parallel and element reduced error diffused block truncation coding block truncation coding btc is an efficient compression technique for its inherent simple coding strategy however the annoying blocking effect and false contour accompanied in high coding gain configurations make the applications relatively limited compares to some up to date compression schemes for this error diffused block truncation coding edbtc is proposed to solve these problems and obtain satisfactory results unfortunately the edbtc sacrifices the parallel advantage of traditional btc moreover the number of diffused directions of edbtc can be reduced to obtain higher efficiency for these the interlaced error diffused block truncation coding iedbtc is proposed in this work to claim back the parallel advantage in addition the diffused elements are also reduced from four to two with the proposed optimization procedure while preserving the image quality

improved block truncation coding based on the void and cluster dithering approach block truncation coding btc is an efficient technology for image compression an improved btc algorithm namely ordered dither block truncation coding odbtc is presented in this study in order to provide better image quality the void and cluster halftoning is combined with the btc the odbtc results show that the image quality is improved when it is operated in high coding gain applications another feature of the odbtc is the dither array look up table lut which significantly reduces the complexity compared to the btc

operator means of probability measures abstract let p be the complete metric space consisting of positive invertible operators on an infinite dimensional hilbert space with the thompson metric we introduce the notion of operator means of probability measures on p in parallel with kubo and ando s definition of two variable operator means and show that every operator mean is contractive for the wasserstein distance by means of a fixed point method we consider deformation of such operator means and show that the deformation of any operator mean becomes again an operator mean in our sense based on this deformation procedure we prove a number of properties and inequalities for operator means of probability measures

riemannian geometry and matrix geometric means the geometric mean of two positive definite matrices has been defined in several ways and studied by several authors including pusz and woronowicz and ando the characterizations by these authors do not readily extend to three matrices and it has been a long standing problem to define a natural geometric mean of three positive definite matrices in some recent papers new understanding of the geometric mean of two positive definite matrices has been achieved by identifying the geometric mean of a and b as the midpoint of the geodesic with respect to a natural riemannian metric joining a and b this suggests some natural definitions for a geometric mean of three positive definite matrices we explain the necessary geometric background and explore the properties of some of these candidates

an effective matrix geometric mean satisfying the ando li mathias properties we propose a new matrix geometric mean satisfying the ten properties given by ando li and mathias linear alg appl 2004 this mean is the limit of a sequence which converges superlinearly with convergence of order 3 whereas the mean introduced by ando li and mathias is the limit of a sequence having order of convergence 1 this makes this new mean very easily computable we provide a geometric interpretation and a generalization which includes as special cases our mean and the ando li mathias mean

image compression by variable block truncation coding with optimal threshold a variable block truncation coding btc algorithm is proposed for image compression it is shown that there exists an optimal threshold for the quantization in btc algorithms fixed and variable that minimizes the errors compared to the fixed btc fbtc the variable btc vbtc gives better performance on all the tested images the use of vbtc with optimal threshold leads to a reduction of the error in the reconstructed images by almost 40 of the error in the reconstructed images obtained by fbtc this enhanced performance suggests that the vbtc with optimal threshold is a better alternative to the fixed block truncation coding

an extended lie trotter formula and its applications abstract in this paper we present a class of lie trotter formulae for hermitian operators including the formulae derived by hiai petz and furuta a lie trotter formula for weighted log euclidean geometric means of several positive definite operators is given in terms of sagae tanabe geometric and spectral geometric means

a riemannian framework for tensor computing tensors are nowadays a common source of geometric information in this paper we propose to endow the tensor space with an affine invariant riemannian metric we demonstrate that it leads to strong theoretical properties the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries null eigenvalues are at the infinity the geodesic between two tensors and the mean of a set of tensors are uniquely defined etc we have previously shown that the riemannian metric provides a powerful framework for generalizing statistics to manifolds in this paper we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation filtering diffusion and restoration of missing data for instance most interpolation and gaussian filtering schemes can be tackled efficiently through a weighted mean computation linear and anisotropic diffusion schemes can be adapted to our riemannian framework through partial differential evolution equations provided that the metric of the tensor space is taken into account for that purpose we provide intrinsic numerical schemes to compute the gradient and laplace beltrami operators finally to enforce the fidelity to the data either sparsely distributed tensors or complete tensors fields we propose least squares criteria based on our invariant riemannian distance which are particularly simple and efficient to solve

applications of geometric means on symmetric cones let v be a euclidean jordan algebra and let omega be the corresponding symmetric cone the geometric mean a b of two elements a and b in omega is defined as a unique solution which belongs to omega of the quadratic equation p x a 1 b where p is the quadratic representation of v in this paper we show that for any a in omega the sequence of n mathrm th iterate f a n x of the function f a omega to omega defined by f a x 1 over 2 x p a x 1 converges to a as applications we obtain that the geometric mean a b of omega can be represented as a limit of successive iteration of arithmetic means and harmonic means and we derive the lowner heinz inequality on the symmetric cone omega 0 leq a leq b mathrm implies a p leq b p mathrm for 0 leq p leq 1 furthermore we obtain a formula mathrm exp x y lim n to infty mathrm exp frac 2 n x exp frac 2 n y n which leads a golden thompson type inequality mathrm exp x y leq p mathrm exp x over 2 mathrm exp y for the spectral norm on v

an elementary proof of arithmetic geometric mean inequality of the weighted riemannian mean of positive definite matrices abstract the weighted riemannian mean of positive definite matrices is a kind of weighted geometric mean of n matrices some properties of the weighted riemannian mean are easily obtained but some of them are not easy in this short paper a simplified proof of arithmetic geometric harmonic mean inequalities will be obtained

sample mean versus sample frechet mean for combining complex wishart matrices a statistical study the space of covariance matrices is a non euclidean space the matrices form a manifold which if equipped with a riemannian metric becomes a riemannian manifold and recently this idea has been used for comparison and clustering of complex valued spectral matrices which at a given frequency are typically modeled as complex wishart distributed random matrices identically distributed sample complex wishart matrices can be combined via a standard sample mean to derive a more stable overall estimator however using the riemannian geometry their so called sample frechet mean can also be found we derive the expected value of the determinant of the sample frechet mean and the expected value of the sample frechet mean itself the population frechet mean is shown to be a scaled version of the true covariance matrix the risk under convex loss functions for the standard sample mean is never larger than for the frechet mean in simulations the sample mean also performs better for the estimation of an important functional derived from the estimated covariance matrix namely partial coherence

an order inequality characterizing invariant barycenters on symmetric cones abstract this paper is concerned with invariant and contractive barycenters on the wasserstein space of probability measures on metric spaces of non positive curvature where the center of gravity also called the cartan barycenter is the canonical barycenter on hadamard spaces we establish an order inequality of probability measures on partially ordered symmetric spaces of non compact type namely symmetric cones self dual homogeneous cones characterizing the cartan barycenter among other invariant and contractive barycenters the derived inequality and partially ordered structures on the probability measure space lead also to significant results on norm inequalities including the ando hiai inequality for probability measures on symmetric cones

variation of discrete spectra a formula see 1 below estimating collectively the variation of eigenvalues of a symmetric matrix under a perturbation is extended to the case of discrete eigenvalues of a selfadjoint operator in hilbert space under the assumption that the perturbation is compact for this purpose the notion of an extended enumeration of discrete eigenvalues is introduced

on some inequalities for the matrix power and karcher means abstract we establish a useful inequality for matrix power means of positive definite matrices it eventually provides not only a new characterization of the karcher mean cartan centroid riemannian geometric mean but also a simple proof of the convergence of power means to the karcher mean some new matrix mean inequalities on the karcher and power means are presented

the noncommutative lowner theorem for matrix monotone functions over operator systems abstract given a function f a b r lowner s theorem states f is monotone when extended to self adjoint matrices via the functional calculus if and only if f extends to a self map of the complex upper half plane in recent years several generalizations of lowner s theorem have been proven in several variables we use the relaxed agler m c carthy and young theorem on locally matrix monotone functions in several commuting variables to generalize results in the noncommutative case specifically we show that a real free function defined over an operator system must analytically continue to a noncommutative upper half plane as map into another noncommutative upper half plane

minimum cross entropy threshold selection thresholding is a common and easily implemented form of image segmentation many methods of automatic threshold selection based on the optimization of some discriminant function have been proposed such functions often take the form of a metric distance or similarity measure between the original image and the segmented result a non metric measure the cross entropy is used here to determine the optimum threshold it is shown that this measure is related to other commonly used measures of distance or similarity under special conditions although it is in some senses more general some typical results using this method are presented together with results using a metric form of the cross entropy

qr images optimized image embedding in qr codes this paper introduces the concept of qr images an automatic method to embed qr codes into color images with bounded probability of detection error these embeddings are compatible with standard decoding applications and can be applied to any color image with full area coverage the qr information bits are encoded into the luminance values of the image taking advantage of the immunity of qr readers against local luminance disturbances to mitigate the visual distortion of the qr image the algorithm utilizes halftoning masks for the selection of modified pixels and nonlinear programming techniques to locally optimize luminance levels a tractable model for the probability of error is developed and models of the human visual system are considered in the quality metric used to optimize the luminance levels of the qr image to minimize the processing time the optimization techniques proposed to consider the mechanics of a common binarization method and are designed to be amenable for parallel implementations experimental results show the graceful degradation of the decoding rate and the perceptual quality as a function the embedding parameters a visual comparison between the proposed and existing methods is presented

on the exponential metric increasing property a short and simple proof is given for the inequality that shows that positive definite matrices constitute a riemannian manifold of negative curvature the idea of the proof leads to generalisations to non riemannian metrics and to connections with some well known inequalities of mathematical physics

the hermite hadamard inequality for convex functions on a global npc space abstract we prove an extension of choquet s theorem to the framework of compact metric spaces with a global nonpositive curvature together with sturm s extension k t sturm probability measures on metric spaces of nonpositive curvature in pascal auscher et al eds heat kernels and analysis on manifolds graphs and metric spaces lecture notes from a quarter program on heat kernels random walks and analysis on manifolds and graphs april 16 july 13 2002 paris france in contemp math vol 338 amer math soc providence ri 2003 pp 357 390 of jensen s inequality this provides a full analogue of the hermite hadamard inequality for the convex functions defined on such spaces

matrix power means and the karcher mean we define a new family of matrix means pt a t 1 1 where and a vary over all positive probability vectors in rn and n tuples of positive definite matrices resp each of these means except t 0 arises as a unique positive definite solution of a non linear matrix equation satisfies all desirable properties of power means of positive real numbers and interpolates between the weighted harmonic and arithmetic means the main result is that the karcher mean coincides with the limit of power means as t 0 this provides not only a sequence of matrix means converging to the karcher mean but also a simple proof of the monotonicity of the karcher mean conjectured by bhatia and holbrook and other new properties which have recently been established by lawson and lim and also bhatia and karandikar using probabilistic methods on the metric structure of positive definite matrices equipped with the trace metric

weighted multivariable operator means of positive definite operators abstract in this paper we present a new weighted multivariable operator mean of positive definite operators over an arbitrary hilbert space which provides us the first generally applicable extension of the classical kubo ando theory of 2 variable operator means the construction is a weighted extension of the bini meini poloni symmetrization process originally given for the matrix geometric mean here to be able to consider such an iterative procedure we need a weighted version of every kubo ando mean in two variables therefore we also give a new construction for two arbitrary positive operators on a possibly infinite dimensional hilbert space that provides weighted counterparts to every not necessarily symmetric kubo ando mean and also agrees with the most well known weighted operator means

jensen and minkowski inequalities for operator means and anti norms abstract jensen inequalities for positive linear maps of choi and hansen pedersen type are established for a large class of operator matrix means such as some p means and some kubo ando means these results are also extensions of the minkowski determinantal inequality to this end we develop the study of anti norms a notion parallel to the symmetric norms in matrix analysis including functionals like schatten q norms for a parameter q 1 and the minkowski functional det 1 n a

differentiators and the geometry of polynomials in 1959 davis introduced the concept of a differentiator of an operator on a finite dimensional hilbert space we prove that every such operator possesses a differentiator we also use the theory of differentiators to solve several problems in the geometry of polynomials for instance we answer in the affirmative a twenty year old unsolved conjecture of schoenberg a related conjecture of katsoprinakis and a fifty year old unsolved conjecture of de bruijn and springer

digital color halftoning with generalized error diffusion and multichannel green noise masks in this paper we introduce two novel techniques for digital color halftoning with green noise stochastic dither patterns generated by homogeneously distributing minority pixel clusters the first technique employs error diffusion with output dependent feedback where unlike monochrome image halftoning an interference term is added such that the overlapping of pixels of different colors can be regulated for increased color control the second technique uses a green noise mask a dither array designed to create green noise halftone patterns which has been constructed to also regulate the overlapping of different colored pixels as is the case with monochrome image halftoning both techniques are tunable allowing for large clusters in printers with high dot gain characteristics and small clusters in printers with low dot gain characteristics

monotonic properties of the least squares mean we settle an open problem of several years standing by showing that the least squares mean for positive definite matrices is monotone for the usual loewner order indeed we show this is a special case of its appropriate generalization to partially ordered complete metric spaces of nonpositive curvature our techniques extend to establish other basic properties of the least squares mean such as continuity and joint concavity moreover we introduce a weighted least squares mean and derive our results in this more general setting

a fixed point mean approximation to the cartan barycenter of positive definite matrices abstract in this paper we define a new weighted inductive geometric mean h n on the riemannian manifold of positive definite matrices of fixed size and present its fixed point mean approximation to the cartan barycenter karcher mean we show that for t 0 1 the weighted geometric mean equation x h n 1 t a 1 a n x where t t 1 t w 1 w n 1 t has a unique positive definite solution that approaches as t 0 to the weighted cartan mean of a 1 a n numerical computations for the stochastic convergence in terms of h n to derive a new contractive barycenter other than the cartan barycenter are presented

contractive barycentric maps and l1 ergodic theorems on the cone of positive definite matrices abstract we are concerned with contractive with respect to the wasserstein metric barycenters of probability measures with bounded support on the convex cone of positive definite matrices equipped with the thompson metric based on the important construction schemes of multivariate matrix means namely the proximal average and the cartan mean the least squares average for the cartan hadamard metric we construct a one parameter family of contractive barycentric maps interpolating continuously and monotonically the harmonic arithmetic and cartan barycenters we show that each contractive barycentric map is monotonic for the stochastic order induced by the cone and establish stochastic approximations and l 1 ergodic theorems for the parameterized contractive barycenters

majorization and refined jensen mercer type inequalities for self adjoint operators abstract in this paper jensen mercer s inequality is generalized by applying the method of pre majorization used for comparing two tuples of self adjoint operators a general result in a matrix setting is established special cases of the main theorem are studied to recover other inequalities of mercer type

equilibrium points of logarithmic potentials induced by positive charge distributions i generalized de bruijn springer relations equilibrium points of logarithmic potentials induced by positive charge distributions i generalized de bruijn springer relations

the hadamard product for the weighted karcher means abstract the weighted karcher mean of positive definite matrices is defined as the unique minimizer of the weighted sum of squares of the riemannian distances to each of given points using the well known connection between the tensor product and the hadamard product we show that the hadamard product of weighted karcher means for permuted tuples with fixed weight is bounded by the hadamard product of given positive definite matrices it generalizes the results for the case of two variable geometric means and sagae tanabe inductive means

monotonicity of the matrix geometric mean an attractive candidate for the geometric mean of m positive definite matrices a1 am is their riemannian barycentre g one of its important operator theoretic properties monotonicity in the m arguments has been established recently by lawson and lim we give an elementary proof of this property using standard matrix analysis and some counting arguments we derive some new inequalities for g one of these says that for any unitarily invariant norm g is not bigger than the geometric mean of a1 am

on the basic representation theorem for convex domination of measures a direct constructive proof is given for the basic representation theorem for convex domination of measures the proof is given in the finitistic case purely atomic measures with a finite number of atoms and a simple argument is then given to extend this result to the general case including both probability measures and finite borel measures on infinite dimensional spaces the infinite dimensional case follows quickly from the finite dimensional case with the use of the approximation property

matrix geometric means based on shuffled inductive sequences abstract we propose a new deterministic sequence converging to the least squares mean of n symmetric positive definite spd matrices by least squares mean we refer to the riemannian barycenter with respect to the natural metric also known as the trace metric or affine invariant metric on the set p n of n n spd matrices in some papers this mean is also referred to as the karcher mean the sequence we propose belongs to the family of inductive sequences obtained by letting a point take successive steps towards the data points with step lengths progressively diminishing to zero we use the word inductive since each point of those sequences can be seen as the inductive mean of a well chosen set of points containing multiple replications of the data points we show that visiting the data points in a cyclic order usually results in a slow convergence and remedy this weakness by reordering repeatedly the data points using a shuffling algorithm we prove the convergence of the resulting inductive sequence to the least squares mean of the data in the general framework of npc spaces complete metric spaces with non positive curvature which includes the set p n we also illustrate numerically on p n that some points of this sequence are fairly accurate estimates of the least squares mean while being considerably cheaper to compute and perform well as initializers for state of the art algorithms to reduce further the computation time we finally exploit inductive sequences to produce a family of parallelizable algorithms for estimating the least squares mean

an approach of majorization in spaces with a curved geometry abstract the hardy littlewood polya majorization theorem is extended to the framework of some spaces with a curved geometry such as the global npc spaces and the wasserstein spaces we also discuss the connection between our concept of majorization and the subject of schur convexity several applications are included

the least squares mean of positive hilbert schmidt operators abstract we show that the least squares mean on the riemannian manifold of positive operators in the extended hilbert schmidt algebra of linear operators on a hilbert space equipped with the canonical trace metric is the unique solution of the corresponding karcher equation this allows us to conclude that the least squares mean is the restriction of the karcher mean on the open cone of all bounded positive definite operators and hence inherits the basic properties of that mean conversely the karcher mean on the positive definite operators is shown to be the unique monotonically strongly continuous extension of the least squares mean on

factorizations and geometric means of positive definite matrices abstract in this paper we provide a new class of metric geometric means of positive definite matrices varying over hermitian unitary matrices we show that each hermitian unitary matrix induces a factorization of the cone p m of m m positive definite hermitian matrices into geodesically convex subsets and a hadamard metric structure on p m an explicit formula for the corresponding metric midpoint operation is presented in terms of the geometric and spectral geometric means and show that the resulting two variable mean is different to the standard geometric mean some basic properties comparable to those of the geometric mean and its extensions to finite number of positive definite matrices are studied

inequalities for the wasserstein mean of positive definite matrices abstract we prove majorization inequalities for different means of positive definite matrices these include the cartan mean the karcher mean the log euclidean mean the wasserstein mean and the power mean

operator means of probability measures and generalized karcher equations abstract in this article we consider means of positive bounded linear operators on a hilbert space we present a complete theory that provides a framework which extends the theory of the karcher mean its approximating matrix power means and a large part of kubo ando theory to arbitrary many variables in fact to the case of probability measures with bounded support on the cone of positive definite operators this framework characterizes each operator mean extrinsically as unique solutions of generalized karcher equations which are obtained by exchanging the matrix logarithm function in the karcher equation to arbitrary operator monotone functions over the positive real half line if the underlying hilbert space is finite dimensional then these generalized karcher equations are riemannian gradients of convex combinations of strictly geodesically convex log determinant divergence functions hence these new means are their global minimizers in analogy to the case of the karcher mean as pointed out our framework is based on fundamental contraction results with respect to the thompson metric which provides us nonlinear contraction semigroups in the cone of positive definite operators that form a decreasing net approximating these operator means in the strong topology from above

strong law of large numbers for the l1 karcher mean abstract sturm s strong law of large numbers in cat 0 spaces has been an influential tool to study the geometric mean or also called karcher barycenter of positive definite matrices it provides an easily computable stochastic approximation based on inductive means convergence of a deterministic version of this approximation has been proved by holbrook providing his nodice theorem for the karcher mean of positive definite matrices the karcher mean has also been extended to the infinite dimensional case of positive operators on a hilbert space by lawson lim and then to probability measures with bounded support by the second author however the cat 0 property of the space is lost and one defines the mean as the unique solution of a nonlinear operator equation on a convex banach finsler manifold the formulations of sturm s strong law of large numbers and holbrook s nodice approximation are natural and both conjectured to converge however all previous techniques of their proofs break down due to the banach finsler nature of the space in this paper we prove both conjectures by establishing the most general l 1 form of sturm s strong law of large numbers and holbrook s nodice theorem in the operator norm by developing a stochastic discrete time resolvent flow for the karcher barycenter using its wasserstein contraction property

loewner s theorem in several variables abstract in this paper we establish a multivariable non commutative generalization of loewner s theorem characterizing operator monotone functions as real functions admitting analytic continuation mapping the upper complex half plane into itself the non commutative several variable theorem proved here represents several variable operator monotone functions of positive operator tuples not a priori assumed to be analytic or continuous as a schur complement of a positive linear matrix pencil to achieve this we establish the abstract integral formula for these functions using only matrix convexity and lmis that represents operator monotone and operator concave free functions as a conditional expectation of a schur complement of a linear matrix pencil on a tensor product operator algebra the results can be applied to any of the various multivariable operator means that have been constructed in the last decades or so including the karcher mean to obtain an explicit closed formula for these operator means

computing the karcher mean of symmetric positive definite matrices we present and analyze an iterative method for approximating the karcher mean of a set ofn n positive denite

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

convex geometric means abstract a class of multivariable weighted geometric means of positive definite matrices admitting jensen type inequalities for geodesically convex functions is considered it is shown that there are infinitely many such geometric means including the weighted inductive bini meini poloni and karcher means and each of these means provides a geometric mean majorization on the space of positive definite matrices some connections between our geometric mean majorizations and classical results of the standard majorization of real numbers are discussed in particular we establish the hardy littlewood polya majorization theorem and also rado s theorem and schur s convexity theorem for the weighted karcher mean

inverse spectral problem for normal matrices and the gauss lucas theorem we establish an analog of the cauchy poincare interlacing theorem for normal matrices in terms of majorization and we provide a solution to the corresponding inverse spectral problem using this solution we generalize and extend the gauss lucas theorem and prove the old conjecture of de bruijn springer on the location of the roots of a complex polynomial and its derivative and an analog of rolle s theorem conjectured by schoenberg

weak supermajorization and families as doubly superstochastic operators on lp i abstract we present necessary and sufficient conditions that a family a a i j i j i of real numbers may be considered as a bounded linear operator on banach spaces l 1 i and l i where i is an arbitrary non empty set moreover we get that these conditions are sufficient for a family to be a bounded linear operator on l p i for each p 1 within this class of operators the notion of doubly superstochastic operator is introduced as an extension of the doubly superstochastic matrix and some of its essentially properties are presented in the second part we extend the notion of weak supermajorization relation on the banach space l p i using doubly superstochastic operators and present close relationship between this relation and superstochastic operators as generalisation well known results in the theory of majorization among others for two functions f g l 1 i we show that relations f w s g and g w s f hold if and only if there exist a permutation p such that g p f

a converse of loewner heinz inequality and applications to operator means abstract let f t be an operator monotone function then a b implies f a f b but the converse implication is not true let a b be the geometric mean of a b 0 if a b then b 1 a i the converse implication is not true either we will show that if f b i 1 f a i i for all sufficiently small 0 then f a i f b i and a b moreover we extend it to multi variable matrices means

a dual interpretation for direct binary search and its implications for tone reproduction and texture quality the direct binary search dbs algorithm employs a search heuristic to minimize the mean squared perceptually filtered error between the halftone and continuous tone original images based on an efficient method for evaluating the effect on the mean squared error of trial changes to the halftone image we show that dbs also minimizes in a pointwise sense the absolute error under the same visual model but at twice the viewing distance associated with the mean squared error metric this dual interpretation sheds light on the convergence properties of the algorithm and clearly explains the tone bias that has long been observed with halftoning algorithms of this type it also demonstrates how tone bias and texture quality are linked via the scale parameter the product of printer resolution and viewing distance finally we show how the tone bias can be eliminated by tone correcting the continuous tone image prior to halftoning it