the z spl pi m algorithm a method for interferometric image reconstruction in sar sas this paper presents an effective algorithm for absolute phase not simply modulo 2 spl pi estimation from incomplete noisy and modulo 2 spl pi observations in interferometric aperture radar and sonar insar insas the adopted framework is also representative of other applications such as optical interferometry magnetic resonance imaging and diffraction tomography the bayesian viewpoint is adopted the observation density is 2 spl pi periodic and accounts for the interferometric pair decorrelation and system noise the a priori probability of the absolute phase is modeled by a compound gauss markov random field cgmrf tailored to piecewise smooth absolute phase images we propose an iterative scheme for the computation of the maximum a posteriori probability map absolute phase estimate each iteration embodies a discrete optimization step z step implemented by network programming techniques and an iterative conditional modes icm step spl pi step accordingly the algorithm is termed z spl pi m where the letter m stands for maximization an important contribution of the paper is the simultaneous implementation of phase unwrapping inference of the 2 spl pi multiples and smoothing denoising of the observations this improves considerably the accuracy of the absolute phase estimates compared to methods in which the data is low pass filtered prior to unwrapping a set of experimental results comparing the proposed algorithm with alternative methods illustrates the effectiveness of our approach

two supercharacter theories for the parabolic subgroups in orthogonal and symplectic groups abstract we construct two supercharacter theories in the sense of p diaconis and i m isaacs for the parabolic subgroups in orthogonal and symplectic groups for each supercharacter theory we obtain a supercharacter analog of the a a kirillov formula for irreducible characters of finite unipotent groups

phase unwrapping via graph cuts phase unwrapping is the inference of absolute phase from modulo 2pi phase this paper introduces a new energy minimization framework for phase unwrapping the considered objective functions are first order markov random fields we provide an exact energy minimization algorithm whenever the corresponding clique potentials are convex namely for the phase unwrapping classical lp norm with pges1 its complexity is kt n 3n where k is the length of the absolute phase domain measured in 2pi units and t n m is the complexity of a max flow computation in a graph with n nodes and m edges for nonconvex clique potentials often used owing to their discontinuity preserving ability we face an np hard problem for which we devise an approximate solution both algorithms solve integer optimization problems by computing a sequence of binary optimizations each one solved by graph cut techniques accordingly we name the two algorithms puma for phase unwrapping max flow min cut a set of experimental results illustrates the effectiveness of the proposed approach and its competitiveness in comparison with state of the art phase unwrapping algorithms

a generalized em algorithm for 3 d bayesian reconstruction from poisson data using gibbs priors a generalized expectation maximization gem algorithm is developed for bayesian reconstruction based on locally correlated markov random field priors in the form of gibbs functions and on the poisson data model for the m step of the algorithm a form of coordinate gradient ascent is derived the algorithm reduces to the em maximum likelihood algorithm as the markov random field prior tends towards a uniform distribution three different gibbs function priors are examined reconstructions of 3 d images obtained from the poisson model of single photon emission computed tomography are presented

graph cut based discrete valued image reconstruction efficient graph cut methods have been used with great success for labeling and denoising problems occurring in computer vision unfortunately the presence of linear image mappings has prevented the use of these techniques in most discrete amplitude image reconstruction problems in this paper we develop a graph cut based framework for the direct solution of discrete amplitude linear image reconstruction problems cast as regularized energy function minimizations we first analyze the structure of discrete linear inverse problem cost functions to show that the obstacle to the application of graph cut methods to their solution is the variable mixing caused by the presence of the linear sensing operator we then propose to use a surrogate energy functional that overcomes the challenges imposed by the sensing operator yet can be utilized efficiently in existing graph cut frameworks we use this surrogate energy functional to devise a monotonic iterative algorithm for the solution of discrete valued inverse problems we first provide experiments using local convolutional operators and show the robustness of the proposed technique to noise and stability to changes in regularization parameter then we focus on nonlocal tomographic examples where we consider limited angle data problems we compare our technique with state of the art discrete and continuous image reconstruction techniques experiments show that the proposed method outperforms state of the art techniques in challenging scenarios involving discrete valued unknowns

super character theory and comparison arguments for a random walk on the upper triangular matrices abstract consider the random walk on the n n upper triangular matrices with ones on the diagonal and elements over f p where we pick a row at random and either add it or subtract it from the row directly above it the main result of this paper is to prove that the dependency of the mixing time on p is p 2 this is proven by combining super character theory and comparison theory arguments

bayesian approaches to phase unwrapping theoretical study the problem of phase unwrapping of two dimensional 2 d phase signals has gained a considerable interest it deals with the problem of estimating reconstructing an absolute phase from the observation of its noisy principal wrapped values this is an ill posed problem since many possible solutions correspond to a given observation many phase unwrapping algorithms have been proposed relying on different constraints for the phase signal sampling process or the nature e g smoothness regularity of the phase signal we look at these algorithms from the bayesian point of view estimation theory and analyze the role of the prior assumptions studying their equivalencies to the regularization constraints already used this study leads to the development of the two new phase unwrapping algorithms which are able to work in quite difficult conditions of aliasing and noise the theoretical study of the analyzed schemes is illustrated by some experiments on synthetic phase signals

absolute phase image reconstruction a stochastic nonlinear filtering approach this paper formulates and proposes solutions to the problem of estimating reconstructing the absolute not simply modulo 2 spl pi phase of a complex random field from noisy observations of its real and imaginary parts this problem is representative of a class of important imaging techniques such as interferometric synthetic aperture radar optical interferometry magnetic resonance imaging and diffraction tomography we follow a bayesian approach then not only a probabilistic model of the observation mechanism but also prior knowledge concerning the phase image to be reconstructed are needed we take as prior a nonsymmetrical half plane autoregressive nshp ar gauss markov random field gmrf based on a reduced order state space formulation of the linear nshp ar model and on the nonlinear observation mechanism a recursive stochastic nonlinear filter is derived the corresponding estimates are compared with those obtained by the extended kalman bucy filter a classical linearizing approach to the same problem a set of examples illustrate the effectiveness of the proposed approach

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

on steepest descent algorithms for discrete convex functions this paper investigates the complexity of steepest descent algorithms for two classes of discrete convex functions m convex functions and l convex functions simple tie breaking rules yield complexity bounds that are polynomials in the dimension of the variables and the size of the effective domain combining the present results with a standard scaling approach leads to an efficient algorithm for l convex function minimization

an efficient algorithm for image segmentation markov random fields and related problems problems of statistical inference involve the adjustment of sample observations so they fit some a priori rank requirements or order constraints in such problems the objective is to minimize the deviation cost function that depends on the distance between the observed value and the modify value in markov random field problems there is also a pairwise relationship between the objects the objective in markov random field problem is to minimize the sum of the deviation cost function and a penalty function that grows with the distance between the values of related pairs separation function we discuss markov random fields problems in the context of a representative application the image segmentation problem in this problem the goal is to modify color shades assigned to pixels of an image so that the penalty function consisting of one term due to the deviation from the initial color shade and a second term that penalizes differences in assigned values to neighboring pixels is minimized we present here an algorithm that solves the problem in polynomial time when the deviation function is convex and separation function is linear and in strongly polynomial time when the deviation cost function is linear quadratic or piecewise linear convex with few pieces where few means a number exponential in a polynomial function of the number of variables and constraints the complexity of the algorithm for a problem on n pixels or variables m adjacency relations or constraints and range of variable values colors u is o t n m n log u where t n m is the complexity of solving the minimum s t cut problem on a graph with n nodes and m arcs furthermore other algorithms are shown to solve the problem with convex deviation and convex separation in running time o mn log n log nu and the problem with nonconvex deviation and convex separation in running time o t nu mu the nonconvex separation problem is np hard even for fixed value of u for the family of problems with convex deviation functions and linear separation function the algorithm described here runs in polynomial time which is demonstrated to be fastest possible

two dimensional phase unwrapping using a minimum spanning tree algorithm phase unwrapping refers to the determination of phase from modulo 2 pi data some of which may not be reliable in 2d this is equivalent to confining the support of the phase function to one or more arbitrarily shaped regions a phase unwrapping algorithm is presented which works for 2d data known only within a set of nonconnected regions with possibly nonconvex boundaries the algorithm includes the following steps segmentation to identify connectivity phase unwrapping within each segment using a taylor series expansion phase unwrapping between disconnected segments along an optimum path and filling of phase information voids the optimum path for intersegment unwrapping is determined by a minimum spanning tree algorithm although the algorithm is applicable to any 2d data the main application addressed is magnetic resonance imaging mri where phase maps are useful

residues cluster based segmentation and outlier detection method for large scale phase unwrapping 2 d phase unwrapping is an important technique in many applications however with the growth of image scale how to tile and splice the image effectively has become a new challenge in this paper the phase unwrapping problem is abstracted as solving a large scale system of inconsistent linear equations with the difficulties of large scale phase unwrapping analyzed l0 norm criterion is found to have potentials in efficient image tiling and splicing making use of the clustering characteristic of residue distribution a tiling strategy is proposed for l0 norm criterion unfortunately l0 norm is an np hard problem which is very difficult to find an exact solution in a polynomial time in order to effectively solve this problem equations corresponding to branch cuts of l0 norm in the inconsistent equation system mentioned earlier are considered as outliers and then an outlier detection based phase unwrapping method is proposed through this method a highly accurate approximate solution to this np hard problem is achieved a set of experimental results shows that the proposed approach can avoid the inconsistency between local and global phase unwrapping solutions caused by image tiling

supercharacters and superclasses for algebra groups we study certain sums of irreducible characters and compatible unions of conjugacy classes in finite algebra groups these groups generalize the unimodular upper triangular groups over a finite field and the supercharacter theory we develop extends results of carlos andre and ning yan that were originally proved in the upper triangular case this theory sometimes allows explicit computations in situations where it would be impractical to work with the full character table we discuss connections with the kirillov orbit method and with gelfand pairs and we give conditions for a supercharacter or a superclass to be an ordinary irreducible character or conjugacy class respectively we also show that products of supercharacters are positive integer combinations of supercharacters

exact optimization for markov random fields with convex priors we introduce a method to solve exactly a first order markov random field optimization problem in more generality than was previously possible the mrf has a prior term that is convex in terms of a linearly ordered label set the method maps the problem into a minimum cut problem for a directed graph for which a globally optimal solution can be found in polynomial time the convexity of the prior function in the energy is shown to be necessary and sufficient for the applicability of the method

model based phase unwrapping of 2 d signals a parametric model and a corresponding parameter estimation algorithm for unwrapping 2 d phase functions are presented the proposed algorithm performs global analysis of the observed signal since this analysis is based on parametric model fitting the proposed phase unwrapping algorithm has low sensitivity to phase aliasing due to low sampling rates and noise as well as to local errors in its first step the algorithm fits a 2 d polynomial model to the observed phase the estimated phase is then used as a reference information that directs the actual phase unwrapping process the phase of each sample of the observed field is unwrapped by increasing decreasing it by the multiple of 2 spl pi which is the nearest to the difference between the principle value of the phase and the estimated phase value at this coordinate in practical applications the entire phase function cannot be approximated by a single 2 d polynomial model hence the observed field is segmented and each segment is fit with its own model once the phase model of the observed field has been estimated we can repeat the model based unwrapping procedure described earlier for the case of a single segment and a single model field

centrally generated primitive ideals of u n for exceptional types abstract let g be a complex semisimple lie algebra let b be a borel subalgebra of g let n be the nilradical of b and let u n be the universal enveloping algebra of n we study primitive ideals of u n almost all primitive ideals are centrally generated i e are generated by their intersections with the center z n of u n we present an explicit characterization of the centrally generated primitive ideals of u n in terms of the dixmier map and the kostant cascade in the case when g is a simple algebra of exceptional type for classical simple lie algebras a similar characterization was obtained by ivan penkov and the first author as a corollary we establish a classification of centrally generated primitive ideals of u n for an arbitrary semisimple algebra g

the inverse problem of the calculus of variations for scalar fourth order ordinary differential equations a simple invariant characterization of the scalar fourth order ordinary differential equations which admit a variational multiplier is given the necessary and sufficient conditions for the existence of a multiplier are expressed in terms of the vanishing of two relative invariants which can be associated with any fourth order equation through the application of cartan s equivalence method the solution to the inverse problem for fourth order scalar equations provides the solution to an equivalence problem for secondorder lagrangians as well as the precise relationship between the symmetry algebra of a variational equation and the divergence symmetry algebra of the associated lagrangian

fast approximate energy minimization via graph cuts many tasks in computer vision involve assigning a label such as disparity to every pixel a common constraint is that the labels should vary smoothly almost everywhere while preserving sharp discontinuities that may exist e g at object boundaries these tasks are naturally stated in terms of energy minimization the authors consider a wide class of energies with various smoothness constraints global minimization of these energy functions is np hard even in the simplest discontinuity preserving case therefore our focus is on efficient approximation algorithms we present two algorithms based on graph cuts that efficiently find a local minimum with respect to two types of large moves namely expansion moves and swap moves these moves can simultaneously change the labels of arbitrarily large sets of pixels in contrast many standard algorithms including simulated annealing use small moves where only one pixel changes its label at a time our expansion algorithm finds a labeling within a known factor of the global minimum while our swap algorithm handles more general energy functions both of these algorithms allow important cases of discontinuity preserving energies we experimentally demonstrate the effectiveness of our approach for image restoration stereo and motion on real data with ground truth we achieve 98 percent accuracy

a robust wrap reduction algorithm for fringe projection profilometry and applications in magnetic resonance imaging in this paper we present an effective algorithm to reduce the number of wraps in a 2d phase signal provided as input the technique is based on an accurate estimate of the fundamental frequency of a 2d complex signal with the phase given by the input and the removal of a dependent additive term from the phase map unlike existing methods based on the discrete fourier transform dft the frequency is computed by using noise robust estimates that are not restricted to integer values then to deal with the problem of a non integer shift in the frequency domain an equivalent operation is carried out on the original phase signal this consists of the subtraction of a tilted plane whose slope is computed from the frequency followed by a re wrapping operation the technique has been exhaustively tested on fringe projection profilometry fpp and magnetic resonance imaging mri signals in addition the performance of several frequency estimation methods has been compared the proposed methodology is particularly effective on fpp signals showing a higher performance than the state of the art wrap reduction approaches in this context it contributes to canceling the carrier effect at the same time as it eliminates any potential slope that affects the entire signal its effectiveness on other carrier free phase signals e g mri is limited to the case that inherent slopes are present in the phase data

statistical mechanics approach to the phase unwrapping problem the use of mean field theory to unwrap principal phase patterns has been recently proposed in this paper we generalize the mean field approach to process phase patterns with arbitrary degree of undersampling the phase unwrapping problem is formulated as that of finding the ground state of a locally constrained finite size spin l ising model under a non uniform magnetic field the optimization problem is solved by the mean field annealing technique synthetic experiments show the effectiveness of the proposed algorithm

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

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

sar image regularization with fast approximate discrete minimization synthetic aperture radar sar images like other coherent imaging modalities suffer from speckle noise the presence of this noise makes the automatic interpretation of images a challenging task and noise reduction is often a prerequisite for successful use of classical image processing algorithms numerous approaches have been proposed to filter speckle noise markov random field mrf modelization provides a convenient way to express both data fidelity constraints and desirable properties of the filtered image in this context total variation minimization has been extensively used to constrain the oscillations in the regularized image while preserving its edges speckle noise follows heavy tailed distributions and the mrf formulation leads to a minimization problem involving nonconvex log likelihood terms such a minimization can be performed efficiently by computing minimum cuts on weighted graphs due to memory constraints exact minimization although theoretically possible is not achievable on large images required by remote sensing applications the computational burden of the state of the art algorithm for approximate minimization namely the alpha expansion is too heavy specially when considering joint regularization of several images we show that a satisfying solution can be reached in few iterations by performing a graph cut based combinatorial exploration of large trial moves this algorithm is applied to joint regularization of the amplitude and interferometric phase in urban area sar images

a lagrangian description of the higher order painleve equations we derive the lagrangians of the higher order painleve equations using jacobi s last multiplier technique some of these higher order differential equations display certain remarkable properties like passing the painleve test and satisfy the conditions stated by juras thus allowing for a lagrangian description

an optimal graph theoretic approach to data clustering theory and its application to image segmentation a novel graph theoretic approach for data clustering is presented and its application to the image segmentation problem is demonstrated the data to be clustered are represented by an undirected adjacency graph g with arc capacities assigned to reflect the similarity between the linked vertices clustering is achieved by removing arcs of g to form mutually exclusive subgraphs such that the largest inter subgraph maximum flow is minimized for graphs of moderate size approximately 2000 vertices the optimal solution is obtained through partitioning a flow and cut equivalent tree of g which can be efficiently constructed using the gomory hu algorithm 1961 however for larger graphs this approach is impractical new theorems for subgraph condensation are derived and are then used to develop a fast algorithm which hierarchically constructs and partitions a partially equivalent tree of much reduced size this algorithm results in an optimal solution equivalent to that obtained by partitioning the complete equivalent tree and is able to handle very large graphs with several hundred thousand vertices the new clustering algorithm is applied to the image segmentation problem the segmentation is achieved by effectively searching for closed contours of edge elements equivalent to minimum cuts in g which consist mostly of strong edges while rejecting contours containing isolated strong edges this method is able to accurately locate region boundaries and at the same time guarantees the formation of closed edge contours

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

super characters of finite unipotent groups of types b n c n and d n abstract we define and study super characters over the complex field of the classical finite unipotent groups of types b n c n and d n under the assumption that the prime is sufficiently large we extend the known results for the unitriangular group proved by the first author in the papers c a m andre basic characters of the unitriangular group j algebra 175 1995 287 319 and c a m andre basic characters of the unitriangular group for arbitrary primes proc amer math soc 130 7 2002 1943 1954 in particular we prove that every irreducible complex character occurs as a constituent of a unique super character we also give a combinatorial description of all the irreducible characters of maximum degree

phase unwrapping of mr phase images using poisson equation the authors have developed a technique based on a solution of the poisson equation to unwrap the phase in magnetic resonance mr phase images the method is based on the assumption that the magnitude of the inter pixel phase change is less than spl pi per pixel therefore the authors obtain an estimate of the phase gradient by wrapping the gradient of the original phase image the problem is then to obtain the absolute phase given the estimate of the phase gradient the least squares ls solution to this problem is shown to be a solution of the poisson equation allowing the use of fast poisson solvers the absolute phase is then obtained by mapping the ls phase to the nearest multiple of 2 k from the measured phase the proposed technique is evaluated using mr phase images and is proven to be robust in the presence of noise an application of the proposed method to the 3 point dixon technique for water and fat separation is demonstrated

phase local approximation phasela technique for phase unwrap from noisy data the local polynomial approximation lpa is a nonparametric regression technique with pointwise estimation in a sliding window we apply the lpa of the argument of cos and sin in order to estimate the absolute phase from noisy wrapped phase data using the intersection of confidence interval ici algorithm the window size is selected as adaptive pointwise varying this adaptation gives the phase estimate with the accuracy close to optimal in the mean squared sense for calculations we use a gauss newton recursive procedure initiated by the phase estimates obtained for the neighboring points it enables tracking properties of the algorithm and its ability to go beyond the principal interval pi pi and to reconstruct the absolute phase from wrapped phase observations even when the magnitude of the phase difference takes quite large values the algorithm demonstrates a very good accuracy of the phase reconstruction which on many occasion overcomes the accuracy of the state of the art algorithms developed for noisy phase unwrap the theoretical analysis produced for the accuracy of the pointwise estimates is used for justification of the ici adaptation algorithm

a new approach to the maximum flow problem all previously known efficient maximum flow algorithms work by finding augmenting paths either one path at a time as in the original ford and fulkerson algorithm or all shortest length augmenting paths at once using the layered network approach of dinic an alternative method based on the preflow concept of karzanov is introduced a preflow is like a flow except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out the method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths the algorithm and its analysis are simple and intuitive yet the algorithm runs as fast as any other known method on dense graphs achieving an o n 3 time bound on an n vertex graph by incorporating the dynamic tree data structure of sleator and tarjan we obtain a version of the algorithm running in o nm log n 2 m time on an n vertex m edge graph this is as fast as any known method for any graph density and faster on graphs of moderate density the algorithm also admits efficient distributed and parallel implementations a parallel implementation running in o n 2 log n time using n processors and o m space is obtained this time bound matches that of the shiloach vishkin algorithm which also uses n processors but requires o n 2 space

a supercharacter theory for the sylow p subgroups of the finite symplectic and orthogonal groups abstract we define the superclasses for a classical finite unipotent group u of type b n q c n q or d n q and show that together with the supercharacters defined in c a m andre a m neto supercharacters of the sylow p subgroups of the finite symplectic and orthogonal groups pacific j math 239 2 2009 201 230 they form a supercharacter theory in the sense of p diaconis i m isaacs supercharacters and superclasses for algebra groups trans amer math soc 360 5 2008 2359 2392 in particular we prove that the supercharacters take a constant value on each superclass and evaluate this value as a consequence we obtain a factorization of any superclass as a product of elementary superclasses in addition we also define the space of superclass functions and prove that it is spanned by the supercharacters as a consequence we re obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters finally we define the supercharacter table of u and prove various orthogonality relations for supercharacters similar to the well known orthogonality relations for irreducible characters