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

an efficient cordic array structure for the implementation of discrete cosine transform we propose a novel implementation of the discrete cosine transform dct and the inverse dct idct algorithms using a cordic coordinate rotation digital computer based systolic processor array structure first we reformulate an n point dct or idct algorithm into a rotation formulation which makes it suitable for cordic processor implementation we then propose to use a pipelined cordic processor as the basic building block to construct l d and 2 d systolic type processor arrays to speed up the dct and idct computation due to the proposed novel rotation formulation we achieve 100 processor utilization in both 1 d and 2 d configurations furthermore we show that for the 2 d configurations the same data processing throughput rate ran be maintained as long as the processor array dimensions are increased linearly with n neither the algorithm formulation or the array configuration need to be modified hence the proposed parallel architecture is scalable to the problem size these desirable features make this novel implementation compare favorably to previously proposed dct implementations

an analysis of real fourier domain based adaptive algorithms implemented with the hartley transform using cosine sine symmetries the least mean squared lms algorithm and its variants have been the most often used algorithms in adaptive signal processing however the lms algorithm suffers from a high computational complexity especially with large filter lengths the fourier transform based block normalized lms fbnlms reduces the computation count by using the discrete fourier transform dft and exploiting the fast algorithms for implementing the dft even though the savings achieved with the fbnlms over the direct lms implementation are significant the computational requirements of fbnlms are still very high rendering many real time applications like audio and video estimation infeasible the hartley transform based bnlms hbnlms is found to have a computational complexity much less than and a memory requirement almost of the same order as that of the fbnlms this paper is based on the cosine and sine symmetric implementation of the discrete hartley transform dht which is the key in reducing the computational complexity of the fbnlms by 33 asymptotically with respect to multiplications the parallel implementation of the discrete cosine transform dct in turn can lead to more efficient implementations of the hbnlms

the algebraic approach to the discrete cosine and sine transforms and their fast algorithms it is known that the discrete fourier transform dft used in digital signal processing can be characterized in the framework of the representation theory of algebras namely as the decomposition matrix for the regular module mathbb c z n mathbb c x x n 1 this characterization provides deep insight into the dft and can be used to derive and understand the structure of its fast algorithms in this paper we present an algebraic characterization of the important class of discrete cosine and sine transforms as decomposition matrices of certain regular modules associated with four series of chebyshev polynomials then we derive most of their known algorithms by pure algebraic means we identify the mathematical principle behind each algorithm and give insight into its structure our results show that the connection between algebra and digital signal processing is stronger than previously understood

efficient computation of the discrete pseudo wigner distribution a description is given of a novel algorithm the fast fourier transform in part fftp for the computation of the discrete pseudo wigner distribution dpwd the fftp computes the cosine and sine parts of the discrete fourier transform dft separately by employing real inverse sinusoidal twiddle factors unlike the conventional methods which directly utilize the complex dft the fftp yields real output since the dpwd is always real in addition the new method reduces the computational cost by making full use of symmetries and removing redundancies in the fftp computation the authors also describe a simple algorithm for computing the discrete hilbert transform dht to produce the nonaliased dpwd a pipeline structure for real time and a bulk processing technique for offline implementations of the method are presented

a novel dct based algorithm for computing the modulated complex lapped transform in this correspondence we propose a novel discrete cosine transform dct based algorithm for the fast computation of the modulated complex lapped transform mclt which we have previously proposed the method is mainly based on two dcts and two stages of butterfly operations for a length m mclt the proposed algorithm requires only mlog2m multiplications for the sine window and mlog2m 2m multiplications for any other window function thus the multiplicative complexity of the proposed algorithm is superior to any other previous algorithm without any penalty on the total number of multiplications and additions

permuted difference coefficient realization of fir digital filters a new realization for fir digital filters using permuted difference coefficients is proposed in this paper its coefficients are obtained as the difference between the successive values of the original coefficients reordered in a sequence with falling magnitude the proposed realization can hold desirable properties in fir filters such as an exactly linear phase characteristic and stable implementation and is effectively applied to a wide range filter response quantization error analysis shows that the internal data word lengths must be somewhat increased to maintain the same roundoff noise as in a direct form realization computational complexity becomes about 23 percent and 18 percent for 99th and 299th order filters taking the excess data word lengths into account compared with the direct form

fast algorithm for computing discrete cosine transform an efficient method for computing the discrete cosine transform dct is proposed based on direct decomposition of the dct the recursive properties of the dct for an even length input sequence is derived which is a generalization of the radix 2 dct algorithm based on the recursive property a new dct algorithm for an even length sequence is obtained the proposed algorithm is very structural and requires fewer computations when compared with others the regular structure of the proposed algorithm is suitable for fast parallel algorithm and vlsi implementation

fractional cosine sine and hartley transforms in previous papers the fourier transform ft has been generalized into the fractional fourier transform frft the linear canonical transform lct and the simplified fractional fourier transform sfrft because the cosine sine and hartley transforms are very similar to the ft it is reasonable to think they can also be generalized by the similar way we introduce several new transforms they are all the generalization of the cosine sine or hartley transform we first derive the fractional cosine sine and hartley transforms frct frst frht they are analogous to the frft then we derive the canonical cosine and sine transforms cct cst they are analogous to the lct we also derive the simplified fractional cosine sine and hartley transforms sfrct sfrst sfrht they are analogous to the sfrft and have the advantage of real input real output we also discuss the properties digital implementation and applications e g the applications for filter design and space variant pattern recognition of these transforms the transforms introduced in this paper are very efficient for digital implementation we can just use one half or one fourth of the real multiplications required for the frft and lct to implement them when we want to process even odd or pure real imaginary functions we can use these transforms instead of the frft and lct besides we also show that the frct frst cct cst and sfrct sfrst are also useful for the one sided t spl isin 0 spl infin signal processing

fast computation of md dct iv md dst iv by md dwt or md dct ii this paper reveals relationships between the type iv multidimensional discrete cosine transform md dct iv and the type ii multidimensional discrete cosine transform md dct ii or the multidimensional discrete w transform md dwt based on the relationships md dct iv can be efficiently computed by using the fast algorithms for md dct ii or md dwt iii or md dwt iii efficient method for implementation of the post addition or pre addition process is also presented compared to the widely used row column method considerable savings on the number of arithmetic operations are achieved for example the number of multiplications for computing an r dimensional dct iv is only about frac 1 r times that needed by the row column method and the number of additions is also reduced numerical experiments are made to test the computational time and roundoff errors they show that the proposed fast algorithms save time

a two dimensional fast cosine transform algorithm based on hou s approach a structured approach is used to generate a fast algorithm to compute two dimensional discrete cosine transforms dct based on hou s method hou s algorithm is extended to the 2 d case using an approach presented in both matrix and diagrammatical forms the matrix approach is discussed and this forms a basis on which a 2 d fast dct algorithm is derived it is shown that this matrix method has a structure similar to that of the 1 d cooley tukey fast fourier transform fft algorithm then the decimation in frequency dif 2 d fast dct algorithm is presented using matrix forms which use the tensor or kronecker product as a construction tool finally the 2 d algorithm is described by logic diagrams which reveal the relationship between the 2 d algorithm and its 1 d counterpart as an example the logic diagram of an 8 point 8 point 2 d dct using the new 2 d dct algorithm is generated through a simple procedure

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

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

a categorification of morelli s theorem we prove a theorem relating torus equivariant coherent sheaves on toric varieties to polyhedrally constructible sheaves on a vector space at the level of k theory the theorem recovers morelli s description of the k theory of a smooth projective toric variety morelli in adv math 100 2 154 182 1993 specifically let x be a proper toric variety of dimension n and let m mathbb r mathrm lie t mathbb r vee cong mathbb r n be the lie algebra of the compact dual real torus t mathbb r vee cong u 1 n then there is a corresponding conical lagrangian t m and an equivalence of triangulated dg categories mathcal p mathrm erf t x cong mathit sh cc m mathbb r lambda where mathcal p mathrm erf t x is the triangulated dg category of perfect complexes of torus equivariant coherent sheaves on x and shcc m is the triangulated dg category of complex of sheaves on m with compactly supported constructible cohomology whose singular support lies in this equivalence is monoidal it intertwines the tensor product of coherent sheaves on x with the convolution product of constructible sheaves on m

two dimensional dct dst universal computational structure for 2 sup m spl times 2 sup n block sizes a refined generalized signal flow graph for the direct two dimensional discrete cosine transform 2 d dct and discrete sine transform 2 d dst computation the so called 2 d dct dst universal computational structure is described it represents a generalized unified approach to the fast 2 d dct and 2 d dst computation for any 2 sup m spl times 2 sup n block sizes i e both square and rectangular blocks including the one dimensional 1 d case the regular structure moderate arithmetic complexity numerical stability and multiple block size capability makes it suitable for vlsi or parallel implementation

existence of a nontrivial solution for a class of superquadratic elliptic problems abstract a nontrivial solution is obtained for a class of superquadratic elliptic problems by variational methods

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

restructured recursive dct and dst algorithms the discrete cosine transform dct and the discrete sine transform dst have found wide applications in speech and image processing as well as telecommunication signal processing for the purpose of data compression feature extraction image reconstruction and filtering in this paper we present new recursive algorithms for the dct and the dst the proposed method is based on certain recursive properties of the dct coefficient matrix and can be generalized to design recursive algorithms for the 2 d dct and the 2 d dst these new structured recursive algorithms are able to decompose the dct and the dst into two balanced lower order subproblems in comparison to previous research works therefore when converting our algorithms into hardware implementations we require fewer hardware components than other recursive algorithms finally we propose two parallel algorithms for accelerating the computation

fast computation of the discrete cosine transform and the discrete hartley transform a new factorization of the discrete hartley transform dht is presented it is used to derive new algorithms for the dht and the discrete cosine transform dct with reduced number of multiplications

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

infinitely many solutions of a symmetric dirichlet problem here q is a bounded domain in r with smooth boundary and f i r is c and satisfies certain growth conditions if m 1 and f is an even function hence f r r is an odd function then ambrosetti and rabinowitz l 21 proved the existence of infinitely many weak solutions of d using a symmetric version of the mountain pass theorem a similar result is due to michalek 3 who considered the case where m 21 rm s c and f is invariant under a free action of the cyclic group z p on c here z p is considered as a subgroup of s c c acting essentially via scalar multiplication on c in this paper we want to investigate which kind of symmetries guarantee the existence of infinitely many solutions of d let g be a compact lie group and p g o m an orthogonal representation of g on lrm then g acts on irm via the map g x ir ir g u y gv p g v we write v for the space rm considered as a g space do there exist infinitely many solutions of d if f is invariant under this symmetry i e f gu f u for all g e g and u e v without further assumptions on p the answer to this question is no in general for example if p g id for all g e g then there is no symmetry at all and one cannot expect infinitely many solutions more generally one should exclude the existence of fixed points vg u e vi gu u for all g e g 0 our first result says that there exist infinitely many solutions of d if the following condition is satisfied

fast imdct and mdct algorithms a matrix approach this paper presents a systematic investigation of the modified discrete cosine transform inverse modified discrete cosine transform mdct imdct algorithm using a matrix representation this approach results in new understanding of the mdct imdct enables the development of new algorithms and makes clear the connection between the algorithms we represent in a matrix form the imdct as the product of the type iv dct with simple scaling sign changing and permutation operations such that fast algorithms for the type iv dct can be simply modified for the imdct and vice versa then the simple symmetry and inversion properties of the type iv dct are used to develop new algorithms and establish the connection between existing fast imdct algorithms this approach also enables us to show that mdct and imdct share common core operation and present an efficient architecture for implementing both the mdct and the imdct in one hardware

prime factor dct algorithms in this correspondence new algorithms are presented for computing the l d and 2 d discrete cosine transform dct of even length by using the discrete fourier transform dft a comparison of the proposed algorithms to other fast ones points out their computational efficiency which is mainly based on the advantages of prime factor decomposition and a proper choice of index mappings

mixed radix discrete cosine transform presents two new fast discrete cosine transform computation algorithms a radix 3 and a radix 6 algorithm these two new algorithms are superior to the conventional radix 3 algorithm as they i require less computational complexity in terms of the number of multiplications per point ii provide a wider choice of the sequence length for which the dct can be realized and iii support the prime factor decomposed computation algorithm to realize the 2 sup m 3 sup n point dct furthermore a mixed radix algorithm is also proposed such that an optimal performance can be achieved by applying the proposed radix 3 and radix 6 and the well developed radix 2 decomposition techniques in a proper sequence

a fast 4 4 dct algorithm for the recursive 2 d dct the authors present an efficient algorithm for the computation of the 4 4 discrete cosine transform dct the algorithm is based on the decomposition of the 4 4 dct into four 4 point 1 d dcts thus only 1 d transformations and some additions are required it is shown that the proposed algorithm requires only 16 multiplications which is half the number needed for the conventional row column method since the 2 sup m 2 sup m dct can be computed using the 4 4 dct recursively for any m the proposed algorithm leads to a fast algorithm for the computation of the 2 d dct

applications of local linking to critical point theory we prove the existence of nontrivial critical points of functionals with a local linking at 0 applications to elliptic boundary value problems and to hamiltonian systems are given c 1995 academic press inc

low complexity multidimensional dct approximations for high order tensor data decorrelation in this paper we introduce low complexity multidimensional discrete cosine transform dct approximations 3d dct approximations are formalized in terms of high order tensor theory the formulation is extended to higher dimensions with arbitrary lengths several multiplierless 8 times 8 times 8 approximate methods are proposed and the computational complexity is discussed for the general multidimensional case the proposed methods complexity cost was assessed presenting considerably lower arithmetic operations when compared with the exact 3d dct the proposed approximations were embedded into 3d dct based video coding scheme and a modified quantization step was introduced the simulation results showed that the approximate 3d dct coding methods offer almost identical output visual quality when compared with exact 3d dct scheme the proposed 3d approximations were also employed as a tool for visual tracking the approximate 3d dct based proposed system performs similarly to the original exact 3d dct based method in general the suggested methods showed competitive performance at a considerably lower computational cost

the theory of toric manifolds toric manifolds the topological analogue of toric varieties are determined by an n dimensional simple convex polytope and a function from the set of codimensionone faces into the primitive vectors of an integer lattice their cohomology was determined by davis and januszkiewicz in 1991 and corresponds with the theorem of danilovjurkiewicz in the toric variety case we use the adams spectral sequence to compute the ko theory of all toric manifolds and certain toric varieties

integer dct based on direct lifting of dct idct for lossless to lossy image coding a discrete cosine transform dct can be easily implemented in software and hardware for the jpeg and mpeg formats however even though some integer dcts intdcts for lossless to lossy image coding have been proposed such transform requires redesigned devices this paper proposes a hardware friendly intdct that can be applied to both lossless and lossy coding our intdct is implemented by direct lifting of dct and inverse dct idct consequently any existing dct device can be directly applied to every lifting block although our method requires a small side information block sib it is validated by its application to lossless to lossy image coding

a new algorithm to compute the dct and its inverse a novel algorithm to convert the discrete cosine transform dct to skew circular convolutions is presented the motivation for developing such an algorithm is the fact that vlsi implementation of distributed arithmetic is very efficient for computing convolutions it is also shown that the inverse dct idct can be computed using the same building blocks which are used for computing the dct a dct idct processor can be designed to compute either the dct or the idct depending on a 1 b control signal

on the computation of the length 2 sup m discrete cosine and sine transforms via the permuted difference coefficient an intrinsic property wherein the set of numbers formed using the magnitudes of a basis vector s elements is the same for all basis vectors in a length 2 sup m type iii discrete sine transform dst and discrete cosine transform dct is proved we also show that the set of numbers formed using the magnitudes of any basis vector s elements in a length 2 sup m type iii dst is the same as that in a length 2 sup m type iii dct the same characteristics exist for the length 2 sup m type iv dst and dct a new combinational vlsi architecture that is only composed of adders for implementing the length 2 sup m type iii dst or dct dst iii dct iii using the intrinsic property and the permuted difference coefficient pdc algorithm is developed the advantages of this new architecture are high structural regularity very high speed and suitability for vlsi realization the other advanced sequential structure which is composed of registers multiplexers and an accumulator is also proposed to give a much lower complexity than the combinational structure this new sequential structure is very suitable for chip level microprocessor based or vlsi realization the quantization error that exhibits the effect of the internal finite word length representations of the input and the coefficient is also analyzed it is shown that if the length of data sequence is quadrupled then to maintain the same signal to noise ratio one additional bit must be added to represent both the input and the coefficient it is also shown that the roundoff error of the coefficients is less sensitive than that of the inputs

fast algorithms for the dft and other sinusoidal transforms a new matrix factorization is proposed for dct iv which is the basis of fast algorithms for many sinusoidal transforms the new fast algorithm for dft based on the new factorization requires the same number of multiplications and far fewer additions than the preuss algorithm

a generalized output pruning algorithm for matrix vector multiplication and its application to compute pruning discrete cosine transform in this correspondence a generalized output pruning algorithm for matrix vector multiplication is proposed it is shown that for a given decomposition of the matrix of the transform kernel and the pruning pattern the unnecessary operations for computing an output pruning discrete cosine transform dct can be eliminated thoroughly by using the proposed algorithm

multiple solutions for a class of semilinear elliptic equations with general potentials abstract in this paper we study the existence of infinitely many nontrivial solutions for a class of semilinear elliptic equations u a x u g x u in a bounded smooth domain of r n n 3 with the dirichlet boundary value where the primitive of the nonlinearity g is of superquadratic growth near infinity in u and the potential a is allowed to be sign changing recent results in the literature are generalized and significantly improved

multiplicity of solutions for a class of elliptic boundary value problems abstract in this paper we study the existence of infinitely many solutions for a class of elliptic boundary value problems the existence results include both superlinear case and an asymptotically linear case our arguments are based on a recently given new fountain theorem due to zou w zou variant fountain theorems and their applications manuscripta math 104 2001 343 358

existence and multiplicity of solutions for a class of elliptic boundary value problems abstract in this paper we investigate the existence and multiplicity of solutions for the following elliptic boundary value problems u a x u g x u in u 0 on where g x u k u x u w u x u by using the symmetric mountain pass theorem we obtain two results about infinitely many solutions when g x u is odd in u k satisfies the pinching condition and w has a super quadratic growth moreover when the condition g x u is odd is not assumed by using the mountain pass theorem we also obtain two existence results of one nontrivial weak solution one of these results generalizes a recent result in mao zhu and luan 2012 10

robin problems with a general potential and a superlinear reaction abstract we consider semilinear robin problems driven by the negative laplacian plus an indefinite potential and with a superlinear reaction term which need not satisfy the ambrosetti rabinowitz condition we prove existence and multiplicity theorems producing also an infinity of smooth solutions using variational tools truncation and perturbation techniques and morse theory critical groups

fast algorithms for multidimensional dct to dct computation between a block and its associated subblocks in this paper we first propose an efficient algorithm for computing one dimensional 1 d discrete cosine transform dct for a signal block given its two adjacent subblocks in the dct domain and then introduce several algorithms for the fast computation of multidimensional m d dct with size n sub 1 spl times n sub 2 spl times spl times n sub m given 2 sup m subblocks of dct coefficients with size n sub 1 2 spl times n sub 2 2 spl times spl times n sub m 2 where n sub i i 1 2 m are powers of 2 obviously the row column method which employs the most efficient algorithms along each dimension reduces the computational complexity considerably compared with the traditional method which employs only the one dimensional 1 d fast dct and inverse dct idct algorithms however when m spl ges 2 the traditional method which employs the most efficient multidimensional dct idct algorithms has lower computational complexity than the row column method besides we propose a direct method by dividing the data into 2 sup m parts for independent fast computation in which only two steps of r dimensional r 1 2 m idct and additional multiplications and additions are required if all the dimensional sizes are the same the number of multiplications required for the direct method is only 2 sup m 1 m2 sup m 1 times of that required for the row column method and if n spl ges 2 sup 2m 1 the computational efficiency of the direct method is surely superior to that of the traditional method which employs the most efficient multidimensional dct idct algorithms

systolic architectures for the computation of the discrete hartley and the discrete cosine transforms based on prime factor decomposition two dimensional systolic array implementations for computing the discrete hartley transform dht and the discrete cosine transform dct when the transform size n is decomposable into mutually prime factors are proposed the existing two dimensional formulations for dht and dct are modified and the corresponding algorithms are mapped into two dimensional systolic arrays the resulting architecture is fully pipelined with no control units the hardware design is based on bit serial left to right msb most significant bit to lsb least significant bit binary arithmetic

on algebras related to the discrete cosine transform abstract an algebraic theory for the discrete cosine transform dct is developed which is analogous to the well known theory of the discrete fourier transform dft whereas the latter diagonalizes a convolution algebra which is a polynomial algebra modulo a product of various cyclotomic polynomials the former diagonalizes a polynomial algebra modulo a product of various polynomials related to the chebyshev types when the dimension of the algebra is a power of 2 the dct diagonalizes a polynomial algebra modulo a product of chebyshev polynomials of the first type in both dft and dct cases the chinese remainder theorem plays a key role in the design of fast algorithms

variational elliptic problems which are nonquadratic at infinity p co if n 1 2 then it is well known that the weak solutions u e h 2 of p are the critical points of the c functional j u ulvul dx jcf x u dx where f x s jif x t dt in this variational setting the literature usually distinguishes between two situations for prob lem p the subquadratic situation where the potential f satisfies lim suplsl f x s s2 i c m and the superquadratic situation where f satisfies limlsl m f x s s2 00 the main goal of this paper is to present a unified approach to both situations by means of a condition of nonquadraticity at infinity on f more precisely our hypotheses are as follows

infinitely many solutions for a class of semilinear elliptic equations abstract in this paper we find some new conditions to ensure the existence of infinitely many nontrivial solutions for the dirichlet boundary value problems of the form u a x u g x u in a bounded smooth domain conditions s 1 s 3 in the present paper are somewhat weaker than the famous ambrosetti rabinowitz type superquadratic condition here we assume that the primitive of the nonlinearity g is either asymptotically quadratic or superquadratic as u

new fast recursive algorithms for the computation of discrete cosine and sine transforms fast recursive algorithms for the computation of the discrete cosine and sine transforms are developed an n point discrete cosine transform dct or discrete sine transform dst can be computed from two n 2 point dcts or dsts compared to the existing algorithms the algorithms have less multiplications by two and add operations are better positioned giving rise to faster computation and easier vlsi implementation

automatic generation of fast discrete signal transforms this paper presents an algorithm that derives fast versions for a broad class of discrete signal transforms symbolically the class includes but is not limited to the discrete fourier and the discrete trigonometric transforms this is achieved by finding fast sparse matrix factorizations for the matrix representations of these transforms unlike previous methods the algorithm is entirely automatic and uses the defining matrix as its sole input the sparse matrix factorization algorithm consists of two steps first the symmetry of the matrix is computed in the form of a pair of group representations second the representations are stepwise decomposed giving rise to a sparse factorization of the original transform matrix we have successfully demonstrated the method by computing automatically efficient transforms in several important cases for the dft we obtain the cooley tukey 1965 fft for a class of transforms including the dct type ii the number of arithmetic operations for our fast transforms is the same as for the best known algorithms our approach provides new insights and interpretations for the structure of these signal transforms and the question of why fast algorithms exist the sparse matrix factorization algorithm is implemented within the software package arep

analysis of pruning in fast cosine transform the discrete cosine transform dct is now the established technique for image and video coding in most image coding set schemes a large number of high frequency transform coefficients can be neglected and need not be calculated therefore significant savings in computation can be obtained by applying pruning in the fast algorithms we analyze pruning in chan and ho s 1990 fast dct algorithm and evaluate the saving of operations

non uniformly asymptotically linear p laplacian problems abstract we relax conditions on uniform convergences of f x s s p 2 s which are essential in many results on multiplicity for asymptotically linear p laplacian problems

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

t duality and homological mirror symmetry for toric varieties abstract let x be a complete toric variety the coherent constructible correspondence of fang et al 2011 14 equates p erf t x with a subcategory sh c c m r of constructible sheaves on a vector space m r the microlocalization equivalence of nadler and zaslow 2009 27 and nadler 2009 25 relates these sheaves to a subcategory fuk t m r of the fukaya category of the cotangent t m r when x is nonsingular taking the derived category yields an equivariant version of homological mirror symmetry dcoh t x dfuk t m r which is an equivalence of triangulated tensor categories the nonequivariant coherent constructible correspondence of treumann preprint 33 embeds p erf x into a subcategory s h c t r of constructible sheaves on a compact torus t r when x is nonsingular the composition of and microlocalization yields a version of homological mirror symmetry dcoh x dfuk t t r which is a full embedding of triangulated tensor categories when x is nonsingular and projective the composition is compatible with t duality in the following sense an equivariant ample line bundle l has a hermitian metric invariant under the real torus whose connection defines a family of flat line bundles over the real torus orbits this data produces a t dual lagrangian brane l on the universal cover t m r of the dual real torus fibration we prove l l in fuk t m r thus equivariant homological mirror symmetry is determined by t duality

an efficient prime factor algorithm for the discrete cosine transform and its hardware implementations the prime factor decomposition is a fast computational technique for many important digital signal processing operations such as the convolution the discrete fourier transform the discrete hartley transform and the discrete cosine transform dct the authors present a new prime factor algorithm for the dct they also design a prime factor algorithm for the discrete sine transform based on the prime factor dct algorithm hardware implementations for the prime factor dct are also studied they are especially interested in the hardware designs which are suitable for the vlsi implementations they show three hardware designs for the prime factor dct including a vlsi circuit fabricated directly according to the signal flow graph a linear systolic array and a mesh connected systolic array these three designs show the trade off between cost and performance the methodology which deals with general n sub 1 spl times n sub 2 point dcts where n sub 1 and n sub 2 are mutually prime is illustrated by converting a 15 point dct problem into a 3 spl times 5 point 2d dct problem

one and two dimensional constant geometry fast cosine transform algorithms and architectures this paper presents general radix one and two dimensional 1 d and 2 d constant geometry fast cosine transform algorithms and architectures suitable for vlsi owing to their regular structures a constant geometry algorithm is obtained by shuffling the rows and columns of each decomposed dct matrix that corresponds to a butterfly stage the 1 d algorithm is derived and then it is extended to the 2 d case based on the derived algorithms the architectures with a flexible degree of parallelism are discussed

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

fast multiplierless approximations of the dct with the lifting scheme we present the design implementation and application of several families of fast multiplierless approximations of the discrete cosine transform dct with the lifting scheme called the bindct these bindct families are derived from chen s 1977 and loeffler s 1989 plane rotation based factorizations of the dct matrix respectively and the design approach can also be applied to a dct of arbitrary size two design approaches are presented in the first method an optimization program is defined and the multiplierless transform is obtained by approximating its solution with dyadic values in the second method a general lifting based scaled dct structure is obtained and the analytical values of all lifting parameters are derived enabling dyadic approximations with different accuracies therefore the bindct can be tuned to cover the gap between the walsh hadamard transform and the dct the corresponding two dimensional 2 d bindct allows a 16 bit implementation enables lossless compression and maintains satisfactory compatibility with the floating point dct the performance of the bindct in jpeg h 263 and lossless compression is also demonstrated

prime factor algorithm for multidimensional discrete cosine transform a prime factor fast algorithm is proposed for the computation of the multidimensional forward and inverse discrete cosine transform dct by using an example of a two dimensional 2 d dct it shows that an r dimensional dct can be obtained from a 2r dimensional dct with a post processing stage efficient method for input output mapping is reported to substantially reduce the computational overhead associated with the prime factor algorithm

fast unified computation of the multi dimensional discrete sinusoidal transforms based on a one to one index mapping approach conversions of two types of m dimensional discrete w transforms into the multiple sums involving a number of one dimensional discrete w transforms 1 d dwts are first presented a unified fast m 1 dimensional polynomial transform m 1 d pt and 1 d dwts algorithm for computing these two types of m d dwts are then proposed by revealing the relationships among the m dimensional discrete w transforms m d dwts the m dimensional generalized discrete fourier transforms m d gdfts the m dimensional discrete cosine transforms m d dcts and the m dimensional discrete sine transforms m d dsts the generalized fast algorithms for 14 types of the m d discrete sinusoidal transforms including four types of m d dwts and the same number of types of m d gdfts and three types of m d dcts and the same number of types of m d dsts are presented the number of multiplications for all 14 types of the m d discrete sinusoidal transforms needed by the proposed algorithm is only 1 m times that of the widely used corresponding row column methods the number of additions required by the proposed algorithm is also reduced considerably finally with the help of the software platform visual c 6 0 for windows 98 the computation time comparisons between the proposed and the reported algorithms are given the numerical experiments show that the proposed algorithms are not only highly efficient but are also very simple in computational structure

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

block based frequency scalable technique for efficient hierarchical coding in this paper we propose a block based frequency scalable technique for efficient hierarchical coding the proposed technique divides an image into its multiple resolution versions based on the spectral properties of discrete cosine transform dct kernels we present that spectral decomposition downsampling and dct operations are performed effectively over input dct coefficients of one dimensional 1 d and two dimensional 2 d signals by using the proposed transform matrices the proposed image coder is observed to reduce the computational complexity and the memory buffer size with a higher peak signal to noise ratio psnr when compared with the traditional hierarchical image coder in addition the proposed architecture can preserve compatibility easily with the previous dct based image coder