Image Analysis and Reconstruction using a Wavelet Transform Constructed from a Reducible Representation of the Euclidean Motion Group

Inspired by the early visual system of many mammalians we consider the construction of-and reconstruction from- an orientation score as a local orientation representation of an image, . The mapping is a wavelet transform corresponding to a reducible representation of the Euclidean motion group onto and oriented wavelet . This wavelet transform is a special case of a recently developed generalization of the standard wavelet theory and has the practical advantage over the usual wavelet approaches in image analysis (constructed by irreducible representations of the similitude group) that it allows a stable reconstruction from one (single scale) orientation score. Since our wavelet transform is a unitary mapping with stable inverse, we directly relate operations on orientation scores to operations on images in a robust manner. Furthermore, by geometrical examination of the Euclidean motion group , which is the domain of our orientation scores, we deduce that an operator Φ on orientation scores must be left invariant to ensure that the corresponding operator on images is Euclidean invariant. As an example we consider all linear second order left invariant evolutions on orientation scores corresponding to stochastic processes on G. As an application we detect elongated structures in (medical) images and automatically close the gaps between them. Finally, we consider robust orientation estimates by means of channel representations, where we combine robust orientation estimation and learning of wavelets resulting in an auto-associative processing of orientation features. Here linear averaging of the channel representation is equivalent to robust orientation estimation and an adaptation of the wavelet to the statistics of the considered image class leads to an auto-associative behavior of the system. The Netherlands Organization for Scientific Research is gratefully acknowledged for financial support. This work has been supported by EC Grant IST-2003-004176 COSPAL.     

Crash failure detection in asynchronous agent communication languages

Agent Communication Languages (ACLs) have been developed to provide a way for agents to communicate with each other supporting cooperation in Multi-Agent Systems (MAS). In the past few years many ACLs have been proposed for MAS and new standards are emerging such as the ACL developed by the Foundation for Intelligent Physical Agents (FIPA). Despite these efforts, an important issue in the research on ACLs is still open and concerns how these languages should deal with failures of agents in asynchronous MAS. The Fault Tolerant Agent Communication Language ( - ) presented in this paper addresses this issue dealing with crash failures of agents. - provides high-level communication primitives which support a fault-tolerant anonymous interaction protocol designed for open MAS. We present a formal semantics for - and a formal specification of the underlying agent architecture. This formal framework allows us to prove that the ACL satisfies a set of well defined knowledge-level programming requirements. To illustrate the language features we show how - can be effectively used to write high-level executable specifications of fault tolerant protocols, such as the Contract Net one.     

Complexity of some arithmetic problems for binary polynomials

We study various combinatorial complexity measures of Boolean functions related to some natural arithmetic problems about binary polynomials, that is, polynomials over . In particular, we consider the Boolean function deciding whether a given polynomial over is squarefree. We obtain an exponential lower bound on the size of a decision tree for this function, and derive an asymptotic formula, having a linear main term, for its average sensitivity. This allows us to estimate other complexity characteristics such as the formula size, the average decision tree depth and the degrees of exact and approximative polynomial representations of this function. Finally, using a different method, we show that testing squarefreeness and irreducibility of polynomials over cannot be done in for any odd prime p. Similar results are obtained for deciding coprimality of two polynomials over as well.     

Cardinal interpolation with periodic polysplines on strips

Abstract  We obtain a multivariate extension of a classical result of Schoenberg on cardinal spline interpolation. Specifically, we prove the existence of a unique function in , polyharmonic of order p on each strip , , and periodic in its last n variables, whose restriction to the parallel hyperplanes , , coincides with a prescribed sequence of n-variate periodic data functions satisfying a growth condition in . The constructive proof is based on separation of variables and on Micchelli’s theory of univariate cardinal -splines. Keywords: cardinal -splines, polyharmonic functions, multivariable interpolation Mathematics Subject Classification (2000): 41A05, 41A15, 41A63     

Quantum Orlicz Spaces in Information Geometry

Let H₀ be a selfadjoint operator such that Tr is of trace class for some , and let denote the set of ε-bounded forms, i.e., for some 0 $$" align="middle" border="0"> . Let χ := Span . Let denote the underlying set of the quantum information manifold of states of the form . We show that if Tr ,

Stability of Linear Difference and Differential Inclusions

Any given n×n matrix A is shown to be a restriction, to the A-invariant subspace, of a nonnegative N×N matrix B of spectral radius (B) arbitrarily close to (A). A difference inclusion , where is a compact set of matrices, is asymptotically stable if and only if can be extended to a set of nonnegative matrices B with or . Similar results are derived for differential inclusions.     

Valiant’s model and the cost of computing integers

Let (n) be the minimum number of arithmetic operations required to build the integer from the constants 1 and 2. A sequence x_n is said to be easy to compute if there exists a polynomial p such that for all It is natural to conjecture that sequences such as or n! are not easy to compute. In this paper we show that a proof of this conjecture for the first sequence would imply a superpolynomial lower bound for the arithmetic circuit size of the permanent polynomial. For the second sequence, a proof would imply a superpolynomial lower bound for the permanent or P PSPACE.     

The complexity of constructing pseudorandom generators from hard functions

We study the complexity of constructing pseudorandom generators (PRGs) from hard functions, focussing on constant-depth circuits. We show that, starting from a function computable in alternating time O(l) with O(1) alternations that is hard on average (i.e. there is a constant such that every circuit of size fails to compute f on at least a 1/poly(l) fraction of inputs) we can construct a computable by DLOGTIME-uniform constant-depth circuits of size polynomial in n. Such a PRG implies under DLOGTIME-uniformity. On the negative side, we prove that starting from a worst-case hard function (i.e. there is a constant such that every circuit of size fails to compute f on some input) for every positive constant there is no black-box construction of a computable by constant-depth circuits of size polynomial in n. We also study worst-case hardness amplification, which is the related problem of producing an average-case hard function starting from a worst-case hard one. In particular, we deduce that there is no blackbox worst-case hardness amplification within the polynomial time hierarchy. These negative results are obtained by showing that polynomialsize constant-depth circuits cannot compute good extractors and listdecodable codes.     

Linear Interval Equations: Midpoint Preconditioning May Produce a 100% Overestimation for Arbitrarily Narrow Data Even in Case <Emphasis Type="Italic">n</Emphasis> = 4

We construct a linear interval system Ax = b with a 4 × 4 interval matrix whose all proper interval coefficients (there are also some noninterval ones) are of the form [–, ]. It is proved that for each > 0, the interval hull and interval hull of the midpoint preconditioned system satisfy and , hence midpoint preconditioning produces a 100% overestimation of independently of in this case. The example was obtained as a result of an extensive MATLAB search.     

An algebraic approach for $${\mathcal{H}}$$-matrix preconditioners

Hierarchical matrices ( -matrices) approximate matrices in a data-sparse way, and the approximate arithmetic for -matrices is almost optimal. In this paper we present an algebraic approach for constructing -matrices which combines multilevel clustering methods with -matrix arithmetic to compute the -inverse, -LU, and the -Cholesky factors of a matrix. Then the -inverse, -LU or -Cholesky factors can be used as preconditioners in iterative methods to solve systems of linear equations. The numerical results show that this method is efficient and greatly speeds up convergence compared to other approaches, such as JOR or AMG, for solving some large, sparse linear systems, and is comparable to other -matrix constructions based on Nested Dissection.     

On the complexity of computing determinants

We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n × n matrix A with integer entries in and bit operations; here denotes the largest entry in absolute value and the exponent adjustment by +o(1) captures additional factors for positive real constants C₁, C₂, C₃. The bit complexity results from using the classical cubic matrix multiplication algorithm. Our algorithms are randomized, and we can certify that the output is the determinant of A in a Las Vegas fashion. The second category of problems deals with the setting where the matrix A has elements from an abstract commutative ring, that is, when no divisions in the domain of entries are possible. We present algorithms that deterministically compute the determinant, characteristic polynomial and adjoint of A with n^3.2+o(1) and O(n^2.697263) ring additions, subtractions and multiplications.To B. David Saunders on the occasion of his 60th birthday     

A Modular Algorithm for Computing Cohomologies of Lie Algebras and Superalgebras

The paper describes an improved algorithm for computing cohomologies of Lie (super)algebras. The original algorithm developed earlier by the author of this paper is based on the decomposition of the entire cochain complex into minimal subcomplexes. The suggested improvement consists in the replacement of the arithmetic of rational or integer numbers by a more efficient arithmetic of modular fields and the use of the relationship dim H ^k( _p) dimH ^k() between the dimensions of cohomologies over an arbitrary modular field _p = /p and the filed of rational numbers . This inequality allows us to rapidly find subcomplexes for which dimH ^k( _p) > 0 (the number of such subcomplexes is usually not great) using computations over an arbitrary _p and, then, carry out all required computations over in these subcomplexes.     

A Time-Splitting Spectral Method for the Generalized Zakharov System in Multi-Dimensions

The generalized Zakharov system (ZS) couples a dispersive field E (scalar or vectorial) and nondispersive fields with a propagating speed of . In this paper, we extend our one-dimensional time-splitting spectral method (TSSP) for the generalized ZS into higher dimension. A main new idea is to reformulate the multi-dimensional wave equations for the nondispersive fields into a first-order system using a change of variable defined in the Fourier space. The proposed scheme TSSP is unconditionally stable, second-order in time and spectrally accurate in space. Moreover, in the subsonic regime, it allows numerical capturing of the subsonic limit without resolving the small parameters . Numerical examples confirm these properties of this method