quantum cohomology of flag manifolds and toda lattices we discuss relations of vafa s quantum cohomology with floer s homology theory introduce equivariant quantum cohomology formulate some conjectures about its general properties and on the basis of these conjectures compute quantum cohomology algebras of the flag manifolds the answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a toda lattice

accurate vessel segmentation with constrained b snake we describean active contour framework with accurate shape and size constraints on the vessel cross sectional planes to produce the vessel segmentation it starts with a multiscale vessel axis tracing in a 3d computed tomography ct data followed by vessel boundary delineation on the cross sectional planes derived from the extracted axis the vessel boundary surface is deformed under constrained movements on the cross sections and is voxelized to produce the final vascular segmentation the novelty of this paper lies in the accurate contour point detection of thin vessels based on the ct scanning model in the efficient implementation of missing contour points in the problematic regions and in the active contour model with accurate shape and size constraints the main advantage of our framework is that it avoids disconnected and incomplete segmentation of the vessels in the problematic regions that contain touching vessels vessels in close proximity to each other diseased portions pathologic structure attached to a vessel and thin vessels it is particularly suitable for accurate segmentation of thin and low contrast vessels our method is evaluated and demonstrated on ct data sets from our partner site and its results are compared with three related methods our method is also tested on two publicly available databases and its results are compared with the recently published method the applicability of the proposed method to some challenging clinical problems the segmentation of the vessels in the problematic regions is demonstrated with good results on both quantitative and qualitative experimentations our segmentation algorithm can delineate vessel boundaries that have level of variability similar to those obtained manually

dct algorithms for composite sequence lengths this paper presents fast algorithms for the type ii and iii discrete cosine transforms of composite sequence length in particular a radix q algorithm where q is an odd integer is derived for uniform or mixed radix decomposition of the discrete cosine transform by combining the radix q and radix 2 algorithms a general decomposition method for any composite length is developed reduction of computational complexity can be achieved for many sequence lengths compared with that needed by the well known radix 2 algorithm furthermore both the proposed and chan and siu s 1993 mixed radix algorithms achieve the same computational complexity for n 3 2 sup p and 9 2 sup p however our algorithm uses a simpler decomposition approach and provides a wider range of choices of sequence lengths

gromov witten classes quantum cohomology and enumerative geometry the paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry in particular it contains an axiomatic treatment of gromov witten classes and a discussion of their properties for fano varieties cohomological field theories are defined and it is proved that tree level theories are determined by their correlation functions application to counting rational curves on del pezzo surfaces and projective spaces are given

shifted fourier transform based tensor algorithms for the 2 d dct in this paper tensor algorithms for calculating the two dimensional 2 d discrete cosine transform dct are presented the tensor approach is based on the concept of the covering revealing the transforms which yields in particular the splitting of the shifted 2 sup r spl times 2 sup r point fourier and cosine transforms into 2 sup r 1 3 one dimensional 1 d incomplete 2 sup r point transforms the multiplicative complexity of the 2 d 2 sup r spl times 2 sup r point discrete cosine transforms in terms of the tensor representation is 4 sup r 3 2 sup r 2 r sup 2 7r 14 which is reduced to 4 sup r 8 3 2 sup r r sup 2 7r 10 20 3 when using the improved tensor algorithm the multiplicative complexity in the general l sup r spl times l sup r case with a prime l 2 as well as in the l sub 1 l sub 2 spl times l sub 1 l sub 2 case with arbitrary co prime l sub 1 l sub 2 1 is provided the examples of the tensor algorithms for calculating the 8 spl times 8 point dct through 104 88 and 84 multiplications are given in detail based on the proposed concept the fast algorithm for calculating the 1 d dct i is also developed the multiplicative complexity of the 2 sup r point dct i is 2 sup r 1 r 2 r 5 2 8 the comparative estimates with the known algorithms are given

mutations of puzzles and equivariant cohomology of two step flag varieties we introduce a mutation algorithm for puzzles that is a three direction analogue of the classical jeu de taquin algorithm for semistandard tableaux we apply this algorithm to prove our conjectured puzzle formula for the equivariant schubert structure constants of two step flag varieties this formula gives an expression for the structure constants that is positive in the sense of graham thanks to the equivariant version of the quantum equals classical result our formula specializes to a littlewood richardson rule for the equivariant quantum cohomology of grassmannians

a fast recursive algorithm for computing the discrete cosine transform the discrete cosine transform dct is widely applied in various fields including image data compression because it operates like the karhunen loeve transform for stationary random data this paper presents a recursive algorithm for dct with a structure that allows the generation of the next higher order dct from two identical lower order dct s as a result the method for implementing this recursive dct requires fewer multipliers and adders than other dct algorithms

quantum cohomology of flag manifolds g b and quantum toda lattices let g be a connected semi simple complex lie group b its borel subgroup t a maximal complex torus contained in b and lie t its lie algebra this setup gives rise to two constructions the generalized nonperiodic toda lattice 28 29 and the flag manifold g b the toda lattice for g b t is the dynamical system on the cotangent bundle t lie t endowed with the canonical holomorphic symplectic form and the holomorphic hamiltonian function we consider in this paper

on the multiplicative complexity of discrete cosine transforms the multiplicative complexity of discrete cosine transforms dcts of arbitrary dimensions on input sizes which are powers of two are obtained new upper bounds on the multiplicative complexity of scaled dcts on input sizes which are powers of two are also obtained

scaled dct s on input sizes that are composite given efficient scaled discrete cosine transforms dct s of size p and q where p and q are relatively prime we present methods for constructing fast algorithms to compute scaled dct s and scaled inverse dct s of size pq extensions to multidimensional transforms are also given

gromov witten invariants on grassmannians we prove that any three point genus zero gromov witten invariant on a type a grassmannian is equal to a classical intersection number on a two step flag variety we also give symplectic and orthogonal analogues of this result in these cases the two step flag variety is replaced by a sub maximal isotropic grassmannian our theorems are applied in type a to formulate a conjectural quantum littlewood richardson rule and in the other classical lie types to obtain new proofs of the main structure theorems for the quantum cohomology of lagrangian and orthogonal grassmannians

on the interrelationships among a class of convolutions in this paper some interrelationships among a class of circular operations are investigated based on matrix formulation it is shown that a class of convolutions representing forward backward and convolution correlation of two periodic sequences may be related to each other in terms of discrete transforms having the circular convolution property the results obtained are useful in efficient realization of adaptive digital filters using fast transforms

fast algorithms for type iii dct of composite sequence lengths this correspondence presents an odd factor algorithm for the type iii discrete cosine transform dct for uniform or mixed radix decomposition by jointly using the old factor and the existing radix 2 algorithms a general decomposition method for arbitrarily composite sequence length is developed a reduction of computational complexity can be achieved compared with that needed by other reported algorithms for m 2 sup m the decomposition approach has a simple computational structure and supports a wider range of choices for different sequence lengths

affine approach to quantum schubert calculus this paper presents a formula for products of schubert classes in the quantum cohomology ring of the grassmannian we introduce a generalization of schur symmetric polynomials for shapes that are naturally embedded in a torus then we show that the coefficients in the expansion of these toric schur polynomials in terms of the regular schur polynomials are exactly the 3 point gromov witten invariants which are the structure constants of the quantum cohomology ring this construction implies three symmetries of the gromov witten invariants of the grassmannian with respect to the groups s3 z nz 2 and z 2z the last symmetry is a certain curious duality of the quantum cohomology which inverts the quantum parameter q our construction gives a solution to a problem posed by fulton and woodward about the characterization of the powers of the quantum parameter q which occur with nonzero coefficients in the quantum product of two schubert classes the curious duality switches the smallest such power of q with the highest power we also discuss the affine nil temperley lieb algebra that gives a model for the quantum cohomology

efficient block based frequency domain wavelet transform implementations subband decompositions for image coding have been explored extensively over the last few decades the condensed wavelet packet cwp transform is one such decomposition that was recently shown to have coding performance advantages over conventional decompositions a special feature of the cwp is that its design and implementation are performed in the cyclic frequency domain while performance gains have been reported efficient implementations of the cwp or more generally efficient implementations of cyclic filter banks have not yet been fully explored in this paper we present efficient block based implementations of cyclic filter banks along with an analysis of the arithmetic complexity block based cyclic filter bank implementations of the cwp coder are compared with conventional subband wavelet image coders whose filter banks are implemented in the time domain it is shown that block based cyclic filter bank implementations can result in cwp coding systems that outperform the popular image coding systems both in terms of arithmetic complexity and coding performance

quantum k theory of grassmannians we show that equivariant k theoretic 3 point gromov witten invariants of genus zero on a grassmann variety are equal to triple intersec tions computed in the equivariant k theory of a two step flag manifold thus generalizing an earlier result of buch kresch and tamvakis in the process we show that the gromov witten variety of curves passing through 3 general points is irreducible and rational our applications include pieri and giambelli formulas for the quantum k theory ring of a grassmannian which determine the multiplication in this ring we also compute the dual schubert basis for this ring and show that its structure constants satisfy s3 symmetry our for mula for gromov witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of chaput manivel and perrin

schubert calculus and equivariant cohomology of grassmannians abstract we give a description of equivariant cohomology of grassmannians that places the theory into a general framework for cohomology theories of grassmannians as a result we obtain a formalism for equivariant cohomology where the basic results of equivariant schubert calculus the basis theorem pieri s formula and giambelli s formula can be obtained from the corresponding results of the general framework by a change of basis in order to show that our formalism reflects the geometry of grassmannians we relate our theory to the treatment of equivariant cohomology of grassmannians by a knutson and t tao

a higher order tensor vessel tractography for segmentation of vascular structures a new vascular structure segmentation method which is based on a cylindrical flux based higher order tensor hot is presented on a vessel structure the hot naturally models branching points which create challenges for vessel segmentation algorithms in a general linear hot model embedded in 3d one has to work with an even order tensor due to an enforced antipodal symmetry on the unit sphere however in scenarios such as in a bifurcation the antipodally symmetric tensor embedded in 3d will not be useful in order to overcome that limitation we embed the tensor in 4d and obtain a structure that can model asymmetric junction scenarios during construction of a higher order tensor e g third or fourth order in 4d the orientation vectors lie on the unit 3 sphere in contrast to the unit 2 sphere in 3d tensor modeling this 4d tensor is exploited in a seed based vessel segmentation algorithm where the principal directions of the 4d hot is obtained by decomposition and used in a hot tractography approach we demonstrate quantitative validation of the proposed algorithm on both synthetic complex tubular structures as well as real cerebral vasculature in magnetic resonance angiography mra datasets and coronary arteries from computed tomography angiography cta volumes

a littlewood richardson rule for factorial schur functions we give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial schur functions it is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial schur function applications to capelli operators and quantum immanants are also given

on equivariant quantum schubert calculus for g p we show a z2 filtered algebraic structure and a quantum to classical principle on the torus equivariant quantum cohomology of a complete flag variety of general lie type generalizing earlier works of leung and the second author we also provide various applications on equivariant quantum schubert calculus including an equivariant quantum pieri rule for partial flag variety fln1 nk n 1 of lie type a

fast algorithms for the discrete cosine transform several fast algorithms for computing discrete cosine transforms dcts and their inverses on multidimensional inputs of sizes which are powers of 2 are introduced because the 1 d 8 point dct and the 2 d 8 8 point dct are so widely used they are discussed in detail algorithms for computing scaled dcts and their inverses are also presented these have applications in compression of continuous tone image data where the dct is generally followed by scaling and quantization

quantum schubert polynomials where in is the ideal generated by symmetric polynomials in x1 xn without constant term another geometric description of the cohomology ring of the flag manifold is based on the decomposition of fln into schubert cells these are even dimensional cells indexed by the elements w of the symmetric group sn the corresponding cohomology classes oa called schubert classes form an additive basis in h fln 2 to relate the two descriptions one would like to determine which elements of 2 xl xn in correspond to the schubert classes under the isomorphism 1 1 this was first done in 2 see also 8 for a general case of an arbitrary complex semisimple lie group later lascoux and schiitzenberger 22 came up with a combinatorial version of this theory for the type a by introducing remarkable polynomial representatives of the schubert classes oa called schubert polynomials and denoted gw recently motivated by ideas that came from the string theory 31 30 mathematicians defined for any kahler algebraic manifold x the small quantum cohomology ring qh x 2 which is a certain deformation of the classical cohomology ring see e g 28 19 14 and references therein the additive structure of qh x 2 is essentially the same as that of ordinary cohomology in particular qh fln z is canonically isomorphic as an abelian group to the tensor product h fln 2 0 z ql qn 1 where the qi are formal variables deformation parameters the multiplicative structure of the quantum cohomology is however

vessels as 4 d curves global minimal 4 d paths to extract 3 d tubular surfaces and centerlines in this paper we propose an innovative approach to the segmentation of tubular structures this approach combines all of the benefits of minimal path techniques such as global minimizers fast computation and powerful incorporation of user input while also having the capability to represent and detect vessel surfaces directly which so far has been a feature restricted to active contour and surface techniques the key is to represent the trajectory of a tubular structure not as a 3 d curve but to go up a dimension and represent the entire structure as a 4 d curve then we are able to fully exploit minimal path techniques to obtain global minimizing trajectories between two user supplied endpoints in order to reconstruct tubular structures from noisy or low contrast 3 d data without the sensitivity to local minima inherent in most active surface techniques in contrast to standard purely spatial 3 d minimal path techniques however we are able to represent a full tubular surface rather than just a curve which runs through its interior our representation also yields a natural notion of a tube s central curve we demonstrate and validate the utility of this approach on magnetic resonance mr angiography and computed tomography ct images of coronary arteries

equivariant quantum schubert calculus abstract we study the t equivariant quantum cohomology of the grassmannian we prove the vanishing of a certain class of equivariant quantum littlewood richardson coefficients which implies an equivariant quantum pieri rule as in the equivariant case this implies an algorithm to compute the equivariant quantum littlewood richardson coefficients

a pieri type formula for h t sln c b the singular cohomology of the grassmann variety of k planes in cn has a basis s indexed by partitions the classical pieri formula is an explicit rule for determining the coefficients in the expansion of the cup products1m s c1m s where 1m is a column of length m and s1m is the mth chern class of the tautological bundle lascoux and schutzenberger c r acad sci paris294 1982 447 450 formulated a generalization of pieri s formula to the cohomology of the flag variety sln c b and briefly indicated an algebraic proof manivel cours specialises 3 1998 provides details of this proof a geometric proof was given by sottile ann inst fourier grenoble 46 1996 89 110 in this paper we state and prove a generalization of this pieri type formula for the t equivariant cohomology of the flag variety we use the algebraic description of the t equivariant cohomology of the flag variety due to kostant and kumar adv math 62 1986 187 237 and arabia bull soc math france117 1989 129 165 and our new formula exposes an equality of certain structure constants in this algebra our proof is an induction based on the original idea in lascoux and schutzenberger

quantum integrability and generalised quantum schubert calculus abstract we introduce and study a new mathematical structure in the generalised quantum cohomology theory for grassmannians namely we relate the schubert calculus to a quantum integrable system known in the physics literature as the asymmetric six vertex model our approach offers a new perspective on already established and well studied special cases for example equivariant k theory and in addition allows us to formulate a conjecture on the so far unknown case of quantum equivariant k theory

littlewood richardson coefficients for reflection groups abstract in this paper we explicitly compute all littlewood richardson coefficients for semisimple and kac moody groups g that is the structure constants also known as the schubert structure constants of the cohomology algebra h g p c where p is a parabolic subgroup of g these coefficients are of importance in enumerative geometry algebraic combinatorics and representation theory our formula for the littlewood richardson coefficients is purely combinatorial and is given in terms of the cartan matrix and the weyl group of g however if some off diagonal entries of the cartan matrix are 0 or 1 the formula may contain negative summands on the other hand if the cartan matrix satisfies a i j a j i 4 for all i j then each summand in our formula is nonnegative that implies nonnegativity of all littlewood richardson coefficients we extend this and other results to the structure coefficients of the t equivariant cohomology of flag varieties g p and bott samelson varieties i g

positivity in equivariant quantum schubert calculus a conjecture of d peterson proved by w graham states that the structure constants of the t equivariant cohomology of a homogeneous space g p satisfy a certain positivity property in this paper we show that this positivity property holds in the more general situation of equivariant quantum cohomology 1 introduction it is well known that the integral cohomology of the homogeneous space x g p for g a connected semisimple complex lie group and p a parabolic subgroup satisfies a positivity property its structure constants are nonnegative integers equal to the number of intersection points of three schubert varieties in general position whose codimensions add up to the dimension of x recently graham gr has proved a conjecture of peterson p asserting that h x the t equivariant cohomology of x where t c r is a maximal torus in g enjoys a more general positivity property

a new algorithm to compute the discrete cosine transform a new algorithm is introduced for the 2m point discrete cosine transform this algorithm reduces the number of multiplications to about half of those required by the existing efficient algorithms and it makes the system simpler

fast algorithm for the 3 d dct ii recently many applications for three dimensional 3 d image and video compression have been proposed using 3 d discrete cosine transforms 3 d dcts among different types of dcts the type ii dct dct ii is the most used in order to use the 3 d dcts in practical applications fast 3 d algorithms are essential therefore in this paper the 3 d vector radix decimation in frequency 3 d vr dif algorithm that calculates the 3 d dct ii directly is introduced the mathematical analysis and the implementation of the developed algorithm are presented showing that this algorithm possesses a regular structure can be implemented in place for efficient use of memory and is faster than the conventional row column frame rcf approach furthermore an application of 3 d video compression based 3 d dct ii is implemented using the 3 d new algorithm this has led to a substantial speed improvement for 3 d dct ii based compression systems and proved the validity of the developed algorithm

flux based anisotropic diffusion applied to enhancement of 3 d angiogram we present a new approach to anisotropic diffusion based on a multidirectional diffusion flux the diffusion flux is decomposed in an orthogonal basis effectively enabling enhancement of contours as well as diffusion along the contours to this end we have selected a three dimensional basis that depicts the directions of principal curvature and has an interesting interpretation in the context of the vessels the diffusion function associated to each vector of the basis depends on the first order derivative of the intensity in this direction instead of the traditional norm of the smoothed gradient accordingly we present the results of a restoration of computed tomography data of the liver

tubular structure segmentation based on minimal path method and anisotropic enhancement we present a new interactive method for tubular structure extraction the main application and motivation for this work is vessel tracking in 2d and 3d images the basic tools are minimal paths solved using the fast marching algorithm this allows interactive tools for the physician by clicking on a small number of points in order to obtain a minimal path between two points or a set of paths in the case of a tree structure our method is based on a variant of the minimal path method that models the vessel as a centerline and surface this is done by adding one dimension for the local radius around the centerline the crucial step of our method is the definition of the local metrics to minimize we have chosen to exploit the tubular structure of the vessels one wants to extract to built an anisotropic metric the designed metric is well oriented along the direction of the vessel admits higher velocity on the centerline and provides a good estimate of the vessel radius based on the optimally oriented flux this measure is required to be robust against the disturbance introduced by noise or adjacent structures with intensity similar to the target vessel we obtain promising results on noisy synthetic and real 2d and 3d images and we present a clinical validation

robust and objective decomposition and mapping of bifurcating vessels computational modeling of human arteries has been broadly employed to investigate the relationships between geometry hemodynamics and vascular disease recent developments in modeling techniques have made it possible to perform such analyses on realistic geometries acquired noninvasively and thus have opened up the possibility to extend the investigation to populations of subjects however for this to be feasible novel methods for the comparison of the data obtained from large numbers of realistic models in the presence of anatomic variability must be developed in this paper we present an automatic technique for the objective comparison of distributions of geometric and hemodynamic quantities over the surface of bifurcating vessels the method is based on centerlines and consists of robustly decomposing the surface into its constituent branches and mapping each branch onto a template parametric plane the application of the technique to realistic data demonstrates how similar results are obtained over similar geometries allowing for proper model to model comparison thanks to the computational and differential geometry criteria adopted the method does not depend on user defined parameters or user interaction it is flexible with respect to the bifurcation geometry and it is readily extendible to more complex configurations of interconnecting vessels

giambelli formulae for the equivariant quantum cohomology of the grassmannian we find presentations by generators and relations for the equivariant quantum cohomology of the grassmannian for these presentations we also find determinantal formulae for the equivariant quantum schubert classes to prove this we use the theory of factorial schur functions and a characterization of the equivariant quantum cohomology ring

equivariant quantum cohomology of the odd symplectic grassmannian the odd symplectic grassmannian mathrm ig mathrm ig k 2n 1 parametrizes k dimensional subspaces of mathbb c 2n 1 which are isotropic with respect to a general necessarily degenerate symplectic form the odd symplectic group acts on mathrm ig with two orbits and mathrm ig is itself a smooth schubert variety in the submaximal isotropic grassmannian mathrm ig k 2n 2 we use the technique of curve neighborhoods to prove a chevalley formula in the equivariant quantum cohomology of mathrm ig i e a formula to multiply a schubert class by the schubert divisor class this generalizes a formula of pech in the case k 2 and it gives an algorithm to calculate any multiplication in the equivariant quantum cohomology ring

vessel tractography using an intensity based tensor model with branch detection in this paper we present a tubular structure segmentation method that utilizes a second order tensor constructed from directional intensity measurements which is inspired from diffusion tensor image dti modeling the constructed anisotropic tensor which is fit inside a vessel drives the segmentation analogously to a tractography approach in dti our model is initialized at a single seed point and is capable of capturing whole vessel trees by an automatic branch detection algorithm developed in the same framework the centerline of the vessel as well as its thickness is extracted performance results within the rotterdam coronary artery algorithm evaluation framework are provided for comparison with existing techniques 96 4 average overlap with ground truth delineated by experts is obtained in addition to other measures reported in the paper moreover we demonstrate further quantitative results over synthetic vascular datasets and we provide quantitative experiments for branch detection on patient computed tomography angiography cta volumes as well as qualitative evaluations on the same cta datasets from visual scores by a cardiologist expert

an algorithm for calculation of the discrete cosine transform by paired transform a new algorithm for splitting the one dimensional 1 d 2 sup r point discrete cosine transform dct into a set of short 2 sup k point type iv dcts k 1 r 1 is introduced the splitting is performed by means of paired transformation that is defined by the paired representation of signals with respect to the cosine transform a proposed method of calculating the 2 sup r point cosine transform requires 2 sup r 1 r multiplications and 2 sup r 1 3r 2 1 additions when r spl ges 2