a stochastic continuation approach to piecewise constant reconstruction we address the problem of reconstructing a piecewise constant 3 d object from a few noisy 2 d line integral projections more generally the theory developed here readily applies to the recovery of an ideal n d signal n ges 1 from indirect measurements corrupted by noise stabilization of this ill conditioned inverse problem is achieved with the potts prior model which leads to a challenging optimization task to overcome this difficulty we introduce a new class of hybrid algorithms that combines simulated annealing with deterministic continuation we call this class of algorithms stochastic continuation sc we first prove that under mild assumptions sc inherits the finite time convergence properties of generalized simulated annealing then we show that sc can be successfully applied to our reconstruction problem in addition we look into the concave distortion acceleration method introduced for standard simulated annealing and we derive an explicit formula for choosing the free parameter of the cost function numerical experiments using both synthetic data and real radiographic testing data show that sc outperforms standard simulated annealing

tilting bundles on toric fano fourfolds this paper constructs tilting bundles obtained from full strong exceptional collections of line bundles on all smooth 4 dimensional toric fano varieties the tilting bundles lead to a large class of explicit calabi yau 5 algebras obtained as the corresponding rolled up helix algebra a database of the full strong exceptional collections can be found in the package quiverstoricvarieties for the computer algebra system macaulay2

ringel duality as an instance of koszul duality abstract in 30 s koenig s ovsienko and the second author showed that every quasi hereditary algebra is morita equivalent to the right algebra i e the opposite algebra of the left dual of a coring let a be an associative algebra and v an a coring whose right algebra r is quasi hereditary in this paper we give a combinatorial description of an associative algebra b and a b coring w whose right algebra is the ringel dual of r we apply our results in small examples to obtain restrictions on the a structure of the ext algebra of standard modules over a class of quasi hereditary algebras related to birational morphisms of smooth surfaces

cooling schedules for optimal annealing a monte carlo optimization technique called simulated annealing is a descent algorithm modified by random ascent moves in order to escape local minima which are not global minima the level of randomization is determined by a control parameter t called temperature which tends to zero according to a deterministic cooling schedule we give a simple necessary and sufficient condition on the cooling schedule for the algorithm state to converge in probability to the set of globally minimum cost states in the special case that the cooling schedule has parametric form tt c log1 t the condition for convergence is that c be greater than or equal to the depth suitably defined of the deepest local minimum which is not a global minimum state

diffusions for global optimizations we seek a global minimum of u 0 1 n to r the solution to d dt x t nabla u x t will find local minima the solution to dx t nabla u x t dt sqrt 2t dw t where w is standard n dimensional brownian motion and the boundaries are reflecting will concentrate near the global minima of u at least when temperature t is small the equilibrium distribution for x t is gibbs with density pi t x alpha exp u x t this suggests setting t t t downarrow 0 to find the global minima of u we give conditions on u x and t t such that the solution to dx t nabla u x t dt sqrt 2t dw t converges weakly to a distribution concentrated on the global minima of u

two dimensional root structures and convergence properties of the separable median filter the root signals invariant to filtering structures of the two dimensional separable median filter are derived and presented in addition it is proved that with rare exception after repetitive passes of the separable median filter any two dimensional signal will be reduced to a signal containing only root structures

simulated annealing and quantum detailed balance the analogue of simulated annealing is considered for time inhomogeneous evolutions of a von neumann algebra of operators whose instantaneous generator at each timet satisfies the quantum detailed balance condition with respect to a faithful normal state which depends on time through a suitable cooling schedule convergence to the nonfaithful limiting state is proved under the usual kinds of assumptions the approach is interesting in view of possible applications to stochastic ising models and to boltzmann machines

rank order smoothers with two dimensional data model abstract development of the theory of rank order smoothers is motivated by hardware implementation as well as statistical considerations a recently proposed class of rank order smoothers based on minimization of polyhedral convex functions involving one dimensional data models employs minimization algorithms derived from dynamic programming computations proceed by storing and manipulating arrays of breakpoints of piecewise linear convex functions such an approach has certain advantages over alternatives foremost among these is the attainment of a fixed point or root signal in a single application of the minimization algorithm other advantages are the intelligibility of the connection between the rough data and the root signal smooth derived from it the absence of any significant problem of how to deal with end values compatibility with real time applications and storage and computation time requirements of the order of the number of pieces of rough data when experimental comparison was made to median smoothers better smoothing was observed on the negative side such smoothers have been largely restricted to types that behave fundamentally like median smoothers i e have the same sorts of fixed points or root signals it is shown that by letting the data model be two dimensional and formulating the dynamic programming method so that variables are eliminated in pairs then the root signal sets of such smoothers are greatly expanded while the various advantages are retained

convergence in probability of compressed annealing we consider combinatorial optimization problems for which the formation of a neighborhood structure of feasible solutions is impeded by a set of constraints neighborhoods are recovered by relaxing the complicating constraints into the objective function within a penalty term we examine a heuristic called compressed annealing that integrates a variable penalty multiplier approach within the framework of simulated annealing we refer to the value of the penalty multiplier as pressure we analyze the behavior of compressed annealing by exploring the interaction between temperature which controls the ability of compressed annealing to climb hills and pressure which controls the height of the hills we develop a necessary and sufficient condition on the joint cooling and compression schedules for compressed annealing to converge in probability to the set of global minima our work generalizes the results of hajek 1988 in the sense that when there are no relaxed constraints our results reduce to his

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

preprojective algebra structure on skew group algebras abstract we give a class of finite subgroups g s l n k for which the skew group algebra k x 1 x n g does not admit the grading structure of a higher preprojective algebra namely we prove that if a finite group g s l n k is conjugate to a subgroup of s l n 1 k s l n 2 k for some n 1 n 2 1 then the skew group algebra k x 1 x n g is not morita equivalent to a higher preprojective algebra this is related to the preprojective algebra structure on the tensor product of two koszul bimodule calabi yau algebras we prove that such an algebra cannot be endowed with a grading structure as required for a higher preprojective algebra moreover we construct explicitly the bound quiver of the higher preprojective algebra over a finite dimensional koszul algebra of finite global dimension we show in addition that preprojective algebras over higher representation infinite koszul algebras are derivation quotient algebras whose relations are given by a superpotential

helices on del pezzo surfaces and tilting calabi yau algebras we study tilting for a class of calabi yau algebras associated to helices on fano varieties we do this by relating the tilting operation to mutations of exceptional collections for helices on del pezzo surfaces the algebras are of dimension three and using an argument of herzog together with results of kuleshov and orlov we obtain a complete description of the tilting process in terms of quiver mutations

the relationship of the multi shell to multistage and standard median filters it is shown that for a particular window size by changing gradually the minimum and maximum operators to the higher and lower order statistics of the shell respectively the multi shell median filter will gradually turn from a multistage max median filter into a 2 d standard median filter

order configuration functions mathematical characterizations and applications to digital signal and image processing we call a function f in n variables an order configuration function if for any x1 xn such that xi1 xin we have f x1 xn xt where t is determined by the n tuple i1 in corresponding to that ordering equivalently it is a function built as a minimum of maxima or a maximum of minima well known examples are the minimum the maximum the median and more generally rank functions or the composition of rank functions such types of functions are often used in nonlinear processing of digital signals or images for example in the median or separable median filter min max filters rank filters etc in this paper we study the mathematical properties of order configuration functions and of a wider class of functions that we call order subconfiguration functions we give several characterization theorems for them we show through various examples how our concepts can be used in the design of digital signal filters or image transformations based on order configuration functions

nonstationary markov chains and convergence of the annealing algorithm we study the asymptotic behavior as timet of certain nonstationary markov chains and prove the convergence of the annealing algorithm in monte carlo simulations we find that in the limitt a nonstationary markov chain may exhibit phase transitions nonstationary markov chains in general and the annealing algorithm in particular lead to biased estimators for the expectation values of the process we compute the leading terms in the bias and the variance of the sample means estimator we find that the annealing algorithm converges if the temperaturet t goes to zero no faster thanc log t t0 ast with a computable constantc andt0 the initial time the bias and the variance of the sample means estimator in the annealing algorithm go to zero likeo t 1 e for some 0 e 1 with e 0 only in very special circumstances our results concerning the convergence of the annealing algorithm and the rate of convergence to zero of the bias and the variance of the sample means estimator provide a rigorous procedure for choosing the optimal annealing schedule this optimal choice reflects the competition between two physical effects a the adiabatic effect whereby if the temperature is loweredtoo abruptly the system may end up not in a ground state but in a nearby metastable state and b the super cooling effect whereby if the temperature is loweredtoo slowly the system will indeed approach the ground state s but may do so extremely slowly

simulated annealing via sobolev inequalities we use sobolev inequalities to study the simulated annealing algorithm this approach takes advantage of the local time reversibility of the process and yields the optimal freezing schedule as well as quantitative information about the rate at which the process is tending to its ground state

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

diffusion for global optimization in r n we seek a global minimum of u mathbb r n to mathbb r the solution to d dt x t nabla u x t will find local minima using the idea of simulated annealing we consider th

robust wigner distribution with application to the instantaneous frequency estimation the wigner distribution wd produces highly concentrated time frequency tf representation of nonstationary signals it may be used as an efficient signal analysis tool including the cases of frequency modulated signals corrupted with the gaussian noise in some applications a significant amount of impulse noise is present then the wd fails to produce satisfactory results the robust periodogram has been introduced for spectral estimation of this kind of noisy signals it can produce good concentration for pure harmonic signals however it is not so efficient in the cases of signals with rapidly varying frequency this is the motivation for introducing the robust wd it is a reliable tf representation tool for wide class of nonstationary signals corrupted with impulse noise this distribution produces good accuracy of the instantaneous frequency if estimation using the huber 1981 loss function a generalization of the wd is presented it includes both the standard and the robust wd as special cases this distribution can be used for tf analysis of signals corrupted with a mixture of impulse and gaussian noise the presented theory is illustrated on examples including applications on the if estimation and time varying filtering of signals corrupted with a mixture of the gaussian and impulse noise the case study analysis of the if estimators accuracy based on the standard and the robust wd forms is performed in order to improve the if estimation a median filter is applied on the obtained if estimate

spherical subcategories in representation theory we introduce a new invariant for triangulated categories the poset of spherical subcategories ordered by inclusion this yields several numerical invariants like the cardinality and the height of the poset we explicitly describe spherical subcategories and their poset structure for derived categories of certain finite dimensional algebras

iterative composite filtering for image restoration an algorithm solution to the noisy image restoration problem under assumptions that the image is nonstationary and that the noise process is a superposition of white and impulsive noises is proposed a composite model is used for the image in order to consider its nonstationarities in the mean and the autocorrelation separating the gross information about the image from its textural information the authors exploit the advantages of median range and levinson filters in restoring the image median statistics are used to estimate the image s gross information and to filter the impulsive noise range statistics are used to segment the textural image into approximately locally stationary images to be filtered by levenson filters the proposed restoration algorithm adapts to the nonstationarity of the image and thus it performs well the algorithm is compared with others based on either median or linear filtering alone

decimated rank order filtering rank order filters are used for a wide range of applications they are noted for their ability to remove impulsive noise and to preserve sharp changes in the signal however there is a tradeoff between noise removing capability and edge retention when impulsive noise and sharp edges in the signal coincide edges are no longer preserved another tradeoff is noise removal versus computational cost the increase in the number of operations is proportional to the noise removal capability the authors introduce a new filter the decimated rank order filter that achieves large amount of noise removal with much less detail and edge smoothing as compared with the standard rank order filter furthermore this is possible at a reduced computational burden the decimated median filter is a particular example of this class the authors investigate its deterministic and statistical properties and show that it compares favorably with the standard median filter in terms of impulsive noise removal especially in the neighborhood of edges they extend the structure of decimated rank order filters they find all rank orders of the signal to obtain the decimated sorter and show that the decimated sorter is the bitonic sorter used in switching networks they point out the connection between their research and concepts in sorting and switching structures

a new class of detail preserving filters for image processing a new class of median type filters for image processing is proposed in the filters linear fir substructures are used in conjunction with the median operation the root signals and noise attenuation properties of the fir median hybrid filters are analyzed and compared to representative edge preserving filtering operations the concept of multilevel median operation is introduced to improve the detail preserving property of conventional median and the fir median hybrid filters in the multilevel filters there exists a tradeoff between noise attenuation and detail preservation the analysis and examples indicate that fir median hybrid filters preserve details better and are computationally much more efficient than the conventional median and the k nearest neighbor averaging filters

a separable median filter for image noise smoothing this paper investigates some properties of the separable filter resulting from successive applications of a one dimensional median filter on the rows and columns of an image although the output of this separable filter is not identical to the corresponding nonseparable two dimensional median filter with a square window its performance in image noise smoothing is close in particular its effectiveness in smoothing noise and its behavior with edges are characterized and compared with those of the two dimensional median filter it is shown that the separable filter has a much simpler implementation in real time hardware at video rates for example

the nearest neighbor median filter some deterministic properties and implementations abstract further analysis of the nearest neighbor median filter nnmf is carried out it is proved that the nnmf has the threshold decomposition property and the stacking property using these properties it is formally shown that the nnmf is a subclass of the weighted median filter wmf and the weight coefficients of the equivalent wmf are derived it is shown that the nnmf is more efficient than the equivalent wmf from the viewpoint of computational complexity two theorems about the effect of repeated operations of nnmf are proved these theorems are described in the style of dealing with local configurations of image moreover hardware implementations of the nnmf are presented

flops and clusters in the homological minimal model programme suppose that f x rightarrow mathrm spec r is a minimal model of a complete local gorenstein 3 fold where the fibres of f are at most one dimensional so by van den bergh duke math j 122 3 423 455 2004 there is a noncommutative ring lambda derived equivalent to x for any collection of curves above the origin we show that this collection contracts to a point without contracting a divisor if and only if a certain factor of lambda is finite dimensional improving a result of donovan and wemyss contractions and deformations arxiv 1511 00406 we further show that the mutation functor of iyama and wemyss invent math 197 3 521 586 2014 6 is functorially isomorphic to the inverse of the bridgeland chen flop functor in the case when the factor of lambda is finite dimensional these results then allow us to jump between all the minimal models of mathrm spec r in an algorithmic way without having to compute the geometry at each stage we call this process the homological mmp this has several applications in git approaches to derived categories and also to birational geometry first using mutation we are able to compute the full git chamber structure by passing to surfaces we say precisely which chambers give the distinct minimal models and also say which walls give flops and which do not enabling us to prove the craw ishii conjecture in this setting second we are able to precisely count the number of minimal models and also give bounds for both the maximum and the minimum numbers of minimal models based only on the dual graph enriched with scheme theoretic multiplicity third we prove a bijective correspondence between maximal modifying r module generators and minimal models and for each such pair in this correspondence give a further correspondence linking the endomorphism ring and the geometry this lifts the auslander mckay correspondence to dimension three

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

asymptotics of the spectral gap with applications to the theory of simulated annealing soit m une variete de riemann compacte connexe a n dimensions de produit interieur et soit u m r 1 une fonction lisse donnee pour 0 on definit 7b l c m c m par 7b l o exp u exp u o ou et sont le gradient et la divergence sur m soit la taille de l ecart entre 0 et le reste du spectre de 7b l on etudie le comportement asymptotique de

efficient computation of local geometric moments local moments have attracted attention as local features in applications such as edge detection and texture segmentation the main reason for this is that they are inherently integral based features so that their use reduces the effect of uncorrelated noise the computation of local moments when viewed as a neighborhood operation can be interpreted as a convolution of the image with a set of masks nevertheless moments computed inside overlapping windows are not independent and convolution does not take this fact into account by introducing a matrix formulation and the concept of accumulation moments this paper presents an algorithm which is computationally much more efficient than convolving and yet as simple

detail preserving ranked order based filters for image processing the theoretical analysis of multistage median filters is developed it is shown that multistage median filters are a combination of max median and min median filters since multistage median filters belong to the class of two dimensional stack filters they have threshold decomposition attributes making their theoretical analysis simple statistical threshold decomposition is applied to derive the statistical characteristics of these filters and the results are used to evaluate the performance of these two types of multistage filters finally a quantitative and qualitative comparison of the multistage filters and of other efficient detail preserving filters is presented the comparisons are made using the mean squared error and the mean absolute error criteria

optimal pivot selection in fast weighted median search weighted median filters are increasingly being used in signal processing applications and thus fast implementations are of importance this paper introduces a fast algorithm to compute the weighted median wm of n samples which has linear time and space complexity as opposed to o n log n which is the time complexity of traditional sorting algorithms a popular selection algorithm often used to find the wm in large data sets is quickselect whose performance is highly dependent on how the pivots are chosen we introduce an optimization based pivot selection strategy which results in significantly improved performance as well as a more consistent runtime compared to traditional approaches the selected pivots are order statistics of subsets in order to find the optimal order statistics as well as the optimal subset sizes a set of cost functions are derived which when minimized lead to optimal design parameters we compare the complexity to floyd and rivest s algorithm select which to date has been the fastest median finding algorithm and we show that the proposed algorithm compared with select requires close to 30 fewer comparisons it is also shown that the proposed selection algorithm is asymptotically optimal for large n

t structures on some local calabi yau varieties abstract let z be a smooth fano variety satisfying the condition that the rank of the grothendieck group of z is one more than the dimension of z let z denote the total space of the canonical line bundle of z considered as a non compact calabi yau variety we use the theory of exceptional collections to describe t structures on the derived category of coherent sheaves on z the combinatorics of these t structures is determined by a natural action of an affine braid group closely related to the well known action of the artin braid group on the set of exceptional collections on z

higher apr tilting preserves n representation infiniteness abstract we show that m apr tilting preserves n representation infiniteness for 1 m n moreover we show that these tilting modules lift to tilting modules for the corresponding higher preprojective algebras which is n 1 cy algebras we also study the interplay of the two kinds of tilting modules

on the convergence and applications of generalized simulated annealing the convergence of the generalized simulated annealing with time inhomogeneous communication cost functions is discussed this study is based on the use of log sobolev inequalities and semigroup techniques in the spirit of a previous article by one of the authors we also propose a natural test set approach to study the global minima of the virtual energy the second part of the paper is devoted to the application of these results we propose two general markovian models of genetic algorithms and we give a simple proof of the convergence toward the global minima of the fitness function finally we introduce a stochastic algorithm that converges to the set of the global minima of a given mean cost optimization problem

cluster analysis by binary morphology an approach to unsupervised pattern classification that is based on the use of mathematical morphology operations is developed the way a set of multidimensional observations can be represented as a mathematical discrete binary set is shown clusters are then detected as well separated subsets by means of binary morphological transformations

separable markov random field model and its applications in low level vision this brief proposes a continuously valued markov random field mrf model with separable filter bank denoted as mrfsepa which significantly reduces the computational complexity in the mrf modeling in this framework we design a novel gradient based discriminative learning method to learn the potential functions and separable filter banks we learn mrfsepa models with 2 d and 3 d separable filter banks for the applications of gray scale color image denoising and color image demosaicing by implementing mrfsepa model on graphics processing unit we achieve real time image denoising and fast image demosaicing with high quality results