towards the classification of scalar nonpolynomial evolution equations quasilinearity we prove that for m 7 scalar evolution equations of the form u t f x t u u m which admit a nontrivial conserved density of order m 1 are linear in u m the existence of such conserved densities is a necessary condition for integrability in the sense of admitting a formal symmetry hence integrable scalar evolution equations of order m 7 admitting nontrivial conserved densities are quasilinear

a compact covering method to exploit embedding capacity for matrix encoding recently a number of covering schemes proposed to exploit embedding capacity in covering images using matrix encoding when binary secret data include a high frequency of repeated bit blocks a preprocessing phase is employed to compress secret data prior to the embedding process regardless of whether the compressed secret data still contain some repeated bit blocks this paper proposes two novel compact covering schemes compact covering scheme ccs and intelligent compact covering scheme iccs which directly compress and conceal the repeated bit blocks of secret data bits into covering images within a single phase the proposed schemes can exploit the embedding capacity and increase the embedding efficiency by reducing the required encoding operations in addition the proposed iccs scheme can indicate up to which degree of repeated bit blocks of secret data it can take advantage of in order to improve the embedding capacity experimental results demonstrate that the proposed iccs method can significantly improve the embedding capacity when the ratio of the repeated bit blocks over the total number of blocks is low the proposed schemes are feasible for actual data hiding applications

classification of third order integrable evolution equations abstract the classification of scalar quasilinear integrable evolution equations of third order is completed under the symmetry approach a nonstandard way to represent an evolution equation in the form of a system is used in order to simplify the calculations the integrable nonlinearizable quasilinear equations are found to be related to the classical kdv and kn equations through differential substitutions we formulate the conjecture that all the third order integrable evolution equations are related to these two equations or to the linear one through miura type differential substitutions

reformulation of the theory of conceptual spaces this paper presents a conceptual system based on two independently developed extensions of gardenfors formulation of conceptual spaces the new approach continues to emphasize the role of properties and their associations in conceptual representation and to recognize the importance of similarity judgments in reasoning tasks in the new theory domains are sets equipped with a measure and a property is a measurable membership function on a domain concepts are sets of properties and their binary associations a new notion of the degree of ambiguity of properties for a given reasoning task is used to focus attention upon properties having the greatest discrimination power the theory is demonstrated on a maritime application involving the categorization of vessel routes

hiding data in images by simple lsb substitution in this paper a data hiding scheme by simple lsb substitution is proposed by applying an optimal pixel adjustment process to the stego image obtained by the simple lsb substitution method the image quality of the stego image can be greatly improved with low extra computational complexity the worst case mean square error between the stego image and the cover image is derived experimental results show that the stego image is visually indistinguishable from the original cover image the obtained results also show a significant improvement with respect to a previous work

footprint of uncertainty for type 2 fuzzy sets in the paper a new and comprehensible definition is proposed for type 2 fuzzy sets t2 fss and the primary and secondary memberships function are defined respectively by using multi valued mapping a new definition and formula for the footprint of uncertainty fou is presented and based on the new definitions the relation between fou and the original definition of t2 fs is discussed finally the partition method of fou is provided to represent the primary membership grade the fou and the interval type 2 fuzzy sets it2 fs when the upper and lower membership functions umf lmf of fou are given

on the calculation of extended max and min operations between convex fuzzy sets of the real line in this paper we will identify the sets of so called sub and pseudo highest intersection points of convex fuzzy sets of the real line and will explore their properties based on the properties of these sets an algorithm for calculating extended max and min operations between two or more convex fuzzy sets of the real line with general membership functions not necessarily continuous is proposed

on type 2 fuzzy relations and interval valued type 2 fuzzy sets this paper introduces new operations on the algebra of fuzzy truth values extended supremum and extended infimum which are generalizations of the extended operations of maximum and minimum between fuzzy truth values for type 2 fuzzy sets respectively using these new operations the properties of type 2 fuzzy relations are discussed especially the compositions of type 2 fuzzy relations on this basis this paper introduces interval valued type 2 fuzzy sets and interval valued type 2 fuzzy relations and discusses their properties

a novel message embedding algorithm using the optimal weighted modulus presenting a novel message embedding algorithm using the optimal weighted modulus systematically producing optimal weights within a controllable maximal range resolving the problem of pixel illegality and achieving the minimal pixel distortion the algorithm is superior to 11 current state of the art works our algorithm is undetectable from the spam and rs steganalyzer this paper proposes a novel message embedding algorithm using the optimal weighted modulus our algorithm referred to as a weighted modulus and abbreviated by wm n m conceals an m ary secret digit in a group of n pixels using an n tuple optimal weight ensures that our algorithm can produce stego pixels with minimal pixel distortion we adopt four steps to pragmatically accomplish the goal of message embedding in the first step we introduce a check weight algorithm to systematically generate all of the valid weights secondly we classify valid weights by referring to theoretically minimal distortion to produce an optimal weight along with the classification we construct a pixel alteration table which simplifies the process of the message concealment through convenient vector operations we take advantage of the homogeneous alteration table that is built in this step to decrease the distortion encountered when resolving the problem of pixel underflow or overflow finally we propose a universal bit conversion scheme which transforms a serial secret bit stream into m ary digits to minimize the transformation loss thereby achieving maximal embedding capacity our algorithm accurately predicts results prior to real message concealment effectively conveys maximal secret bits and speedily produces stego images with theoretically minimal distortion the experimental results demonstrate that our algorithm outperforms 11 current state of the art competitors it is not detectable by the spam and rs steganalytic attacks

a model for the measurement of membership and the consequences of its empirical implementation representation and uniqueness theorems are presented for the fundamental measurement of the membership of a fuzzy set when the domain of discourse is order dense the conclusion that membership is on an interval scale is justified by the inapplicability of extensive measurement to fuzziness and the lack of a natural origin for membership some descriptive statistics are also presented from an empirical study currently in progress to construct membership functions they suggest that subjects differ in their perception of fuzziness and that the models in the literature for certain linguistic operators such as not very and antonym are not sufficiently general to account for the range of individual perceptions of vague attributes a more general mathematical model is proposed the question of the meaningfulness of statements about membership functions is also discussed in the light of the assertion that membership can be fundamentally measured at best on an interval scale some of the difficulties which arise in fuzzy set theory due to this property are illustrated and a solution is suggested involving the use of a function which is derived from membership but is on an absolute scale

enhanced interval approach for encoding words into interval type 2 fuzzy sets and its convergence analysis construction of interval type 2 fuzzy set models is the first step in the perceptual computer which is an implementation of computing with words the interval approach ia has so far been the only systematic method to construct such models from data intervals that are collected from a survey however as pointed out in this paper it has some limitations and its performance can be further improved this paper proposes an enhanced interval approach eia and demonstrates its performance on data that are collected from a web survey the data part of the eia has more strict and reasonable tests than the ia and the fuzzy set part of the eia has an improved procedure to compute the lower membership function we also perform a convergence analysis to answer two important questions 1 does the output interval type 2 fuzzy set from the eia converge to a stable model as increasingly more data intervals are collected and 2 if it converges then how many data intervals are needed before the resulting interval type 2 fuzzy set is sufficiently similar to the model obtained from infinitely many data intervals we show that the eia converges in a mean square sense and generally 30 data intervals seem to be a good compromise between cost and accuracy

on the integrability of homogeneous scalar evolution equations abstract we determine the existence of infinitely many symmetries for equations of the formut uk f u uk 1 when they are homogeneous with respect to the scalinguk k with 0 algorithms are given to determine whether a system has a symmetry also independent oftandx if it has one generalized symmetry we prove it has infinitely many and these can be found using recursion operators or master symmetries the method of proof uses the symbolic method and results from diophantine approximation theory we list the 10 integrable hierarchies the methods can in principle be applied to the 0 case as we illustrate for one example with 0 which seems to be new in principle they can also be used for systems of evolution equations but so far this has only been demonstrated for one class of examples

local prediction based difference expansion reversible watermarking this paper investigates the use of local prediction in difference expansion reversible watermarking for each pixel a least square predictor is computed on a square block centered on the pixel and the corresponding prediction error is expanded the same predictor is recovered at detection without any additional information the proposed local prediction is general and it applies regardless of the predictor order or the prediction context for the particular cases of least square predictors with the same context as the median edge detector gradient adjusted predictor or the simple rhombus neighborhood the local prediction based reversible watermarking clearly outperforms the state of the art schemes based on the classical counterparts experimental results are provided

fuzzy weighted averages and implementation of the extension principle abstract this paper addresses the computational aspect of the extension principle when the principle is applied to algebraic mappings and in particular to weighted average operations in risk and decision analysis a computational algorithm based on the cut representation of fuzzy sets and interval analysis is described the method provides a discrete but exact solution to extended algebraic operations in a very efficient and simple manner examples are given to illustrate the method and its relation to other discrete methods and the exact approach by non linear programming the algorithm has been implemented in a computer program which can handle very general extended algebraic operations on fuzzy numbers

reversible data hiding with oriented and minimized distortions using cascading trellis coding a novel cascading trellis coding algorithm is proposed a novel data hiding scheme with oriented and minimized distortions is proposed an application scheme that develops the dom algorithm in dwt domain is proposed reversible data hiding technique usually generates low quality images when the relative payload is high this paper proposes a novel reversible data hiding scheme that provides improved performances first a distortion oriented minimized dom embedding algorithm is proposed by using a cascading trellis coding algorithm the overall modifications to the host coefficients are minimal and some specified host coefficients are kept intact following that the proposed dom algorithm is used in the discrete wavelet transform dwt domain to increase the payload both the scaling coefficients and the wavelet coefficients are involved in computation however by using the dom algorithm the scaling coefficients are kept unchanged by these approaches the proposed scheme provides high quality to the stego image and reversibility to the host image without the problems of overflow and underflow furthermore by using a multi round embedding operation a high payload is achieved experimental results show that the proposed scheme outperforms the state of the art reversible hiding schemes

level grading a new graded algebra structure on differential polynomials application to the classification of scalar evolution equations we define a new grading which we call the level grading on the algebra of polynomials generated by the derivatives uk i over the ring of c functions of x t u u1 uk where this grading has the property that the total derivative and the integration by parts with respect to x are filtered algebra maps in addition if u satisfies the evolution equation ut f u where f is a polynomial of order m k p and of level p then the total derivative with respect to t dt is also a filtered algebra map furthermore if the separant belongs to then the canonical densities i are polynomials of level 2i 1 and dt i is of level 2i 1 m we define kdv like evolution equations as those equations for which all the odd canonical densities are non trivial we use the properties of level grading to obtain a preliminary classification of scalar evolution equations of orders m 7 9 11 13 up to their dependence on x t u u1 and u2 these equations have the property that the canonical density 1 is where and are functions of x t u u1 u2 this form of 1 is shared by the essentially nonlinear class of third order equations and a new class of fifth order equations

multivariate modeling and type 2 fuzzy sets this paper explores the link between type 2 fuzzy sets and multivariate modeling elements of a space x are treated as observations fuzzily associated with values in a multivariate feature space a category or class is likewise treated as a fuzzy allocation of feature values possibly dependent on values in x we observe that a type 2 fuzzy set on x generated by these two fuzzy allocations captures imprecision in the class definition and imprecision in the observations in practice many type 2 fuzzy sets are in fact generated in this way and can therefore be interpreted as the output of a classification task we then show that an arbitrary type 2 fuzzy set can be so constructed by taking as a feature space a set of membership functions on x this construction presents a new perspective on the representation theorem of mendel and john the multivariate modeling underpinning the type 2 fuzzy sets can also constrain realizable forms of membership functions because averaging operators such as centroid and subsethood on type 2 fuzzy sets involve a search for optima over membership functions constraining this search can make computation easier and tighten the results we demonstrate how the construction can be used to combine representations of concepts and how it therefore provides an additional tool alongside standard operations such as intersection and subsethood for concept fusion and computing with words

adaptive image data hiding in edges using patched reference table and pair wise embedding technique reference table rt based embedding method embeds secret digits into pixel pairs under the guidance of a reference table most of the existing rt based methods either require elaborate conversions among different bases or have limited embedding capacity in this paper we use a patched reference table prt as a guide and propose a prt method to provide a better image quality and extendable embedding capacity we also exploit the concept of pixel value differencing pvd and propose another method prt pvd in the traditional pvd based methods the shape of the difference histograms of the stego images is significantly altered and thus vulnerable to some steganalyzers prt pvd adopts the prt method and uses a specially designed embedding sequence to preserve the difference histogram shape experimental results reveal that the proposed prt and prt pvd methods not only have better embedding efficiency over the existing methods but also are robust to detection by modern steganalysis tools

optimizing least significant bit substitution using cat swarm optimization strategy embedding secret data into a cover image using simple least significant bit substitution can degrade the image quality dramatically especially when a large number of bits are substituted the exhaustive least significant bit substitution method is proposed to solve this problem however the idea has no practical application due to its long computation time this paper adopts the cat swarm optimization cso strategy to obtain the optimal or near optimal solution of the stego image quality problem the cso strategy is generated by observing the behavior of cats which has been proved to achieve better performance on finding the best global solutions we revised the cso strategy in our proposed scheme to make it practicable and suitable to solve the mentioned problem the experimental results show that the proposed scheme can obtain a better solution with less computation time

a virtual image cryptosystem based upon vector quantization we propose a new image cryptosystem to protect image data it encrypts the original image into another virtual image since both original and virtual images are significant our new cryptosystem can confuse illegal users besides the camouflage this new cryptosystem has three other benefits first our cryptosystem is secure even if the illegal users know that our virtual image is a camouflage second this cryptosystem can compress image data finally our method is more efficient than a method that encrypts the entire image directly

writing on wet paper in this paper we show that the communication channel known as writing in memory with defective cells is a relevant information theoretical model for a specific case of passive warden steganography when the sender embeds a secret message into a subset c of the cover object x without sharing c with the recipient the set c which is also called the selection channel could be arbitrary determined by the sender from the cover object using a deterministic pseudo random or a truly random process we call this steganography writing on wet paper and realize it using a simple variable rate random linear code that gives the sender a convenient flexibility and control over the embedding process and is thus suitable for practical implementation the importance of the wet paper scenario for covert communication is discussed within the context of adaptive steganography and perturbed quantization steganography heuristic arguments supported by tests using blind steganalysis indicate that the wet paper steganography provides improved steganographic security and is less vulnerable to steganalytic attacks compared with existing methods with shared selection channels

on the integrability of non polynomial scalar evolution equations we show the existence of infinitely many symmetries for homogeneous equations when 0 if the equation has one generalized symmetry we prove that it has infinitely many and these can be produced by recursion operators identifying equations under homogeneous transformations we find that the only integrable equations in this class are the potential burgers potential modified korteweg de vries and potential kupershmidt equations we can draw some conclusions from these results for the case 1 which although theoretically incomplete seem to cover the known integrable systems for this case

fuzzy weighted average an improved algorithm abstract this paper suggests a modification on the fuzzy weighted average fwa method developed by dong and wong 1987 a steepest descend ascend method is proposed to facilitate steps 3 and 4 of the fwa to find the minimum maximum of f x1 x2 xn w1 w2 wn i 1n wixi i 1n wi an example used by dong and wong is again employed to illustrate the new method the same results are obtained but the new method requires much less evaluations and computations the advantages and disadvantages of the two algorithms are discussed also several relevant theorems are stated and discussed in this paper and the proofs of these theorems are listed in the appendix

general framework to histogram shifting based reversible data hiding histogram shifting hs is a useful technique of reversible data hiding rdh with hs based rdh high capacity and low distortion can be achieved efficiently in this paper we revisit the hs technique and present a general framework to construct hs based rdh by the proposed framework one can get a rdh algorithm by simply designing the so called shifting and embedding functions moreover by taking specific shifting and embedding functions we show that several rdh algorithms reported in the literature are special cases of this general construction in addition two novel and efficient rdh algorithms are also introduced to further demonstrate the universality and applicability of our framework it is expected that more efficient rdh algorithms can be devised according to the proposed framework by carefully designing the shifting and embedding functions

steganography using reversible texture synthesis we propose a novel approach for steganography using a reversible texture synthesis a texture synthesis process resamples a smaller texture image which synthesizes a new texture image with a similar local appearance and an arbitrary size we weave the texture synthesis process into steganography to conceal secret messages in contrast to using an existing cover image to hide messages our algorithm conceals the source texture image and embeds secret messages through the process of texture synthesis this allows us to extract the secret messages and source texture from a stego synthetic texture our approach offers three distinct advantages first our scheme offers the embedding capacity that is proportional to the size of the stego texture image second a steganalytic algorithm is not likely to defeat our steganographic approach third the reversible capability inherited from our scheme provides functionality which allows recovery of the source texture experimental results have verified that our proposed algorithm can provide various numbers of embedding capacities produce a visually plausible texture images and recover the source texture

an interval type 2 fuzzy model of computing with words as a methodology computing with words allows the use of words instead of numbers or symbols in the process of computing and reasoning and thus conforms more to humans inference when it is used to describe real world problems motivated by zadeh s paradigm of computing with words the literature has proposed a kind of type 1 fuzzy automata as a formal model of computing with words which takes type 1 fuzzy subsets of symbols as input however type 1 fuzzy representation provides a limited platform for approximating the meaning of words since it is not able to capture linguistic uncertainty in this paper we develop a formal interval type 2 fuzzy model of computing with words by generalizing the existing type 1 fuzzy sets based model to an interval type 2 fuzzy environment concretely we take interval type 2 fuzzy automata i e interval type 2 fuzzy finite automata and interval type 2 fuzzy pushdown automata which combine interval type 2 fuzzy set theory and automaton theory as a computational model of computing with words furthermore we develop the extension principles to extend from computing with values to computing with words and show that computing with words can be implemented with computing with values in the interval type 2 fuzzy setting at an extra computational cost

finding optimal least significant bit substitution in image hiding by dynamic programming strategy abstract the processing of simple least significant bit lsb substitution embeds the secret image in the least significant bits of the pixels in the host image this processing may degrade the host image quality so significantly that grabbers can detect that there is something going on in the image that interests them to overcome this drawback an exhaustive least significant bit substitution scheme was proposed by wang et al but it takes huge computation time wang et al then proposed another method that uses a genetic algorithm to search approximate optimal solutions and computation time is no longer so huge in this paper we shall use the dynamic programming strategy to get the optimal solution the experimental results will show that our method consumes less computation time and also gets the optimal solution

on the equivalence of linearization and formal symmetries as integrability tests for evolution equations gurses integrability test 1991 consists of the compatibility of the linearized equation with an eigenvalue equation and leads to the recursion operator this test is applied to quasilinear fifth order equations and the same classification as the formal symmetry method of mikhailov et al 1990 is obtained the same classification for polynomial equations is obtained using fokas test 1987 i e the existence of one higher order symmetry it is shown that the recursion operators of a specific form can be constructed using symmetries and conserved covariants and the recursion operators for polynomial equations are obtained with this method

operations on type 2 fuzzy sets in this paper we discuss set operations on type 2 fuzzy sets including join and meet under minimum product t norm algebraic operations properties of membership grades of type 2 sets and type 2 relations and their compositions all this is needed to implement a type 2 fuzzy logic system fls

on the classification of scalar evolution equations with non constant separant the separant of the evolution equation u t f where f is some differentiable function of the derivatives of u up to order m is the partial derivative where as an integrability test we use the formal symmetry method of mikhailov shabat sokolov which is based on the existence of a recursion operator as a formal series the solvability of its coefficients in the class of local functions gives a sequence of conservation laws called the conserved densities we apply this method to the classification of scalar evolution equations of orders for which and are non trivial i e they are not total derivatives and is not linear in its highest order derivative we obtain the top level parts of these equations and their top dependencies with respect to the level grading that we defined in a previous paper as a grading on the algebra of polynomials generated by the derivatives u b i over the ring of functions of in this setting b and i are called base and level respectively we solve the conserved density conditions to show that if depends on then these equations are level homogeneous polynomials in furthermore we prove that if is non trivial then with while if is trivial then where and and are functions of we show that the equations that we obtain form commuting flows and we construct their recursion operators that are respectively of orders 2 and 6 for non trivial and trivial respectively omitting lower order dependencies we show that equations with non trivial and b 3 are symmetries of the essentially non linear third order equation for trivial the equations with b 5 are symmetries of a non quasilinear fifth order equation obtained in previous work while for b 3 4 they are symmetries of quasilinear fifth order equations

aggregation using the fuzzy weighted average as computed by the karnik mendel algorithms by connecting work from two different problems the fuzzy weighted average fwa and the generalized centroid of an interval type 2 fuzzy set a new alpha cut algorithm for solving the fwa problem has been obtained one that is monotonically and superexponentially convergent this new algorithm uses the karnik mendel km algorithms to compute the fwa cut end points it appears that the km cut algorithms approach for computing the fwa requires the fewest iterations to date and may therefore be the fastest available fwa algorithm to date

centroid of a type 2 fuzzy set abstract in this paper we introduce the centroid and generalized centroid of a type 2 fuzzy set both of which are essential for implementing a type 2 fuzzy logic system and explain how to compute them for practical use we show how to compute the centroid of interval and gaussian type 2 fuzzy sets an exact computation procedure is provided for an interval type 2 set whereas an approximation is provided for both interval and gaussian type 2 sets examples are given that compare the exact computational results with the approximate results

generalized extended fuzzy implications this paper is devoted to investigate generalized extended fuzzy implications in accordance with the generalized extension principle the algebraic properties of generalized extended fuzzy implications are discussed on different algebras of fuzzy truth values on this basis fuzzy valued fuzzy implications are constructed from arbitrary fuzzy implications and t norms moreover a fuzzy valued fuzzy implication is identical with a generalized extended fuzzy implication on the algebra of fuzzy values when they are induced by exactly the same continuous fuzzy implication and same upper semicontinuous t norm

wireless sensor network lifetime analysis using interval type 2 fuzzy logic systems extending the lifetime of the energy constrained wireless sensor networks is a crucial challenge in sensor network research in this paper we present a novel approach based on fuzzy logic systems to analyze the lifetime of a wireless sensor network we demonstrate that a type 2 fuzzy membership function mf i e a gaussian mf with uncertain standard deviation std is most appropriate to model a single node lifetime in wireless sensor networks in our research we study two basic sensor placement schemes square grid and hex grid two fuzzy logic systems flss a singleton type 1 fls and an interval type 2 fls are designed to perform lifetime estimation of the sensor network we compare our fuzzy approach with other nonfuzzy schemes in previous papers simulation results show that fls offers a feasible method to analyze and estimate the sensor network lifetime and the interval type 2 fls in which the antecedent and the consequent membership functions are modeled as gaussian with uncertain std outperforms the singleton type 1 fls and the nonfuzzy schemes

a further study of large payloads matrix embedding a further study of large payloads me by improving the embedding efficiency is proposed a new matrix is constructed for me a sub optimal search strategy is exploited to reduce the computational complexity effective and practically feasible large payloads me can be realized by the proposed method matrix embedding me is a well known steganographic method that can improve the embedding efficiency in me the sender and recipient agree in advance on a parity check matrix pcm of a binary linear code and utilize the pcm for data embedding at the decoder side the embedded message can be extracted by the recipient as the syndrome of the received stego data the pcm can be taken as any full ranked matrix and its selection is actually crucial to the embedding performance in this paper by extending some previous works we propose a novel scheme to further improve the embedding efficiency of large payloads me first we utilize a new and specifically designed matrix for me then instead of finding a coset leader as the modification to the cover which is time consuming we turn to finding a vector in the coset with relatively small hamming weight by the proposed approach effective and practically feasible large payloads me can be realized and extensive experiments show that a significant increase of embedding efficiency is achieved compared with some state of the art works

steganalysis of lsb matching based on statistical modeling of pixel difference distributions this paper presents a least significant bit lsb matching steganography detection method based on statistical modeling of pixel difference distributions previous research indicates that natural images are highly correlated in a local neighborhood and that the value zero appears most frequently in intensity differences between adjacent pixels the statistical model of the distribution of pixel difference can be established using the laplace distribution lsb matching steganography randomly increases or decreases the pixel value by 1 when the message is embedded thus the frequency of occurrence of the value zero in pixel differences changes most dramatically during message embedding based on the laplacian model of pixel difference distributions this paper proposes a method to estimate the number of the zero difference value using the number of non zero difference values from stego images and uses the relative estimation error between the estimated and actual values of the number of the zero difference value as the classification feature experimental results indicate that the proposed algorithm is effective in detecting lsb matching steganography and can achieve better detection performance than the local extreme method under most circumstances

an optimal data hiding scheme with tree based parity check reducing distortion between the cover object and the stego object is an important issue for steganography the tree based parity check method is very efficient for hiding a message on image data due to its simplicity based on this approach we propose a majority vote strategy that results in least distortion for finding a stego object the lower embedding efficiency of our method is better than that of previous works when the hidden message length is relatively large

image hiding by optimal lsb substitution and genetic algorithm in this paper we develop a novel method to embed important data in the host image so that the interceptors will not notice about the existence of the data the basic concept of the proposed method is by simple lsb substitution to prevent illicit access of data and increase the system performance the method of randomized process and the optimal lsb substitution are proposed the e ectiveness of the optimal lsb substitution in the worst case is proved we also develop a genetic algorithm to solve the problem of hiding important data in the rightmost k lsbs of the host image which may involve a huge computation time to nd the optimal result when k is large finally based on the concept of perceptual modeling an improved hiding technique is developed to obtain a high quality embedding result experimental results will reveal that the quality of the embedding result will not be seriously degraded and is near optimal by utilizing the proposed method 2001 pattern recognition society published by elsevier science ltd all rights reserved

the collapsing method of defuzzification for discretised interval type 2 fuzzy sets this paper proposes a new approach for defuzzification of interval type 2 fuzzy sets the collapsing method converts an interval type 2 fuzzy set into a type 1 representative embedded set res whose defuzzified values closely approximates that of the type 2 set as a type 1 set the res can then be defuzzified straightforwardly the novel representative embedded set approximation resa to which the method is inextricably linked is expounded stated and proved within this paper it is presented in two forms simple resa this approximation deals with the most simple interval fou in which a vertical slice is discretised into 2 points interval resa this approximation concerns the case in which a vertical slice is discretised into 2 or more points the collapsing method simple resa version was tested for accuracy and speed with excellent results on both criteria the collapsing method proved more accurate than the karnik mendel iterative procedure kmip for an asymmetric test set for both a symmetric and an asymmetric test set the collapsing method outperformed the kmip in relation to speed