some results on almost square banach spaces abstract we study almost square banach spaces under a topological point of view indeed we prove that the class of banach spaces which admits an equivalent norm to be asq is that of those banach spaces which contain an isomorphic copy of c 0 we also prove that the symmetric projective tensor products of an almost square banach space have the strong diameter two property

optimization of stack filters based on mirrored threshold decomposition an adaptive optimization algorithm for the design of a new class of stack filters is presented unlike stack smoothers this new class of stack filters based on mirrored threshold decomposition has been empowered not only with lowpass filtering characteristics but with bandpass and highpass filtering characteristics as well therefore these filters can be effectively used in applications where frequency selection is critical an adaptive optimization approach is introduced where the positive boolean function pbf that characterizes the stack filter in the binary domain of mirrored threshold decomposition is represented by a soft truth table where each possible binary input sequence is mapped to a real number in the interval 1 1 at each iteration of the adaptive algorithm the probability that the pbf makes the correct decision when a given input sequence is presented is incremented by suitably changing the entries of the soft truth table the proposed adaptive algorithm is simple to implement since it requires only increment decrement and local comparison operations the performance of optimal stack filters is illustrated by several simulations

optimal median type filtering under structural constraints this paper presents a method for the design of median type filters that achieve the maximum noise attenuation under structural constraints imposed by the requirement of preserving certain signal or image features as compared with the design of optimal weighted median wm filters that calls for solving a set of linear inequalities this method is extremely simple yet general enough the filter is obtained by modifying directly the boolean function of a median filter and an analytic closed form representation of its boolean function can be obtained furthermore it is proven theoretically that under the same set of structural constraints the filter designed in this way will never do worse in removing i i d noise with any distribution than the optimal wm filter improvements as high as 41 46 and 52 have been achieved in simulations in a 2 d case for the uniform gaussian and double exponential distributions respectively also the improvement in the impulsive noise environment is very significant as is demonstrated by an image enhancement application

a training framework for stack and boolean filtering fast optimal design procedures and robustness case study a training framework is developed in this paper to design optimal nonlinear filters for various signal and image processing tasks the targeted families of nonlinear filters are the boolean filters and stack filters the main merit of this framework at the implementation level is perhaps the absence of constraining models making it nearly universal in terms of application areas we develop fast procedures to design optimal or close to optimal filters based on some representative training set furthermore the training framework shows explicitly the essential part of the initial specification and how it affects the resulting optimal solution symmetry constraints are imposed on the data and consequently on the resulting optimal solutions for improved performance and ease of implementation the case study is dedicated to natural images the properties of optimal boolean and stack filters when the desired signal in the training set is the image of a natural scene are analyzed specifically the effect of changing the desired signal using various natural images and the characteristics of the noise the probability distribution function the mean and the variance is analyzed elaborate experimental conditions were selected to investigate the robustness of the optimal solutions using a sensitivity measure computed on data sets a remarkably low sensitivity and consequently a good generalization power of boolean and stack filters are revealed boolean based filters are thus shown to be not only suitable for image restoration but also robust making it possible to build libraries of optimal filters which are suitable for a set of applications

almost square dual banach spaces abstract we show that finite dimensional banach spaces fail to be uniformly non locally almost square moreover we construct an equivalent almost square bidual norm on l as a consequence we get that every dual banach space containing c 0 has an equivalent almost square dual norm finally we characterize separable real almost square spaces in terms of their position in their fourth duals

octahedral norms and convex combination of slices in banach spaces abstract we study the relation between octahedral norms daugavet property and the size of convex combinations of slices in banach spaces we prove that the norm of an arbitrary banach space is octahedral if and only if every convex combination of w slices in the dual unit ball has diameter 2 which answers an open question as a consequence we get that the banach spaces with the daugavet property and its dual spaces have octahedral norms also we show that for every separable banach space containing l 1 and for every e 0 there is an equivalent norm so that every convex combination of w slices in the dual unit ball has diameter at least 2 e

optimal weighted median filtering under structural constraints a new expression for the output moments of weighted median filtered data is derived the noise attenuation capability of a weighted median filter can now be assessed using the l vector and m vector parameters in the new expression the second major contribution of the paper is the development of a new optimality theory for weighted median filters this theory is based on the new expression for the output moments and combines the noise attenuation and some structural constraints on the filter s behavior in certain special cases the optimal weighted median filter can be obtained by merely solving a set of linear inequalities this leads in some cases to closed form solutions for optimal weighted median filters some applications of the theory developed in this paper in 1 d signal processing and image processing are discussed throughout the analysis some striking similarities are pointed out between linear fir filters and weighted median filters

octahedral norms in spaces of operators abstract we study octahedral norms in the space of bounded linear operators between banach spaces in fact we prove that l x y has octahedral norm whenever x and y have octahedral norm as a consequence the space of operators l l 1 x has octahedral norm if and only if x has octahedral norm these results also allow us to get the stability of strong diameter 2 property for projective tensor products of banach spaces which is an improvement of the known results about the size of nonempty relatively weakly open subsets in the unit ball of the projective tensor product of banach spaces

unconditional almost squareness and applications to spaces of lipschitz functions abstract we introduce an unconditional concept of almost squareness in order to provide a partial negative answer to the problem of existence of almost square dual banach spaces we also take advantage of this notion to provide some criteria of non duality of some subspaces of scalar as well as vector valued lipschitz functions

a new class of nonlinear filters neural filters a class of nonlinear filters based on threshold decomposition and neural networks is defined it is shown that these neural filters include all filters defined either by continuous functions such as linear finite impulse response fir filters or by boolean functions such as generalized stack filters adaptive least mean absolute error and adaptive least mean square error algorithms are derived for determining optimal neural filters as special cases adaptive generalized stack and adaptive generalized weighted order statistic filtering algorithms under both error criteria are derived experimental results in 1d and 2d signal processing are presented to compare the performances of the adaptive neural filters and other widely used filters

a class of rank order smoothers smoothers or nonlinear filters that make estimates purely on the basis of rank and temporal order determinations among members of the input data are inherently robust and compatible with vlsi implementations the classic example is the moving median smoother but it is possible to define others which have significantly different attributes both in relation to the statistical estimation problem and in relation to hardware implications attributes offered in the latter category which are especially desirable in systolic vlsi include essentially recursive operation and elimination of the interactions between rank and temporal ordering requirements which are a feature of median smoothers experimental results indicate significantly improved response to noisy steps for these smoothers in comparison to median smoothers of comparable rectangular pulse detection capabilities

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

stack filters the median and other rank order operators possess two properties called the threshold decomposition and the stacking properties the first is a limited superposition property which leads to a new architecture for these filters the second is an ordering property which allows an efficient vlsi implementation of the threshold decomposition architecture motivated by the success of rank order filters in a wide variety of applications and by the ease with which they can now be implemented we consider in this paper a new class of filters called stack filters they share the threshold decomposition and stacking properties of rank order filters but are otherwise unconstrained they are shown to form a very large class of easily implemented nonlinear filters which includes the rank order operators as well as all compositions of morphological operators the convergence properties of these filters are investigated using techniques similar to those used to determine root signal behavior of median filters the results obtained include necessary conditions for a stack filter to preserve monotone regions or edges in signals the output distribution for these filters is also found all the stack filters of window width 3 are determined along with their convergence properties among these filters are found two which we have named asymmetric median filters they share all the properties of median filters except that they remove impulses of one sign only that is one removes only positive going edges the other removes only negative going edges while the median filter removes impulses of both signs this investigation of the properties of stack filters thus produces several new useful and easily implemented filters

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

a generalized gaussian image model for edge preserving map estimation the authors present a markov random field model which allows realistic edge modeling while providing stable maximum a posterior map solutions the model referred to as a generalized gaussian markov random field ggmrf is named for its similarity to the generalized gaussian distribution used in robust detection and estimation the model satisfies several desirable analytical and computational properties for map estimation including continuous dependence of the estimate on the data invariance of the character of solutions to scaling of data and a solution which lies at the unique global minimum of the a posteriori log likelihood function the ggmrf is demonstrated to be useful for image reconstruction in low dosage transmission tomography

big slices versus big relatively weakly open subsets in banach spaces abstract we study the unknown differences between the size of slices and relatively weakly open subsets of the unit ball in banach spaces we show that every banach space containing c 0 isomorphically can be equivalently renormed so that every slice of its unit ball has diameter 2 and still its unit ball contains nonempty relatively weakly open subsets with diameter arbitrarily small which answers an open question and stresses the differences between the size of slices and relatively weakly open subsets of the unit ball of banach spaces

extreme differences between weakly open subsets and convex combinations of slices in banach spaces abstract we show that every banach space containing isomorphic copies of c 0 can be equivalently renormed so that every nonempty relatively weakly open subset of its unit ball has diameter 2 and however its unit ball still contains convex combinations of slices with diameter arbitrarily small which improves in an optimal way the known results about the size of this kind of subsets in banach spaces

state description for the root signal set of median filters median filtering is a simple digital technique for smoothing signals one main characteristic of the filter is that it maps the input signal space into a root signal space where signals invariant to median filters are called roots of the signal in this paper we develop the theory for the root signal set of median filters a tree structure for the root signal set is obtained for binary signals the number of roots r n for a signal of length n and window size filter 2s 1 is exactly represented by the difference equation r n r n 1 r n s a general solution is obtained in a z domain approach finally a method for faster one dimensional median filter operation is introduced

on the robustness of the class of stack filters a widely held view in the nonlinear signal processing community is that the class of stack filters is robust although this is supported by extensive experimental evidence no systematic theoretical justification exists despite the availability of analytical tools for studying robustness of individual stack filters we focus on rank selection probabilities rsps as measures of robustness as it is well known that other statistical characterizations of stack filters such as output distributions breakdown probabilities and output distributional influence functions can be represented in terms of rsps we show in a very general sense that the class of stack filters is highly robust it is also shown that almost all stack filters have very similar output distributions for independent and identically distributed i i d input signals and thus very similar statistical behavior

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

weakly open sets in the unit ball of the projective tensor product of banach spaces abstract a banach space is said to have the diameter two property if every non empty relatively weakly open subset of its unit ball has diameter two we prove that the projective tensor product of two banach spaces whose centralizer is infinite dimensional has the diameter two property the same statement also holds for x y if the centralizer of x is infinite dimensional and the unit sphere of y contains an element of numerical index one we provide examples of classical banach spaces satisfying the assumptions of the results if k is any infinite compact hausdorff topological space then c k y has the diameter two property for any nonzero banach space y we also provide a result on the diameter two property for the injective tensor product

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

a fast algorithm for designing stack filters stack filters are a class of nonlinear filters with excellent properties for signal restoration unfortunately present algorithms for designing stack filters can only be used for small window sizes because of either their computational overhead or their serial nature this paper presents a new adaptive algorithm for determining a stack filter that minimizes the mean absolute error criterion the new algorithm retains the iterative nature of many current adaptive stack filtering algorithms but significantly reduces the number of iterations required to converge to an optimal filter this algorithm is faster than all currently available stack filter design algorithms is simple to implement and is shown in this paper to always converge to an optimal stack filter extensive comparisons between this new algorithm and all existing algorithms are provided the comparisons are based both on the performance of the resulting filters and upon the time and space complexity of the algorithms they demonstrate that the new algorithm has three advantages it is faster than all other available algorithms it can be used on standard workstations sparc 5 with 48 mb to design filters with windows containing 20 or more points and its highly parallel structure allows very fast implementations on parallel machines this new algorithm allows cascades of stack filters to be designed stack filters with windows containing 72 points have been designed in a matter of minutes under this new approach

rank conditioned rank selection filters for signal restoration a class of nonlinear filters called rank conditioned rank selection rcrs filters is developed and analyzed in this paper the rcrs filters are developed within the general framework of rank selection rs filters which are filters constrained to output an order statistic from the observation set many previously proposed rank order based filters can be formulated as rs filters the only difference between such filters is in the information used in deciding which order statistic to output the information used by rcrs filters is the ranks of selected input samples hence the name rank conditioned rank selection filters the number of input sample ranks used is referred to as the order of the rcrs filter the order can range from zero to the number of samples in the observation window giving the filters valuable flexibility low order filters can give good performance and are relatively simple to optimize and implement if improved performance is demanded the order can be increased but at the expense of filter simplicity in this paper many statistical and deterministic properties of the rcrs filters are presented a procedure for optimizing over the class of rcrs filters is also presented finally extensive computer simulation results that illustrate the performance of rcrs filters in comparison with other techniques in image restoration applications are presented

adaptive stack filtering with application to image processing with the aid of threshold decomposition it is shown that optimal stack filters under the mean absolute error mae criterion are equal to optimal or bayesian classifiers subject to stacking constraints under the mean classification error mce criterion nonadaptive and adaptive constrained least mean absolute lma algorithms are developed for the esti mation of stack filters through the linearization of the unit step function in the objective function the convergence of the al gorithms is proven under certain conditions although the methods do not generally give optimal stack filters under the mae criterion these algorithms have several distinct merits compared to other stack filter optimization methods 1 the op timization problem has a unique solution that approximates the optimal stack filters in the least mean square sense 2 the meth ods can be implemented in the binary and real domains and 3 for stack filters defined by linearly separable positive boolean functions pbf s or weighted order statistic wos filters the number of the parameters to be estimated is reduced to the window width of the filter a comparison between images re stored by the new algorithms and the stack filtering algorithm optimal under the mae criterion confirms the effectiveness of the proposed algorithms

fir median hybrid filters this paper introduces a new generalized class of median type filters which we call fir median hybrid fmh filters the input signal x n is filtered with m linear phase fir filters and the output of the fmh filter is the median of the outputs of the fir filters the statistical properties of the fmh filters are analyzed for input signals with gaussian double exponential and uniform density functions it is shown that fmh filters have essentially the same statistical properties and same type of root signals as the standard median sm filters fmh filters preserve corner points and ramps and attenuate impulsive type noise components effectively an interesting subclass of the fmh filters requiring only one scaling multiplier two additions and three compare swap operations irrespective of the filter length is introduced in the examples speed improvement by a factor of 450 is obtained over an otherwise approximately equivalent sm filter

spl lscr m s filters for image restoration applications a new filtering architecture is proposed generalizing some previously introduced multilevel median filters an efficient design procedure for the new filtering architecture is demonstrated for image restoration application simulation results show a good noise rejection performance combined with a fine detail preservation capability

extended permutation filters and their application to edge enhancement extended permutation ep filters are defined and analyzed in particular we focus on extended permutation rank selection eprs filters these filters are constrained to output an order statistic from an extended observation vector this extended vector includes n observation samples and k statistics that are functions of the observation samples the rank permutations from selected samples in this extended observation vector are used as the basis for selecting an order statistic output we show that by including the sample mean in the extended observation vector the filters exhibit excellent edge enhancement properties we also show that several previously defined classes of rank order based edge enhancers cs lum and wmmr sharpeners can be formulated as subclasses of eprs filters these sharpening subclasses are in addition to the smoothing subclasses which include rank conditioned rank selection permutation stack and weighted order statistic filters thus this novel class of filters provides a broad framework within which many rank order based smoothers and edge enhancers can be unified edge enhancement properties are developed and an l sub n norm eprs filter optimization procedure is presented finally extensive computer simulation results are presented comparing the performance of eprs and other sharpening filters in edge enhancement applications

adaptive multistage weighted order statistic filters based on the backpropagation algorithm as a concise representation of stack filters multistage weighted order statistic mwos filters are introduced which correspond to multistage threshold logic gates or multilayer perceptrons in the binary domain two adaptive algorithms are derived for finding optimal mwos filters under the mean absolute error criterion and the mean square error criterion respectively experimental results from image enhancement are provided to compare the performance of adaptive mwos filters and adaptive stack filters

an objective comparison of 3 d image interpolation methods to aid in the display manipulation and analysis of biomedical image data they usually need to be converted to data of isotropic discretization through the process of interpolation traditional techniques consist of direct interpolation of the grey values when user interaction is called for in image segmentation as a consequence of these interpolation methods the user needs to segment a much greater typically 4 10 spl times amount of data to mitigate this problem a method called shape based interpolation of binary data was developed resides significantly reducing user time this method has been shown to provide more accurate results than grey level interpolation the authors proposed an approach for the interpolation of grey data of arbitrary dimensionality that generalized the shape based method from binary to grey data this method has characteristics similar to those of the binary shape based method in particular the authors showed preliminary evidence that it produced more accurate results than conventional grey level interpolation methods in this paper concentrating on the three dimensional 3 d interpolation problem the authors compare statistically the accuracy of 8 different methods nearest neighbor linear grey level grey level cubic spline grey level modified cubic spline goshtasby et al 1992 and 3 methods from the grey level shape based class a population of patient magnetic resonance and computed tomography images corresponding to different parts of the human anatomy coming from different 3 d imaging applications are utilized for comparison each slice in these data sets is estimated by each interpolation method and compared to the original slice at the same location using 3 measures mean squared difference number of sites of disagreement and largest difference the methods are statistically compared pairwise based on these measures the shape based methods statistically significantly outperformed all other methods in all measures in all applications considered here with a statistical relevance ranging from 10 to 32 mean 15 for mean squared difference

almost square banach spaces abstract we single out and study a natural class of banach spaces almost square banach spaces in an almost square space we can find given a finite set x 1 x 2 x n in the unit sphere a unit vector y such that x i y is almost one these spaces have duals that are octahedral and finite convex combinations of slices of the unit ball of an almost square space have diameter 2 we provide several examples and characterizations of almost square spaces we prove that non reflexive spaces which are m ideals in their biduals are almost square we show that every separable space containing a copy of c 0 can be renormed to be almost square a local and a weak version of almost square spaces are also studied

weakly open sets in the unit ball of some banach spaces and the centralizer abstract we show that every banach space x whose centralizer is infinite dimensional satisfies that every non empty weakly open set in b y has diameter 2 where y n s x n fold symmetric projective tensor product of x endowed with the symmetric projective norm for every natural number n we provide examples where the above conclusion holds that includes some spaces of operators and infinite dimensional c algebras we also prove that every non empty weak open set in the unit ball of the space of n homogeneous and integral polynomials on x has diameter two for every natural number n whenever the cunningham algebra of x is infinite dimensional here we consider the space of n homogeneous integral polynomials as the dual of the space n s e x n fold symmetric injective tensor product of x endowed with the symmetric injective norm for instance every infinite dimensional l 1 satisfies that its cunningham algebra is infinite dimensional we obtain the same result for every non reflexive l embedded space and so for every predual of an infinite dimensional von neumann algebra

stack filters and the mean absolute error criterion a method to determine the stack filter which minimizes the mean absolute error between its output and a desired signal given noisy observations of this desired signal is presented specifically an optimal window width b stack filter can be determined with a linear program with o b2 sup b variables this algorithm is efficient since the number of different inputs to a window width b filter is m sup b if the filter has m valued input and the number of stack filters grows faster than 2 raised to the 2 sup b 2 power it is shown that optimal stack filtering under the mean absolute error criterion is analogous to optimal linear filtering under the mean squared error criterion both linear filters and stack filters are defined by superposition properties both classes are implementable and both have tractable procedures for finding the optimal filter under an appropriate error criterion

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

linear time domain model of the acoustic potential field a new time domain formulation of the acoustic wave is developed to avoid approximating assumptions of the linearized scalar wave equation that limit its validity to low mach particle velocity modeling or to a smooth potential field in a stationary medium the proposed model offers precision of the moving frame while retaining the form of the widely used linearized scalar wave equation although with respect to modified coordinates it is applicable to field calculations involving transient waves with unlimited particle velocity propagating in inhomogenous fluids or in those with time varying density the model is based on the exact flux continuity equation and the equation of motion both using the moving reference frame the resulting closed form free space scalar wave equation employing total derivatives is converted back to the partial differential form by using modified independent variables the modified variables are related to the common coordinates of space and time following integral expressions involving transient particle velocity representing wave radiated by each point of a stationary source consequently transient field produced by complex surface velocity sources can be calculated following existing surface integrals of the radiation theory although using modified coordinates the use of the proposed model is presented in a numerical simulation of a transient velocity source vibrating at selected magnitudes leading to the determination of the propagating pressure and velocity wave at any point

finding of optimal stack filter by graphic searching methods an efficient process to filter noise via an optimal stack filter is proposed the graphic search based techniques are employed to speed up the finding of the optimal stack filter experimental results and performance evaluation are demonstrated to show the efficiency of our proposed method

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