morphological shape decomposition a technique for decomposing a binary shape into a union of simple binary shapes is presented the decomposition is shown to be unique and invariant to translation rotation and scaling the techniques used in the decomposition are based on mathematical morphology the shape description produced can be used in object recognition and in binary image coding

iteratively regularized gauss newton method for nonlinear inverse problems with random noise we study the convergence of regularized newton methods applied to nonlinear operator equations in hilbert spaces if the data are perturbed by random noise it is shown that the expected square error is bounded by a constant times the minimax rates of the corresponding linearized problem if the stopping index is chosen using a priori knowledge of the smoothness of the solution for unknown smoothness the stopping index can be chosen adaptively based on lepskib s balancing principle for this stopping rule we establish an oracle inequality which implies order optimal rates for deterministic errors and optimal rates up to a logarithmic factor for random noise the performance and the statistical properties of the proposed method are illustrated by monte carlo simulations

morphological decomposition of 2 d binary shapes into modestly overlapped octagonal and disk components one problem with several leading morphological shape representation algorithms is the heavy overlapping among the representative disks of the same size a shape component formed by grouping connected disk centers may use many heavily overlapping disks to represent a simple shape part sometimes these representative disks form complicated structures a generalized skeleton transform was recently introduced which allows a shape to be represented as a collection of modestly overlapped octagonal shape parts however the generalized skeleton transform needs to be applied many times furthermore an octagonal component is not easily matched up with another octagonal component in this paper we describe a octagon fitting algorithm which identifies a special maximal octagon for each image point in a given shape this transform leads to the development of two new shape decomposition algorithms these algorithms are more efficient to implement the octagon fitting algorithm only needs to be applied once the components generated are better characterized mathematically the disk components used in the second decomposition algorithm are more primitive than octagons and easily matched up with other disk components from another shape the experiments show that the new decomposition algorithms produce as efficient representations as the old algorithm for both exact and approximate cases a simple shape matching algorithm using disk components is also demonstrated

optimal discretization of inverse problems in hilbert scales regularization and self regularization of projection methods we study the efficiency of the approximate solution of ill posed problems based on discretized noisy observations which we assume to be given beforehand a basic purpose of the paper is the consideration of stochastic noise but deterministic noise is also briefly discussed we restrict ourselves to problems which can be formulated in hilbert scales within this framework we shall quantify the degree of ill posedness provide general conditions on projection schemes to achieve the best possible order of accuracy we pay particular attention on the problem of self regularization vs tikhonov regularization moreover we study the information complexity asymptotically any method which achieves the best possible order of accuracy must use at least such amount of noisy observations

error estimates for regularization methods in hilbert scales in this paper we study regularization methods to reconstruct the solution x of the linear ill posed problem ax y a x to y from noisy data y delta in y the regularization methods are of the general form x alpha delta bar x g alpha b s a a b s a y delta a bar x where b denotes an unbounded self adjoint strictly positive definite operator in the hilbert space x assuming ax sim x a and x bar x p le e for some bar x in x a ge 0 and p ge 0 x r b r 2 x is the norm in a hilbert scale x r r in rnus we derive error estimates which show that the accuracy of the above regularization methods is order optimal provided that the function g alpha lambda the regularization parameter alpha and the parameter s are chosen properly

symbolic powers of sums of ideals let i and j be nonzero ideals in two noetherian algebras a and b over a field k let i j denote the ideal generated by i and j in a otimes k b we prove the following expansion for the symbolic powers begin aligned i j n sum i j n i i j j end aligned if a and b are polynomial rings and if mathrm char k 0 or if i and j are monomial ideals we give exact formulas for the depth and the castelnuovo mumford regularity of i j n which depend on the interplay between the symbolic powers of i and j the proof involves a result of independent interest which states that the induced map mathrm tor i a k i n rightarrow mathrm tor i r k i n 1 is zero for any homogeneous ideal i and i ge 0 n ge 0 we also investigate other properties and invariants of i j n such as the equality between ordinary and symbolic powers the waldschmidt constant and the cohen macaulayness

optimal structuring elements for the morphological pattern restoration of binary images in this paper we derive the optimal structuring elements of morphological filters in image restoration the expected pattern transformation of random sets is presented an estimation theory framework for random sets is subsequently proposed this framework is based on the least mean difference lmd estimator the lmd estimator is defined to minimize the cardinality of the expected pattern transformation of the set difference of the parameter and the estimate several important results for the determination of the lmd estimator are derived the lmd structuring elements of morphological filters in image restoration are finally derived

combinatorial decompositions of a class of rings in this paper we introduce some combinatorial methods for the study of graded rings and use these methods to analyze a class of rings that are of importance in combinatorics our methods are based on a unique representation theorem which is shown section 2 to hold for any graded algebra and which in a sense is an extension of the direct sum decomposition for cohen macaulay algebras this unique representation is reminiscent of a similar result obtained by rees 20 both results yield a decomposition of the algebra however our result differs from that of rees in that it carries more information about the algebra our main results are concerned with a class of rings see section 3 for the definition associated to simplicial complexes there are several natural operations that can be performed on simplicial complexes to obtain new complexes which in turn have an interpretation for the corresponding rings we show sections 4 5 and 6 that our basic decompositions can be transferred during these operations from the original rings to the newly constructed ones the operations studied here are the chain transform or barycentric subdivision section 4 rank selection both for partially ordered sets and for simplicial complexes section s and localization section 6 as we develop these tools we give some applications in section 5 we give a new topological characterization of the cohen macaulay property for partially ordered sets an immediate consequence of which is the rankselection theorem baclawski 2 this characterization is closely related to those obtained by garsia in 14

buchsbaumness of ordinary powers of two dimensional square free monomial ideals abstract let s k x 1 x 2 x n be a polynomial ring let i be a stanley reisner ideal in s of a pure simplicial complex of dimension one in this paper we study the buchsbaum property of s i r for any integer r 0 our first purpose is giving a characterization of ext modules ext s p s m t s j for any monomial ideal j where m t x 1 t x 2 t x n t in terms of certain simplicial complexes then we consider the buchsbaum property of s i r the main tool to check the buchsbaumness is the surjectivity criterion we see the behavior of the canonical map from ext s p s m t s i r to h m p s i r from the view point of reduced cohomology groups of simplicial complexes

numerical differentiation from a viewpoint of regularization theory in this paper we discuss the classical ill posed problem of numerical differentiation assuming that the smoothness of the function to be differentiated is unknown using recent results on adaptive regularization of general ill posed problems we propose new rules for the choice of the stepsize in the finite difference methods and for the regularization parameter choice in numerical differentiation regularized by the iterated tikhonov method these methods are shown to be effective for the differentiation of noisy functions and the order optimal convergence results for them are proved

cohen macaulayness of monomial ideals and symbolic powers of stanley reisner ideals abstract we present criteria for the cohen macaulayness of a monomial ideal in terms of its primary decomposition these criteria allow us to use tools of graph theory and of linear programming to study the cohen macaulayness of monomial ideals which are intersections of prime ideal powers we can characterize the cohen macaulayness of the second symbolic power or of all symbolic powers of a stanley reisner ideal in terms of the simplicial complex these characterizations show that the simplicial complex must be very compact if some symbolic power is cohen macaulay in particular all symbolic powers are cohen macaulay if and only if the simplicial complex is a matroid complex we also prove that the cohen macaulayness can pass from a symbolic power to another symbolic powers in different ways

a family of rules for parameter choice in tikhonov regularization of ill posed problems with inexact noise level we consider tikhonov regularization of linear ill posed problems with noisy data the choice of the regularization parameter by classical rules such as discrepancy principle needs exact noise level information these rules fail in the case of an underestimated noise level and give large error of the regularized solution in the case of very moderate overestimation of the noise level we propose a general family of parameter choice rules which includes many known rules and guarantees convergence of approximations quasi optimality is proved for a sub family of rules many rules from this family work well also in the case of many times under or overestimated noise level in the case of exact or overestimated noise level we propose to take the regularization parameter as the minimum of parameters from the post estimated monotone error rule and a certain new rule from the proposed family the advantages of the new rules are demonstrated in extensive numerical experiments

bayesian inverse problems with gaussian priors the posterior distribution in a nonparametric inverse problem is shown to contract to the true parameter at a rate that depends on the smoothness of the parameter and the smoothness and scale of the prior correct combinations of these characteristics lead to the minimax rate the frequentist coverage of credible sets is shown to depend on the combination of prior and true parameter with smoother priors leading to zero coverage and rougher priors to conservative coverage in the latter case credible sets are of the correct order of magnitude the results are numerically illustrated by the problem of recovering a function from observation of a noisy version of its primitive

sparse recovery by the standard tikhonov method it is a common belief that the tikhonov scheme with the cdot l 2 penalty fails to reconstruct a sparse structure with respect to a given system i however in this paper we present a procedure for the sparse recovery which is totally based on the standard tikhonov method this procedure consists of two steps at first the tikhonov scheme is used as a sieve to find the coefficients near i which are suspected to be non zero within this step the performance of the standard tikhonov method is controlled in some sparsity promoting space rather than in the original hilbert one in the second step of the proposed procedure the coefficients with indices selected in the previous step are estimated by means of the data functional strategy the choice of the regularization parameter is a crucial issue for both steps we show that a recently developed parameter choice rule called the balancing principle can be effectively used here we also present the results of computational experiments giving the evidence of the reliability of our approach

morphological representation of order statistics filters we propose a comprehensive theory for the morphological bounds on order statistics filters and their repeated iterations conditions are derived for morphological openings and closings to serve as bounds lower and upper respectively on order statistics filters and their repeated iterations under various assumptions morphological open closings and close openings are also shown to serve as tighter bounds lower and upper respectively on iterations of order statistics filters simulations of the application of the results presented to image restoration are finally provided

convergence of regularization methods with filter functions for a regularization parameter chosen with gsure and mildly ill posed inverse problems abstract in this work we show that the regularization methods based on filter functions with a regularization parameter chosen with the gsure principle are convergent for mildly ill posed inverse problems and under some smoothness source condition the convergence rate of the methods is not optimal for very ill posed problems but the efficiency increases with the smoothness of the solution

map estimation of finite gray scale digital images corrupted by supremum infimum noise finite gray scale digital images are modeled as realizations of discrete random functions drf and then the estimation of realizations of drf corrupted by a supremum infimum noise model is considered it is proved that morphological operators such as openings closings supremum of openings and infimum of closings are optimal maximum a posteriori map estimators under an appropriate and minimal set of assumptions relating to the structural and statistical constraints on image drf and noise drf these results are obtained for independent identically distributed i i d noise for single and multiframe observation scenarios next the assumption of i i d noise is relaxed and the map optimality and strong consistency of morphological filters for filtering image drf degraded by morphologically smooth noise i e colored noise is proved simulations on actual image data are carried out in support of the validity of theoretical results presented

a generalized discrete morphological skeleton transform with multiple structuring elements for the extraction of structural shape components a common problem shared by several leading morphological shape representation algorithms is that there is much overlapping among the representative disks of the same size a shape component represented by a group of connected disk centers sometimes uses many heavily overlapping representative disks to represent a relatively simple shape part a shape component may also contain a large number of representative disks that form a complicated structure we introduce a generalized discrete morphological skeleton transform that uses eight structuring elements to generate skeleton subsets so that no two skeletal points from the same skeleton subset are adjacent to each other each skeletal point represents a shape part that is in general an octagon with four pairs of parallel opposing sides the number of representative points needed to represent a given shape is significantly lower than that in the standard skeleton transform a collection of shape components needed to build a structural representation is easily derived from the generalized skeleton transform each shape component covers a significant area of the given shape and severe overlapping is avoided the given shape can also be accurately approximated using a small number of shape components

morphological reversible contour representation in this paper a novel morphological reversible contour representation of discrete binary images is proposed a binary image is represented by a set of nonoverlapping multilevel contours and a residual image in this proposed representation the total number of pixels representing an image is far less than the total number of pixels obtained by the seed based morphological contour skeleton lossless representation the proposed contour representation is simple unique and general without restrictions on the binary image to be represented the resulting multicontour image component is robust to noise an efficient differential chain contour coding scheme is employed to further compress the represented image the proposed method yields very low bit rates compared to the existing morphological techniques an automatic filling procedure which properly fills a proper multicontour image according to its topological structure without need of seed points is proposed

interpolation of 3 d binary images based on morphological skeletonization the morphological skeleton interpolation msi algorithm is presented it is an efficient shape based interpolation method used for interpolating slices in a three dimensional 3 d binary object it is based on morphological skeletonization which is used for two dimensional 2 d slice representation the proposed morphological skeleton matching process provides translation rotation and scaling information at the same time the interpolated slices preserve the shape of the original object slices when the slices have similar shapes it can also modify the shape of an object when the successive slices do not have similar shapes applications on artificial and real data are also presented

equality of ordinary and symbolic powers of stanley reisner ideals abstract this paper studies properties of simplicial complexes with the equality i m i m for a given m 2 the main results are combinatorial characterizations of such complexes in the two dimensional case it turns out that there exists only a finite number of complexes with this property and that these complexes can be described completely as a consequence we are able to determine all complexes for which i m is cohen macaulay for some m 2 in particular there are complexes with i 2 i 2 or i 3 i 3 but i m i m for all m 4 and that if i m i m for some m 4 then i m i m for all m 1 similarly there are complexes for which i 2 is cohen macaulay but i m is not cohen macaulay for all m 3 and if i m is cohen macaulay for some m 3 then i is a complete intersection

depth functions of symbolic powers of homogeneous ideals this paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal our investigation is focused on the behavior of the function mathrm depth r i t dim r mathrm pd i t 1 where i t denotes the t th symbolic power of a homogeneous ideal i in a noetherian polynomial ring r and mathrm pd denotes the projective dimension it has been an open question whether the function mathrm depth r i t is non increasing if i is a squarefree monomial ideal we show that mathrm depth r i t is almost non increasing in the sense that mathrm depth r i s ge mathrm depth r i t for all s ge 1 and t in e s where begin aligned e s bigcup i ge 1 t in mathbb n i s 1 1 le t le is end aligned which contains all integers t ge s 1 2 1 the range e s is the best possible since we can find squarefree monomial ideals i such that mathrm depth r i s mathrm depth r i t for t not in e s which gives a negative answer to the above question another open question asks whether the function mathrm depth r i t is always constant for t gg 0 we are able to construct counter examples to this question by monomial ideals on the other hand we show that if i is a monomial ideal such that i t is integrally closed for t gg 0 e g if i is a squarefree monomial ideal then mathrm depth r i t is constant for t gg 0 with begin aligned lim t rightarrow infty mathrm depth r i t dim r dim oplus t ge 0 i t mathfrak m i t end aligned our last result which is the main contribution of this paper shows that for any positive numerical function phi t which is periodic for t gg 0 there exist a polynomial ring r and a homogeneous ideal i such that mathrm depth r i t phi t for all t ge 1 as a consequence for any non negative numerical function psi t which is periodic for t gg 0 there is a homogeneous ideal i and a number c such that mathrm pd i t psi t c for all t ge 1

morphological representation of discrete and binary images a general theory for the morphological representation of discrete and binary images is presented the basis of this theory relies on the generation of a set of nonoverlapping segments of an image via repeated erosions and set transformations which in turn produces a decomposition that guarantees exact reconstruction the relationship between the proposed representation and some existing shape analysis tools e g discrete size transform pattern spectrum skeletons is investigated thus introducing the representation as the basis of a unified theory for geometrical image analysis particular cases of the general representation scheme are shown to yield a number of useful image decompositions which are directly related to various forms of morphological skeletons the relationship between the representation and the various forms of morphological skeletons is studied as a result of this study a unified theory is developed for the mathematical description of the morphological skeleton decomposition of discrete and binary images

cohen macaulayness of large powers of stanley reisner ideals abstract we prove that for m 3 the symbolic power i m of a stanley reisner ideal is cohen macaulay if and only if the simplicial complex is a matroid similarly the ordinary power i m is cohen macaulay for some m 3 if and only if i is a complete intersection these results solve several open questions on the cohen macaulayness of ordinary and symbolic powers of stanley reisner ideals moreover they have interesting consequences on the cohen macaulayness of symbolic powers of facet ideals and cover ideals

almost gorenstein homogeneous rings and their h vectors abstract in this paper for the development of the study of almost gorenstein graded rings we discuss some relations between almost gorensteinness of cohen macaulay homogeneous rings and their h vectors concretely for a cohen macaulay homogeneous ring r we give a sufficient condition for r to be almost gorenstein in terms of the h vector of r theorem 3 1 and we also characterize almost gorenstein homogeneous domains with small socle degrees in terms of the h vector of r theorem 4 4 moreover we also provide the examples of almost gorenstein homogeneous domains arising from lattice polytopes

regularization of some linear ill posed problems with discretized random noisy data for linear statistical ill posed problems in hilbert spaces we introduce an adaptive procedure to recover the unknown so lution from indirect discrete and noisy data this procedure is shown to be order optimal for a large class of problems smooth ness of the solution is measured in terms of general source condi tions the concept of operator monotone functions turns out to be an important tool for the analysis

spatially variant morphological restoration and skeleton representation the theory of spatially variant sv mathematical morphology is used to extend and analyze two important image processing applications morphological image restoration and skeleton representation of binary images for morphological image restoration we propose the sv alternating sequential filters and sv median filters we establish the relation of sv median filters to the basic sv morphological operators i e sv erosions and sv dilations for skeleton representation we present a general framework for the sv morphological skeleton representation of binary images we study the properties of the sv morphological skeleton representation and derive conditions for its invertibility we also develop an algorithm for the implementation of the sv morphological skeleton representation of binary images the latter algorithm is based on the optimal construction of the sv structuring element mapping designed to minimize the cardinality of the sv morphological skeleton representation experimental results show the dramatic improvement in the performance of the sv morphological restoration and sv morphological skeleton representation algorithms in comparison to their translation invariant counterparts

efficient morphological shape representation mathematical morphology is well suited to capturing geometric information hence morphology based approaches have been popular for object shape representation the two primary morphology based approaches the morphological skeleton and the morphological shape decomposition msd each represent an object as a collection of disjoint sets a practical shape representation scheme though should give a representation that is computationally efficient to use unfortunately little work has been done for the morphological skeleton and the msd to address efficiency we propose a flexible search based shape representation scheme that typically gives more efficient representations than the morphological skeleton and msd our method decomposes an object into a number of simple components based on homothetics of a set of structuring elements to form the representation the components are combined using set union and set difference operations we use three constituent component types and a thorough cost based search strategy to find efficient representations we also consider allowing object representation error which may yield even more efficient representations

pattern spectrum and multiscale shape representation the results of a study on multiscale shape description smoothing and representation are reported multiscale nonlinear smoothing filters are first developed using morphological opening and closings g matheron 1975 used openings and closings to obtain probabilistic size distributions of euclidean space sets continuous binary images these distributions are used to develop a concept of pattern spectrum a shape size descriptor a pattern spectrum is introduced for continuous graytone images and arbitrary multilevel signals as well as for discrete images by developing a discrete size family of patterns large jumps in the pattern spectrum at a certain scale indicate the existence of major protruding or intruding substructures of the signal at the scale an entropy like shape size complexity measure is also developed based on the pattern spectrum for shape representation a reduced morphological skeleton transform is introduced for discrete binary and graytone images this transform is a sequence of skeleton components sparse images which represent the original shape at various scales it is shown that the partially reconstructed images from the inverse transform on subsequences of skeleton components are the openings of the image at a scale determined by the number of eliminated components in addition two way correspondences are established among the degree of shape smoothing via multiscale openings or closings the pattern spectrum zero values and the elimination or nonexistence of skeleton components at certain scales

efficient morphological shape representation with overlapping disk components this paper proposes a new morphological shape representation algorithm in which a two dimensional 2 d binary shape is represented as a union of certain disks contained in the given shape the representative disks of different sizes may overlap but excessive overlapping between them is avoided the algorithm combines the advantages of the morphological skeleton transform mst and the morphological shape decomposition msd the representative disks have simple and well defined mathematical characterizations the algorithm is simple and efficient to implement the experimental results show that the number of representative disks used by our algorithm is significant lower than that used by the msd the overlapping level between the representative disks is much lower than that of the mst a simple procedure can be used to combine the representative disks into more meaningful shape components these shape components seem to correspond better to the natural shape parts than those generated by the msd it is also possible to build a good approximation for a given shape using only a small number of major components

boundary constrained morphological skeleton minimization and skeleton reconstruction a new algorithm for minimizing a morphological skeleton entitled boundary constrained skeleton minimization bcsm as well as a new algorithm for reconstructing an original image from its minimized skeletal structure termed boundary constrained skeleton reconstruction bcsr are proposed the new algorithms are shown to reduce data storage requirements from n 1 binary images represented as separate skeleton subsets with their corresponding indices to 2 binary images composed of a binary morphological skeleton and its corresponding morphological boundary structure in addition to a reduction in memory storage bcsm and bcsr result in substantial savings in computational complexity the proposed algorithms are evaluated in the context of image analysis and coding and their performance is compared to previous algorithms proposed by serra 1982 1988 and by maragos and schafer 1986 sample evaluations indicate a greater than 22 fold savings in computational requirements and 11 fold reduction in memory requirements

convergence rates of general regularization methods for statistical inverse problems and applications previously the convergence analysis for linear statistical inverse problems has mainly focused on spectral cut off and tikhonov type estimators spectral cut off estimators achieve minimax rates for a broad range of smoothness classes and operators but their practical usefulness is limited by the fact that they require a complete spectral decomposition of the operator tikhonov estimators are simpler to compute but still involve the inversion of an operator and achieve minimax rates only in restricted smoothness classes in this paper we introduce a unifying technique to study the mean square error of a large class of regularization methods spectral methods including the aforementioned estimators as well as many iterative methods such as nu methods and the landweber iteration the latter estimators converge at the same rate as spectral cut off but require only matrix vector products our results are applied to various problems in particular we obtain precise convergence rates for satellite gradiometry l 2 boosting and errors in variable problems

hilbert functions and level algebras abstract the authors consider certain quotients a r i of the polynomial ring r k x1 xr over an arbitrary field k they first determine upper and lower bounds on the hilbert functions of any algebra having the form a r v where v is the largest ideal of r agreeing in degrees at least j with the ideal v generated by a vector subspace v rj of degree j forms these bounds extend the macaulay bounds theorem 1 4 a level artinian quotient of a r i has socle soc a 0 m m x1 xr in a single degree j the authors next determine the extremal hilbert functions hmax t j r and hmin t j r that occur for level graded artin algebra quotients of r having socle degree j and type t dimk soc a and they describe the extremal strata theorem 1 8 they next give a natural upper bound for the hilbert function of level algebra quotients of the coordinate ring o z of a punctual subscheme z p n n r 1 in terms of the hilbert function h z and j t this bound was known and known to be sharp in the case z is locally gorenstein and t 1 finally they show that there are no level algebras of hilbert function t 1 3 4 5 6 2 proposition 2 7 given that there are smooth punctual subschemes z of p 2 with hilbert function h z 1 3 4 5 6 6 this result shows that the sharpness of the natural upper bound in the case where z is gorenstein and t 1 does not extend to level algebra quotients of type 2 even when z is smooth the extremality results for the hilbert functions of level algebras and some counterexamples use the extremality theorem of f h s macaulay 1927 proc london math soc 26 531 555 and g gotzmann s 1978 math z 158 61 70 results on the hilbert scheme

statistical inverse estimation in hilbert scales the recovery of signals from indirect measurements blurred by random noise is considered under the assumption that prior knowledge regarding the smoothness of the signal is avialable for greater flexibility the general problem is embedded in an abstract hilbert scale in the applications sobolev scales are used for the construction of estimators we employ preconditioning along with regularized operator inversion in the appropriate inner product where the operator is bounded but not necessarily compact a lower bound to certain minimax rates is included and it is shown that in generic examples the proposed estimators attain the asymptotic minimax rate examples include errors in variables deconvolution and indirect nonparametric regression special instances of the latter are estimation of the source term in a differential equation and the estimation of the initial state in the heat equation

analysis of profile functions for general linear regularization methods the stable approximate solution of ill posed linear operator equations in hilbert spaces requires regularization tight bounds for the noise free part of the regularization error are constitutive for bounding the overall error norm bounds of the noise free part which decrease to zero along with the regularization parameter are called profile functions and are the subject of our analysis the interplay between properties of the regularization and certain smoothness properties of solution sets which we shall describe in terms of sourcewise representations is crucial for the decay of associated profile functions on the one hand we show that a given decay rate is possible only if the underlying true solution has appropriate smoothness on the other hand if smoothness fits the regularization then decay rates are easily obtained if smoothness does not fit then we will measure this in terms of some distance function tight bounds for these allow us to obtain profile functions finally we study the most realistic case when smoothness is measured with respect to some operator which is related to the one governing the original equation only through a link condition in many parts the analysis is done on geometric basis extending classical concepts of linear regularization theory in hilbert spaces we emphasize intrinsic features of linear ill posed problems which are frequently hidden in the classical analysis of such problems

a representation theory for morphological image and signal processing a unifying theory for many concepts and operations encountered in or related to morphological image and signal analysis is presented the unification requires a set theoretic methodology where signals are modeled as sets systems signal transformations are viewed as set mappings and translational invariant systems are uniquely characterized by special collections of input signals this approach leads to a general representation theory in which any translation invariant increasing upper semicontinuous system can be presented exactly as a minimal nonlinear superposition of morphological erosions or dilations the theory is used to analyze some special cases of image signal analysis systems such as morphological filters median and order statistic filters linear filters and shape recognition transforms although the developed theory is algebraic its prototype operations are well suited for shape analysis hence the results also apply to systems that extract information about the geometrical structure of signals

dilational interpolatory inequalities they also wish to acknowledge the support of the radon institute of computational and applied mathematics where the initial draft of this paper was finalized

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

symbolic powers of monomial ideals and vertex cover algebras abstract we introduce and study vertex cover algebras of weighted simplicial complexes these algebras are special classes of symbolic rees algebras we show that symbolic rees algebras of monomial ideals are finitely generated and that such an algebra is normal and cohen macaulay if the monomial ideal is squarefree for a simple graph the vertex cover algebra is generated by elements of degree 2 and it is standard graded if and only if the graph is bipartite we also give a general upper bound for the maximal degree of the generators of vertex cover algebras

standard graded vertex cover algebras cycles and leaves the aim of this paper is to characterize simplicial complexes which have standard graded vertex cover algebras this property has several nice consequences for the squarefree monomial ideals defining these algebras it turns out that such simplicial complexes are closely related to a range of hypergraphs which generalize bipartite graphs and trees these relationships allow us to obtain very general results on standard graded vertex cover algebras which cover previous major results on rees algebras of squarefree monomial ideals

level property of ordinary and symbolic powers of stanley reisner ideals abstract in this paper we prove that the t th ordinary and or symbolic power of a stanley reisner ideal is level for some positive integer t 3 if and only if i is a complete intersection and equi generated for t 2 we give a characterization of level property of the second symbolic power i 2 when is a matroid complex of dimension one

adaptive estimation of linear functionals in hilbert scales from indirect white noise observations we consider adaptive estimating the value of a linear functional from indirect white noise observations for a flexible approach the problem is embedded in an abstract hilbert scale we develop an adaptive estimator that is rate optimal within a logarithmic factor simultaneously over a wide collection of balls in the hilbert scale it is shown that the proposed estimator has the best possible adaptive properties for a wide range of linear functionals the case of discretized indirect white noise observations is studied and the adaptive estimator in this setting is developed

licci squarefree monomial ideals generated in degree two or with deviation two abstract we study squarefree monomial ideals which are in the linkage class of a complete intersection we call such an ideal licci firstly we classify all licci edge ideals secondly we prove that any cohen macaulay almost complete intersection squarefree monomial ideal is licci thirdly we characterize licci squarefree monomial ideals of deviation 2 in terms of hypergraphs associated to the ideals as an application of the classification in kimura terai and yoshida 2009 15

level rings and algebras with straightening laws abstract stanley 13 defined the concept of level rings intermediate between cohen macaulay and gorenstein which is of interest from viewpoints of both commutative algebra and combinatorics via the theory of canonical modules the study of level rings turns out to be one of the most important foundations of the theory of algebras with straightening laws the purpose of this paper is to state some basic results on level rings which are either stanley reisner rings or algebras with straightening laws

stability of depths of powers of edge ideals abstract let g be a graph and let i i g be its edge ideal in this paper we provide an upper bound of n from which depth r i g n is stationary and compute this limit explicitly this bound is always achieved if g has no cycles of length 4 and every its connected component is either a tree or a unicyclic graph

on the associated graded modules of canonical modules the canonical modules of cohen macaulay rings give a useful technical tool in commutative algebra algebraic geometry and singularity theory via various duality theorems but not much is known about the canonical modules themselves in this paper we investigate the structure of the canonical moduies expecially their associated graded modules we ask the following two questions let k be a canonical module over a cohenmacaulay local ring r then

efficient morphological shape representation by varying overlapping levels among representative disks abstract the morphological skeleton transform the morphological shape decomposition and the overlapped morphological shape decomposition are three basic morphological shape representation schemes in this paper we propose a new way of generalizing these basic representation algorithms to improve representational efficiency in all three basic algorithms a fixed overlapping policy is used to control the overlapping relationships among representative disks of different sizes in our new algorithm different overlapping policies are used to generate shape components that have different overlapping relationships among themselves the overlapping policy is selected dynamically according to local shape features experiments show that compared to the three basic algorithms our algorithm produces more efficient representations with lower numbers of representative points

radial symmetries based decomposition of cell clusters in binary and gray level images the segmentation of structures of complex cytological and histological images is a necessary intermediate step for image analysis that give rise to binary images in many cases these binary images can be rather far away from a subsequent object specific quantification because biological structures digitized by optoelectronic devices may situated close together so that they appear as one fused object in the projective image such fusions of objects may become complex so that large clusters of biological structures emerge to quantify individual objects of a cluster they must be separated the shape size and intensity variation of cells in complex organs like the brain may breed planar configurations that can be splitted only inadequate by common techniques e g watershed separation or basic morphological processing of images considering iteratively object contours suitable features of saliency can be accumulated that give rise to markers of singular objects such significant markers may drive a separation process more effective than common approaches the determination of markers by an iterative method should be scale translation and rotation invariant and robust with regard to noise due to the variability of biological specimen we realize a technique that splits cell clumps consisting of different cell sizes and shapes into meaningful parts the multiscale method applied here is based on the analysis of the contour shape and the object area by iterative voting using oriented kernels these cone shaped kernels vote iteratively for the local center of mass of the components of an aggregation the voting is performed along the gradient of the distance transformation of the binarized image of aggregates iterative voting is initialized by voting along the gradient direction where at each iteration the voting direction and shape of the kernel is refined resp the kernel topography is refined and reoriented iteratively it turned out that the kernel topography is unique because it votes for the most likely set of grid points where the gravity center of an individual cluster component may be located furthermore a new procedure is realized to use the local intensities of aggregations for kernel voting the last voted iteration provides gravitation centers resp centers of mass of the clumped cells these are extracted and used as markers to determine individual cell boundaries by a marker based watershed postprocessing the subject of this paper is to highlight the basic algorithm of iterative kernel voting and expanding it to process intensities within clusters as well as contour information the approach is applied to synthetic images that were modified systematically with regard to object topology natural aggregates of cells at the light microscopic level and cell clusters derived from high resolution flat bed scanning were splitted in addition to these examples images from a benchmark databases were investigated the splittings generated by the iterative voting approach were compared with expected splittings of test persons and with results of the watershed method especially the gray level based iterative voting method provides superior results for cell cluster separation in comparison to the watershed procedure

a fast thresholded linear convolution representation of morphological operations in this correspondence we present a fast thresholded linear convolution representation of morphological operations the thresholded linear convolution representation of dilation and erosion is first proposed a comparison of the efficiency of the direct implementation of morphological operations and the thresholded linear convolution representation of morphological operations is subsequently conducted mathematical morphology has emerged as a powerful new tool for image processing

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

cohen macaulayness for symbolic power ideals of edge ideals abstract let s k x 1 x n be a polynomial ring over a field k let i g s denote the edge ideal of a graph g we show that the lth symbolic power i g l is a cohen macaulay ideal i e s i g l is cohen macaulay for some integer l 3 if and only if g is a disjoint union of finitely many complete graphs when this is the case all the symbolic powers i g l are cohen macaulay ideals similarly we characterize graphs g for which s i g l has flc as an application we show that an edge ideal i g is complete intersection provided that s i g l is cohen macaulay for some integer l 3 this strengthens the main theorem in crupi et al 2010 3

simplicial complexes and macaulay s inverse systems let be a simplicial complex on v x 1 x n with stanley reisner ideal i delta subseteq r k x 1 ldots x n the goal of this paper is to investigate the class of artinian algebras a a delta a 1 ldots a n r i delta x 1 a 1 ldots x n a n where each a i 2 by utilizing the technique of macaulay s inverse systems we can explicitly describe the socle of a in terms of as a consequence we determine the simplicial complexes that we will call levelable for which there exists a tuple a 1 a n such that a a 1 a n is a level algebra

comparison between the morphological skeleton and morphological shape decomposition the morphological skeleton and morphological shape decomposition msd are two popular approaches for morphological shape representation each method represents an object as an algebraic combination of a number of components where each component is given by a locus of points dilated by a specified structuring element homothetic this correspondence develops a theoretical comparison between the two methods combining the theoretical results with several representation cost measures we make a concrete comparison of the efficiency of the two methods the results indicate that for complex objects i e objects requiring a full range of homothetic sizes in the morphological skeleton representation the msd represents objects more efficiently than the morphological skeleton for three of four suggested cost measures

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

filtered legendre expansion method for numerical differentiation at the boundary point with application to blood glucose predictions let f 1 1 r be continuously differentiable we consider the question of approximating f 1 from given data of the form t j f t j j 1 m where the points t j are in the interval 1 1 it is well known that the question is ill posed and there is very little literature on the subject known to us we consider a summability operator using legendre expansions together with high order quadrature formulas based on the points t j s to achieve the approximation we also estimate the effect of noise on our approximation the error estimates both with or without noise improve upon those in the existing literature and appear to be unimprovable the results are applied to the problem of short term prediction of blood glucose concentration yielding better results than other comparable methods

optimal morphological pattern restoration from noisy binary images a theoretical analysis of morphological filters for the optimal restoration of noisy binary images is presented the problem is formulated in a general form and an optimal solution is obtained by using fundamental tools from mathematical morphology and decision theory consideration is given to the set difference distance function as a measure of comparison between images this function is used to introduce the mean difference function as a quantitative measure of the degree of geometrical and topological distortion introduced by morphological filtering it is proved that the class of alternating sequential filters is a set of parametric smoothing morphological filters that best preserve the crucial structure of input images in the least mean difference sense

morphological representation of 2 d binary shapes using rectangular components abstract the morphological skeleton transform is a shape representation scheme that decomposes a shape into a union of all maximal homothetics of a structuring element contained in the shape in this paper we develop an algorithm that generalizes the skeleton transform by allowing many different rectangles of different sizes and aspect ratios to be used as shape components the shape components in our representations still have simple and well defined mathematical characterizations the representation is uniquely defined and the algorithm still is simple and efficient to implement experiments show that our representations use significantly less shape components than those produced by the skeleton transform we also describe different ways to derive a new set of rectangular shape components with less overlapping from the original set of rectangles from our representation algorithm

a generalized fuzzy mathematical morphology and its application in robust 2 d and 3 d object representation in this paper the generalized fuzzy mathematical morphology gfmm is proposed based on a novel definition of the fuzzy inclusion indicator fii fii is a fuzzy set used as a measure of the inclusion of a fuzzy set into another that is proposed to be a fuzzy set it is proven that the fii obeys a set of axioms which are proposed to be extensions of the known axioms that any inclusion indicator should obey and which correspond to the desirable properties of any mathematical morphology operation the gfmm provides a very powerful and flexible tool for morphological operations the binary and grayscale mathematical morphologies can be considered as special cases of the proposed gfmm an application for robust skeletonization and shape decomposition of two dimensional 2 d and three dimensional 3 d objects is presented simulation examples show that the object reconstruction from their skeletal subsets that can be achieved by using the gfmm is better than by using the binary mathematical morphology in most cases furthermore the use of the gfmm for skeletonization and shape decomposition preserves the shape and the location of the skeletal subsets and spines

morphological shape representation abstract a new approach for shape representation is described which provides a general scheme for object description and unifies some of the existing representation techniques the proposed method has the advantage that it is based on the use of simple geometric objects which are intuitively used by humans in their perception of shapes furthermore its unifying framework can be used both in computer graphics and computer vision thus providing a tool to close the existing gap between object modeling and object recognition in certain applications

cohen macaulayness of powers of two dimensional squarefree monomial ideals abstract two dimensional squarefree monomial ideals can be seen as the stanley reisner ideals of graphs the main results of this paper are combinatorial characterizations for the cohen macaulayness of ordinary and symbolic powers of such an ideal in terms of the associated graph