using sampling and simplex derivatives in pattern search methods in this paper we introduce ways of making a pattern search more efficient by reusing previous evaluations of the objective function based on the computation of simplex derivatives e g simplex gradients at each iteration one can attempt to compute an accurate simplex gradient by identifying a sampling set of previously evaluated points with good geometrical properties this can be done using only past successful iterates or by considering all past function evaluations the simplex gradient can then be used to reorder the evaluations of the objective function associated with the directions used in the poll step or to update the mesh size parameter according to a sufficient decrease criterion neither of which requires new function evaluations we present these procedures in detail and apply them to a set of problems from the cuter collection numerical results show that these procedures can enhance significantly the practical performance of pattern search methods

new look on q2 r point fast fourier transforms in this paper the method of splitting orparallelization of calculation of the n point discrete fourier transform dft by the dfts of smaller orders is described for that the concept of partitions revealing the one dimensional 1 d dft of order q2 r where r 1and q 1 is a positive odd number is described two different partitions are considered and the corresponding effective algorithms of calculation of the q2 r point 1 d dft are described for the q 3 case the calculation of the transform is based on the paired representations when the signal is represented as a set of 1 d signals which define the 1 d dft in disjoint subsets of frequency points which cover the set of all frequencies the splittings of the q2 r point 1 d dft are performed by the 1 d discrete 2and qpaired transforms which allow for calculating with a minimum number of operations the examples of the paired transforms and computational complexity of the proposed algorithms for the n 6 and 12 cases 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

recent progress in unconstrained nonlinear optimization without derivatives we present an introduction to a new class of derivative free methods for unconstrained optimization we start by discussing the motivation for such methods and why they are in high demand by practitioners we then review the past developments in this field before introducing the features that characterize the newer algorithms in the context of a trust region framework we focus on techniques that ensure a suitable geometric quality of the considered models we then outline the class of algorithms based on these techniques as well as their respective merits we finally conclude the paper with a discussion of open questions and perspectives

euler characters and submanifolds of constant positive curvature this article develops methods for studying the topology of submanifolds of constant positive curvature in euclidean space it proves that if m is an n dimensional compact connected riemannian submanifold of constant positive curvature in e 2n 1 then m n must be simply connected it also gives a conformal version of this theorem

reducing the number of function evaluations in mesh adaptive direct search algorithms the mesh adaptive direct search mads class of algorithms is designed for nonsmooth optimization where the objective function and constraints are typically computed by launching a time consuming computer simulation each iteration of a mads algorithm attempts to improve the current best known solution by launching the simulation at a finite number of trial points common implementations of mads generate 2n trial points at each iteration where n is the number of variables in the optimization problem the objective of the present work is to dynamically reduce that number we present an algorithmic framework that reduces the number of simulations to exactly n 1 without impacting the theoretical guarantees from the convergence analysis numerical experiments are conducted for several different contexts the results suggest that these strategies allow the new algorithms to reach a better solution with fewer function evaluations

orthomads a deterministic mads instance with orthogonal directions the purpose of this paper is to introduce a new way of choosing directions for the mesh adaptive direct search mads class of algorithms the advantages of this new orthomads instantiation of mads are that the polling directions are chosen deterministically ensuring that the results of a given run are repeatable and that they are orthogonal to each other which yields convex cones of missed directions at each iteration that are minimal in a reasonable measure convergence results for orthomads follow directly from those already published for mads and they hold deterministically rather than with probability one as is the case for ltmads the first mads instance the initial numerical results are quite good for both smooth and nonsmooth and constrained and unconstrained problems considered here

on the convergence of the multidirectional search algorithm this paper presents the convergence analysis for the multidirectional search algorithm a direct search method for unconstrained minimization the analysis follows the classic lines of proofs of convergence for gradient related methods the novelty of the argument lies in the fact that explicit calculation of the gradient is unnecessary although it is assumed that the function is continuously differentiable over some subset of the domain the proof can be extended to treat most nonsmooth cases of interest the argument breaks down only at points where the derivative exists but is not continuous finally it is shown how a general convergence theory can be developed for an entire class of direct search methods which includes such methods as the factorial design algorithm and the pattern search algorithm that share a key feature of the multidirectional search algorithm

hex splines a novel spline family for hexagonal lattices this paper proposes a new family of bivariate nonseparable splines called hex splines especially designed for hexagonal lattices the starting point of the construction is the indicator function of the voronoi cell which is used to define in a natural way the first order hex spline higher order hex splines are obtained by successive convolutions a mathematical analysis of this new bivariate spline family is presented in particular we derive a closed form for a hex spline of arbitrary order we also discuss important properties such as their fourier transform and the fact they form a riesz basis we also highlight the approximation order for conventional rectangular lattices hex splines revert to classical separable tensor product b splines finally some prototypical applications and experimental results demonstrate the usefulness of hex splines for handling hexagonally sampled data

the fractional fourier transform and time frequency representations the functional fourier transform frft which is a generalization of the classical fourier transform was introduced a number of years ago in the mathematics literature but appears to have remained largely unknown to the signal processing community to which it may however be potentially useful the frft depends on a parameter spl alpha and can be interpreted as a rotation by an angle spl alpha in the time frequency plane an frft with spl alpha spl pi 2 corresponds to the classical fourier transform and an frft with spl alpha 0 corresponds to the identity operator on the other hand the angles of successively performed frfts simply add up as do the angles of successive rotations the frft of a signal can also be interpreted as a decomposition of the signal in terms of chirps the authors briefly introduce the frft and a number of its properties and then present some new results the interpretation as a rotation in the time frequency plane and the frft s relationships with time frequency representations such as the wigner distribution the ambiguity function the short time fourier transform and the spectrogram these relationships have a very simple and natural form and support the frft s interpretation as a rotation operator examples of frfts of some simple signals are given an example of the application of the frft is also given

efficient computation of the dft with only a subset of input or output points ways of efficiently computing the discrete fourier transform dft when the number of input and output data points differ are discussed the two problems of determining whether the length of the input sequence or the length of the output sequence is reduced can be found to be duals of each other and the same methods can to a large extent be used to solve both the algorithms utilize the redundancy in the input or output to reduce the number of operations below those of the fast fourier transform fft algorithms the usual pruning method is discussed and an efficient algorithm called transform decomposition is introduced it is based on a mixture of a standard fft algorithm and the horner polynomial evaluation scheme equivalent to the one in goertzel s algorithms it requires fewer operations and is more flexible than pruning the algorithm works for power of two and prime factor algorithms as well as for real input data

comments on split manageable efficient algorithm for fourier and hadamard transforms the paper split manageable efficient algorithm for fourier and hadamard transforms in ieee transactions on signal processing vol 48 no 1 pp 172 183 january 2000 presents a novel paired transform fast fourier transform fft algorithm that is claimed to require fewer total real additions and multiplications than previous algorithms we show this claim to be false although the algorithm correctly computes the discrete fourier transform an accurate count shows that it requires more real arithmetic operations than pre existing ffts

on the convergence of pattern search algorithms we introduce an abstract definition of pattern search methods for solving nonlinear unconstrained optimization problems our definition unifies an important collection of optimization methods that neither compute nor explicitly approximate derivatives we exploit our characterization of pattern search methods to establish a global convergence theory that does not enforce a notion of sufficient decrease our analysis is possible because the iterates of a pattern search method lie on a scaled translated integer lattice this allows us to relax the classical requirements on the acceptance of the step at the expense of stronger conditions on the form of the step and still guarantee global convergence

parallel space decomposition of the mesh adaptive direct search algorithm this paper describes a parallel space decomposition psd technique for the mesh adaptive direct search mads algorithm mads extends a generalized pattern search for constrained nonsmooth optimization problems the objective of the present work is to obtain good solutions to larger problems than the ones typically solved by mads the new method psd mads is an asynchronous parallel algorithm in which the processes solve problems over subsets of variables the convergence analysis based on the clarke calculus is essentially the same as for the mads algorithm a practical implementation is described and some numerical results on problems with up to 500 variables illustrate the advantages and limitations of psd mads

efficient algorithms for computing the 2 d hexagonal fourier transforms in this paper representations of the two dimensional 2 d signals are presented that reduce the computation of 2 d discrete hexagonal fourier transforms 2 d dhfts the representations are based on the concept of the covering that reveals the mathematical structure of the transforms specifically a set of unitary paired transforms is derived that splits the 2 d dhft into a number of smaller one dimensional 1 d dfts examples for the 8 spl times 4 and 16 spl times 8 hexagonal lattices are described in detail the number of multiplications required for computing the 8 spl times 4 and 16 spl times 8 point dhfts are equal 20 and 136 respectively in the general n spl ges 8 case the number of multiplications required to compute the 2n spl times n point dhft by the paired transforms equals n sup 2 log n 1 n

geometric transformations on the hexagonal grid the hexagonal grid has long been known to be superior to the more traditional rectangular grid system in many aspects in image processing and machine vision related fields however systematic developments of the mathematical backgrounds for the hexagonal grid are conspicuously lacking the purpose of this paper is to study geometric transformations on the hexagonal grid formulations of the transformation matrices are carried out in a symmetrical hexagonal coordinate frame a trio of new trigonometric functions are defined in this paper to facilitate the rotation transformations a fast algorithm for rounding an arbitrary point to the nearest hexagonal grid point is also presented

on the number of multiplications necessary to compute a length 2 n dft the number of multiplications necessary and sufficient to compute a length 2ndft is determined the method of derivation is shown to apply to the multiplicative complexity results of winograd for a length pndft for p an odd prime number the multiplicative complexity of the one dimensional dft is summarized for many possible lengths

design and evaluation of more accurate gradient operators on hexagonal lattices digital two dimensional images are usually sampled on square lattices while the receptors of the human eye are following a hexagonal structure it is the main motivation for adopting hexagonal lattices the fundamental operation in many image processing algorithms is to extract the gradient information as such various gradient operators have been proposed for square lattices and have been thoroughly optimized accurate gradient operators for hexagonal lattices have however not been researched well enough while the distance between neighbor pixels is constant we therefore derive consistent gradient operators on hexagonal lattices and compare them with the existing optimized filters on square lattices the results show that the derived filters on hexagonal lattices achieve a better signal to noise ratio than those on square lattices results on artificial images also show that the derived filters on hexagonal lattices outperform the square ones with respect to accuracy of gradient intensity and orientation detection

duhamel hollmann like discrete fourier transform algorithm with the smallest multiplicative complexity over a finite field the new method for the discrete fourier transform computation over a finite field is introduced this method is a nontrivial generalization of the duhamel hollmann algorithm with replacement of the toeplitz convolution calculation by the normalized cyclic convolution calculation both algorithms have the smallest multiplicative complexity

split manageable efficient algorithm for fourier and hadamard transforms in this paper a general efficient manageable split algorithm to compute one dimensional 1 d unitary transforms by using the special partitioning in the frequency domain is introduced the partitions determine fast transformations that split the n point unitary transform into a set of n sub i point transforms i 1 n n sub 1 n sub n n here we introduce a class of splitting transformations the so called paired transforms based on these transforms the decompositions of the fourier transforms of arbitrary orders are given and the corresponding algorithms are considered comparative estimates revealing the efficiency of the proposed algorithms with respect to the known ones are given in particular a proposed method of calculating the 2 sup r point fourier transform requires 2 sup r 1 r 3 2 multiplications and 2 sup r 1 r 9 r sup 2 3r 6 additions in terms of the paired transforms the splitting of the 2 sup r point hadamard transform is described as a result the proposed algorithm for computing this transform uses on the average no more than six operations of additions per sample

transform coding with integer to integer transforms a new interpretation of transform coding is developed that downplays quantization and emphasizes entropy coding allowing a comparison of entropy coding methods with different memory requirements with conventional transform coding based on computing karhunen loeve transform coefficients and then quantizing them vector entropy coding can be replaced by scalar entropy coding without an increase in rate thus the transform coding advantage is a reduction in memory requirements for entropy coding this paper develops a transform coding technique where the source samples are first scalar quantized and then transformed with an integer to integer approximation to a nonorthogonal linear transform among the possible advantages is to reduce the memory requirement further than conventional transform coding by using a single common scalar entropy codebook for all components the analysis shows that for high rate coding of a gaussian source this reduction in memory requirements comes without any degradation of rate distortion performance

geometry of interpolation sets in derivative free optimization we consider derivative free methods based on sampling approaches for nonlinear optimization problems where derivatives of the objective function are not available and cannot be directly approximated we show how the bounds on the error between an interpolating polynomial and the true function can be used in the convergence theory of derivative free sampling methods these bounds involve a constant that reflects the quality of the interpolation set the main task of such a derivative free algorithm is to maintain an interpolation sampling set so that this constant remains small and at least uniformly bounded this constant is often described through the basis of lagrange polynomials associated with the interpolation set we provide an alternative more intuitive definition for this concept and show how this constant is related to the condition number of a certain matrix this relation enables us to provide a range of algorithms whilst maintaining the interpolation set so that this condition number or the geometry constant remain uniformly bounded we also derive bounds on the error between the model and the function and between their derivatives directly in terms of this condition number and of this geometry constant

scaled radix 2 8 algorithm for efficient computation of length n 2 m dfts this paper presents a scaled radix 2 8 fast fourier transform fft sr28fft algorithm for computing length n 2m discrete fourier transforms dfts scaled by complex number rotating factors the idea of the sr28fft algorithm is from the modified split radix fft msrfft algorithm and its purpose is to furnish other algorithms with high efficiency but without shortcomings of the msrfft algorithm a novel radix 2 4 fft nr24fft algorithm and a novel radix 2 8 fft nr28fft algorithm are proposed these two algorithms use sr28fft to calculate their sub dfts of odd indexed terms several aspects of the two algorithms such as computational complexity computation accuracy and coefficient evaluations or accesses to the lookup table all are improved the bit reverse method can be used for their order permutation and no extra memory is required to store their extra coefficients by the two novel algorithms which contribute significantly to the performance of the fft algorithms the sr28fft algorithm can also be applied to other algorithms whose decomposition contains sub dfts of powers of two the appendix presents an algorithm named sr28fft 2 for further reducing the number of arithmetic operations and nr24fft and nr28fft algorithms based on sr28fft 2 requires fewer real operations than that required by the msrfft algorithm

a fast algorithm with less operations for length n q times 2 m dfts discrete fourier transform dft is widely used in almost all fields of science and engineering fast fourier transform fft is an efficient tool for computing dft in this paper we present a fast fourier transform fft algorithm for computing length q times 2 m dfts the algorithm transforms all q points sub dfts into three parts in the second part the operations of subtransformation contain only multiplications by real constant factors by transformation length 2 m scaled dfts sdft are obtained an extension of scaled radix 2 8 fft sr28fft is presented for computing these sdfts in which the real constant factors of sdfts are attached to the coefficients of sub dfts to simplify multiplication operations the proposed algorithm achieves reduction of arithmetic complexity over the related algorithms it can achieve a further reduction of arithmetic complexity for computing a length n q times 2 m idft by 2n 4m real multiplications in addition the proposed algorithm is applied to real data fft and is extended to 6 m dfts

inexact restoration method for nonlinear optimization without derivatives a derivative free optimization method is proposed for solving a general nonlinear programming problem it is assumed that the derivatives of the objective function and the constraints are not available the new method is based on the inexact restoration scheme where each iteration is decomposed in two phases in the first one the violation of the feasibility is reduced in the second one the objective function is minimized onto a linearization of the nonlinear constraints at both phases polynomial interpolation models are used in order to approximate the objective function and the constraints at the first phase a derivative free solver for box constrained optimization can be used for the second phase we propose a new method ad hoc based on trust region strategy that uses the projection of the simplex gradient on the tangent space under suitable assumptions the algorithm is well defined and convergence results are proved a numerical implementation is described and numerical experiments are presented to validate the theoretical results

tensor transform based quaternion fourier transform algorithm we present a concept of the tensor representation of color images in quaternion algebra we present concepts of direction components composing the color image we propose a new method of the 2 d quaternion discrete fourier transform qdft we calculate the 2 d qdft with minimum number of operations of multiplication we examine the proposed method in rgb color image model the traditional fourier transform is a fundamental tool in signal and image processing which in color imaging transforms images either in monochrome or as separate color channels thereby ignoring interactions between the color channels the quaternion discrete fourier transform qdft a generalization of the fourier transform represents the three color channels of an image in the rgb color space as a vector field in the quaternion arithmetic allowing for simultaneous analysis of all color data it is expected that this new color imaging model which captures the interactions between the different color channels will improve upon existing image processing and recognition systems efficient qdft algorithms are derived by combining classical dft transforms enabling standard fast fourier transform fft software to provide a fast qdft numerical implementation we developed an open source quaternion fourier transform tool that calculates the 2 d qdft by the column row method which is performed by calculating multiple 1 d qdfts where each 1 d transform is calculated by two complex 1 d dfts the matrices involved in these calculations necessitate the inclusion of a discussion on symplectic decomposition our main contribution is the presentation a new algorithm for the tensor representation of the qdft that is structurally simpler and uses less multiplications than its predecessors for instance for the images of size n i n where n 2 r r 2 the tensor 2 d qdft saves about 2 n 2 r real multiplications

comments on method of flow graph simplification for the 16 point discrete fourier transform the paper method of flow graph simplification for the 16 point discrete fourier transform by grigoryan and bhamidipati presents a paired transform fast fourier transform fft algorithm that is claimed to perform the size 8 and size 16 complex data discrete fourier transform dft with 44 and 140 arithmetic operations respectively if true this count would be less than the 56 and 168 operations achieved in the best pre existing split radix methods grigoryan and bhamidipati s count of real additions is erroneous however and this comment shows that their algorithm actually has arithmetic complexity identical to that of standard split radix algorithms

optimization by direct search new perspectives on some classical and modern methods direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis nonetheless users remained loyal to these methods most of which were easy to program some of which were reliable in the past fifteen years these methods have seen a revival due in part to the appearance of mathematical analysis as well as to interest in parallel and distributed com puting this review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results the underlying principles allow general ization to handle bound constraints and linear constraints we also discuss extensions to problems with nonlinear constraints

direct search methods on parallel machines direct search methods are methods designed to solve unconstrained minimization problems of the form mathop min limits x in mathbb r n f x where f n these methods are distinguished by the fact that they neither use nor require explicit derivative information the search for a local minimizer is driven solely by function information popular methods in this class include the factorial design algorithm of box 1 the pattern search algorithm of hooke and jeeves 4 and the simplex method of neider and mead 5

efficient discrete spatial techniques for blur support identification in blind image deconvolution this paper proposes two discrete spatial techniques for identifying blur support in blind image deconvolution blur identification is a challenging problem in blind image deconvolution in particular if the blur support can be estimated reliably in the beginning of restoration the computational cost of many blind deconvolution schemes can be reduced significantly and their convergence performance improved this paper proposes two methods called maximum average square difference masd and maximum average absolute difference maad they are derived from the autoregressive ar model of the underlying images the efficiency and validity of the techniques are also analyzed in this paper a main advantage of the proposed techniques is their algorithmic and implementation simplicity experimental results show that they are effective in identifying the blur support reliably thus providing a sound foundation for further blind image deconvolution

novel reversible integer fourier transform with control bits this paper presents a novel concept of the reversible integer discrete fourier transform ridft of order 2r r 2 when the transform is split by the paired representation into a minimum set of short transforms i e transforms of orders 2k k r by means of the paired transform the signal is represented as a set of short signals which carry the spectral information of the signal at specific and disjoint sets of frequencies the paired transform based fast fourier transform fft involves a few operations of multiplication that can be approximated by integer transforms examples of 1 point transforms with one control bit are described control bits allow us to invert such approximations two control bits are required to perform the 8 point ridft and 12 or even 8 bits for the 16 point ridft of real inputs the proposed forward and inverse ridfts are fast and the computational complexity of these transforms is comparative with the complexity of the fft the 8 point direct and inverse ridfts are described in detail

finding optimal algorithmic parameters using derivative free optimization the objectives of this paper are twofold we devise a general framework for identifying locally optimal algorithmic parameters algorithmic parameters are treated as decision variables in a problem for which no derivative knowledge or existence is assumed a derivative free method for optimization seeks to minimize some measure of performance of the algorithm being fine tuned this measure is treated as a black box and may be chosen by the user examples are given in the text the second objective is to illustrate this framework by specializing it to the identification of locally optimal trust region parameters in unconstrained optimization the derivative free method chosen to guide the process is the mesh adaptive direct search a generalization of pattern search methods we illustrate the flexibility of the latter and in particular make provision for surrogate objectives locally optimal parameters with respect to overall computational time on a set of test problems are identified each function call may take several hours and may not always return a predictable result a tailored surrogate function is used to guide the search towards a local solution the parameters thus identified differ from traditionally used values and allow one to solve a problem that remained otherwise unsolved in a reasonable time using traditional values

on the determination of functions from their integral values along certain manifolds when one integrates a function of two variables x y a point function f p in the plane subject to suitable regularity conditions along an arbitrary straight line g then one obtains in the integral values f g a line function in part a of the present paper the problem which is solved is the inversion of this linear functional transformation that is the following questions are answered can every line function satisfying suitable regularity conditions be regarded as constructed in this way if so is f uniquely known from f and how can f be calculated in part b a solution of the dual problem of calculating a line function f g from its point mean values f p is solved in a certain sense finally in part c certain generalizations are discussed prompted by consideration of non euclidean manifolds as well as higher dimensional spaces the treatment of these problems themselves of interest gains enhanced importance through the numerous relationships that exist between this topic and the theory of logarithmic and newtonian potentials these are mentioned at appropriate places in the text

a unified treatment of cooley tukey algorithms for the evaluation of the multidimensional dft in this paper the cooley tukey fast fourier transform fft algorithm is generalized to the multidimensional case in a natural way which allows for the evaluation of discrete fourier transforms of rectangularly or hexagonally sampled signals or of signals which are sampled on an arbitrary periodic grid in either the spatial or fourier domain this general algorithm incorporates both the traditional rectangular row column and vector radix algorithms as special cases this fft algorithm is shown to result from the factorization of an integer matrix for each factorization of that matrix a different algorithm can be developed this paper presents the general algorithm discusses its computational efficiency and relates it to existing multi dimensional fft algorithms

algebraic signal processing theory 2 d spatial hexagonal lattice we develop the framework for signal processing on a spatial or undirected 2 d hexagonal lattice for both an infinite and a finite array of signal samples this framework includes the proper notions of z transform boundary conditions filtering or convolution spectrum frequency response and fourier transform in the finite case the fourier transform is called discrete triangle transform like the hexagonal lattice this transform is nonseparable the derivation of the framework makes it a natural extension of the algebraic signal processing theory that we recently introduced namely we construct the proper signal models given by polynomial algebras bottom up from a suitable definition of hexagonal space shifts using a procedure provided by the algebraic theory these signal models in turn then provide all the basic signal processing concepts the framework developed in this paper is related to mersereau s early work on hexagonal lattices in the same way as the discrete cosine and sine transforms are related to the discrete fourier transform a fact that will be made rigorous in this paper

some results in fixed point error analysis of the bruun ftt algorithm results for a fixed point error analysis of the bruun fast fourier transform fft algorithm are presented the noise to signal ratio of the bruun algorithm is evaluated theoretically as well as experimentally and is compared to that of the decimation in time fft the noise to signal ratio of the bruun fft is found to be significantly higher if safe scaling is applied in both cases

a derivative free nonmonotone line search technique for unconstrained optimization a tolerant derivative free nonmonotone line search technique is proposed and analyzed several consecutive increases in the objective function and also nondescent directions are admitted for unconstrained minimization to exemplify the power of this new line search we describe a direct search algorithm in which the directions are chosen randomly the convergence properties of this random method rely exclusively on the line search technique we present numerical experiments to illustrate the advantages of using a derivative free nonmonotone globalization strategy with approximated gradient type methods and also with the inverse sr1 update that could produce nondescent directions in all cases we use a local variation finite differences approximation to the gradient

split radix algorithms for length p sup m dft s the split radix algorithm for the discrete fourier transform dft of length 2 sup m is considered first the reason why the split radix algorithm is better than any single radix algorithm on length 2 sup m dfts is given then the split radix approach is generalized to length p sup m dfts it is shown that whenever a radix p sup 2 outperforms a radix p algorithm then a radix p p sup 2 algorithm will outperform both of them as an example a radix 3 9 algorithm is developed for length 3 sup m dfts

integer dcts and fast algorithms a method is proposed to factor the type ii discrete cosine transform dct ii into lifting steps and additions after approximating the lifting matrices we get a new type ii integer discrete cosine transform intdct ii that is float point multiplication free based on the relationships among the various types of dcts we can generally factor any dcts into lifting steps and additions and then get four types of integer dcts which need no float point multiplications by combining the polynomial transform and the one dimensional 1 d integer cosine transform a two dimensional 2 d integer discrete cosine transform is proposed the proposed transform needs only integer operations and shifts furthermore it is nonseparable and requires a far fewer number of operations than that used by the corresponding row column 2 d integer discrete cosine transform

pattern search algorithms for mixed variable programming many engineering optimization problems involve a special kind of discrete variable that can be represented by a number but this representation has no significance such variables arise when a decision involves some situation like a choice from an unordered list of categories this has two implications the standard approach of solving problems with continuous relaxations of discrete variables is not available and the notion of local optimality must be defined through a user specified set of neighboring points we present a class of direct search algorithms to provide limit points that satisfy some appropriate necessary conditions for local optimality for such problems we give a more expensive version of the algorithm that guarantees additional necessary optimality conditions a small example illustrates the differences between the two versions a real thermal insulation system design problem illustrates the efficacy of the user controls for this class of algorithms

fast algorithms for the multidimensional discrete fourier transform in this paper the prime factor algorithm for the evaluation of a one dimensional discrete fourier transform is generalized to the evaluation of multidimensional discrete fourier transforms defined on arbitrary periodic sampling lattices it is shown that such an algorithm is equivalent in computational complexity to the evaluation of a rectangular discrete fourier transform as a sidelight to the derivation of the algorithm a chinese remainder theorem is derived for integer lattices

kronecker products unitary matrices and signal processing applications discrete unitary transforms are extensively used in many signal processing applications and in the development of fast algorithms kronecker products have proved quite useful in this semitutorial paper we briefly review properties of kronecker products and direct sums of matrices which provide a compact notation in treating patterned matrices a generalized matrix product which inherits some useful algebraic properties from the standard kronecker product and allows a large class of discrete unitary transforms to be generated from a single recursion formula is then introduced the notation is intimately related to sparse matrix factorizations and examples are included illustrating the utility of the new notation in signal processing applications finally some novel characteristics of hadamard transforms and polyadic permutations are derived in the framework of kronecker products

basis vector decomposition based two stage computational algorithms for dft and dht discrete fourier transform dft discrete hartley transform dht algorithms based on the basis vector decomposition of the corresponding transform matrices are derived the computations of dft are divided into two stages an add subtract preprocessing and a block diagonal postprocessing both stages can be computed effectively it can be proved that the computational complexity of the proposed dft algorithm is identical to that of the most popular split radix fft yet requires real number arithmetics only generation and storage of the real multiplicative coefficients are easier than that in conventional ffts connections of the proposed approach with several well known dft algorithms are included furthermore many variations of the proposed algorithm are also pointed out

fast computation of the discrete walsh and hadamard transforms the discrete walsh and hadamard transforms are often used in image processing tasks such as image coding pattern recognition and sequency filtering a new discrete walsh transform dwt algorithm is derived in which a modified form of the dwt relation is decomposed into smaller sized transforms using vectorized quantities a new sequency ordered discrete hadamard transform dhat algorithm is also presented the proposed approach results in more regular algorithms requiring no independent data swapping and fewer array index updating and bit reversal operations an analysis of the computational complexity and the execution time performance are provided the results are compared with those of the existing algorithms

transform based image enhancement algorithms with performance measure this paper presents a new class of the frequency domain based signal image enhancement algorithms including magnitude reduction log magnitude reduction iterative magnitude and a log reduction zonal magnitude technique these algorithms are described and applied for detection and visualization of objects within an image the new technique is based on the so called sequency ordered orthogonal transforms which include the well known fourier hartley cosine and hadamard transforms as well as new enhancement parametric operators a wide range of image characteristics can be obtained from a single transform by varying the parameters of the operators we also introduce a quantifying method to measure signal image enhancement called eme this helps choose the best parameters and transform for each enhancement a number of experimental results are presented to illustrate the performance of the proposed algorithms

2 d and 1 d multipaired transforms frequency time type wavelets a concept of multipaired unitary transforms is introduced these kinds of transforms reveal the mathematical structure of fourier transforms and can be considered intermediate unitary transforms when transferring processed data from the original real space of signals to the complex or frequency space of their images considering paired transforms we analyze simultaneously the splitting of the multidimensional fourier transform as well as the presentation of the processed multidimensional signal in the form of the short one dimensional 1 d signals that determine such splitting the main properties of the orthogonal system of paired functions are described and the matrix decompositions of the fourier and hadamard transforms via the paired transforms are given the multiplicative complexity of the two dimensional 2 d 2 sup r spl times 2 sup r point discrete fourier transform by the paired transforms is 4 sup r 2 r 7 3 8 3 12 r 3 which shows the maximum splitting of the 5 d fourier transform into the number of the short 1 d fourier transforms the 2 d paired transforms are not separable and represent themselves as frequency time type wavelets for which two parameters are united frequency and time the decomposition of the signal is performed in a way that is different from the traditional haar system of functions

the integer transforms analogous to discrete trigonometric transforms the integer transform such as the walsh transform is the discrete transform that all the entries of the transform matrix are integer it is much easier to implement because the real number multiplication operations can be avoided but the performance is usually worse on the other hand the noninteger transform such as the dft and dct has a good performance but real number multiplication is required w derive the integer transforms analogous to some popular noninteger transforms these integer transforms retain most of the performance quality of the original transform but the implementation is much simpler especially for the two dimensional 2 d block transform in image video the saving can be huge using integer operations in 1989 cham had derived the integer cosine transform here we will derive the integer sine hartley and fourier transforms we also introduce the general method to derive the integer transform from some noninteger transform besides the integer transform derived by cham still requires real number multiplication for the inverse transform we modify the integer transform introduced by cham and introduce the complete integer transform it requires no real number multiplication operation no matter what the forward or inverse transform the integer transform we derive would be more efficient than the original transform for example for the 8 point dft and idft there are in total four real numbers and eight fixed point multiplication operations required but for the forward and inverse 8 point complete integer fourier transforms there are totally 20 fixed point multiplication operations required however for the integer transform the implementation is simpler and many of the properties of the original transform are kept

16 point reversible integer discrete fourier transform with 12 control bits this correspondence discusses the reversible integer 16 point discrete fourier transform ridft which uses integer operations with control bits the decomposition of the ridft is based on the paired representation when the fourier transform is split recursively into a set of short transforms of orders 8 4 2 and 1 control bits allow for inverting the integer approximations of multiplications by twiddle factors the proposed 16 point ridft uses 16 operations of real multiplication and 62 additions the integer approximation of the transform with eight control bits with additional two lifting schemes which requires two more multiplications is also considered

integer fast fourier transform a concept of integer fast fourier transform intfft for approximating the discrete fourier transform is introduced unlike the fixed point fast fourier transform fxpfft the new transform has the properties that it is an integer to integer mapping is power adaptable and is reversible the lifting scheme is used to approximate complex multiplications appearing in the fft lattice structures where the dynamic range of the lifting coefficients can be controlled by proper choices of lifting factorizations split radix fft is used to illustrate the approach for the case of 2 sup n point fft in which case an upper bound of the minimal dynamic range of the internal nodes which is required by the reversibility of the transform is presented and confirmed by a simulation the transform can be implemented by using only bit shifts and additions but no multiplication a method for minimizing the number of additions required is presented while preserving the reversibility the intfft is shown experimentally to yield the same accuracy as the fxpfft when their coefficients are quantized to a certain number of bits complexity of the intfft is shown to be much lower than that of the fxpfft in terms of the numbers of additions and shifts finally they are applied to noise reduction applications where the intfft provides significantly improvement over the fxpfft at low power and maintains similar results at high power

method of paired transforms for reconstruction of images from projections discrete model an effective method of discrete image reconstruction from its projections is introduced the method is based on the vector and paired representations of the two dimensional 2d image with respect to the 2d discrete fourier transform such representations yield algorithms for image reconstruction by a minimal number of attenuation measurements in certain projections the proposed algorithms are described in detail for an n spl times n image when n 2 sup r r 1 the inverse formulas for image reconstruction are given the efficiency of the algorithms is expressed in the fact that they require a minimal number of multiplications or can be implemented without such at all the problem of discrete image reconstruction is also considered in three dimensional 3d space namely on the 3d torus where the reconstruction is performed by means of the nonlinear projections that are integral over 3d spirals on the torus

split vector radix fast fourier transform the split radix approach for computing the discrete fourier transform dft is extended for the vector radix fast fourier transform fft to two and higher dimensions it is obtained by further splitting the n 2 n 2 transforms with twiddle factors in the radix 2 2 fft algorithm the generalization of this split vector radix fft algorithm to higher radices and higher dimensions is also presented by introducing a general approach for constructing the fast hartley transform fht from the corresponding fft new vector and split vector radix fht algorithms with the same desirable properties as their fft counterparts are obtained

the construction of low dispersive fdtd on hexagon based on the sampling theorem with arbitrary geometry the spatial discretization can be carried out on hexagonal lattices grids instead of rectangular lattices in the two dimensional case with the concept of the directional derivative in aid a new finite difference time domain fdtd method based on hexagonal lattices h fdtd is introduced the coefficients of the h fdtd method are chosen with aim to obtain the low numerical dispersion final numerical experiment shows that the numerical anisotropy of the h fdtd method is lower than that of the conventional yee fdtd method based on rectangular lattices r fdtd in addition some numerical examples are provided to verify its good performance on the low numerical anisotropy

fast fourier transform using matrix decomposition to reduce both the multiplicative complexity and total number of operations this paper introduces a modeling scheme of the fast fourier transform fft to decompose the discrete fourier transform dft matrix recursively into a set of sparse matrices integrating three orthogonal transforms the hadamard modified haar and hybrid transforms the proposed scheme is able to obtain different fft representations with less computation operations than state of the arts to investigate the applications of the proposed fft scheme a multi stage image encryption algorithm is also introduced experimental results and security analysis are provided to show its encryption performance

automatic generation of prime length fft programs describes a set of programs for circular convolution and prime length fast fourier transforms ffts that are relatively short possess great structure share many computational procedures and cover a large variety of lengths the programs make clear the structure of the algorithms and clearly enumerate independent computational branches that can be performed in parallel moreover each of these independent operations is made up of a sequence of suboperations that can be implemented as vector parallel operations this is in contrast with previously existing programs for prime length ffts they consist of straight line code no code is shared between them and they cannot be easily adapted for vector parallel implementations the authors have also developed a program that automatically generates these programs for prime length ftts this code generating program requires information only about a set of modules for computing cyclotomic convolutions

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

method of flow graph simplification for the 16 point discrete fourier transform an efficient method for the realization of the paired algorithm for calculation of the one dimensional 1 d discrete fourier transform dft by simplifying the signal flow graph of the transform is described the signal flow graph is modified by separating the calculation for real and imaginary parts of all inputs and outputs in the signal flow graph and using properties of the transform the examples for calculation of the eight and 16 point dfts are considered in detail the calculation of the 16 point dft of real data requires 12 real multiplications and 58 additions two multiplications and 20 additions are used for the eight point dft

pattern search methods for user provided points application to molecular geometry problems this paper deals with the application of pattern search methods to the numerical solution of a class of molecular geometry problems with important applications in molecular physics and chemistry the goal is to find a configuration of a cluster or a molecule with minimum total energy the minimization problems in this class of molecular geometry problems have no constraints and the objective function is smooth the difficulties arise from the existence of several local minima and especially from the expensive function evaluation total energy and the possible nonavailability of first order derivatives we introduce a pattern search approach that attempts to exploit the physical nature of the problem by using energy lowering geometrical transformations and to take advantage of parallelism without the use of derivatives numerical results for a particular instance of this new class of pattern search methods are presented showing the promise of our approach the new pattern search methods can be used in any other context where there is a user provided scheme to generate points leading to a potential objective function decrease