fuzzy geometry an updated overview abstract this paper briefly reviews work on the fuzzy topology and geometry of image subsets including adjacency separation and connectedness distance and relative position area perimeter and diameter convexity medial axes and thinning as well as some applications of these concepts in image processing and analysis

fuzzy plane geometry i points and lines abstract we introduce a comprehensive study of fuzzy geometry in this paper by first defining a fuzzy point and a fuzzy line in fuzzy plane geometry we consider the fuzzy distance between fuzzy points and show it is a weak fuzzy metric we study various definitions of a fuzzy line develop their basic properties and investigate parallel fuzzy lines

improving jacobi and gauss seidel iterations abstract when convergent jacobi or gauss seidel iterations can be applied to solve systems of linear equations a natural question is how convergence rates are affected if the original system is modified by performing some gaussian elimination we prove that if the initial iteration matrix is nonnegative then such elimination improves convergence our results extend those contained in 4

connectedness of coherent fuzzy pictures abstract a fuzzy digital picture is called connected if it has only one connected component of constant fuzzy level that is maximal i e not adjacent to any component of higher fuzzy level in this paper we propose two acceptors defined over fuzzy digital pictures one is a one pebble acceptor and another is a deterministic one way nonwriting parallel sequential acceptor we also prove that both acceptors can determine whether a given coherent fuzzy digital picture is connected

fuzzy plane geometry crisp plane geometry starts with points then lines and parallel lines circles triangles rectangles etc in fuzzy plane geometry we will do the same our fuzzy points lines circles etc will all be fuzzy subsets of r r we assume the standard xy rectangular coordinate system in the plane since fuzzy subsets of r r will be surfaces in r 3 we can not easily present graphs of their membership functions however cuts of fuzzy subsets of r r will be crisp subsets of the plane using an xy coordinate system we may draw pictures of cuts of fuzzy points in r r also for fuzzy lines etc in this way we can see what the membership functions of fuzzy subsets of r r look like

on parameter dependent connected components of gray images abstract we introduce a new definition of the connected components of gray images which takes into account both the gray values of the pixels and the differences of the gray values of the neighbouring pixels and which depends on two parameters e and we may adjust the values of the parameters for different types of images or in different levels of image analysis we analyse a gray image in an intermediate level through the inter relationships of its e components rather than only in the pixel level

more on modifications and improvements of classical iterative schemes for m matrices in the last four decades many articles have been devoted to the modifications and improvements of classes of preconditioners for linear systems whose matrix coefficient is an m matrix in order to improve on the convergence rates of the classical iterative schemes jacobi gauss seidel etc the present work is a contribution towards the generalization of the most common preconditioners used so far

grayscale level connectivity theory and applications a novel notion of connectivity for grayscale images is introduced defined by means of a binary connectivity assigned at image level sets in this framework a grayscale image is connected if all level sets below a prespecified threshold are connected the proposed notion is referred to as grayscale level connectivity and includes as special cases other well known notions of grayscale connectivity such as fuzzy grayscale connectivity and grayscale blobs in contrast to those approaches the present framework does not require all image level sets to be connected moreover a connected grayscale object may contain more than one regional maximum grayscale level connectivity is studied in the rigorous framework of connectivity classes the use of grayscale level connectivity in image analysis applications such as object extraction image segmentation object based filtering and hierarchical image representation is discussed and illustrated

the shortest path problem with discrete fuzzy arc lengths in the past the fuzzy shortest path problem in a network has attracted attention from many researchers for its importance to various applications in this paper we propose a new algorithm to deal with the fuzzy shortest path problem it is composed of fuzzy shortest path length procedure and similarity measure the former is presented to determine the fuzzy shortest path length from source node to the destination node in the network and the latter is used to measure the similarity degree between fuzzy length sets this algorithm not only can yield shortest length but also can offer the actual shortest path to decision makers an illustrative example is also included to demonstrate our proposed algorithm

nonlinear image labeling for multivalued segmentation we describe a framework for multivalued segmentation and demonstrate that some of the problems affecting common region based algorithms can be overcome by integrating statistical and topological methods in a nonlinear fashion we address the sensitivity to parameter setting the difficulty with handling global contextual information and the dependence of results on analysis order and on initial conditions we develop our method within a theoretical framework and resort to the definition of image segmentation as an estimation problem we show that thanks to an adaptive image scanning mechanism there is no need of iterations to propagate a global context efficiently the keyword multivalued refers to a result property which spans over a set of solutions the advantage is twofold first there is no necessity for setting a priori input thresholds secondly we are able to cope successfully with the problem of uncertainties in the signal model to this end we adopt a modified version of fuzzy connectedness which proves particularly useful to account for densitometric and topological information simultaneously the algorithm was tested on several synthetic and real images the peculiarities of the method are assessed both qualitatively and quantitatively

incorporating local image structure in normalized cut based graph partitioning for grouping of pixels abstract graph partitioning for grouping of image pixels has been explored a lot with normalized cut based graph partitioning being one of the popular ones in order to have a credible allegiance to the perceptual grouping taking place in early human vision we propose and study in this paper the incorporation of local image structure context in normalized cut based graph partitioning for grouping of image pixels similarity and proximity which have been studied earlier for grouping of image pixels are only two among many perceptual cues that act during grouping in early human vision in addition to the said two cues we study three other such cues namely common fate common region and continuity and find indications of local image structure utilization during grouping of image pixels appropriate incorporation of local image structure context is achieved by representing it using neighborhood in the form of histogram and fuzzy set we demonstrate both qualitatively and quantitatively through experimental results that the incorporation of local image structure improves performance of grouping of image pixels

improving aor method for consistent linear systems abstract milaszewicz j p milaszewicz improving jacobi and gauss seidel iterations linear algebra appl 93 1987 161 170 and gunawardena et al a d gunawardena s k jain l snyder modified iteration methods for consistent linear systems linear algebra appl 154 156 1991 123 143 presented preconditioned methods for linear system in order to improve the convergence rates of jacobi and gauss seidel iterative schemes respectively in this paper we apply aor and sor iterative schemes to the preconditioned linear systems and provide two theorems to improve the convergence rates of these iterative methods meanwhile the methods in this paper can be used to more class of linear systems

fuzzy normal subgroups and fuzzy cosets abstract we introduce here the notion of 1 i a fuzzy normal subgroup and 2 ii a fuzzy coset we prove that for a group g a fuzzy subgroup is fuzzy normal if and only if it is constant on the conjugate classes of g further we show that the level subgroups of a fuzzy normal subgroup are all normal we obtain several results which are analogs of some basic theorems of group theory in particular a fuzzy version of lagrange s theorem is established

fuzzy geometry in image analysis an attempt is made here to demonstrate the way of implementing the concept of fuzzy geometry in image processing analysis problems some new measures are also provided in this context four algorithms are described for object background classification and skeleton extraction in both fuzzy and nonfuzzy crisp forms their effectiveness is demonstrated on various types of images the crisp nonfuzzy outputs of the proposed algorithms are compared with the results of some of the recently developed conventional techniques

bounds for membership functions a correlation based approach the paper describes a mathematical formulation for defining bounds for s type membership functions using correlation 1 these bounds impose restrictions on variation in the values of membership functions the significance of these bounds in the image processing analysis problems has been extensively studied on various images it has been observed that membership functions satisfying the bounds can provide desirable results it is also found that zadeh s standard s function 2 satisfies these bounds

fuzzy spatial relation ontology for image interpretation the semantic interpretation of images can benefit from representations of useful concepts and the links between them as ontologies in this paper we propose an ontology of spatial relations in order to guide image interpretation and the recognition of the structures it contains using structural information on the spatial arrangement of these structures as an original theoretical contribution this ontology is then enriched by fuzzy representations of concepts which define their semantics and allow establishing the link between these concepts which are often expressed in linguistic terms and the information that can be extracted from images this contributes to reducing the semantic gap and it constitutes a new methodological approach to guide semantic image interpretation this methodological approach is illustrated on a medical example dealing with knowledge based recognition of brain structures in 3d magnetic resonance images using the proposed fuzzy spatial relation ontology

fuzzy topological vector spaces over r in this paper the concepts of fuzzy vector fuzzy vector space and fuzzy topological vector space are investigated a topology called sendograph topology which is compatible with the algebraic structure is given

the diameter of a fuzzy set abstract in pattern recognition one often wants to measure geometric properties of imprecisely defined subsets of an image this paper proposes definitions of intrinsic and extrinsic diameter for fuzzy subsets which reduce to the ordinary definitions when the subsets are crisp we also define height and width for a fuzzy subset and show how they relate to the area i e integral of membership for convex fuzzy subsets the intrinsic diameter cannot exceed the extrinsic diameter but it can be smaller finally for piecewise constant convex fuzzy subsets the intrinsic diameter cannot exceed half the fuzzy perimeter but this need not be true in the nonconvex case

sketching a grey tone pattern from its distance transform we describe a procedure to create an abstraction of a grey tone pattern by sketching its regions which have locally higher intensities the sketch is a set of simple digital lines qualitatively analogous to the skeleton representation computed in the case of a single valued pattern the grey tone pattern is regarded as constituted by a number of regions with constant grey value and the skeleton is found by detecting suitable pixels on the distance transform of the pattern computation of the distance transform is accomplished according to the city block distance by ordered propagation over regions with increasing grey values neighbourhood conditions are used to detect the set of the skeletal pixels which is subsequently reduced to unit thickness finally the skeleton undergoes a pruning process which removes a part or all of some of its branches

comparison results on preconditioned sor type iterative method for z matrices linear systems in this paper we present some comparison theorems on preconditioned iterative method for solving z matrices linear systems comparison results show that the rate of convergence of the gauss seidel type method is faster than the rate of convergence of the sor type iterative method

fuzzy hypervector spaces based on fuzzy singletons we introduce and study fuzzy hypervector spaces based on fuzzy singletons in this regard by considering the notion of fuzzy singletons we characterize a fuzzy hypervector space fuzzily spanned by a fuzzy subset then we use these results to introduce the concept of fuzzy freeness of a fuzzy subset m of a hypervector space v and finally we characterize it in terms of linear independence in the usual sense

human based spatial relationship generalization through neural fuzzy approaches abstract spatial relationships exhibited among regions in an image play an important role in the interpretation of a scene while humans have an innate ability to recognize spatial relations it has been difficult to produce algorithms to model these relationships there have been several attempts to define spatial relationships between objects in a digital image most recently with the use of fuzzy set theory unfortunately no method can work effectively in all cases in this paper we introduce neural network structures combined with the choquet fuzzy integration to generalize spatial relationship membership functions to match the spatial relationship membership value with human intuition we tested people s perceptions of spatial relations and applied the human test results as the desired outputs of neural networks finally some experimental results on synthetic and real images are shown

degree of adjacency or surroundedness abstract definitions of the degree of adjacency of two regions in the plane and the degree of surroundedness of one region by another are proposed some elementary properties of these concepts are established and it is also shown that they have natural generalizations to fuzzy subsets of the plane applications of the proposed measures to digital polygons are demonstrated and fast algorithms for computing these measures are given

similarity measure between fuzzy sets and between elements abstract two similarity measures are proposed one for the similarity between fuzzy sets and the other between elements in fuzzy sets with and example it is shown that the proposed measures can be used in the behavior analysis in an organization

representation and reconstruction of fuzzy disks by moments in this paper we analyze the representation and reconstruction of fuzzy disks by using moments both continuous and digital fuzzy disks are considered a fuzzy disk is a convex fuzzy spatial set where the membership of a point to the fuzzy disk depends only on the distance of the point to the centre of the disk we show that for a certain class of membership functions defining a fuzzy disk there exists a one to one correspondence between the set of fuzzy disks and the set of their generalized moment representations theoretical error bounds for the accuracy of the estimation of generalized moments of a continuous fuzzy disk from the generalized moments of its digitization and in connection with that the accuracy of an approximate reconstruction of a continuous fuzzy disk from the generalized moments of its digitization are derived defuzzification reduction of a continuous fuzzy disk to a crisp representative is also considered a statistical study of generated synthetic objects complements the theoretical results

segmentation of blurred objects by classification of isolabel contours abstract segmentation of objects with blurred boundaries is an important and challenging problem especially in the field of medical image analysis a new approach to segmentation of homogeneous blurred objects in grayscale images is described in this paper the proposed algorithm is based on building of an isolabel contour map of the image and classification of closed isolabel contours by the svm each closed isolabel contour is described by the feature vector that can include intensity based features of the image area enclosed by the contour as well as geometrical features of the contour shape the image labeling procedure for construction of the training base becomes very fast and convenient because it is reduced to clicking on isolabel contours delineating the objects of interest on the isolabel contour map the proposed algorithm was applied to the problem of brain lesion segmentation in mri and demonstrated performance figures above 98 on real data both in sensitivity and in specificity

fuzzy homogeneity approach to multilevel thresholding the spatial ambiguity among pixels has inherent vagueness rather than randomness therefore the conventional methods might not work well we propose fuzzy homogeneity vectors to handle the greyness and spatial uncertainties among pixels and to perform multilevel thresholding the experimental results prove that the proposed approach works better than the histogram based algorithms

on fuzzy ideals of a ring 1 abstract the aim of this paper is to study fuzzy ideals of a ring to characterise regular rings and noetherian rings by fuzzy ideals and to determine all fuzzy prime ideals of the ring z of integers

unified formulation of a class of image thresholding techniques in this paper we show that otsu s image thresholding kittler and illingworth s minimum error thresholding and huang and wang s fuzzy thresholding methods can be derived under a similar mathematical formulation the difference among the three methods is the choice of different weighting functions for computing a criterion function that can be considered as a weighted summation of the image gray level histogram we can have a better understanding of the three thresholding techniques and derive other thresholding methods based on this unified formulation

a kernel problem with fuzzy visibility this paper proposes a new visibility concept fuzzy visibility reflecting that the visibility property varies according to the condition of application areas the concept of visibility is defined on the basis of the fuzzy theory for two points p and q p is visible from q if the visibility degree between them is no less than 0 1 if there exists a point from which all points in a polygon are visible then it is said that the visibility value of the polygon is the kernel problem is constructing the region such that all points inside simple polygon can be visible from any point of the region an optimal algorithm is given for the kernel problem with o n time complexity for a given simple polygon we present o n algorithms for computing the maximum and minimum visibility values of the polygon

a note on the modified sor method for z matrices abstract in this note some errors in a recent article by huang et al t z huang g h cheng x y cheng modified sor type iterative method for z matrices appl math comput 175 2006 258 268 are pointed out and new proofs for the corresponding results in huang et al are presented

quantitative analysis of properties and spatial relations of fuzzy image regions properties of objects and spatial relations between objects play an important role in rule based approaches for high level vision the partial presence or absence of such properties and relationships can supply both positive and negative evidence for region labeling hypotheses similarly fuzzy labeling of a region can generate new hypotheses pertaining to the properties of the region its relation to the neighboring regions and finally hypotheses pertaining to the labels of the neighboring regions a unified methodology that can be used to characterize both properties and spatial relationships of object regions in a digital image is presented the methods proposed for computing the properties and relations of image regions can be used to arrive at more meaningful decisions about the contents of the scene

on correlation between two fuzzy sets in this paper we investigated the correlation between two fuzzy sets defined on the same universal support it is shown that spearman s rank correlation coefficient can be applied if the members of the supports are ranked according to the fuzzy membership values of each set next a membership value based fuzzy correlation measure is proposed and its properties are explained the proposed measure is compared with an earlier definition 4

analysis of data generated from multidimensional type 1 and type 2 fuzzy membership functions due to the numerous applications that utilize different types of fuzzy membership functions mfs it may sometimes be difficult to choose an appropriate mf for a particular application in this paper we establish preliminary guidelines to direct this selection by proposing a three stage method in the forward stage different mfs such as crisp mfs type 1 t1 fuzzy mfs and type 2 t2 fuzzy mfs are generated from multidimensional data sets next in the reverse stage data is generated back from these mfs by considering different bin sizes in doing so various data sets may be generated for different applications which require fuzzy data finally for the similarity analysis stage we propose an iterative algorithm that makes use of the results of wilcoxon signed rank wsr and wilcoxon rank sum wrs tests to compare the original data and the generated data from the results of these tests recommendations concerning the suitability of mfs for a specific application may be suggested by observing the accuracy of representation and the requirements of the application with this analysis the objective is to gain insight on when t2 fuzzy sets may be considered to outperform t1 fuzzy sets and vice versa several examples are provided using synthetic and real data to validate the iterative algorithm for data sets in various dimensions

fuzzy ideals and fuzzy prime ideals of a ring abstract besides obtaining certain ring theoretic analogues some peculiarities of the fuzzy setting are also discussed in particular a study of i the conditions under which a given fuzzy ideal can or cannot be expressed as a union of two proper fuzzy ideals ii the algebraic nature of images and preimages of fuzzy prime ideals under homomorphisms and iii the ring of all the fuzzy cosets of a fuzzy prime fuzzy maximal ideal is carried out

on the dimension of a fuzzy subspace abstract isomorphism between two fuzzy subspaces is defined and some properties of two isomorphic fuzzy subspaces are discussed in terms of their image sets level subspaces and dimensions the dimension of a fuzzy quotient space is also studied

histogram thresholding by minimizing graylevel fuzziness the problem of histogram sharpening and thresholding by minimizing greyness ambiguity using the measures of fuzziness in a set is considered this work provides a complete mathematical formulation of the said problem in order that both fuzzy and crisp segmentation result the criteria regarding the choices of membership functions and the window sizes are established they involve symmetry in ambiguity around the crossover point and bound functions for restricting the variation of membership function values the segmentation thus obtained provides not only flexibility in methodology but also effective results experimental results further establish the same conclusions

on the perimeter and area of fuzzy sets abstract the note of area comes naturally as a measure of a fuzzy set recently a rosenfeld defined the permeter diameter height and width of a fuzzy set athough these do not generalize the corresponding notions for crisp sets they lack inner conformity in that they do not satisfy simple interrelations we are used to in the case of crisp sets in the present note we modify definitions and show that our perimeter diameter and height fit more comfortably with the notion of area we also define a projection operator that should be used along with the multiplicative conjunction in the process of fuzzy reasoning

an adaptable threshold detector a data set often comprises some data classes for example a gray scale image may consist of some objects each of which has similar pixels gray scales the threshold obtained by otsu s thresholding method otm is biased towards certain data class with larger variance or larger number of data when the variances or the numbers of data among classes are quite different in this paper adaptable threshold detector atd is proposed to improve the effectiveness of otm in determining proper thresholds by dividing class variance by class interval atd is more versatile at selecting application dependent thresholds by changing two parameter values which describe the relative importance among data size standard deviation and class interval of a class in this paper atd is applied to crop the expected objects from images to verify its effect upon thresholding experimental results demonstrate that atd is able to perform better than otm in segmenting objects from images besides excelling over the valley emphasis method vem and the minimum class variance thresholding method mcvtm atd is also suitable for separating objects from serialized video images i e computerized tomography

analytical fuzzy plane geometry iii in this study we attempt to construct fuzzy circles in a fuzzy geometrical plane we provide a comprehensive study where we find a fuzzy number with a predetermined fuzzy distance from a given fuzzy number we formulate fuzzy circles using the conventional basic definitions of crisp circles two different formulations of fuzzy circles are proposed which depend on the information known about a fuzzy circle the interrelations between the evaluated fuzzy circles are investigated we show that the center of a fuzzy circle may not be a fuzzy point however the radius of a fuzzy circle that passes through three given fuzzy points is always a fuzzy number the concepts of same and inverse points and a fuzzy number along a line are used to define and analyze the proposed ideas our discussions and studies are supported by suitable numerical and pictorial examples

topological median filters this paper describes the definition and testing of a new type of median filter for images the topological median filter implements some existing ideas and some new ideas on fuzzy connectedness to improve over a conventional median filter the extraction of edges in noise the concept of spl alpha connectivity is defined and used to create an algorithm for computing the degree of connectedness of a pixel to all the other pixels in an arbitrary neighborhood the resulting connectivity map of the neighborhood effectively disconnects peaks in the neighborhood that are separated from the center pixel by a valley in the brightness topology the median of the connectivity map is an estimate of the median of the peak or plateau to which the center pixel belongs unlike the conventional median filter the topological median is relatively unaffected by disconnected features in the neighborhood of the center pixel four topological median filters are defined qualitative and statistical analyses of the four filters are presented it is demonstrated that edge detection can be more accurate on topologically median filtered images than on conventionally median filtered images

image segmentation using fuzzy correlation a concept of correlation between two properties fuzzy representations of an image is introduced a set of algorithms for image segmentation both fuzzy and nonfuzzy has been formulated the spatial information is taken care of by the following measures transitional correlation and within class correlation a relation between the correlation coefficient and the index of fuzziness is theoretically established and experimentally verified the effectiveness of the algorithms is illustrated on images having different types of histograms

watershed cuts thinnings shortest path forests and topological watersheds we recently introduced watershed cuts a notion of watershed in edge weighted graphs in this paper our main contribution is a thinning paradigm from which we derive three algorithmic watershed cut strategies the first one is well suited to parallel implementations the second one leads to a flexible linear time sequential implementation whereas the third one links the watershed cuts and the popular flooding algorithms we state that watershed cuts preserve a notion of contrast called connection value on which several morphological region merging methods are implicitly based we also establish the links and differences between watershed cuts minimum spanning forests shortest path forests and topological watersheds finally we present illustrations of the proposed framework to the segmentation of artwork surfaces and diffusion tensor images

fuzzy medial axis transformation fmat practical feasibility abstract the fuzzy medial axis transformation fmat of a fuzzy set f is a set of fuzzy disks whose sup is f unfortunately specifying the fmat sometimes requires more storage space than specifying f itself the present paper describes some techniques to improve the compact representation of fmat thereby making it practically feasible and useful the algorithms involve reduction of redundancy in fmat its approximation and reduction of the searching spaces for its computation the computational aspects for the convenience of writing an efficient program have also been described a concept of sponsoring capability of a pixel is described in order to view the principle of removing redundancy in fmat and to generate its approximation two kinds of approximations are made namely distributed approximation and optimum fuzzy compactness the aforesaid features have been demonstrated on a gray image

on fuzzy inner product spaces and fuzzy co inner product spaces abstract the notions of fuzzy co norm and fuzzy co orthogonality are introduced by formulating the notion of a fuzzy co inner product space and their basic properties are studied

directed connected operators asymmetric hierarchies for image filtering and segmentation connected operators provide well established solutions for digital image processing typically in conjunction with hierarchical schemes in graph based frameworks such operators basically rely on symmetric adjacency relations between pixels in this article we introduce a notion of directed connected operators for hierarchical image processing by also considering non symmetric adjacency relations the induced image representation models are no longer partition hierarchies i e trees but directed acyclic graphs that generalize standard morphological tree structures such as component trees binary partition trees or hierarchical watersheds we describe how to efficiently build and handle these richer data structures and we illustrate the versatility of the proposed framework in image filtering and image segmentation

on fuzzy ideals and fuzzy bi ideals in semigroups abstract in this paper we give some properties of fuzzy ideals and fuzzy bi ideals of semigroups and characterize semigroups that are left duo left simple and semilattices of subsemigroups in terms of fuzzy ideals and fuzzy bi ideals

on the direct product of fuzzy subgroups abstract anthony and sherwood 1979 1980 redefined the fuzzy subgroup the idea introduced by rosenfeld 1971 foster 1979 introduced the concept of product fuzzy subgroups using rosenfeld s definition in this paper the product fuzzy subgroups will be redefined according to anthony and sherwood s definition

fast computation of a contrast invariant image representation this paper sets out a new representation of an image which is contrast independent the image is decomposed into a tree of shapes based on connected components of level sets which provides a full and nonredundant representation of the image a fast algorithm to compute the tree the fast level lines transform fllt is explained in detail some simple and direct applications of this representation are shown

level subgroups and union of fuzzy subgroups abstract in this paper we investigate the conditions under which a given fuzzy subgroup of a given group can or can not be realized as a union of two proper fuzzy subgroups moreover we formulate the notion of fuzzy subgroup generated by a given fuzzy subset by level subgroups we also give characterizations of fuzzy conjugate subgroups and fuzzy characteristic subgroups by their level subgroups also we discuss the relationship between level subgroups of a given fuzzy subgroup and the level subgroups of its homomorphic image

on connectivity properties of grayscale pictures abstract a grayscale digital picture is called connected if it has only one connected component of constant gray level that is maximal i e not adjacent to any component of higher gray level this note establishes some equivalent conditions for connectedness and also defines a grayscale generalization of the genus in terms of sums of local property values

fuzzy prime ideals of rings abstract the notions of fuzzy ideal and fuzzy prime ideal of a ring with truth values in a complete lattice satisfying the infinite meet distributive law are introduced which generalize the existing notions with truth values in the unit interval of real numbers a procedure to construct any fuzzy ideal from a family of ideals with certain conditions is established all fuzzy prime maximal ideals of a given ring are determined by establishing a one to one correspondence between fuzzy prime maximal ideals and the pairs p where p is a prime maximal ideal of the ring and is a prime element dual atom in the complete lattice

fuzzy invariant subgroups and fuzzy ideals abstract the purpose of this paper is to introduce some basic concepts of fuzzy algebra as fuzzy invariant subgroups fuzzy ideals and to prove some fundamental properties in particular it will give a characteristic of a field by a fuzzy ideal

fuzzy cosets and some fuzzy radicals abstract an attempt is made to formulate fuzzy versions of jacobson and prime radicals of a ring certain basic results pertaining to the notions of jacobson and prime radicals of a ring are extended to the fuzzy setting in particular the following is obtained the jacobson radical prime radical of the ring of fuzzy cosets of in r is zero where r is a commutative ring with identity and is the fuzzy jacobson radical fuzzy prime radical of r

modeling and measuring the spatial relation along regions contours and fuzzy sets the analysis of spatial relations among objects in an image is an important vision problem that involves both shape analysis and structural pattern recognition in this paper we propose a new approach to characterize the spatial relation along an important feature of spatial configurations in space that has been overlooked in the literature up to now we propose a mathematical definition of the degree to which an object a is along an object b based on the region between a and b and a degree of elongatedness of this region in order to better fit the perceptual meaning of the relation distance information is included as well in order to cover a more wide range of potential applications both the crisp and fuzzy cases are considered in the crisp case the objects are represented in terms of 2d regions or 1d contours and the definition of the alongness between them is derived from a visibility notion and from the region between the objects however the computational complexity of this approach leads us to the proposition of a new model to calculate the between region using the convex hull of the contours on the fuzzy side the region based approach is extended experimental results obtained using synthetic shapes and brain structures in medical imaging corroborate the proposed model and the derived measures of alongness thus showing that they agree with the common sense

fuzzy semiprime ideals in semigroups abstract in this paper we shall introduce the notion of fuzzy semiprimality in a semigroup which is an extension of semiprimality in it and characterize a semigroup that is a semilattice of simple semigroups in terms of fuzzy semiprimality

fuzzy plane geometry ii circles and polygons abstract this paper continues our research in fuzzy plane geometry previously we studied fuzzy points and fuzzy lines now we investigate fuzzy circles and fuzzy polygons we show that the fuzzy area of a fuzzy circle or a fuzzy polygon is a fuzzy number we also argue that the fuzzy perimeter of a fuzzy circle or a fuzzy polygon is a fuzzy number

thinning approach for noisy digital patterns abstract many problems difficult to solve by traditional computer algorithms can be explored by the principles of human visual perception from the hint of human visual symmetry a thinning approach for noisy digital patterns is presented in this paper it includes three main algorithms algorithm 1 is designed to find the effective circular range for each pixel containing the maximum effective information algorithm 2 is designed to compute the symmetry score of the pixel distribution in the circular range and convert the symmetry information into a gray scale image algorithm 3 provides a modified gray scale thinning algorithm for thinning the gray scale image to obtain a connected thin line experiments show that the thinning results obtained by the proposed method are properly acceptable whensnr 2 2 db and the connectedness of the thin line can be preserved

image segmentation based on fuzzy connectedness using dynamic weights traditional segmentation techniques do not quite meet the challenges posed by inherently fuzzy medical images image segmentation based on fuzzy connectedness addresses this problem by attempting to capture both closeness based on characteristic intensity and hanging togetherness based on intensity homogeneity of image elements to the target object this paper presents a modification and extension of previously published image segmentation algorithms based on fuzzy connectedness which is computed as a linear combination of an object feature based and a homogeneity based component using fixed weights we provide a method called fuzzy connectedness using dynamic weights dyw to introduce directional sensitivity to the homogeneity based component and to dynamically adjust the linear weights in the functional form of fuzzy connectedness dynamic computation of the weights relieves the user of the exhaustive search process to find the best combination of weights suited to a particular application this is critical in applications such as analysis of cardiac cine magnetic resonance mr images where the optimal combination of affinity component weights can vary for each slice each phase and each subject in spite of data being acquired from the same mr scanner with identical protocols we present selected results of applying dyw to segment phantom images and actual mr computed tomography and infrared data the accuracy of dyw is assessed by comparing it to two different formulations of fuzzy connectedness our method consistently achieves accuracy of more than 99 15 for a range of image complexities contrast 5 65 noise to contrast ratio of 6 18 and bias field of four types with maximum gain factor of up to 10

image segmentation using fuzzy homogeneity criterion abstract the approach proposed here is using fuzzy homogeneity vectors and the fuzzy cooccurrence matrix to segment images homogeneity vectors could be used to represent the homogeneous feature between a pixel and its neighbors and the degree of homogeneity could be determined by using a fuzzy membership function combining the homogeneity vectors and the fuzzy membership function we could extract the feature of an image which determines the fuzzy region width with the fuzzy region width we could use the fuzzy cooccurrence matrix to measure the fuzzy geometry properties of an image which is the fuzzy entropy values that can be employed for determinine the thresholds to segment images the advantages of this approach are as follows first homogeneity vectors take account of the spatial gray tone dependence thus using homogeneity vectors has a better noise tolerance second from the extracted feature the fuzzy region width could be determined automatically third because the fuzzy region width is decided by the feature of the image it could be adjusted according to the nature of the image fourth the dimension of the fuzzy cooccurrence matrix representing the image is significantly decreased but the properties of images are faithfully preserved a large number of experiments have been carried out on different kinds of images and good results have been achieved by the proposed method this method will have wide application in image processing

the dual of a fuzzy subspace abstract the concept of the dual of a fuzzy subspace is defined double dual fuzzy subspaces and linear maps of fuzzy subspaces are also studied

prime l fuzzy ideals and primary l fuzzy ideals abstract in this paper the concepts of a primary l fuzzy ideal and a prime l fuzzy ideal and p primary are introduced and some fundamental propositions proved in particular a structural theorem for a primary l fuzzy ideal belonging to a prime l fuzzy ideal is given

watershed cuts minimum spanning forests and the drop of water principle we study the watersheds in edge weighted graphs we define the watershed cuts following the intuitive idea of drops of water flowing on a topographic surface we first establish the consistency of these watersheds they can be equivalently defined by their catchment basinsrdquo through a steepest descent property or by the dividing linesrdquo separating these catchment basins through the drop of water principle then we prove through an equivalence theorem their optimality in terms of minimum spanning forests afterward we introduce a linear time algorithm to compute them to the best of our knowledge similar properties are not verified in other frameworks and the proposed algorithm is the most efficient existing algorithm both in theory and in practice finally the defined concepts are illustrated in image segmentation leading to the conclusion that the proposed approach improves on the tested images the quality of watershed based segmentations

comparison results of the preconditioned aor methods for l matrices abstract in this paper we first provide comparison results of several types of the preconditioned aor paor methods for solving a linear system whose coefficient matrix is an l matrix satisfying some weaker conditions than those used in the recent literature next we propose an application of paor method to a preconditioner of krylov subspace method lastly numerical results are provided to show that krylov subspace method with the paor preconditioner performs quite well as compared with the ilu 0 preconditioner

fuzzy vector spaces and fuzzy cosets abstract a fuzzy subspace is characterized by its level subspaces the algebraic nature of fuzzy subspaces under homomorphisms is studied certain conditions are obtained under which a given fuzzy subspace can or cannot be realized as a union of two fuzzy subspaces such that none is contained in the other the construction of a fuzzy subspace generated by a given fuzzy subset of a vector space is given the set of all the fuzzy cosets of a fuzzy subspace is shown to be a vector space and the dimension of the same is obtained a fuzzy quotient space is defined and an analogue of a consequence of the fundamental theorem of homomorphisms is obtained

a novel visual codebook model based on fuzzy geometry for large scale image classification the codebook model has been developed as an effective means for image classification however the inherent operation of assigning visual words to image feature vectors in traditional codebook approaches causes serious ambiguities in image classification in particular the nearest word may not be the best fit to a feature and multiple words may be equally appropriate for one specific feature to resolve these ambiguities we propose a novel visual codebook model based on the n dimensional fuzzy geometry n d fg theory where all visual words and features are modeled as fuzzy points in the n d fg space and appropriate uncertainty is introduced to each fuzzy point to enhance the representation capacity this n d fg codebook model not only inherits advantages from the fuzzy set theory but also facilitates the analysis and determination of the relationship between visual words and features in geometric form by explicitly taking into account the ambiguities we propose a novel measure of similarity between the visual words and fuzzy features following the proposed codebook model and the novel similarity measure we develop two useful image classification algorithms by modifying popular image coding algorithms i e spm and llc finally experimental results demonstrate that the classification accuracy of the proposed algorithms is dramatically improved for a standard large scale image database for example with a codebook size of 256 the proposed algorithms achieve similar performance as traditional algorithms with a codebook size of 1024 indicating that the proposed algorithms reduce the computational cost by 75 while achieving almost identical classification accuracy to traditional algorithms thus the proposed algorithms represent a more efficient and appropriate scheme for big image data this paper aims to overcome the drawbacks of traditional visual words fuzzy visual word and fuzzy feature are defined using n dimensional fuzzy geometry a new similarity measure between fuzzy features and fuzzy visual words is designed two modified image classification frameworks based on fuzzy codebook are proposed experimental results demonstrated their advantages against traditional algorithms

fuzzy hv submodules abstract in this paper we define the concept of fuzzy h v submodule of an h v module and study a few results in this respect we also consider the fundamental relation e defined on an h v module m and prove a fundamental theorem concerning the module m e

modified sor type iterative method for z matrices abstract in this paper we present the convergence analysis for some modified sor type iterative method for solving linear systems and provide a comparison of spectral radius among the sor iterative method and these modified methods comparison results and the numerical example show that the rate of convergence of the gauss seidel method is faster than the rate of convergence of the sor iterative method

constrained connectivity for hierarchical image partitioning and simplification this paper introduces an image partitioning and simplification method based on the constrained connectivity paradigm according to this paradigm two pixels are said to be connected if they satisfy a series of constraints defined in terms of simple measures such as the maximum gray level differences over well defined pixel paths and regions the resulting connectivity relation generates a unique partition of the image definition domain the simplification of the image is then achieved by setting each segment of the partition to the mean value of the pixels falling within this segment fine to coarse partition hierarchies and therefore images of increasing degree of simplification are produced by varying the threshold value associated with each connectivity constraint the paper also includes a generalization to multichannel images application examples a review of related image segmentation techniques and pseudocode for an implementation based on queue and stack data structures

a new algorithm for the discrete fuzzy shortest path problem in a network in a network the arc lengths may represent time or cost in practical situations it is reasonable to assume that each arc length is a discrete fuzzy set we called it the discrete fuzzy shortest path problem there were several methods reported to solve this kind of problem in the open literature in these methods they can obtain either the fuzzy shortest length or the shortest path in this paper we propose a new algorithm that can obtain both of them the discrete fuzzy shortest length method is proposed to find the fuzzy shortest length and the fuzzy similarity measure is utilized to get the shortest path an illustrative example is given to demonstrate our proposed algorithm

the perimeter of a fuzzy set abstract in pattern recognition one often wants to measure the perimeters of regions in images this is straightforward if the region is crisply defined but if it is fuzzy it is not obvious how its perimeter can be measured this paper proposes a definition of perimeter for fuzzy subsets of the plane and shows that it reduces to the standard definition if the fuzzy subset is an ordinary subset the isoperimetric inequality does not generalize to fuzzy subsets but certain properties of the perimeters of convex sets do generalize to fuzzy perimeters of convex fuzzy subsets

fuzzy vector spaces under triangular norms abstract the object of this paper is to redefine fuzzy subspaces using the notion of a t norm and to obtain several results some of which are generalisations of the results of katsaras and liu 3

correlation between two fuzzy membership functions the need for a measure of correlation between two membership functions is stated the properties that the correlation measure may possess are examined a measure of correlation is defined on the basis of those properties the definition is extended to the cases when a the domain is finite and b the domain is a subset of r n the problems encountered while extending the definition are also discussed