a generalized depth estimation algorithm with a single image a depth estimation algorithm proposed by a p pentland 1987 is generalized in the proposed algorithm the raw image data in the vicinity of the edge is used to estimate the depth from defocus since no differentiation operation on the image data is required before the optimization process the method is less sensitive to the noise disturbance of measurements furthermore the edge orientation that was critical in pentland s approach will not be required in the case this algorithm is then applied to synthetic images containing various amounts of noise to test its performance experimental results indicate that the depth estimation errors are kept within 5 of true values on the average when it is applied to real images

singular riemannian foliations on nonpositively curved manifolds we prove the nonexistence of a proper singular riemannian foliation admitting section in compact manifolds of nonpositive curvature then we give a global description of proper singular riemannian foliations admitting sections on hadamard manifolds in addition by using the theory of taut immersions we provide a short proof of this result in the special case of a polar action

moments of utility functions and their applications this paper presents moments and cross moments of utility functions and measures of utility dependence we start with an interpretation of the nth moment of a utility function and describe methods for its assessment in practice and consistency checks that need to be satisfied for any assessed moments we then show how moments of a utility function i provide a new method to determine the parameters of a given functional form of a utility function and ii to derive the functional form of a utility function that satisfies some given moment assessments next we derive a fundamental formula that relates the expected utility of a joint distribution to the expected utility of the marginal distributions for multiattribute utility functions we use this formulation to provide an intuitive interpretation for cross moments of utility functions and illustrate their use in i constructing multiattribute utility functions that incorporate utility dependence and ii in providing necessary conditions for utility independence in decisions with multiple attributes we end with a new measure of utility dependence for multiattribute utility functions and work through several examples to illustrate the approach

bivariate distributions with exponential conditionals abstract it is frequently easier to visualize conditional distributions of experimental variables rather than joint distributions in this article we consider the most general class of bivariate distributions such that both sets of conditional densities are exponential the class proves to be remarkably simple to describe the joint density must be proportional to exp x y xy where the constant of proportionality depends on the classical exponential integral the joint distribution has marginals that are not exponential and a negative correlation coefficient except in the special case of independence after deriving some distributional results we develop methods for parameter estimation and simulation a simple method of moments estimator appears to give reasonable results we also briefly discuss generalizations to higher dimensions and to distributions with conditionals in a general exponential family

the pearson system of utility functions this paper describes a parametric family of utility functions for decision analysis the parameterization embeds the hara class in a four parameter representation for the risk aversion function the resulting utility functions can have only four shapes concave convex s shaped and reverse s shaped this makes the family suited for both expected utility and prospect theory the paper also describes an alternative technique to estimate the four parameters from elicited utilities which is simpler than standard fitting by minimization of the mean quadratic error

solutions to the overdetermined boundary problem for semilinear equations with position dependent nonlinearities abstract we show that a wide range of overdetermined boundary problems for semilinear equations with position dependent nonlinearities admits nontrivial solutions the result holds true both on r n and on compact riemannian manifolds as a byproduct of the proofs we also obtain some rigidity or partial symmetry results for solutions to overdetermined problems on riemannian manifolds of nonconstant curvature

reasoning about edges in scale space explores the role of reasoning in early vision processing in particular the problem of detecting edges is addressed the authors do not try to develop another edge detector but rather they study an edge detector rigorously to understand its behavior well enough to formulate a reasoning process that allow appliance of the detector judiciously to recover useful information they present a multiscale reasoning algorithm for edge recovery reasoning about edges in scale space ress the knowledge in ress is acquired from the theory of edge behavior in scale space and represented by a number of procedures ress recovers desired edge curves through a number of reasoning processes on zero crossing images at various scales the knowledge of edge behavior in scale space enables ress to select proper scale parameters recover missing edges eliminate noise or false edges and correct the locations of edges a brief evaluation of ress is performed by comparing it with two well known multistage edge detection algorithms

toward a symbolic representation of intensity changes in images the symbolic representation of gray value variations is studied with emphasis on the gradient of the image function the goal is to relate the results of this analysis to the structure of the picture which is determined by the physics of the image generation process candidates for contour points are the maximal magnitudes of the gray value gradient for different scales in the direction of the gradient based on the output of such a bank of gradient filters a procedure is proposed to select automatically a suitable scale and with that the size of the right convolution kernel the application of poorly adapted filters which make the exact localization of gray value corners or t x and y junctions more difficult is thus avoided possible gaps at such junctions are discussed for images of real scenes and possibilities for the closure of some of these gaps are demonstrated when the extrema of the magnitudes of the gray value gradients are used

modeling edges at subpixel accuracy using the local energy approach in this paper we describe a new technique for id and 2d edge feature extraction to subpixel accuracy using edge models and the local energy approach a candidate edge is modeled as one of a number of parametric edge models and the fit is refined by a least squared error fitting technique

singularity detection and processing with wavelets the mathematical characterization of singularities with lipschitz exponents is reviewed theorems that estimate local lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed it is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their lipschitz exponents the wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately the local frequency of such oscillations is measured from the wavelet transform modulus maxima it has been shown numerically that one and two dimensional signals can be reconstructed with a good approximation from the local maxima of their wavelet transform modulus as an application an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales in two dimensions the wavelet transform maxima indicate the location of edges in images

submanifold geometry in symmetric spaces the classical local invariants of a submanifold in a space form are the first fundamental form the shape operators and the induced normal connection and they determine the submanifold up to ambient isometry one of the main topics in differential geometry is to study the relation between the local invariants and the global geometry and topology of submanifolds many remarkable results have been developed for submanifolds in space forms whose local invariants satisfy certain natural conditions the study of focal points of a submanifold in an arbitrary riemannian manifold arises from the morse theory of the energy functional on the space of paths in the riemannian manifold joining a fixed point to the submanifold the morse index theorem relates the geometry of a submanifold to the topology of this path space the focal structure is intimately related to the local invariants of the submanifold in the case of space forms one can go backwards and reconstruct the local invariants from the focal structure so it is not too surprising that most of the structure theory of submanifolds can be reformulated in terms of their focal structure what is perhaps surprising is a fact that became increasingly evident to the authors from their individual and joint research over the past decade while extending the theory of submanifolds to ambient spaces more general than space forms proves quite difficult if one tries to use the same approach as for the space forms at least for symmetric spaces it has proved possible to develop an elegant theory based on focal structure that reduces to the classical theory in the case of space forms this paper is an extended report on this theory and the authors believe that the methods developed herein provide important tools for a continuing study of the submanifold geometry in symmetric spaces first we set up some notations let n g be a riemannian manifold m an immersed submanifold of n and m the normal bundle of m the end point map m n of m is the restriction of the exponential map exp to m if v m x is a singular point of and the dimension of the kernel of d v is m then v is called a multiplicity m focal normal and exp v is called a multiplicity m focal point of m with respect to m in n the focal data m is defined to be the set of all pairs v m such that v is a multiplicity m focal normal of m the focal variety v m is the set of all pairs v m with v m m the main purpose of our paper is to study the global geometry and topology

coisotropic and polar actions on complex grassmannians the main result of the paper is the complete classification of the compact connected lie groups acting coisotropically on complex grassmannians this is used to determine the polar actions on the same manifolds

multi scale blur estimation and edge type classification for scene analysis signatures in this work are multi scale representations of local gray level information tied to places in gray scale images where regional differences are locally maximal the information may involve the regional differences themselves called gaussian differences or signed normalized gradient magnitudes korn 1988 or distance relations between edges apparent width measurements or absence of edges in pulse edge pairs at coarser scales using signatures involves the classical problem mentioned by marr and others of relating information across scales a novel result is that a fruitful way of doing this is to build scale paths from coarse to fine exploiting edge focusing and associate with pixel positions along these paths the three quantities gaussian differences apparent width and the binary information absence presence of edges in edge pairs such a structure if used together with proper conditional tests serves the purpose of classifying edges with respect to profile type and can also be used for measuring global contrast degree of diffuseness deblurred line width and qualitative labels such as diffuse versus sharp the structure is used simultaneously for labelling tasks and quantitative measurements theory on apparent widths absence presence of edges in pulse edge pairs is developed for measuring diffuseness and global contrast from gaussian difference signatures a linear least squares approach is suggested extensive experimental results are presented possible applications are in image segementation junction analysis and depth from defocus for the purpose of distinguishing between objects and illumination phenomena such as diffuse shadow edges classification of contours with respect to diffuseness seems useful

the design and use of steerable filters the authors present an efficient architecture to synthesize filters of arbitrary orientations from linear combinations of basis filters allowing one to adaptively steer a filter to any orientation and to determine analytically the filter output as a function of orientation steerable filters may be designed in quadrature pairs to allow adaptive control over phase as well as orientation the authors show how to design and steer the filters and present examples of their use in the analysis of orientation and phase angularly adaptive filtering edge detection and shape from shading one can also build a self similar steerable pyramid representation the same concepts can be generalized to the design of 3 d steerable filters

parallel focal structure and singular riemannian foliations we give a necessary and sufficient condition for a submanifold with parallel focal structure to give rise to a global foliation of the ambient space by parallel and focal manifolds we show that this is a singular riemannian foliation with complete orthogonal transversals for this object we construct an action on the transversals that generalizes the weyl group action for polar actions

polar coordinates induced by actions of compact lie groups let g be a connected lie subgroup of the real orthogonal group 0 n for the action of g on r we construct linear subspaces a that intersect all orbits we determine for which g there exists such an a meeting all the g orbits orthogo nally groups that act transitively on spheres are obvious examples with few exceptions all possible g arise as the isotropy subgroups of riemannian symmetric spaces introduction let g be a compact lie group acting on a real vectorspace v and let k be a g invariant inner product on v having a linear cross section a c v of

a classification of multiplicity free actions abstract the action of a complex algebraic groupgon an affine varietyvis said to be multiplicity free if each irreducible representation ofgoccurs at most once in the ring c v of regular functions onv this paper concerns the classical setting wherevis a vector space andgacts linearly onv the irreducible linear multiplicity free actions have been classified by v kac the focus here is on nonirreducible linear multiplicity free actions we exhibit many such actions which do not decompose as direct products of irreducible actions our main results provide a classification for all linear multiplicity free actions on a vector space

polar actions on compact rank one symmetric spaces are taut we prove that the orbits of a polar action of a compact lie group on a compact rank one symmetric space are tautly embedded with respect to z2 coefficients

coisotropic and polar actions on compact irreducible hermitian symmetric spaces we obtain the full classification of coisotropic and polar isometric actions of compact lie groups on irreducible hermitian symmetric spaces

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

hyperpolar homogeneous foliations on symmetric spaces of noncompact type a foliation f on a riemannian manifold m is homogeneous if its leaves coincide with the orbits of an isometric action on m a foliation f is polar if it admits a section that is a connected closed totally geodesic submanifold of m which intersects each leaf of f and intersects orthogonally at each point of intersection a foliation f is hyperpolar if it admits a flat section these notes are related to joint work with jose carlos diaz ramos and hiroshi tamaru about hyperpolar homogeneous foliations on riemannian symmetric spaces of noncompact type apart from the classification result which we proved in 1 we present here in more detail some relevant material about symmetric spaces of noncompact type and discuss the classification in more detail for the special case m slr 1 r sor 1 1 isometric actions on the real hyperbolic plane the special linear group g sl2 r a b c d a b c d r ad bc 1 acts on the upper half plane h z c z 0 by linear fractional transformations of the form a b c d z az b cz d by equipping h with the riemannian metric ds dx dy y2 dzdz z 2 z x iy

polar and coisotropic actions on kahler manifolds the main result of the paper is that a polar action on a compact irreducible homogeneous kahler manifold is coisotropic this is then used to give new examples of polar actions and to classify coisotropic and polar actions on quadrics

robust decision making using a general utility set abstract we address the problem of ambiguity and inconsistency in a decision maker s dm assessed utility function by using a maxmin framework in this framework dm s utility function belongs to a set of functions the member functions of the set are nondecreasing and satisfy additional boundary and auxiliary conditions the maxmin framework provides robustness in decision making alternatively it allows us to perform sensitivity or parametric analysis for the optimal decision when a reference utility function is known for this purpose we use a cost of ambiguity concept and show that this cost is increasing and concave when extent of ambiguity is parametrically increased next we develop a lagrangian based solution approach for the decision problem we show that under suitable conditions a sample average approximation saa of the lagrangian model can be solved using a mixed integer linear program milp we also show that the set of optimal solutions of the saa converges to that of its true counterpart and the optimum objective value of the saa converges to the true objective value at an exponential rate we use this convergence property to develop a heuristic for identifying a solution of the saa milp with increasing sample size we illustrate the properties of the maxmin model using two examples of portfolio investment and streaming bandwidth we provide a discussion on the performance of a commercial solver used to solve saa milps and the quality of solutions generated by the proposed heuristic

a classification of hyperpolar and cohomogeneity one actions an isometric action of a compact lie group on a riemannian manifold is called hyperpolar if there exists a closed connected submanifold that is flat in the induced metric and meets all orbits orthogonally in this article a classification of hyperpolar actions on the irreducible riemannian symmetric spaces of compact type is given since on these symmetric spaces actions of cohomogeneity one are hyperpolar i e normal geodesics are closed we obtain a classification of the homogeneous hypersurfaces in these spaces by computing the cohomogeneity for all hyperpolar actions this result implies a classification of the cohomogeneity one actions on compact strongly isotropy irreducible homogeneous spaces

cohomogeneity one actions on noncompact symmetric spaces of rank one we classify up to orbit equivalence all cohomogeneity one actions on the hyperbolic planes over the complex quaternionic and cayley numbers and on the complex hyperbolic spaces ch n 3 for the quaternionic hyperbolic spaces mh n n 3 we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results for real hyperbolic spaces this classification problem was essentially solved by elie cartan

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

curvature and symmetry of milnor spheres since milnor s discovery of exotic spheres mi one of the most intriguing problems in riemannian geometry has been whether there are exotic spheres with positive curvature it is well known that there are exotic spheres that do not even admit metrics with positive scalar curvature hi on the other hand there are many examples of exotic spheres with positive ricci curvature cf ch1 he po and na and this work recently culminated in wr where it is shown that every exotic sphere that bounds a parallelizable manifold has a metric of positive ricci curvature this includes all exotic spheres in dimension 7 so far however no example of an exotic sphere with positive sectional curvature has been found in fact until now only one example of an exotic sphere with nonnegative sectional curvature was known the so called gromoll meyer sphere gm in dimension 7 as one of our main results we prove

polar actions on complex hyperbolic spaces we classify polar actions on complex hyperbolic spaces up to orbit equivalence 1 introduction and main results a proper isometric lie group action on a riemannian manifold is called polar if there exists an immersed submanifold that meets every orbit orthogonally such a submanifold is then called a section of the action in the special case where the section is flat in its induced riemannian metric the action is called hyperpolar in this article we classify polar actions on complex hyperbolic spaces the motivation for our work can be traced back to the work of dadok 12 who classified polar representations on euclidean spaces and the paper of palais and terng 30 who proved fundamental properties of polar actions on riemannian manifolds several years later the problem of classifying hyperpolar actions on symmetric spaces of compact type was posed in 18 hyperpolar actions on irreducible symmetric spaces of compact type have been classified by the third named author in 20 the classification of polar actions on compact symmetric spaces of rank one was obtained by podesta and thorbergsson 31 this classification shows that there are finitely many examples of polar non hyperpolar actions on each compact symmetric space of rank one after having completed the classification of polar actions on irreducible hermitian sym metric spaces of compact type biliotti 11 formulated the following conjecture a polar action on an irreducible symmetric space of compact type and higher rank is hyperpolar the third author proved that the conjecture holds for symmetric spaces with simple isom etry group 21 and for the exceptional simple lie groups 22 in recent work lytchak 26 obtained a decomposition theorem for the more general case of singular polar foliations on nonnegatively curved not necessarily irreducible riemannian symmetric spaces in particular his result shows that polar actions on irreducible symmetric spaces of higher rank are hyperpolar if the cohomogeneity is at least three kollross and lytchak 24 then completed the proof that biliotti s conjecture holds and hence that the classification of polar actions on irreducible symmetric spaces of compact type follows from 20 and 31 note that the classification of polar actions on reducible symmetric spaces cannot be ob tained from the corresponding classification in irreducible ones however by the structural

certainty equivalents for three point discrete distribution approximations three point discrete distribution approximations are often used in decision and risk analyses to represent probability distributions for continuous random variables e g as probability nodes in decision or probability trees performance evaluations of such approximations have generally been based on their accuracies in estimating moments for the underlying distributions however moment based comparisons for recently proposed approximations have been limited moreover very little research has addressed the accuracies of any such approximations in representing expected utilities or certainty equivalents of the underlying distributions this is of potential concern since recent research shows three point discrete distribution approximations exist that match the first several moments of an underlying distribution exactly while counterintuitively approximating its certainty equivalents poorly this paper compares the best two approximations for estimating means and variances identified in an earlier study with promising approximations proposed more recently specifically it examines how accurately six simple general purpose three point approximations represent certainty equivalents for continuous random variables as the level of risk aversion is varied as well as how accurately they estimate means and variances the results show that several of these are quite accurate over a variety of test distributions when the level of risk aversion and the characteristics of the distributions are within reasonable bounds their robust performance is significant for decision analysis practice

maximum entropy utility this paper presents a method to assign utility values when only partial information is available about the decision makers preferences we introduce the notion of a utility density function and a maximum entropy principle for utility assignment the maximum entropy utility solution embeds a large family of utility functions that includes the most commonly used functional forms we discuss the implications of maximum entropy utility on the preference behavior of the decision maker and present an application to competitive bidding situations where only previous decisions are observed by each party we also present minimum cross entropy utility which incorporates additional knowledge about the shape of the utility function into the maximum entropy formulation and work through several examples to illustrate the approach

a general theory of canonical forms si g est un groupe de lie compact et m une g variete riemannienne d orbites principales de codimension k alors une section ou une forme canonique pour m est une sous variete fermee lisse a k dimensions de m qui rencontre toutes les orbites de m orthogonalement