an exposition of the ahp in reply to the paper remarks on the analytic hierarchy process it is a fact that people make decisions and have been making decisions for a very long time contrary to what some of us who are interested in decision making may like to believe most people do not take seriously the existence of theories which purport to set their thinking and feeling right they claim to know their own value system and what they want they may wonder how anyone else can know well enough to tell them how best to organize their thinking in order to make better choices yet research has shown that complex decisions are beyond the capacity of the brain to synthesize intuitively and efficiently since decision making is a natural characteristic of people how do we describe what they do so that an ordinary mortal can understand what we are saying we do not wish to legislate the method with which people should make decisions but only to describe it even when it is prescribed by some method in the process we may learn things that can help people make better decisions how the analytic hierarchy process ahp forman et al harker 1986 harker and vargas 1987 saaty 1986 1988a b saaty and vargas 1987 xu 1988 golden et al 1989 saaty and alexander 1989 is a theory of measurement when applied in decision making it assists one to describe the general decision operation by decomposing a complex problem into a multi level hierarchic structure of objectives criteria subcriteria and alternatives the ahp provides a fundamental scale of relative magnitudes expressed in dominance units to represent judgments in the form of paired comparisons a ratio scale of relative magnitudes expressed in priority units is then derived from each set of comparisons an overall rcatio scale of priorities is then synthesized to obtain a ranking of the alternatives from its axioms to its procedures the ahp has turned out to be historically and theoretically a different and independent theory of decision making from utility theory much as a dialogue evolved in mathematics around the consistency of different geometries and around absolute and relative space and time in physics both to dispel absolute notions those who believe that only utility theory can tell us the absolute truth about man s decision making might take a close look at the ahp it has found varied and serious applications it also has a particular way of generating ratio scales and dealing with inconsistency in judgment that have contributed to its effectiveness in resource allocation and in the setting of priorities by a group of decision makers utility theory is a normative process the ahp as a descriptive theory encompasses procedures leading to outcomes as would be ranked by a normative theory but it must go beyond to deal with outcomes not accounted for by the demanding assumptions of a normative theory we must briefly describe the ahp to enable the reader to see that a practicable theory based on ratio scales need not dilute itself to satisfy expectations of people who derive their understanding from a theory based on interval scales this is particularly true if the rival theory in aspiring for generality also makes unrealistic assumptions for example about the transitivity and consistency of preferences and the difficult use of lotteries

on global and local convergence of half quadratic algorithms this paper provides original results on the global and local convergence properties of half quadratic hq algorithms resulting from the geman and yang gy and geman and reynolds gr primal dual constructions first we show that the convergence domain of the gy algorithm can be extended with the benefit of an improved convergence rate second we provide a precise comparison of the convergence rates for both algorithms this analysis shows that the gr form does not benefit from a better convergence rate in general moreover the gy iterates often take advantage of a low cost implementation in this case the gy form is usually faster than the gr form from the cpu time viewpoint

the mathematical equivalence of the spanning tree and row geometric mean preference vectors and its implications for preference analysis abstract pairwise comparison is a widely used approach to elicit comparative judgements from a decision maker dm and there are a number of methods that can be used to then subsequently derive a consistent preference vector from the dm s judgements while the most widely used method is the eigenvector method the row geometric mean approach has gained popularity due to its mathematical properties and its ease of implementation in this paper we discuss a spanning tree method and prove the mathematical equivalence of its preference vector to that of the row geometric mean approach this is an important finding due to the fact that it identifies an approach for generating a preference vector which has the mathematical properties of the row geometric mean preference vector and yet in its entirety the spanning tree method has more to offer than the row geometric mean method in that it is inherently applicable to incomplete sets of pairwise comparison judgements and also facilitates the use of statistical and visual techniques to gain insights into inconsistency in the dm s judgements

a heuristic method to rectify intransitive judgments in pairwise comparison matrices abstract this paper investigates the effects of intransitive judgments on the consistency of pairwise comparison matrices statistical evidence regarding the occurrence of intransitive judgements in pairwise matrices of acceptable consistency is gathered by using a monte carlo simulation which confirms that relatively high percentage of comparison matrices satisfying saaty s cr criterion are ordinally inconsistent it is also shown that ordinal inconsistency does not necessarily decrease in the group aggregation process in contrast with cardinal inconsistency a heuristic algorithm is proposed to improve ordinal consistency by identifying and eliminating intransitivities in pairwise comparison matrices the proposed algorithm generates near optimal solutions and outperforms other tested approaches with respect to computation time

a consistency improving method in the analytic hierarchy process in this paper we propose a method to modify a given comparison matrix by which the consistency ratio cr value of the modified matrix is less than that of the original one and give an algorithm to derive a positive reciprocal matrix with acceptable consistency i e cr 0 1 then the convergence theorem for the given algorithm is established and its practicality is shown by some examples

on the minimum violations ranking of a tournament this paper examines the problem of rank ordering a set of players or objects on the basis of a set of pairwise comparisons arising from a tournament the criterion for deriving this ranking is to have as few cases as possible where player i is ranked above j while i was actually defeated by j in the tournament such a situation is referred to as a violation the objective therefore is to determine the minimum violations ranking mvr while there are situations where this ranking would be allowed to contain ties among subsets of objects we will concern ourselves herein with linear ordering no ties a series of examples are given where this requirement would seem to be appropriate in order to put the mvr problem into proper perspective we introduce the concept of a distance on the set of tournaments a set of natural axioms is presented which any such distance measure should obey and it is proven that in the presence of these axioms a unique such measure exists it is then shown that the mvr problem is equivalent to the minimum distance problem which can be represented in several forms in particular as a problem of determining the minimum feedback edge set in a graph and as a mixed integer generalized network problem this opens up a wide scope of possible solution procedures for the mvr problem an optimal algorithm is presented along with computational results in addition various heuristics are discussed including an improved heuristic referred to as the iterated kendall method

on the uncertainty in the regularized solution regularization is a fundamental technique in the processing of measurement data these data are typically noisy and sparse and in some cases only a projection of the data on a low dimensional subspace is available very often direct analysis of measured data yields unphysical results due to sensitivity to noise and neglect of the underlying distribution of the measured physical phenomena therefore regularization is extensively applied in physics one of the most common regularization methods was developed by a few researchers most notably tikhonov and phillips during the 1940s 1960s in the early 1980s geman and geman reformulated regularization in a bayesian framework in which every candidate function f x is assigned a probability with the expectation of the random variable f x0 equal to the value of the regularized interpolating function at x0 taking this probabilistic approach to second order the variance of this random variable which reflects the amount of uncertainty inherent in the solution is computed when estimating physical phenomena not only the estimated value is important but also the uncertainty associated with it for example more measurements should yield a more certain solution this review provides a formal definition of this uncertainty explores some numerical issues with its computation and extends it to compute the uncertainties of the integral and derivative with the intriguing result that in some cases the uncertainty of the function and the derivative are highly uncorrelated for example there are cases in which the point at which the function is least reliable is the one at which the derivative is most reliable lastly a related definition of optimal sampling is discussed the relevance of optimal sampling to physics is that it offers a method to determine the location of the measuring devices which yields the most reliable restoration of the measured physical phenomenon

iterative reweighted least squares design of fir filters develops a new iterative reweighted least squares algorithm for the design of optimal l sub p approximation fir filters the algorithm combines a variable p technique with a newton s method to give excellent robust initial convergence and quadratic final convergence details of the convergence properties when applied to the l sub p optimization problem are given the primary purpose of l sub p approximation for filter design is to allow design with different error criteria in pass and stopband and to design constrained l sub 2 approximation filters the new method can also be applied to the complex chebyshev approximation problem and to the design of 2d fir filters

hesitant analytic hierarchy process abstract in traditional analytic hierarch process ahp decision makers dms are required to provide crisp judgments over paired comparisons of objectives to construct comparison matrices to enhance the modeling ability of traditional ahp we propose hesitant ahp h ahp that can consider the hesitancy experienced by the dms in decision h ahp is characterized by hesitant judgments where each hesitant judgment can be represented by several possible values different probability distributions can be used to further describe hesitant judgments according to the dms preferences based on a hesitant comparison matrix hcm that consists of hesitant judgments we define two indices to measure the consistency degree and the consensus degree of the hcm respectively from a stochastic point of view a new prioritization method is developed to derive priorities from hcms where the results are with probability interpretations we provide a step by step procedure for h ahp and demonstrate this new method with a real life decision making problem

a majorize minimize strategy for subspace optimization applied to image restoration this paper proposes accelerated subspace optimization methods in the context of image restoration subspace optimization methods belong to the class of iterative descent algorithms for unconstrained optimization at each iteration of such methods a stepsize vector allowing the best combination of several search directions is computed through a multidimensional search it is usually obtained by an inner iterative second order method ruled by a stopping criterion that guarantees the convergence of the outer algorithm as an alternative we propose an original multidimensional search strategy based on the majorize minimize principle it leads to a closed form stepsize formula that ensures the convergence of the subspace algorithm whatever the number of inner iterations the practical efficiency of the proposed scheme is illustrated in the context of edge preserving image restoration

grouped coordinate ascent algorithms for penalized likelihood transmission image reconstruction presents a new class of algorithms for penalized likelihood reconstruction of attenuation maps from low count transmission scans we derive the algorithms by applying to the transmission log likelihood a version of the convexity technique developed by de pierro for emission tomography the new class includes the single coordinate ascent sca algorithm and lange s convex algorithm for transmission tomography as special cases the new grouped coordinate ascent gca algorithms in the class overcome several limitations associated with previous algorithms 1 fewer exponentiations are required than in the transmission maximum likelihood expectation maximization ml em algorithm or in the sca algorithm 2 the algorithms intrinsically accommodate nonnegativity constraints unlike many gradient based methods 3 the algorithms are easily parallelizable unlike the sca algorithm and perhaps line search algorithms we show that the gca algorithms converge faster than the sca algorithm even on conventional workstations an example from a low count positron emission tomography pet transmission scan illustrates the method

two component eigenfunction expansion for open systems described by the wave equation i completeness of expansion the concept of eigenfunction expansions for the wave equation is generalized to open systems in which waves escape to the outside these non conservative systems are non hermitian in the usual sense it is shown that the natural framework is an eigenfunction expansion within a two component formalism that treats the wavefunction and its conjugate momentum together provided the system approaches spatial infinity rapidly without tails and possesses spatial discontinuities the expansion in terms of the eigenfunctions which are now quasinormal modes is shown to be valid

derivatives of the perron root of a positive reciprocal matrix with application to the analytic hierarchy process the derivatives of the perron root of an essentially nonnegative matrix have been thoroughly studied in the literature this paper presents a refinement of these results to a special class of essentially nonnegative matrices namely positive reciprocal matrices this class of matrices arises in saaty s analytic hierarchy process a decision analysis method which is based upon pairwise comparisons among a set of alternatives over various criteria formulas for the first and second partial derivatives of the perron root of a positive reciprocal matrix are developed and the use of these formulas in conjunction with saaty s method is illustrated

deriving priorities from pairwise comparison matrices with a novel consistency index abstract it is important to measure the inconsistency level of a pairwise comparison matrix pcm and derive the priority vector in the analytic hierarchy process ahp in the present study a new consistency index is proposed by using the cosine similarity measures of two row column vectors in a pcm it is called as the double cosine similarity consistency index dcsci since the row and column vectors are all considered some interesting properties of dcsci are investigated and the thresholds for inconsistency tolerance level are discussed in detail then following the idea of dcsci we provide a new method for obtaining the priority vector from a pcm through maximizing the sum of the cosine similarity measures between the priority vector and the row column vectors the priority vector is derived by solving the constructed optimization problem by analyzing the proposed double cosine similarity maximization dcsm method it is found that the existing cosine maximization cm method can be retrieved by carrying out numerical examples some comparisons with the existing methods show that the proposed index and method are effective and flexible

the geometric consistency index approximated thresholds abstract crawford and williams journal of mathematical psychology 29 1985 387 suggested for the row geometric mean method rgmm one of the most extended ahp s priorization procedure a measure of the inconsistency based on stochastic properties of a subjacent model in this paper we formalize this inconsistency measure hereafter called the geometric consistency index gci and provide the thresholds associated with it these thresholds allow us an interpretation of the inconsistency tolerance level analogous to that proposed by saaty multicriteria decision making the analytic hierarchy process new york 1980 for the consistency ratio cr used with the right eigenvector method in conventional ahp

fast algorithms for l p deconvolution this paper discusses two algorithms for l p 1 leq p leq 2 deconvolution one iterative reweighted least squares irls is discussed with an eye toward improving its computational efficiency while the other residual steepest descent rsd is considered in an attempt to improve its convergence properties in the first case fast fourier transforms are used to reduce the number of computations in the second a new rescaling procedure which enhances rsd convergence is introduced the lack of stability of l 1 filters and its implications are discussed simple examples are used to illustrate pertinent concepts

a generalized gaussian image model for edge preserving map estimation the authors present a markov random field model which allows realistic edge modeling while providing stable maximum a posterior map solutions the model referred to as a generalized gaussian markov random field ggmrf is named for its similarity to the generalized gaussian distribution used in robust detection and estimation the model satisfies several desirable analytical and computational properties for map estimation including continuous dependence of the estimate on the data invariance of the character of solutions to scaling of data and a solution which lies at the unique global minimum of the a posteriori log likelihood function the ggmrf is demonstrated to be useful for image reconstruction in low dosage transmission tomography

two component eigenfunction expansion for open systems described by the wave equation ii linear space structure for a broad class of open systems described by the wave equation the eigenfunctions which are quasinormal modes provide a complete basis for simultaneously expanding outgoing wavefunctions in this paper the linear space structure associated with this expansion is developed under a modified inner product the time evolution operator is self adjoint even though energy is not conserved for the system alone thus the eigenfunctions are mutually orthogonal consequently the usual tools of eigenfunction expansions can be transcribed to these open systems as an example the time independent perturbation theory is developed in straightforward analogy with quantum mechanics giving the shift in both the real part and the imaginary part of the eigenvalues

the theory of ratio scale estimation saaty s analytic hierarchy process the analytic hierarchy process developed by saaty saaty t l 1980 the analytic hierarchy process mcgraw hill new york has proven to be an extremely useful method for decision making and planning however some researchers in these areas have raised concerns over the theoretical basis underlying this process this paper addresses currently debated issues concerning the theoretical foundations of the analytic hierarchy process we also illustrate through proof and through examples the validity or fallaciousness of these criticisms

a geometric averaging procedure for constructing supertransitive approximation to binary comparison matrices abstract the usefulness of encoding the fuzzy evaluations of alternatives and the importance weights of criteria in a multiple objective decision problem through binary comparison matrices or pairwise judgment matrices is receiving considerable attention the methodology for identifying the best alternative in a given decision problem involves the computation of the principal eigenvectors of the binary comparison matrices the eigenvectors transform the fuzzy evaluations of the importance of the criteria and the ratings of the alternatives into a ratio scale a difficulty that is often experienced in using this approach in practice is the inconsistency of the binary evaluations this paper proposes a simple averaging procedure to construct a supertransitive approximation to a binary comparison matrix where inconsistency is a problem it is further suggested that such an adjustment might be necessary to more closely reflect the inherent fuzziness of the evaluations contained in a binary comparison matrix the procedure is illustrated by means of examples

three dimensional edge preserving image enhancement for computed tomography computed tomography ct images exhibit a variable amount of noise and blur depending on the physical characteristics of the apparatus and the selected reconstruction method standard algorithms tend to favor reconstruction speed over resolution thereby jeopardizing applications where accuracy is critical in this paper we propose to enhance ct images by applying half quadratic edge preserving image restoration or deconvolution to them this approach may be used with virtually any ct scanner provided the overall point spread function can be roughly estimated in image restoration markov random fields mrfs have proven to be very flexible a priori models and to yield impressive results with edge preserving penalization but their implementation in clinical routine is limited because they are often viewed as complex and time consuming for these practical reasons we focused on numerical efficiency and developed a fast implementation based on a simple three dimensional mrf model with convex edge preserving potentials the resulting restoration method provides good recovery of sharp discontinuities while using convex duality principles yields fairly simple implementation of the optimization further reduction of the computational load can be achieved if the point spread function is assumed to be separable synthetic and real data experiments indicate that the method provides significant improvements over standard reconstruction techniques and compares well with convex potential markov based reconstruction while being more flexible and numerically efficient

a weak order solution to a group ranking and consistency driven pairwise comparisons a novel non numerical algorithm based on a partial order for solving a group ranking problem is compared with the more traditional consistency driven approach both approaches are based on the fundamental concept of pairwise comparisons potential applications are discussed

robust estimation of priorities in the ahp abstract a pairwise comparison matrix of the analytic hierarchy process ahp is considered as a contingency table that helps to identify unusual or false data elicited from a judge special techniques are suggested for robust estimation of priority vectors they include transformation of a saaty matrix to matrix of shares of preferences and solving an eigenproblem designed for the transformed matrices we also introduce an optimizing objective that produces robust priority estimation numerical results are compared using the ahp with these differing approaches the comparison demonstrates that robust estimations yield priority vectors not prone to influence of possible errors among the elements of a pairwise comparison matrix

efficient schemes for total variation minimization under constraints in image processing this paper presents new fast algorithms to minimize total variation and more generally l 1 norms under a general convex constraint such problems are standards of image processing the algorithms are based on a recent advance in convex optimization proposed by yurii nesterov depending on the regularity of the data fidelity term we solve either a primal problem or a dual problem first we show that standard first order schemes allow one to get solutions of precision epsilon in o frac 1 epsilon 2 iterations at worst we propose a scheme that allows one to obtain a solution of precision epsilon in o frac 1 epsilon iterations for a general convex constraint for a strongly convex constraint we solve a dual problem with a scheme that requires o frac 1 sqrt epsilon iterations to get a solution of precision epsilon finally we perform some numerical experiments which confirm the theoretical results on various problems of image processing

determination of a finite range potential from discrete phase shift data by inverse scattering a practical method of inverse scattering is proposed to determine a finite range scattering potential from phase shift data evaluated at a set of discrete energies by using the asymptotic behaviour of the phase shifts and the connection between short and long wavelength scales a relatively few input data are found to be adequate the method works well for repulsive and moderately attractive potentials useful in many applications

on the transitivity and consistency approximated thresholds of some consistency indices for pairwise comparison matrices abstract several measures have been proposed in the literature to verify the rationality and consistency of judgements expressed by means of pairwise comparisons however some of these indices lack a meaningful interpretation because of the absence of thresholds associated with them in this paper we first introduce the consistency approximated thresholds for the salo hamalainen index by deriving an inconsistency tolerance level which is analogous to that proposed for the consistency ratio by saaty for the geometric consistency index by aguaron moreno jimenez and for the consistency measure by koczkodaj then via a simulation we propose a transitivity threshold for n n matrices which may give meaningful information about the degree of misclassification and the reliability of preferences while avoiding the need to revise the judgements on varying n we derive the transitivity thresholds for salo hamalainen and for the other consistency indices proposed in literature

remarks on the analytic hierarchy process the analytic hierarchy process ahp is flawed as a procedure for ranking alternatives in that the rankings produced by this procedure are arbitrary this paper provides a brief review of several areas of operational difficulty with the ahp and then focuses on the arbitrary rankings that occur when the principle of hierarchic composition is assumed this principle requires that the weights on the higher levels of a hierarchy can be determined independently of the weights on the lower levels virtually all of the published examples of the use of the ahp to evaluate alternatives relative to a set of criteria have assumed this principle the key to correcting this flaw is the synthesis of the ahp with the concepts of multiattribute utility theory

convex half quadratic criteria and interacting auxiliary variables for image restoration this paper deals with convex half quadratic criteria and associated minimization algorithms for the purpose of image restoration it brings a number of original elements within a unified mathematical presentation based on convex duality firstly the geman and yang 1995 and geman and reynolds 1992 constructions are revisited with a view to establishing the convexity properties of the resulting half quadratic augmented criteria when the original nonquadratic criterion is already convex secondly a family of convex gibbsian energies that incorporate interacting auxiliary variables is revealed as a potentially fruitful extension of the geman and reynolds construction

on some useful properties of the perron eigenvalue of a positive reciprocal matrix in the context of the analytic hierarchy process abstract a positive n x n matrix r rij is said to be reciprocal if its entries verify fji 1 rij 0 for all 1 2 decision alternatives on an arbitrary ratio scale saaty in journal of mathematical psychology 1977 proposed to use z ama x n n 1 0 a linear transform of the perron eigenvalue max of r as a measure of the cardinal consistency in an agent s responses and posed the problem of determining how it might vary as a function of the rij s he also suggested that an upper bound could be found for that consistency index when the entries of r are restricted to take their values in a bounded set both of these questions are answered here using classical results from linear algebra keywords analytic hierarchy process decision theory pairwise comparisons perron frobenius theo rem reciprocal matrices i introduction saaty 20 has introduced a procedure for pri oritising decision alternatives based on an n x n positive matrix r

inconsistency in the ordinal pairwise comparisons method with and without ties abstract comparing alternatives in pairs is a well known method used to create ranking experts are asked to perform a series of binary comparisons and then using mathematical methods the final ranking is constructed experts conduct a series of single assessments however they may not always be consistent the level of inconsistency among individual assessments is widely accepted as a measure of the ranking quality the higher the ranking quality the higher its credibility one of the earliest and most widespread inconsistency indices is the consistency coefficient defined by kendall and babington smith in their work the authors consider binary pairwise comparisons i e those where the result of an individual comparison can only be better or worse in the presented work the maximal number of inconsistent triads in the set of ordinal pairwise comparisons with ties of arbitrary size is determined formula 14 this in turn opens the possibility of effectively extending the kendall and babington smith index to pairwise comparisons where the result of an individual comparison can be better worse or equal hence this effectively extends the use of this index to the analytic hierarchy process and other quantitative methods based on comparing alternatives in pairs the work also introduces the notions of a generalized tournament and a double tournament as graphs that model ordinal pairwise comparisons with ties and the maximally inconsistent set of pairwise comparisons with ties respectively the relationship between the most inconsistent set of pairwise comparisons with ties and the set cover problem is also shown

the equivalence of half quadratic minimization and the gradient linearization iteration a popular way to restore images comprising edges is to minimize a cost function combining a quadratic data fidelity term and an edge preserving possibly nonconvex regularization term mainly because of the latter term the calculation of the solution is slow and cumbersome half quadratic hq minimization multiplicative form was pioneered by geman and reynolds 1992 in order to alleviate the computational task in the context of image reconstruction with nonconvex regularization by promoting the idea of locally homogeneous image models with a continuous valued line process they reformulated the optimization problem in terms of an augmented cost function which is quadratic with respect to the image and separable with respect to the line process hence the name half quadratic since then a large amount of papers were dedicated to hq minimization and important results including edge preservation along with convex regularization and convergence have been obtained in this paper we show that hq minimization multiplicative form is equivalent to the most simple and basic method where the gradient of the cost function is linearized at each iteration step in fact both methods give exactly the same iterations furthermore connections of hq minimization with other methods such as the quasi newton method and the generalized weiszfeld s method are straightforward

vectorial additive half quadratic minimization for isotropic regularization we propose the vectorial additive half quadratic vahq algorithm to minimize the isotropic regularized cost function with the general edge preserving potential functions pfs in image restoration by introducing an auxiliary vectorial variable the cost function is changed into an augmented one which can be alternately minimized one minimization is solved with an explicit expression the other is implemented by fast fourier transform fft vahq is shown to globally converge to a stationary point for nonconvex pfs providing all stationary points are isolated and to a unique minimum for convex pfs without any isolation assumption on an extended domain of parameters what is more the linear convergence rate of vahq is proved to be less than 1 the explicit expression of the optimal parameters and the optimal bound of the convergence rate are present for convex pfs image restoration examples show the restoration performance of nonconvex pfs the lower computation cost of our algorithm compared to majorize minimize memory gradient mmmg algorithm with the help of fft and some interesting phenomena confirming our conclusions on the convergence domain and the convergence rate

incomplete pairwise comparison and consistency optimization this paper proposes a new method for calculating the missing elements of an incomplete matrix of pairwise comparison values for a decision problem the matrix is completed by minimizing a measure of global inconsistency thus obtaining a matrix which is optimal from the point of view of consistency with respect to the available judgements the optimal values are obtained by solving a linear system and unicity of the solution is proved under general assumptions some other methods proposed in the literature are discussed and a numerical example is presented

on the choice of the regularization parameter in nonlinear inverse problems this paper focuses on regularization techniques for nonlinear ill posed inverse problems tikhonov regularization and regularization due to the use of norm constraints are analyzed a model function technique is proposed to iteratively determine an optimal regularization parameter or the parameter characterizing the norm constraint and to estimate the error in the data if it is not known a priori

jordan blocks and generalized bi orthogonal bases realizations in open wave systems dissipation can sometimes be described by a non hermitian hamiltonian h whose left and right eigenvectors f j fj form a bi orthogonal basis bb for waves in a class of open systems this is known to lead to exact complete bb expansions if f j fj 0 for all j if not normalization seems impossible and many familiar formulae fail examples are given the problem is related to the merging of eigenmodes so that h can only be diagonalized to jordan blocks the resolution involves a generalized bb containing extra vectors whose dynamics are modified by polynomials in the time t the splitting of merged modes under a perturbation is also treated one thus obtains a non trivial extension of the bb formalism for open systems

transitive calibration of the ahp verbal scale abstract one of the strengths of the analytic hierarchy process ahp is that it allows decision makers to specify their preferences using a verbal scale yet as is well known a strict reliance on the corresponding saaty 1 9 numeric scale can induce some inconsistency hence we argue that in certain situations it may be appropriate to calibrate the verbal scale the result of this calibration is a geometric scale based on a single parameter we present some limited evidence that this geometric scale marginally outperforms the saaty 1 9 scale moreover we suggest that this calibration can be used to do a simple sensitivity analysis in cases where judgements are uncertain

constrained restoration and the recovery of discontinuities the linear image restoration problem is to recover an original brightness distribution x sup 0 given the blurred and noisy observations y kx sup 0 b where k and b represent the point spread function and measurement error respectively this problem is typical of ill conditioned inverse problems that frequently arise in low level computer vision a conventional method to stabilize the problem is to introduce a priori constraints on x sup 0 and design a cost functional h x over images x which is a weighted average of the prior constraints regularization term and posterior constraints data term the reconstruction is then the image x which minimizes h a prominent weakness in this approach especially with quadratic type stabilizers is the difficulty in recovering discontinuities the authors therefore examine prior smoothness constraints of a different form which permit the recovery of discontinuities without introducing auxiliary variables for marking the location of jumps and suspending the constraints in their vicinity in this sense discontinuities are addressed implicitly rather than explicitly

image recovery using partitioned separable paraboloidal surrogate coordinate ascent algorithms iterative coordinate ascent algorithms have been shown to be useful for image recovery but are poorly suited to parallel computing due to their sequential nature this paper presents a new fast converging parallelizable algorithm for image recovery that can be applied to a very broad class of objective functions this method is based on paraboloidal surrogate functions and a concavity technique the paraboloidal surrogates simplify the optimization problem the idea of the concavity technique is to partition pixels into subsets that can be updated in parallel to reduce the computation time for fast convergence pixels within each subset are updated sequentially using a coordinate ascent algorithm the proposed algorithm is guaranteed to monotonically increase the objective function and intrinsically accommodates nonnegativity constraints a global convergence proof is summarized simulation results show that the proposed algorithm requires less elapsed time for convergence than iterative coordinate ascent algorithms with four parallel processors the proposed algorithm yields a speedup factor of 3 77 relative to single processor coordinate ascent algorithms for a three dimensional 3 d confocal image restoration problem

deterministic edge preserving regularization in computed imaging many image processing problems are ill posed and must be regularized usually a roughness penalty is imposed on the solution the difficulty is to avoid the smoothing of edges which are very important attributes of the image in this paper we first give conditions for the design of such an edge preserving regularization under these conditions we show that it is possible to introduce an auxiliary variable whose role is twofold first it marks the discontinuities and ensures their preservation from smoothing second it makes the criterion half quadratic the optimization is then easier we propose a deterministic strategy based on alternate minimizations on the image and the auxiliary variable this leads to the definition of an original reconstruction algorithm called artur some theoretical properties of artur are discussed experimental results illustrate the behavior of the algorithm these results are shown in the field of 2d single photon emission tomography but this method can be applied in a large number of applications in image processing

image reconstruction and restoration overview of common estimation structures and problems developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined particular attention is paid to common estimation structures and to practical problems not properly solved yet the problem of image reconstruction and restoration is first formulated some of the current regularization approaches used to solve the problem are then described the concepts of a priori information and compound criterion are introduced a bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated the practical aspects of computing the solution first when the hyperparameters are known and second when they must be estimated are then considered conclusions are drawn and points that still need to be investigated are outlined

monotonic algorithms for transmission tomography we present a framework for designing fast and monotonic algorithms for transmission tomography penalized likelihood image reconstruction the new algorithms are based on paraboloidal surrogate functions for the log likelihood due to the form of the log likelihood function it is possible to find low curvature surrogate functions that guarantee monotonicity unlike previous methods the proposed surrogate functions lead to monotonic algorithms even for the nonconvex log likelihood that arises due to background events such as scatter and random coincidences the gradient and the curvature of the likelihood terms are evaluated only once per iteration since the problem is simplified at each iteration the cpu time is less than that of current algorithms which directly minimize the objective yet the convergence rate is comparable the simplicity monotonicity and speed of the new algorithms are quite attractive the convergence rates of the algorithms are demonstrated using real and simulated pet transmission scans

an intrinsic consistency threshold for reciprocal matrices abstract in this paper we present a new intrinsic and scale independent consistency threshold for reciprocal matrices in the analytic hierarchy process ahp its derivation is based upon the relation between the consistency measure of the ahp and the reduction of weight of a new alternative we compare this standard with an existing one

multi exponential transverse relaxation times estimation from magnetic resonance images under rician noise and spatial regularization relaxation signal inside each voxel of magnetic resonance images mri is commonly fitted by a multi exponential decay curve the estimation of a discrete multi component relaxation model parameters from magnitude mri data is a challenging nonlinear inverse problem since it should be conducted on the entire image voxels under non gaussian noise statistics this paper proposes an efficient algorithm allowing the joint estimation of relaxation time values and their amplitudes using different criteria taking into account a rician noise model combined with a spatial regularization accounting for low spatial variability of relaxation time constants and amplitudes between neighboring voxels the rician noise hypothesis is accounted for either by an adapted nonlinear least squares algorithm applied to a corrected least squares criterion or by a majorization minimization approach applied to the maximum likelihood criterion in order to solve the resulting large scale non negativity constrained optimization problem with a reduced numerical complexity and computing time an optimization algorithm based on a majorization approach ensuring separability of variables between voxels is proposed the minimization is carried out iteratively using an adapted levenberg marquardt algorithm that ensures convergence by imposing a sufficient decrease of the objective function and the non negativity of the parameters the importance of the regularization alongside the rician noise incorporation is shown both visually and numerically on a simulated phantom and on magnitude mri images acquired on fruit samples

waves in open systems via a biorthogonal basis dissipative quantum systems are sometimes phenomenologically described in terms of a non hermitian hamiltonian h with different left and right eigenvectors forming a biorthogonal basis it is shown that the dynamics of waves in open systems can be cast exactly into this form thus providing a well founded realization of the phenomenological description and at the same time placing these open systems into a well known framework the formalism leads to a generalization of norms and inner products for open systems which in contrast to earlier works is finite without the need for regularization the inner product allows transcription of much of the formalism for conservative systems including perturbation theory and second quantization

the harmonic consistency index for the analytic hierarchy process a new consistency measure the harmonic consistency index is obtained for any positive reciprocal matrix in the analytic hierarchy process we show how this index varies with changes in any matrix element a tight upper bound is provided for this new consistency measure when the entries of matrix are at most 9 as is often recommended using simulation the harmonic consistency index is shown to give numerical values similar to the standard consistency index but it is easier to compute and interpret in addition new properties of the column sums of reciprocal matrices are obtained

a unified approach to statistical tomography using coordinate descent optimization over the past years there has been considerable interest in statistically optimal reconstruction of cross sectional images from tomographic data in particular a variety of such algorithms have been proposed for maximum a posteriori map reconstruction from emission tomographic data while map estimation requires the solution of an optimization problem most existing reconstruction algorithms take an indirect approach based on the expectation maximization em algorithm we propose a new approach to statistically optimal image reconstruction based on direct optimization of the map criterion the key to this direct optimization approach is greedy pixel wise computations known as iterative coordinate decent icd we propose a novel method for computing the icd updates which we call icd newton raphson we show that icd newton raphson requires approximately the same amount of computation per iteration as em based approaches but the new method converges much more rapidly in our experiments typically five to ten iterations other advantages of the icd newton raphson method are that it is easily applied to map estimation of transmission tomograms and typical convex constraints such as positivity are easily incorporated

hybrid l 1 l 2 minimization with applications to tomography least squares or l2 solutions of seismic inversion and tomography problems tend to be very sensitive to data points with large errors the lp minimization for 1 p 2 gives more robust solutions but usually with higher computational cost iteratively reweighted least squares irls gives efficient approximate solutions to these lp problems we apply irls to a hybrid l2 l2 minimization problem that behaves like an l2 fit for small residuals and like an l1 fit for large residuals the smooth transition from l2 to l1 behavior is controlled by a parameter that we choose using an estimate of the standard deviation of the data error for linear problems of full rank the hybrid objective function has a unique minimum and irls can be proven to converge to it we obtain a robust efficient method for nonlinear problems a version of the gauss newton algorithm can be applied synthetic crosswell tomography examples and a field data vsp tomography example demonstrate the improvement of the hybrid method over le

a simple method to improve the consistency ratio of the pair wise comparison matrix in anp tests of consistency for the pair wise comparison matrices have been studied extensively since ahp was introduced by saaty in 1970s however existing methods are either too complicated to be applied in the revising process of the inconsistent comparison matrix or are difficult to preserve most of the original comparison information due to the use of a new pair wise comparison matrix those methods might work for ahp but not for anp as the comparison matrix of anp needs to be strictly consistent to improve the consistency ratio this paper proposes a simple method which combines the theorem of matrix multiplication vectors dot product and the definition of consistent pair wise comparison matrix to identify the inconsistent elements the correctness of the proposed method is proved mathematically the experimental studies have also shown that the proposed method is accurate and efficient in decision maker s revising process to satisfy the consistency requirements of ahp anp

contribution of individual judgments toward inconsistency in pairwise comparisons pairwise comparison pc is a well established method to assist decision makers in estimating their preferences in pcs the acquired judgments are used to construct a pc matrix pcm that is used to check whether the inconsistency in judgments is acceptable or requires revision the use of consistency ratio cr a widely used measure for inconsistency has been widely debated and the literature survey has identified a need for a more appropriate measure considering this need a new measure termed congruence is proposed in this paper the measure is shown to be useful in finding the contribution of individual judgments toward overall inconsistency of a pcm and therefore can be used to detect and correct cardinally inconsistent judgments the proposed measure is applicable to incomplete sets of pc judgments without modification unlike cr which requires a complete set of pc judgments to address ordinal inconsistency another measure termed dissonance is proposed as a supplement to the congruence measure the two measures appear useful in detecting both outliers and the phenomenon of consistency deadlock where all judgments equally contribute toward the overall inconsistency

numerical algorithm for the determination of the potential of a conservative system from its normal mode spectra an efficient numerical algorithm is developed for determining the potential of a conservative system described by a klein gordon or schrodinger equation from its normal mode spectra the algorithm is based on first solving the linear problem of perturbative inversion by means of tikhonov regularization and then using a series of perturbative steps to approach the full potential by newton s method the algorithm is stable and convergence is rapid

representing the strengths and directions of pairwise comparisons we consider the problem of finding weights that well represent a set of pairwise multiplicative comparisons of a set of objects as in the ahp and other methods our main contribution is a method for deriving such weights that takes into consideration not only the strengths of the pairwise comparisons but also their directions for example if the comparison directions satisfy transitivity then the weights produced by our method also satisfy transitivity this is not always true for other methods we also present a set of reasonable axioms for which our method is the essentially unique solution our method and axioms are closely related to those of cook and kress eur j oper res 37 1988 335 our method like theirs reduces to solving a linear program hence it is different from the approach used in the ahp for the special case that the comparison directions satisfy transitivity our method is quite simple and reduces to performing a forward pass as in the critical path method

axiomatic foundation of the analytic hierarchy process this paper contains an axiomatic treatment of the analytic hierarchy process ahp the set of axioms corresponding to hierarchic structures are a special case of axioms for priority setting in systems with feedback which allow for a wide class of dependencies the axioms highlight 1 the reciprocal property that is basic in making paired comparisons 2 homogeneity that is characteristic of people s ability for making comparisons among things that are not too dissimilar with respect to a common property and hence the need for arranging them within an order preserving hierarchy 3 dependence of a lower level on the adjacent higher level 4 the idea that an outcome can only reflect expectations when the latter are well represented in the hierarchy the ahp neither assumes transitivity or the stronger condition of consistency nor does it include strong assumptions of the usual notions of rationality a number of facts are derived from these axioms providing an operational basis for the ahp

globally convergent edge preserving regularized reconstruction an application to limited angle tomography we introduce a generalization of a deterministic relaxation algorithm for edge preserving regularization in linear inverse problems this algorithm transforms the original possibly nonconvex optimization problem into a sequence of quadratic optimization problems and has been shown to converge under certain conditions when the original cost functional being minimized is strictly convex we prove that our more general algorithm is globally convergent i e converges to a local minimum from any initialization under less restrictive conditions even when the original cost functional is nonconvex we apply this algorithm to tomographic reconstruction from limited angle data by formulating the problem as one of regularized least squares optimization the results demonstrate that the constraint of piecewise smoothness applied through the use of edge preserving regularization can provide excellent limited angle tomographic reconstructions two edge preserving regularizers one convex the other nonconvex are used in numerous simulations to demonstrate the effectiveness of the algorithm under various limited angle scenarios and to explore how factors such as the choice of error norm angular sampling rate and amount of noise affect the reconstruction quality and algorithm performance these simulation results show that for this application the nonconvex regularizer produces consistently superior results

a local update strategy for iterative reconstruction from projections a method for bayesian reconstruction which relies on updates of single pixel values rather than the entire image at each iteration is presented the technique is similar to gauss seidel gs iteration for the solution of differential equations on finite grids the computational cost per iteration of the gs approach is found to be approximately equal to that of gradient methods for continuously valued images gs is found to have significantly better convergence at modes representing high spatial frequencies in addition gs is well suited to segmentation when the image is constrained to be discretely valued it is shown that bayesian segmentation using gs iteration produces useful estimates at much lower signal to noise ratios than required for continuously valued reconstruction the convergence properties of gradient ascent and gs for reconstruction from integral projections are analyzed and simulations of both maximum likelihood and maximum a posteriori cases are included

majorization minimization algorithms in signal processing communications and machine learning this paper gives an overview of the majorization minimization mm algorithmic framework which can provide guidance in deriving problem driven algorithms with low computational cost a general introduction of mm is presented including a description of the basic principle and its convergence results the extensions acceleration schemes and connection to other algorithmic frameworks are also covered to bridge the gap between theory and practice upperbounds for a large number of basic functions derived based on the taylor expansion convexity and special inequalities are provided as ingredients for constructing surrogate functions with the pre requisites established the way of applying mm to solving specific problems is elaborated by a wide range of applications in signal processing communications and machine learning

scaling binary comparison matrices a comment on narasimhan s proposal and other methods abstract recently narasimhan has proposed an adjustment to saaty s solution for scaling square judgment matrices consisting of pairwise ratio comparisons he recommends that prior to the application of saaty s solution the judgment matrix be adjusted to satisfy the supertransitivity requirement the present note shows that the proposed adjustment is equivalent to an alternative solution to this problem based upon calculation of the geometric means of the matrix rows thus the proposed adjustment should be regarded as an alternative to saaty s solution rather than a preliminary stage finally four solutions to this scaling problem are compared and the need for further research in this area is reiterated

parallelizable bayesian tomography algorithms with rapid guaranteed convergence bayesian tomographic reconstruction algorithms generally require the efficient optimization of a functional of many variables in this setting as well as in many other optimization tasks functional substitution fs has been widely applied to simplify each step of the iterative process the function to be minimized is replaced locally by an approximation having a more easily manipulated form e g quadratic but which maintains sufficient similarity to descend the true functional while computing only the substitute we provide two new applications of fs methods in iterative coordinate descent for bayesian tomography the first is a modification of our coordinate descent algorithm with one dimensional 1 d newton raphson approximations to an alternative quadratic which allows convergence to be proven easily in simulations we find essentially no difference in convergence speed between the two techniques we also present a new algorithm which exploits the fs method to allow parallel updates of arbitrary sets of pixels using computations similar to iterative coordinate descent the theoretical potential speed up of parallel implementations is nearly linear with the number of processors if communication costs are neglected

on the convergence of the lagged diffusivity fixed point method in total variation image restoration in this paper we show that the lagged diffusivity fixed point algorithm introduced by vogel and oman in siam j sci comput 17 1996 pp 227 238 to solve the problem of total variation denoising proposed by rudin osher and fatemi in phys d 60 1992 pp 259 268 is a particular instance of a class of algorithms introduced by voss and eckhardt in computing 25 1980 pp 243 251 whose origins can be traced back to weiszfeld s original work for minimizing a sum of euclidean lengths tohoku math j 43 1937 pp 355 386 there have recently appeared several proofs for the convergence of this algorithm g aubert et al technical report 94 01 informatique signaux et systemes de sophia antipolis 1994 a chambolle and p l lions technical report 9509 ceremade 1995 and d c dobson and c r vogel siam j numer anal 34 1997 pp 1779 1791 here we present a proof of the global and linear convergence using the framework introduced in h voss and u eckhart computing 25 1980 pp 243 251 and give a bound for the convergence rate of the fixed point iteration that agrees with our experimental results these results are also valid for suitable generalizations of the fixed point algorithm

a weak order approach to group ranking a new approach to the problem of a group ranking is presented the solution satisfies four of arrow s six axioms 1 2 and is sufficient for a wide range of applications a constructive algorithm is proposed for finding a group ranking on the basis of individual rankings of which some may collide with others

structure of concurrency abstract noninterleaving models of concurrency assume that behavioural properties of systems can be adequately modelled in terms of causal partial orders we claim that the structure of concurrency is richer with causality being only one of the invariants generated by a set of closely related executions or observations the model we propose supports three levels of abstraction the observation level invariant level and system level and we will proceed from the bottom observation level to the top system level this is in contrast to the way other models for concurrency are introduced as they essentially support two levels of abstraction the system level and behavioural level which includes both observations and invariants with the direction of development going from the system to behavioural level in this paper we first discuss the notion of an observation of a concurrent behaviour in particular we investigate the role played by interval partial orders we then introduce a general framework for dealing with invariants generated by sets of closely related observations this leads to the formulation of the notion of a concurrent history whose structural properties are subsequently studied