on constructing matrices with prescribed singular values and diagonal elements abstract similar to the well known schur horn theorem that characterizes the relationship between the diagonal entries and the eigenvalues of a hermitian matrix the sing thompson theorem characterizes the relationship between the diagonal entries and the singular values of an arbitrary matrix it is noted in this paper that based on the induction principle such a matrix can be constructed numerically by a fast recursive algorithm provided that the given singular values and diagonal elements satisfy the sing thompson conditions

image compression using wavelet transform and multiresolution decomposition schemes for image compression of black and white images based on the wavelet transform are presented the multiresolution nature of the discrete wavelet transform is proven as a powerful tool to represent images decomposed along the vertical and horizontal directions using the pyramidal multiresolution scheme the wavelet transform decomposes the image into a set of subimages called shapes with different resolutions corresponding to different frequency bands hence different allocations are tested assuming that details at high resolution and diagonal directions are less visible to the human eye the resultant coefficients are vector quantized vq using the lgb algorithm by using an error correction method that approximates the reconstructed coefficients quantization error we minimize distortion for a given compression rate at low computational cost several compression techniques are tested in the first experiment several 512 spl times 512 images are trained together and common table codes created using these tables the training sequence black and white images achieve a compression ratio of 60 65 and a psnr of 30 33 to investigate the compression on images not part of the training set many 480 spl times 480 images of uncalibrated faces are trained together and yield global tables code images of faces outside the training set are compressed and reconstructed using the resulting tables the compression ratio is 40 psnrs are 30 36 images from the training set have similar compression values and quality finally another compression method based on the end vector bit allocation is examined

a fast recursive algorithm for constructing matrices with prescribed eigenvalues and singular values the weyl horn theorem characterizes a relationship between the eigenvalues and the singular values of an arbitrary matrix based on that characterization a fast recursive algorithm is developed to construct numerically a matrix with prescribed eigenvalues and singular values besides being of theoretical interest the technique could be employed to create test matrices with desired spectral features numerical experiment shows this algorithm to be quite efficient and robust

schur majorization inequalities for symmetrized sums with applications to tensor products we show that if w y and x z are four vectors in rn then a number of schur majorizations hold between symmetrized vector functions of w x y and z e g wi xj i j yi zj i j where the left hand expression means the vector of dimension n2 consisting of all sums wi xj of the co ordinates of w and x arranged in lexicographic order among other things we get vector and matrix versions of muirhead s theorem for scalar inequalities from the vector inequalities follow many scalar inequalities for symmetrized sums some of which are scattered through the inequality literature in section 2 applications are given to matrix inequalities for tensor products e g if a b and c are hermitian and a b then a c b c

image compression based on fuzzy algorithms for learning vector quantization and wavelet image decomposition this paper evaluates the performance of an image compression system based on wavelet based subband decomposition and vector quantization the images are decomposed using wavelet filters into a set of subbands with different resolutions corresponding to different frequency bands the resulting subbands are vector quantized using the linde buzo gray 1980 algorithm and various fuzzy algorithms for learning vector quantization falvq these algorithms perform vector quantization by updating all prototypes of a competitive neural network through an unsupervised learning process the quality of the multiresolution codebooks designed by these algorithms is measured on the reconstructed images belonging to the training set used for multiresolution codebook design and the reconstructed images from a testing set

fuzzy vector quantization algorithms and their application in image compression this paper presents the development and evaluation of fuzzy vector quantization algorithms these algorithms are designed to achieve the quality of vector quantizers provided by sophisticated but computationally demanding approaches while capturing the advantages of the frequently used in practice k means algorithm such as speed simplicity and conceptual appeal the uncertainty typically associated with clustering tasks is formulated in this approach by allowing the assignment of each training vector to multiple clusters in the early stages of the iterative codebook design process a training vector assignment strategy is also proposed for the transition from the fuzzy mode where each training vector can be assigned to multiple clusters to the crisp mode where each training vector can be assigned to only one cluster such a strategy reduces the dependence of the resulting codebook on the random initial codebook selection the resulting algorithms are used in image compression based on vector quantization this application provides the basis for evaluating the computational efficiency of the proposed algorithms and comparing the quality of the resulting codebook design with that provided by competing techniques

a trace inequality with a subtracted term abstract for fixed real or complex matrices a and b the well known von neumann trace inequality identifies the maximum of tr uavb z sfnc as u and v range over the unitary group the maximum being a bilinear expression in the singular values of a and b this paper establishes the analogue of this inequality for real matrices a and b when u and v range over the proper real orthogonal group the maximum is again a bilinear expression in the singular values but there is a subtracted term when a and b have determinants of opposite sign

positive and doubly stochastic maps and majorization in euclidean jordan algebras abstract a positive map between euclidean jordan algebras is a symmetric cone order preserving linear map we show that the norm of such a map is attained at the unit element thus obtaining an analog of the operator matrix theoretic russo dye theorem a doubly stochastic map between euclidean jordan algebras is a positive unital and trace preserving map we relate such maps to jordan algebra automorphisms and majorization

cone orderings group majorizations and similarly separable vectors abstract we generalize some results on majorization in papers by wu and debnath s wu l debnath inequalities for convex sequences and their applications comput math appl 54 2007 525 534 and marshall et al a w marshall i olkin f proschan monotonicity of ratios of means and other applications of majorization in o shisha ed inequalities academic press new york 1967 pp 177 190 we present sufficient conditions for some vector inequalities to hold in the case of cone orderings and group induced cone orderings the framework used is based on results for similarly separable vectors given in paper m niezgoda bifractional inequalities and convex cones discrete math 306 2006 231 243

a methodology for constructing fuzzy algorithms for learning vector quantization this paper presents a general methodology for the development of fuzzy algorithms for learning vector quantization falvq the design of specific falvq algorithms according to existing approaches reduces to the selection of the membership function assigned to the weight vectors of an lvq competitive neural network which represent the prototypes the development of a broad variety of falvq algorithms can be accomplished by selecting the form of the interference function that determines the effect of the nonwinning prototypes on the attraction between the winning prototype and the input of the network the proposed methodology provides the basis for extending the existing falvq 1 falvq 2 and falvq 3 families of algorithms this paper also introduces two quantitative measures which establish a relationship between the formulation that led to falvq algorithms and the competition between the prototypes during the learning process the proposed algorithms and competition measures are tested and evaluated using the iris data set the significance of the proposed competition measure is illustrated using falvq algorithms to perform segmentation of magnetic resonance images of the brain

group majorization the convex hulls of sets of matrices and the diagonal element singular value inequalities abstract a group induced preorder on matrix space permits the description of the convex hulls of sets of proper and improper real matrices with given singular values and also permits a new proof of the necessity of the known relations between the diagonal elements and singular values of a real matrix

image coding using wavelet transform a scheme for image compression that takes into account psychovisual features both in the space and frequency domains is proposed this method involves two steps first a wavelet transform used in order to obtain a set of biorthogonal subclasses of images the original image is decomposed at different scales using a pyramidal algorithm architecture the decomposition is along the vertical and horizontal directions and maintains constant the number of pixels required to describe the image second according to shannon s rate distortion theory the wavelet coefficients are vector quantized using a multiresolution codebook to encode the wavelet coefficients a noise shaping bit allocation procedure which assumes that details at high resolution are less visible to the human eye is proposed in order to allow the receiver to recognize a picture as quickly as possible at minimum cost a progressive transmission scheme is presented it is shown that the wavelet transform is particularly well adapted to progressive transmission

fuzzy algorithms for learning vector quantization this paper presents the development of fuzzy algorithms for learning vector quantization falvq these algorithms are derived by minimizing the weighted sum of the squared euclidean distances between an input vector which represents a feature vector and the weight vectors of a competitive learning vector quantization lvq network which represent the prototypes this formulation leads to competitive algorithms which allow each input vector to attract all prototypes the strength of attraction between each input and the prototypes is determined by a set of membership functions which can be selected on the basis of specific criteria a gradient descent based learning rule is derived for a general class of admissible membership functions which satisfy certain properties the falvq 1 falvq 2 and falvq 3 families of algorithms are developed by selecting admissible membership functions with different properties the proposed algorithms are tested and evaluated using the iris data set the efficiency of the proposed algorithms is also illustrated by their use in codebook design required for image compression based on vector quantization

topology of the cone of positive maps on qubit systems an alternative geometrical proof of a known theorem concerning the decomposition of positive maps of the matrix algebra has been presented the premise of the proof is the identification of positive maps with operators preserving the lorentz cone in four dimensions and it allows to decompose the positive maps with respect to those preserving the boundary of the cone in addition useful conditions implying complete positivity of a map of have been given together with a sufficient condition for complete positivity of an extremal schwarz map of lastly following the same geometrical approach a description in topological terms of maps that are simultaneously completely positive and completely copositive has been presented

vector quantization of image subbands a survey subband and wavelet decompositions are powerful tools in image coding because of their decorrelating effects on image pixels the concentration of energy in a few coefficients their multirate multiresolution framework and their frequency splitting which allows for efficient coding matched to the statistics of each frequency band and to the characteristics of the human visual system vector quantization vq provides a means of converting the decomposed signal into bits in a manner that takes advantage of remaining inter and intraband correlation as well as of the more flexible partitions of higher dimensional vector spaces since 1988 a growing body of research has examined the use of vq for subband wavelet transform coefficients we present a survey of these methods

g majorization with applications to matrix orderings abstract a vector x is said to g majorize a vector y if y lies in the convex hull of the orbit of x under a group g the present paper contains a straightforward account with two important statements equivalent to g majorization in certain cases for example when g is a finite reflection group one equivalent condition reduces to a finite set of linear inequalities representing a cone ordering in the fundamental region of the group the other condition is that every convex g invariant function of y is less than the same function of x upper and lower weak majorizations gw majorizations are introduced by combining g with a second ordering compatible with it the results are applied where possible to matrix orderings where a g b is g majorization when g is a subgroup of the orthogonal group o n acting by congruence i e g b qbq t with q the matrix representation for an element of o n when g is the symmetric group of permutation matrices this defines a new ordering and generalizes a proposition of kiefer in the design of experiments

second order symmetric duality for nonlinear multiobjective mixed integer programming we formulate two pairs of second order symmetric duality for nonlinear multiobjective mixed integer programs for arbitrary cones by using the concepts of efficiency and second order invexity we establish weak strong converse and self duality theorems for the dual models several known results are obtained as special cases

group majorization and schur type inequalities abstract the problem of the reduction of a group majorization induced by an infinite group to a finite group majorization is discussed a necessary and sufficient condition for a possibility of the reduction is given the condition generalizes some known inequalities in the matrix theory involving eigenvalues or singular values of a matrix a corresponding result for a skew symmetric matrix is derived

subband coding of images subband coding has become quite popular for the source encoding of speech this paper presents a simple yet efficient extension of this concept to the source coding of images we specify the constraints for a set of two dimensional quadrature mirror filters qmf s for a particular frequency domain partition and show that these constraints are satisfied by a separable combination of one dimensional qmf s bits are then optimally allocated among the subbands to minimize the mean squared error for dpcm coding of the subbands also an adaptive technique is developed to allocate the bits within each subband by means of a local variance mask optimum quantization is employed with quantizers matched to the laplacian distribution subband coded images are presented along with their signal to noise ratios snr s the snr performance of the subband coder is compared to that of the adaptive discrete cosine transform dct vector quantization and differential vector quantization for bit rates of 0 67 1 0 and 2 0 bits per pixel for 256 256 monochrome images the adaptive subband coder has the best snr performance

on lei miranda and thompson s result on singular values and diagonal elements abstract thomspon and sing s result on the singular values and the diagonal elements of a complex n n matrix has been recently extended to arbitrary m matrices by lei and by miranda and thompson independently the real case in which the determinant of the product of the m real matrices is nonnegative or nonpositive was raised by miranda and thompson in this note we provide an answer to the question and we also extend some other results of lei

polynomial sized semidefinite representations of derivative relaxations of spectrahedral cones we give explicit polynomial sized in n n and k k semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree k k in n n variables these convex cones form a family of non polyhedral outer approximations of the non negative orthant that preserve low dimensional faces while successively discarding high dimensional faces more generally we construct explicit semidefinite representations polynomial sized in k m k m and n n of the hyperbolicity cones associated with k kth directional derivatives of polynomials of the form p x det sum i 1 n a i x i p x det i 1naixi where the a i ai are m times m m m symmetric matrices these convex cones form an analogous family of outer approximations to any spectrahedral cone our representations allow us to use semidefinite programming to solve the linear cone programs associated with these convex cones as well as their less well understood dual cones

an extension of schur ostrowski s condition weak eaton triples and generalized ai functions abstract in this paper for a given eaton triple v g d we present a characterization of g increasing real functions by using the maximality property of their directional derivatives it generalizes schur ostrowski s condition related to schur convexity of a function it is also useful in establishing theorems of von neumann davis type regarding the form of group invariant real functions with certain kinds of convexity we introduce and investigate weak eaton triples and their chains we show a method for constructing new eaton triples we apply it to define generalized arrangement increasing ai functions we prove that the mentioned derivatives of g increasing maps are generalized ai functions

on lattice property of group induced cone orderings abstract the problem of existence and uniqueness of greatest lower bound glb and least upper bound lub of a set in group induced cone gic ordering is discussed explicit formula for the glb is given the results are applied to obtain the best possible upper bound ingic ordering for values of linear and nonnegative linear maps

miranda and thompson s trace inequality and a log convexity result abstract a geometric proof of miranda and thompson s trace inequality is given via thompson s singular value diagonal element inequalities miranda and thompson s trace inequality is associated with the unitary group we then deal with the cases associated with the orthogonal group and the special unitary group we also discuss the convexity of the set of the diagonal elements of complex matrices with fixed singular values and determinant some question are asked a log convexity result related to gram schmidt decomposition is obtained

recursive quadrature mirror filters criteria specification and design method criteria are found for a pair of recursive infinite impluse response iir filters which have mirror image amplitude responses and whose outputs are in phase quadrature at all frequencies a method is suggested for designing filters with these characteristics and their performance and processing requirements are compared with a typical nonrecursive finite impluse response fir realization a particular filter is described in more detail and it is shown how it could be incorporated in a multiband processing scheme it is concluded that recursive quadrature mirror filters could offer considerable savings in terms of signal processing compared with nonrecursive filters with similar performances additionally it is suggested that because they can be designed to have relatively short absolute delays the performance of the iir filters may prove superior for certain applications

on the dual of a result of miranda and thompson abstract for a and b real matrices with prescribed singular values miranda and thompson characterized max tr aubv when u and v range over so n the real proper orthogonal group motivated by this result we investigate the location of det a ubv for a b u v as previously the corresponding problem for a and b complex matrices and u and v ranging over the unitary group is also discussed analogies are drawn between the determinantal problems and their tracial versions

g majorization group induced cone orderings and reflection groups a vector y is g majorized by a vector x if y is an element of the convex hull of the orbit of x under the action of a group g it is known that if g is a finite reflection group then g majorization is equivalent to a group induced cone ordering in this paper it is established that if for a finite subgroup g of the orthogonal group g majorization is equivalent to a group induced cone ordering then g must be a reflection group

subband coding of images using vector quantization a novel two dimensional subband coding technique is presented that can be applied to images as well as speech a frequency band decomposition of the image is carried out by means of 2d separable quadrature mirror filters which split the image spectrum into 16 equal rate subbands these 16 parallel subband signals are regarded as a 16 dimensional vector source and coded as such using vector quantization in the asymptotic case of high bit rates a theoretical analysis yields that a lower bound to the gain is attainable by choosing this approach over scalar quantization of each subband with an optimal bit allocation it is shown that vector quantization in this scheme has several advantages over coding the subbands separately experimental results are given and it is shown the scheme has a performance that is comparable to that of more complex coding techniques

image coding using vector quantization a review a review of vector quantization techniques used for encoding digital images is presented first the concept of vector quantization is introduced then its application to digital images is explained spatial predictive transform hybrid binary and subband vector quantizers are reviewed the emphasis is on the usefulness of the vector quantization when it is combined with conventional image coding techniques or when it is used in different domains

on the norm property of g c radii and eaton triples abstract let v be a euclidean space and o v be the orthogonal group the norm criterion of the g c radius is studied in the framework of an eaton triple v g d

cases of equality for certain inequalities involving a g doubly stochastic map abstract we study inner product inequalities for a g doubly stochastic map with respect to a group induced cone ordering we give necessary and sufficient conditions for the inequalities to become equalities a special condition is obtained if a birkhoff type theorem holds for a finite reflection group related to the ordering the conditions are applied to derive some classical and new results on equality cases for majorization inequalities connected with certain group actions

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed