on numerical cubatures of singular surface integrals in boundary element methods we present and analyze methods for the accurate and efficient evaluation of weakly cauchy and hypersingular integrals over piecewise analytic curved surfaces in 3 the class of admissible integrands includes all kernels arising in the numerical solution of elliptic boundary value problems in three dimensional domains by the boundary integral equation method the possibly not absolutely integrable kernels of boundary integral operators in local coordinates are pseudohomogeneous with analytic characteristics depending on the local geometry of the surface at the source point this rules out weighted quadrature approaches with a fixed singular weight for weakly singular integrals it is shown that duffy s triangular coordinates leadalways to a removal of the kernel singularity also asymptotic estimates of the integration error are provided as the size of the boundary element patch tends to zero these are based on the rabinowitz richter estimates in connection with an asymptotic estimate of domains of analyticity in 2 it is further shown that the modified extrapolation approach due to lyness is in the weakly singular case always applicable corresponding error and asymptotic work estimates are presented for the weakly singular as well as for cauchy and hypersingular integrals which e g arise in the study of crack problems we analyze a family of product integration rules in local polar coordinates these rules are hierarchically constructed from finite part integration formulas in radial and gaussian formulas in angular direction again we show how the rabinowitz richter estimates can be applied providing asymptotic error estimates in terms of orders of the boundary element size

refinement equations and distributional fixed points abstract connections between l 1 solutions of the refinement equation of the form f x l f lx m dp and distributional fixed points of a special random affine map are investigated using results on convergence of perpetuities the direct form of the fourier transform of l 1 solutions of the above refinement equation is obtained

convergence of an iterative method for total variation denoising in total variation denoising one attempts to remove noise from a signal or image by solving a nonlinear minimization problem involving a total variation criterion several approaches based on this idea have recently been shown to be very effective particularly for denoising functions with discontinuities this paper analyzes the convergence of an iterative method for solving such problems the iterative method involves a lagged diffusivity approach in which a sequence of linear diffusion problems are solved global convergence in a finite dimensional setting is established and local convergence properties including rates and their dependence on various parameters are examined

on the unification of line processes outlier rejection and robust statistics with applications in early vision the modeling of spatial discontinuities for problems such as surface recovery segmentation image reconstruction and optical flow has been intensely studied in computer vision while line process models of discontinuities have received a great deal of attention there has been recent interest in the use of robust statistical techniques to account for discontinuities this paper unifies the two approaches to achieve this we generalize the notion of a line process to that of an analog outlier process and show how a problem formulated in terms of outlier processes can be viewed in terms of robust statistics we also characterize a class of robust statistical problems for which an equivalent outlier process formulation exists and give a straightforward method for converting a robust estimation problem into an outlier process formulation we show how prior assumptions about the spatial structure of outliers can be expressed as constraints on the recovered analog outlier processes and how traditional continuation methods can be extended to the explicit outlier process formulation these results indicate that the outlier process approach provides a general framework which subsumes the traditional line process approaches as well as a wide class of robust estimation problems examples in surface reconstruction image segmentation and optical flow are presented to illustrate the use of outlier processes and to show how the relationship between outlier processes and robust statistics can be exploited an appendix provides a catalog of common robust error norms and their equivalent outlier process formulations

refinement type equations and grincevicjus series abstract we consider l 1 solutions of the following refinement type equations f x n z c n 1 f k x n n z c n 1 f k x n where k 2 is an integer and for all n z reals c n 1 c n 1 are non negative with n z c n 1 c n 1 k and n z log n c n 1 c n 1 necessary and sufficient conditions for the existence of non trivial l 1 solutions in several special cases are given

globally convergent iterative numerical schemes for nonlinear variational image smoothing and segmentation on a multiprocessor machine we investigate several iterative numerical schemes for nonlinear variational image smoothing and segmentation implemented in parallel a general iterative framework subsuming these schemes is suggested for which global convergence irrespective of the starting point can be shown we characterize various edge preserving regularization methods from the image processing literature involving auxiliary variables as special cases of this general framework as a by product global convergence can be proven under conditions slightly weaker than these stated in the literature efficient krylov subspace solvers for the linear parts of these schemes have been implemented on a multiprocessor machine the performance of these parallel implementations has been assessed and empirical results concerning convergence rates and speed up factors are reported

on l1 solutions of a two direction refinement equation abstract in this paper we examine l 1 solutions of the following two direction refinement equation f x n n n c n 1 f k x n n n n c n 1 f k x n we prove that the vector space of all l 1 solutions of the above equation is at most one dimensional and consists of compactly supported functions of constant sign we also show that in many interesting cases any l 1 solution of the two direction refinement equation is either positive or negative on its support next we present sufficient conditions easy for verification for the existence of nontrivial l 1 solutions of the two direction refinement equation as well as for the nonexistence of such solutions

the euler maclaurin expansion and finite part integrals in this paper we compare g p the mellin transform together with its analytic continuation and overline overline g p the related hadamard finite part integral of a function g x which decays exponentially at infinity and has specified singular behavior at the origin except when p is a nonpositive integer these coincide when p is a nonpositive integer overline overline g p is well defined but g p has a pole we show that the terms in the laurent expansion about this pole can be simply expressed in terms of the hadamard finite part integral of a related function this circumstance is exploited to provide a conceptually uniform proof of the various generalizations of the euler maclaurin expansion for the quadrature error functional

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

numerical evaluation of hypersingular integrals abstract boundary integral equations which employ integrals which exist only if defined in the cauchy principal value sense or as the hadamard finite part are currently used with success to solve many two and three dimensional problems of applied mechanics we will recall definitions and main properties of these integrals examine some numerical approaches for their evaluation and present several new results

definitions properties and applications of finite part integrals by using standard calculus we discuss some definitions and properties of finite part integrals point out the essence of this concept and show how these integrals naturally arise in some integral equation applications a couple of new examples are also described

transformation of hypersingular integrals and black box cubature in this paper we will consider hypersingular integrals as they arise by transforming elliptic boundary value problems into boundary integral equations first local representations of these integrals will be derived these representations contain so called finite part integrals in the second step these integrals are reformulated as improper integrals we will show that these integrals can be treated by cubature methods for weakly singular integrals as they exist in the literature

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

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

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

the euler maclaurin expansion for the cauchy principal value integral the classical euler maclaurin summation formula expresses the difference between a definite integral over 0 1 and its approximation using the trapezoidal rule with step lengthh 1 m as an asymptotic expansion in powers ofh together with a remainder term many variants of this exist some of which form the basis of extrapolation methods such as romberg integration in this paper a variant in which the integral is a cauchy principal value integral is derived the corresponding variant of the fourier coefficient asymptotic expansion is also derived the possible role of the former in numerical quadrature is discussed

parallel and deterministic algorithms from mrfs surface reconstruction deterministic approximations to markov random field mrf models are derived one of the models is shown to give in a natural way the graduated nonconvexity gnc algorithm proposed by a blake and a zisserman 1987 this model can be applied to smooth a field preserving its discontinuities a class of more complex models is then proposed in order to deal with a variety of vision problems all the theoretical results are obtained in the framework of statistical mechanics and mean field techniques a parallel iterative algorithm to solve the deterministic equations of the two models is presented together with some experiments on synthetic and real images

asymptotic expansions for two dimensional hypersingular integrals we define and examine two dimensional hypersingular integrals on 0 1 2 and on 0 2 and relate their hadamard finite part hfp values to mellin transforms these integrands have algebraic singularities of a possibly unintegrable nature on the axes and at the origin extending our work on one dimensional integrals reported in 1998 we obtain variants of the classical euler maclaurin expansion for various two dimensional integrals in many cases the constant term in the expansion which is not necessarily the leading term provides the value of the hfp integral these expansions may be used as the basis for the numerical evaluation of a class of hfp integrals by extrapolation

an error expansion for cubature with an integrand with homogeneous boundary singularities a common strategy in the numerical integration over ann dimensional hypercube or simplex is to consider a regular subdivision of the integration domain intom n subdomains and to approximate the integral over each subdomain by means of a cubature formula an asymptotic error expansion whenm is derived in case of an integrand with homogeneous boundary singularities the error expansion also copes with the use of different cubature formulas for the boundary subdomains and for the interior subdomains

probability distribution functions of the grincevicjus series abstract given a sequence n n of independent identically distributed vectors of random variables we consider the grincevicjus series n 1 n k 1 n 1 k and a functional integral equation connected with it we prove that the equation characterizes all probability distribution functions of the grincevicjus series moreover some application of this characterization to a continuous refinement equation is presented

quadrature error expansions we continue the work of part i treating in detail the theory of numerical quadrature over a square 0 1 2 using anm 2 copy q m of a one point quadrature rule as before we determine the nature of an asymptotic expansion for the quadrature error functionalq m f if in inverse powers ofm and related functions valid for specified classes of the integrand functionf the extreme case treated here is one in which the integrand function has a full corner algebraic singularity this has the formx y r x y here and need not be integer andr is x 2 y 2 2 or some other similar homogeneous function the error expansion forms the theoretic basis for the use of extrapolation for this kind of integrand