dictates the influences of these data on the output this filter is thus adaptive and data driven it provides a mechanism for a removing impulsive noise b improved smoothing of nonimpulsive noise and c preserving edges comparisons with gaussian scale space filtering and median filters are made using real images using the architecture of the laplacian pyramid and this nonlinear filter for interpolation we construct a nonlinear multiresolution system that has two features 1 edges are well preserved at low resolutions and 2 difference signals are small and spatially localized this filter implicitly presents a new mechanism for detecting discontinuities differing from techniques based on local gradients and line processes this work shows that scale space filtering nonlinear filtering and scale space clustering are closely related and provides a framework within which further image processing image coding and computer vision problems can be investigated

min max operators in texture analysis a signature is generated for a given picture by operating on it with different masks the operations are gray level generalizations of shrink and expand for binary pictures using serra s morphological methods 10 the signature is a set of numbers each corresponding to an application of an operator at a certain scale and direction and can be used to analyze the discriminate textures it is shown that this family of signatures includes as special cases several currently used texture descriptors

a generalization of median filtering using linear combinations of order statistics we consider a class of nonlinear filters whose output is given by a linear combination of the order statistics of the input sequence assuming a constant signal in white noise the coefficients in the linear combination are chosen to minimize the output mse for several noise distributions it is shown that the optimal order statistic filter osf tends toward the median filter as the noise becomes more impulsive the optimal osf is applied to an actual noisy image and is shown to perform well combining properties of both the averaging and median filters a more general design scheme for applications involving nonconstant signals is also given

theoretical analysis of the max median filter median filtering has been used successfully for extracting features from noisy one dimensional signals however the extension of the one dimensional case to higher dimensions has not always yielded satisfactory results although noise suppression is obtained too much signal distortion is introduced and many features of interest are lost in this paper we introduce a multidimensional filter based on a combination of one dimensional median estimates it is shown that threshold decomposition holds for this class of filters making the deterministic analysis simpler invariant signals to the filter called root signals consist of very low resolution features making this filter much more attractive than conventional median filters

further results on map optimality and strong consistency of certain classes of morphological filters morphological openings and closings can be viewed as consistent map estimators of smooth random binary image signals immersed in i i d clutter or suffering from i i d random dropouts we revisit this viewpoint under much more general assumptions and show that quite surprisingly the above interpretation is still valid

mathematical morphology a modern approach in image processing based on algebra and geometry mathematical morphology is a theory of image transformations and image functionals which is based on set theoretical geometrical and topological concepts the methodology is particularly useful for the analysis of the geometrical structure in an image the main goal of this paper is to give an impression of the underlying philosophy and the mathematical theories which are relevant to this field the following topics are discussed introduction to mathematical morphology generalization to complete lattices morphological filters and their construction by iteration geometrical aspects of morphology e g convexity distance geodesic operators granulometries metric dilations distance transform cost functions and extension of binary operators to grey scale images

tight geodesics in the curve complex the curve graph mathcal g associated to a compact surface is the 1 skeleton of the curve complex defined by harvey masur and minsky showed that this graph is hyperbolic and defined the notion of a tight geodesic therein we prove some finiteness results for such geodesics for example we show that a slice of the union of tight geodesics between any pair of points has cardinality bounded purely in terms of the topological type of we deduce some consequences for the action of the mapping class group on mathcal g in particular we show that it satisfies an acylindricity condition and that the stable lengths of pseudoanosov elements are rational with bounded denominator

m band nonlinear subband decompositions with perfect reconstruction we investigate nonlinear multirate filterbanks with maximal decimation and perfect reconstruction definitions of the desired properties of such structures are given for general nonlinear filterbanks we then consider a triangular representation of linear filterbanks and see that it may be easily extended to the nonlinear case furthermore general nonlinear filterbanks are presented for which perfect reconstruction is either inherently guaranteed or ensured subject to an easily verified condition extensions to bidimensional filters are also discussed and an application for nonlinear multiresolution schemes to feature sieves is shown

lum filters a class of rank order based filters for smoothing and sharpening a new class of rank order based filters called lower upper middle lum filters is introduced the output of these filters is determined by comparing a lower and an upper order statistic to the middle sample in the filter window these filters can be designed for smoothing and sharpening or outlier rejection the level of smoothing done by the filter can range from no smoothing to that of the median filter this flexibility allows the lum filter to be designed to best balance the tradeoffs between noise smoothing and signal detail preservation lum filters for enhancing edge gradients can be designed to be insensitive to low levels of additive noise and to remove impulsive noise furthermore lum filters do not cause overshoot or undershoot some statistical and deterministic properties of the lum filters are developed and a number of experimental results are presented to illustrate the performance these experiments include applications to 1d signals and to images

a review of the hydrodynamic lyapunov modes of hard disk systems a review of the results obtained for hard disk fluids confined to a quasi one dimensional qod system is presented one of the main achievements in recent years has been determining the hydrodynamic lyapunov modes hlms as covariant stable and unstable manifolds defined by k vector analogues of the zero exponent subspace of conserved quantities the tangent space dynamics allows an interpretation of the hlms as reduced hydrodynamic fields over the perturbations and is expanded upon in this paper the time evolution of these modes is governed by the dynamics of conjugate pairs of stable and unstable directions each pair is seen to interact in a subspace almost completely separated from other vectors the modes undergo a rotation until they are collinear with the stable or unstable manifold the angle distributions between these covariant stable and unstable hlms are also determined and tangencies are not observed for a generic chaotic trajectory this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

measuring the fractal dimension of signals morphological covers and iterative optimization fractals can model many classes of time series data the fractal dimension is an important characteristic of fractals that contains information about their geometrical structure at multiple scales the covering methods are a class of efficient approaches to measure the fractal dimension of an arbitrary fractal signal by creating multiscale covers around the signal s graph in this paper we develop a general method that uses multiscale morphological operations with varying structuring elements to unify and extend the theory and digital implementations of covering methods it is theoretically estab lished that for the fractal dimension computation covering one dimensional signals with planar sets is equivalent to mor phologically transforming the signal by one dimensional func tions which reduces the computational complexity from quad ratic in the signal s length to linear then a morphological covering algorithm is developed and applied to discrete time signals synthesized from weierstrass functions fractal inter polation functions and fractional brownian motion further for deterministic parametric fractals depending on a single pa rameter related to their dimension we develop an optimization method that starts from an initial estimate and iteratively con verges to the true fractal dimension by searching in the param eter space and minimizing a distance between the original sig nal and all such signals from the same class experimental results are also provided to demonstrate the good performance of the developed methods

adaptive stack filtering under the mean absolute error criterion an adaptive filter algorithm is developed for the class of stack filters which is a class of nonlinear filters obeying a weak superposition property the adaptation algorithm can be interpreted as a learning algorithm for a group of decision making units the decisions of which are subject to a set of constraints called the stacking constraints under a rather weak statistical assumption on the training inputs the decision strategy adopted by the group which evolves according to the proposed learning algorithm is shown to converge to an optimal strategy in the sense that it corresponds to an optimal stack filter under the mean absolute error criterion this adaptive algorithm requires only increment decrement and comparison operations and only local interconnections between the learning units implementation of the algorithm in hardware is therefore very feasible an example is provided to show how the adaptive stack filtering algorithm can be used in an application in image processing

the digital morphological sampling theorem morphological sampling reduces processing time and cost and yet produces results sufficiently close to the result of full processing a morphological sampling theorem is described which states 1 how a digital image must be morphologically filtered before sampling in order to preserve the relevant information after sampling 2 to what precision an appropriate morphologically filtered image can be reconstructed after sampling and 3 the relationship between morphologically operating before sampling and the more computationally efficient scheme of morphologically operating on the sampled image with a sampled structuring element the digital sampling theorem is developed first for the case of binary morphology and then it is extended to gray scale morphology through the use of the umbra homomorphism theorems

strong consistency of the lad l sub 1 estimator of parameters of stationary autoregressive processes with zero mean strong consistency almost sure convergence to the true parameters of the lad least absolute deviations ar autoregressive parameter estimator has been proven by s gross and w l steiger 1979 under the condition that i i d noise driving a stationary autoregressive process has zero median this work extends their proof to include the case when the driving noise has zero mean thus when the noise pdf probability density function is asymmetric with distinct mean and median the lad estimator will be strongly consistent with the pdf centered with either mean or median at the origin the results of this work extend computer simulations which further indicate that under these conditions the lad estimator is ms consistent mean squared convergence to the true parameters the importance of these results in lad signal processing applications is discussed

map signal estimation in noisy sequences of morphologically smooth images sidiropoulos et al 1994 demonstrated that morphological openings and closings can be viewed as maximum a posteriori map estimators of morphologically smooth signals in signal independent i i d noise the present authors extend these results to the m fold independent observation case and show that the aforementioned estimators are strongly consistent we also demonstrate the validity of a thresholding conjecture sidiropoulos et al 1994 by simulation and use it to evaluate estimator performance taken together these results can help determine the least upper bound m on m which guarantees virtually error free reconstruction of morphologically smooth images

a new image smoothing method based on a simple model of spatial processing in the early stages of human vision the difficulty of preserving edges is central to the problem of smoothing images the main problem is that of distinguishing between meaningful contours and noise so that the image can be smoothed without loss of details substantial efforts have been devoted to solving this difficult problem and a plethora of filtering methods have been proposed in the literature non linear filters have proved to be more efficient than their linear counterparts here a new nonlinear filter for noise smoothing is introduced this filter is based on the psychophysical phenomenon of human visual contrast sensitivity results on real images are presented to demonstrate the validity of our approach compared to other known filtering methods

morphological pyramids with alternating sequential filters the aim of this paper is to find a relationship between alternating sequential filters asf and the morphological sampling theorem mst developed by haralick et al 1987 the motivation behind this approach is to take advantage of the computational efficiency offered by the mst to implement morphological operations first we show alternative proofs for opening and closing in the sampled and unsampled domain using the basis functions these proofs are important because they show that it possible to obtain any level of a morphological pyramid in one step rather than the traditional two step procedure this decomposition is then used to show the relationship of the open closing in the sampled and unsampled domain an upper and a lower bound for the above relationships are presented under certain circumstances an equivalence is shown for open closing between the sampled and the unsampled domain an extension to more complicated algorithms using a union of openings and an intersection of closings is also proposed using the hausdorff metric it is shown that a morphologically reconstructed image cannot have a better accuracy than twice the radius of the reconstruction structuring element binary and gray scale examples are presented

shape analysis and reduction of the morphological basis for digital moving average filters a digital moving average filter with nonnegative mask weights that sum to unity is a gray scale morphological filter as such it possesses a representation as a maximum of erosions over its morphological basis the present paper investigates the special properties of the digital moving average morphological basis these properties being for the most part combinatoric in nature shapes of digital structuring elements are related to the filter weights by means of shape classes this is accomplished by defining a characteristic of a signal relative to the filter weights and then characterizing basis elements by means of the characteristic the notion of shape is then extended to a more general class of morphological filters the concept or a morphological quasi average is introduced and the power of such filters to approximate morphological filters is addressed the effect of basis reduction is studied for a rather wide class of morphological filters the aim being to quantify the degree to which a reduce

morphological filters part ii their relations to median order statistic and stack filters this paper extends the theory of median order statistic os and stack filters by using mathematical morphology to analyze them and by relating them to those morphological erosions dilations openings closings and open closings that commute with thresholding the max min representation of os filters is introduced by showing that any median or other os filter is equal to a maximum of erosions moving local minima and also to a minimum of dilations moving local maxima thus os filters can be computed by a closed formula that involves a max min on prespecified sets of numbers and no sorting stack filters are established as the class of filters that are composed exactly of a finite number of max min operations the kernels of median os and stack filters are collections of input signals that uniquely represent these filters due to their translation invariance the max min functional definitions of these nonlinear iliters is shown to be equivalent to a maximum of erosions by minimal with respect to a signal ordering kernel elements and also to a minimum of dilations by minimal kernel elements of dual filters the representation of stack filters based on their minimal kernel elements is proven to be equivalent to their representation based on irreducible sum of products expressions of boolean functions it is also shown that median filtering and its iterations of any signal by convex 1 d windows is bounded below by openings and above by closings a signal is a root fixed point of the median iff it is a root of both an opening and a closing the open closing and clos opening yield median roots in one pass suppress impulse noise similarly to the median can discriminate between positive and negative noise impulses and are computationally less complex than the median some similar results are obtained for 2 d median filtering

computation of the lyapunov spectrum for continuous time dynamical systems and discrete maps in this paper we describe in detail a method of computing lyapunov exponents for a continuous time dynamical system and extend the method to discrete maps using this method a partial lyapunov spectrum can be computed using fewer equations as compared to the computation of the full spectrum there is no difficulty in evaluating degenerate lyapunov spectra the equations are straightforward to generalize to higher dimensions and the minimal set of dynamical variables is used explicit proofs and other details not given in previous work are included here

new fusion operations for digitized binary images and their applications this paper proposes a new class of fusion operations which are useful for binary image processing to begin with an algorithm for the classical fusion operations is presented they are sequentially composed of two basic operations that is expansion and contraction the strategy of the proposed algorithm is as follows the first operation is performed with the distance transformation and the second operation utilizes the distance information obtained in the first operation next a topological contraction and a topological expansion are defined the topological contraction preserves the euler number of an input image except that some simply connected components of 1 pixels are erased the topological expansion preserves the euler number of an input image except that some simply connected holes are filled then several new fusion operations are obtained by combining the classical contraction and the topological expansion or the classical expansion and the topological contraction sequentially the algorithms for them use the same strategy as the proposed algorithm for the classical fusion operations finally experimental results show that the new fusion operations and their combinations are effectively used in the binary image processing such as shape smoothing merging and separating connected components and structure analysis

optimal filtering of digital binary images corrupted by union intersection noise we model digital binary image data as realizations of a uniformly bounded discrete random set or discrete random set for short which is a mathematical object that can be directly defined on a finite lattice we consider the problem of estimating realizations of discrete random sets distorted by a degradation process that can be described by a union intersection noise model two distinct optimal filtering approaches are pursued the first involves a class of mask filters which arises quite naturally from the set theoretic analysis of optimal filters the second approach involves a class of morphological filters we prove that under i i d noise morphological openings closings unions of openings and intersections of closings can be viewed as map estimators of morphologically smooth signals then we show that by using an appropriate under a given degradation model expansion of the optimal filter we can obtain universal characterizations of optimality that do not rely on strong assumptions regarding the spatial interaction of geometrical primitives of the signal and the noise the results generalize to gray level images in a fairly straightforward manner

residuated semigroups and morphological aspects of translation invariant systems abstract the main goal of this paper is to verify classical properties of morphological operators within the general model of translation invariant ti systems in this model ti operators are defined on the space of lg fuzzy sets i e a g in which g is an abelian group and is a complete lattice ordered group a ti operator y is an operator on which is invariant under translation on g and as groups we consider the generalization of minkowski addition d on and we emphasize that is an involutive residuated topological monoid we verify all properties of traditional set theoretic morphological operators as well as classical representations matheron 1967 for openings closures and granulometries in this general setting we also study spectral skeleton decompositions in this model using the same techniques as in the crisp case goutsias and schonfeld 1991 this formal approach not only clarifies the role of morphological operators but also it gives rise to simpler and clearer proofs using standard results of the theory of residuated semigroups and categories

segmentation of overlapping elliptical objects in silhouette images segmentation of partially overlapping objects with a known shape is needed in an increasing amount of various machine vision applications this paper presents a method for segmentation of clustered partially overlapping objects with a shape that can be approximated using an ellipse the method utilizes silhouette images which means that it requires only that the foreground objects and background can be distinguished from each other the method starts with seedpoint extraction using bounded erosion and fast radial symmetry transform extracted seedpoints are then utilized to associate edge points to objects in order to create contour evidence finally contours of the objects are estimated by fitting ellipses to the contour evidence the experiments on one synthetic and two different real data sets showed that the proposed method outperforms two current state of art approaches in overlapping objects segmentation

nonlinear multiresolution signal decomposition schemes ii morphological wavelets interest in multiresolution techniques for signal processing and analysis is increasing steadily an important instance of such a technique is the so called pyramid decomposition scheme this paper presents a general theory for constructing linear as well as nonlinear pyramid decomposition schemes for signal analysis and synthesis the proposed theory is based on the following ingredients 1 the pyramid consists of a finite or infinite number of levels such that the information content decreases toward higher levels and 2 each step toward a higher level is implemented by an information reducing analysis operator whereas each step toward a lower level is implemented by an information preserving synthesis operator one basic assumption is necessary synthesis followed by analysis yields the identity operator meaning that no information is lost by these two consecutive steps several examples of pyramid decomposition schemes are shown to be instances of the proposed theory a particular class of linear pyramids morphological skeleton decompositions the morphological haar pyramid median pyramids etc furthermore the paper makes a distinction between single scale and multiscale decomposition schemes i e schemes without or with sample reduction finally the proposed theory provides the foundation of a general approach to constructing nonlinear wavelet decomposition schemes and filter banks

using lyapunov exponents to predict the onset of chaos in nonlinear oscillators an analytic technique for predicting the emergence of chaotic instability in nonlinear nonautonomous dissipative oscillators is proposed the method is based on the lyapunov type stability analysis of an arbitrary phase trajectory and the standard procedure of calculating the lyapunov characteristic exponents the concept of temporally local lyapunov exponents is then utilized for specifying the area in the phase space where any trajectory is asymptotically stable and therefore the existence of chaotic attractors is impossible the procedure of linear coordinate transform optimizing the linear part of the vector field is developed for the purpose of maximizing the stability area in the vicinity of a stable fixed point by considering the inverse conditions of asymptotic stability this approach allows formulating a necessary condition for chaotic motion in a broad class of nonlinear oscillatory systems including many cases of practical interest the examples of externally excited one and two well duffing oscillators and a planar pendulum demonstrate efficiency of the proposed method as well as a good agreement of the theoretical predictions with the results of numerical experiments the comparison of the proposed method with melnikov s criterion shows a potential advantage of using the former one at high values of dissipation parameter and or multifrequency type of excitation in dynamical systems

median filtering by threshold decomposition median filters are a special class of ranked order filters used for smoothing signals repeated application of the filter on a quantized signal of finite length ultimately results in a sequence termed a root signal which is invariant to further passes of the median filter in this paper it is shown that median filtering an arbitrary level signal to its root is equivalent to decomposing the signal into binary signals filtering each binary signal to a root with a binary median filter and then reversing the decomposition this equivalence allows problems in the analysis and the implementation of median filters for arbitrary level signals to be reduced to the equivalent problems for binary signals since the effects of median filters on binary signals are well understood this technique is a powerful new tool

anisotropic morphological filters with spatially variant structuring elements based on image dependent gradient fields this paper deals with the theory and applications of spatially variant discrete mathematical morphology we review and formalize the definition of spatially variant dilation erosion and opening closing for binary and gray level images using exclusively the structuring function without resorting to complement this theoretical framework allows to build morphological operators whose structuring elements can locally adapt their shape and orientation across the dominant direction of the structures in the image the shape and orientation of the structuring element at each pixel are extracted from the image under study the orientation is given by means of a diffusion process of the average square gradient field which regularizes and extends the orientation information from the edges of the objects to the homogeneous areas of the image and the shape of the orientated structuring elements can be linear or it can be given by the distance to relevant edges of the objects the proposed filters are used on binary and gray level images for enhancement of anisotropic features such as coherent flow like structures results of spatially variant erosions dilations and openings closings based filters prove the validity of this theoretical sound and novel approach