stochastic gradient identification of polynomial wiener systems analysis and application this paper presents analytical numerical and experimental results for a stochastic gradient adaptive scheme that identifies a polynomial type nonlinear system with memory for noisy output observations the analysis includes the computation of the stationary points the mean square error surface and the stability regions of the algorithm for gaussian data convergence of the mean is studied using l sub 2 and euclidian norms monte carlo simulations confirm the theoretical predictions that show a small sensitivity to the observation noise an application is presented for the identification of a nonlinear time delayed feedback system

on lie groups and toda lattices we extend the construction of the relativistic toda chains as integrable systems on the poisson submanifolds in lie groups beyond the case of the a series for the simply laced case this is just a direct generalization of the well known relativistic toda chains procedure and we construct explicitly the set of ad invariant integrals of motion on symplectic leaves which can be described using poisson quivers which are just blown up dynkin diagrams we also demonstrate how to get the set of minimal integrals of motion using the co multiplication rules for the corresponding lie algebras in the non simply laced case the corresponding bogoyavlensky coxeter toda systems are constructed using the fock goncharov folding of the corresponding poisson submanifolds we discuss also how this procedure can be extended for the affine case beyond the a series and consider explicitly an example from the affine d series

simultaneous homographic and comparametric alignment of multiple exposure adjusted pictures of the same scene an approach is presented that can simultaneously align multiple exposure adjusted pictures of the same scene both in their spatial coordinates as well as in their pixel values the approach is featureless and produces an image mosaic at a common spatial and exposure reference and also addresses the misalignment problem common to methods that compose mosaics from only pair wise registered image pairs the objective function considered minimizes the sum of the collective variance over pixels of a global coordinate grid on which to create the final image the models employed relate images spatially by homographic transformations and tonally by comparametric functions the importance of performing joint spatial and tonal registration on exposure adjusted images is emphasized by providing two examples in which spatial only registration fails the performance between pair wise and simultaneous registration under both spatial only and joint registration procedures is discussed

the limiting behavior of lms a realization oriented analysis is given of the gradient noise misadjustment and lag misadjustment performance of the lms least mean square algorithm new formulas are given for both of these components of excess mean square error it is shown that the traditional formula for lag misadjustment needs to be modified by adding further terms due to gradient noise and noise variance to perform the analysis it is necessary to study the convergence with probability one of the noise free fixed parameter lms algorithm convergence is found under simple conditions that improve on those previously obtained

the lms algorithm with delayed coefficient adaptation the behavior of the delayed least mean square dlms algorithm is studied it is found that the step size in the coefficient update plays a key role in the convergence and stability of the algorithm an upper bound for the step size is derived that ensures the stability of the dlms the relationship between the step size and the convergence speed and the effect of the delay on the convergence speed are also studied the analytical results are supported by computer simulations

the dual bootstrap iterative closest point algorithm with application to retinal image registration motivated by the problem of retinal image registration this paper introduces and analyzes a new registration algorithm called dual bootstrap iterative closest point dual bootstrap icp the approach is to start from one or more initial low order estimates that are only accurate in small image regions called bootstrap regions in each bootstrap region the algorithm iteratively 1 refines the transformation estimate using constraints only from within the bootstrap region 2 expands the bootstrap region and 3 tests to see if a higher order transformation model can be used stopping when the region expands to cover the overlap between images steps 1 and 3 the bootstrap steps are governed by the covariance matrix of the estimated transformation estimation refinement step 2 uses a novel robust version of the icp algorithm in registering retinal image pairs dual bootstrap icp is initialized by automatically matching individual vascular landmarks and it aligns images based on detected blood vessel centerlines the resulting quadratic transformations are accurate to less than a pixel on tests involving approximately 6000 image pairs it successfully registered 99 5 of the pairs containing at least one common landmark and 100 of the pairs containing at least one common landmark and at least 35 image overlap

convergence of the delayed normalized lms algorithm with decreasing step size in several practical applications of the lms algorithm including certain vlsi implementations the coefficient adaptation can be performed only after some fixed delay the resulting algorithm is known as the delayed lms dlms algorithm in the literature previous published analyses of this algorithm are based on mean and moment convergence under the independence assumption between successive input vectors these analyses are interesting and give valuable insights into the convergence properties but from a practical viewpoint they do not guarantee the correct performance of the particular realization with which the user must live we consider a normalized version of this algorithm with a decreasing step size spl mu n and prove the almost sure convergence of the nonhomogeneous algorithm assuming a mixing input condition and the satisfaction of a certain law of large numbers

jointly registering images in domain and range by piecewise linear comparametric analysis this paper describes an approach whereby comparametric analysis is used in jointly registering image pairs in their domain and range i e in their spatial coordinates and pixel values respectively this is accomplished by approximating a camera s nonlinear comparametric function with a constrained piecewise linear one the optimal fitting of this approximation to comparagram data is then used in a re parameterized version of the camera s comparametric function to estimate the exposure difference between images doing this allows the inherently nonlinear problem of joint domain and range registration to be performed using a computationally attractive least squares formalism the paper first presents the range registration process and then describes the strategy for performing the joint registration the models used allow for the pair wise registration of images taken from a camera that can automatically adjust its exposure as well as tilt pan rotate and zoom about its optical center results concerning the joint registration as well as range only registration are provided to demonstrate the method s effectiveness

integrable cluster dynamics of directed networks and pentagram maps abstract the pentagram map was introduced by r schwartz more than 20 years ago in 2009 v ovsienko r schwartz and s tabachnikov established liouville complete integrability of this discrete dynamical system in 2011 m glick interpreted the pentagram map as a sequence of cluster transformations associated with a special quiver using compatibility of poisson and cluster structures and poisson geometry of directed networks on surfaces we generalize glick s construction to include the pentagram map into a family of discrete integrable maps and we give these maps geometric interpretations the appendix relates the simplest of these discrete maps to the toda lattice and its tri hamiltonian structure

lagrangian multiform structure for the lattice gel fand dikii hierarchy the lattice gel fand dikii hierarchy was introduced by nijhoff et al 1992 inverse problems 8 597 621 as the family of partial difference equations generalizing to higher rank the lattice korteweg de vries systems and includes in particular the lattice boussinesq system we present a lagrangian for the generic member of the lattice gel fand dikii hierarchy and show that it can be considered as a lagrangian 2 form when embedded in a higher dimensional lattice obeying a closure relation thus the multiform structure proposed by lobb and nijhoff 2009 j phys a math theor 42 454013 is extended to a multi component system

on the integrability of the shift map on twisted pentagram spirals in this paper we prove that the shift map defined on the moduli space of twisted pentagram spirals of type possesses a non standard lax representation with an associated monodromy whose conjugation class is preserved by the map we prove this by finding a coordinate system in the moduli space of twisted spirals writing the map in terms of the coordinates and associating a natural parameter free non standard lax representation we then show that the map is invariant under the action of a one parameter group on the moduli space of twisted spirals which allows us to construct the lax pair we also show that the monodromy defines an associated riemann surface that is preserved by the map we use this fact to generate invariants of the shift map

in defense of the eight point algorithm the fundamental matrix is a basic tool in the analysis of scenes taken with two uncalibrated cameras and the eight point algorithm is a frequently cited method for computing the fundamental matrix from a set of eight or more point matches it has the advantage of simplicity of implementation the prevailing view is however that it is extremely susceptible to noise and hence virtually useless for most purposes this paper challenges that view by showing that by preceding the algorithm with a very simple normalization translation and scaling of the coordinates of the matched points results are obtained comparable with the best iterative algorithms this improved performance is justified by theory and verified by extensive experiments on real images

analysis of stochastic gradient tracking of time varying polynomial wiener systems this paper presents analytical and monte carlo results for a stochastic gradient adaptive scheme that tracks a time varying polynomial wiener 1958 system i e a linear time invariant lti filter with memory followed by a time varying memoryless polynomial nonlinearity the adaptive scheme consists of two phases 1 estimation of the lti memory using the lms algorithm and 2 tracking the time varying polynomial type nonlinearity using a second coupled gradient search for the polynomial coefficients the time varying polynomial nonlinearity causes a time varying scaling for the optimum wiener filter for phase 1 these time variations are removed for phase 2 using a novel coupling scheme to phase 1 the analysis for gaussian data includes recursions for the mean behavior of the lms algorithm for estimating and tracking the optimum wiener filter for phase 1 for several different time varying polynomial nonlinearities and recursions for the mean behavior of the stochastic gradient algorithm for phase 2 the polynomial coefficients are shown to be accurately tracked monte carlo simulations confirm the theoretical predictions and support the underlying statistical assumptions

spherical mosaics with quaternions and dense correlation we describe an algorithm for generating spherical mosaics from a collection of images acquired from a common optical center the algorithm takes as input an arbitrary number of partially overlapping images an adjacency map relating the images initial estimates of the rotations relating each image to a specified base image and approximate internal calibration information for the camera the algorithm s output is a rotation relating each image to the base image and revised estimates of the camera s internal parameters our algorithm is novel in the following respects first it requires no user input our image capture instrumentation provides both an adjacency map for the mosaic and an initial rotation estimate for each image second it optimizes an objective function based on a global correlation of overlapping image regions third our representation of rotations significantly increases the accuracy of the optimization finally our representation and use of adjacency information guarantees globally consistent rotation estimates the algorithm has proved effective on a collection of nearly four thousand images acquired from more than eighty distinct optical centers the experimental results demonstrate that the described global optimization strategy is superior to non global aggregation of pair wise correlation terms and that it successfully generates high quality mosaics despite significant error in initial rotation estimates

automatic panoramic image stitching using invariant features this paper concerns the problem of fully automated panoramic image stitching though the 1d problem single axis of rotation is well studied 2d or multi row stitching is more difficult previous approaches have used human input or restrictions on the image sequence in order to establish matching images in this work we formulate stitching as a multi image matching problem and use invariant local features to find matches between all of the images because of this our method is insensitive to the ordering orientation scale and illumination of the input images it is also insensitive to noise images that are not part of a panorama and can recognise multiple panoramas in an unordered image dataset in addition to providing more detail this paper extends our previous work in the area brown and lowe 2003 by introducing gain compensation and automatic straightening steps

second order convergence analysis of stochastic adaptive linear filtering the convergence of an adaptive filtering vector is studied when it is governed by the mean square error gradient algorithm with constant step size we consider the mean square deviation between the optimal filter and the actual one during the steady state this quantity is known to be essentially proportional to the step size of the algorithm however previous analyses were either heuristic or based upon the assumption that successive observations were independent which is far from being realistic actually in most applications two successive observation vectors share a large number of components and thus they are strongly correlated in this work we deal with the case of correlated observations and prove that the mean square deviation is actually of the same order or less than the step size of the algorithm this result is proved without any boundedness or barrier assumption for the algorithm as it has been done previously in the literature to ensure the nondivergence our assumptions are reduced to the finite strong memory assumption and the finite moments assumption for the observation they are satisfied in a very wide class of practical applications

convergence of an adaptive linear estimation algorithm in this work we prove the almost sure convergence of an adaptive linear estimator governed by a stochastic gradient algorithm with decreasing step size in the presence of correlated observations two complementary contributions are added to the famous 1977 ljung theorem first we drop the condition of nondivergence of the algorithm assumed by ljung while that condition can be ensured by adding a barrier the convergence of the suitably bounded algorithm itself is not established even on the basis of ljung theorem here the barrier problem is overcome by proving that it is not necessary for the convergence our second contribution is to generalize the model describing the correlated observations no state space model is used and no linear relationship between the observations and the signal to be estimated needs to be assumed instead we use a decreasing covariance model that agrees with a very wide class of practical applications

fluctuation analysis of stochastic gradient identification of polynomial wiener systems this correspondence presents analytical results and monte carlo simulations for the fluctuation behavior of a stochastic gradient adaptive identification scheme this scheme identifies a polynomial wiener system linear fir filter followed by a static polynomial nonlinearity for noisy output observations the analysis includes 1 bounds and a recursion for the misadjustment matrix and 2 algorithm mean square stability regions a diagonal step size matrix for the adaptive coefficients is introduced to speed up convergence the theoretical predictions of the fluctuation analysis are supported by monte carlo simulations

a systolic array realization of an lms adaptive filter and the effects of delayed adaptation a systolic array design for an adaptive filter is presented the filter is based on the least mean square algorithm but due to the problems in implementation of the systolic array a modified algorithm a special case of the delayed lms dlms is used the dlms algorithm introduces a delay in the updating of the filter coefficients the convergence and steady state behavior of the systolic array are analyzed it is shown that the performance of the systolic array is similar to that of a conventional lms implementation for a wide range of practical conditions

almost sure convergence of adaptive algorithms by projections a convergence proof is discussed for the normalized least mean square lms algorithm for ergodic inputs the approach is based on interpreting the algorithm as a sequence of relaxed projection operators by which the key contraction property is derived the proof technique is strongly motivated by physical intuition and provides additional insight into lms type algorithms under ergodic inputs embedded in the development is a slight generalization to a random time varying gain parameter this allows the incorporation of variations such as the lms and signed lms algorithms

image registration by super curves we solve the 2 d affine image registration problem by curve matching and alignment our approach starts with a super curve which is formed by superimposing two affine related curves in one coordinate system we use b spline fusion technique to find a single b spline approximation of the super curve and a registration between the two curves simultaneously this approach achieves superior accuracy and efficiency in curve matching and alignment we then address the occlusion problem by finding partial match between the curves segmented using inflections and cusps which are affine invariant the combination of edge detection and curve alignment lead to accurate image registration

nonmetric calibration of camera lens distortion differential methods and robust estimation this paper addresses the problem of calibrating camera lens distortion which can be significant in medium to wide angle lenses our approach is based on the analysis of distorted images of straight lines we derive new distortion measures that can be optimized using nonlinear search techniques to find the best distortion parameters that straighten these lines unlike the other existing approaches we also provide fast closed form solutions to the distortion coefficients we prove that including both the distortion center and the decentering coefficients in the nonlinear optimization step may lead to instability of the estimation algorithm our approach provides a way to get around this and at the same time it reduces the search space of the calibration problem without sacrificing the accuracy and produces more stable and noise robust results in addition while almost all existing nonmetric distortion calibration methods needs user involvement in one form or another we present a robust approach to distortion calibration based on the least median of squares estimator our approach is thus able to proceed in a fully automatic manner while being less sensitive to erroneous input data such as image curves that are mistakenly considered projections of three dimensional linear segments experiments to evaluate the performance of this approach on synthetic and real data are reported

the stability of adaptive minimum mean square error equalizers using delayed adjustment the necessary and sufficient conditions for the stability and convergence of adaptive minimum mean square error equalizers are developed the analysis includes the effect of delays in the adjustment path as occur for instance when the decoder in a data transmission system delays its output decisions

liouville arnold integrability of the pentagram map on closed polygons the pentagram map is a discrete dynamical system defined on the moduli space of polygons in the projective plane this map has recently attracted a considerable interest mostly because its connection to a number of different domains such as classical projective geometry algebraic combinatorics moduli spaces cluster algebras and integrable systems integrability of the pentagram map was conjectured by schwartz and proved by the present authors for a larger space of twisted polygons in this article we prove the initial conjecture that the pentagram map is completely integrable on the moduli space of closed polygons in the case of convex polygons in the real projective plane this result implies the existence of a toric foliation on the moduli space the leaves of the foliation carry affine structure and the dynamics of the pentagram map is quasiperiodic our proof is based on an invariant poisson structure on the space of twisted polygons we prove that the hamiltonian vector fields corresponding to the monodromy invariants preserve the space of closed polygons and define an invariant affine structure on the level surfaces of the monodromy invariants

integrability of higher pentagram maps we define higher pentagram maps on polygons in mathbb p d for any dimension d which extend r schwartz s definition of the 2d pentagram map we prove their integrability by presenting lax representations with a spectral parameter for scale invariant maps the corresponding continuous limit of the pentagram map in dimension d is shown to be the 2 d 1 equation of the kdv hierarchy generalizing the boussinesq equation in 2d we also study in detail the 3d case where we prove integrability for both closed and twisted polygons and describe the spectral curve first integrals the corresponding tori and the motion along them as well as an invariant symplectic structure

moment forms invariant to rotation and blur in arbitrary number of dimensions we present the construction of combined blur and rotation moment invariants in arbitrary number of dimensions moment invariants to convolution with an arbitrary centrosymmetric filter are derived first and then their rotationally invariant forms are found by means of group representation theory to achieve the desired combined invariance several examples of the invariants are calculated explicitly to illustrate the proposed procedure their invariance robustness and capability of using in template matching and in image registration are demonstrated on 3d mri data and 2d indoor images

groupwise geometric and photometric direct image registration image registration consists in estimating geometric and photometric transformations that align two images as best as possible the direct approach consists in minimizing the discrepancy in the intensity or color of the pixels the inverse compositional algorithm has been recently proposed by baker et al for the direct estimation of groupwise geometric transformations it is efficient in that it performs several computationally expensive calculations at a pre computation phase photometric transformations act on the value of the pixels they account for effects such as lighting change jointly estimating geometric and photometric transformations is thus important for many tasks such as image mosaicing we propose an algorithm to jointly estimate groupwise geometric and photometric transformations while preserving the efficient pre computation based design of the original inverse compositional algorithm it is called the dual inverse compositional algorithm it uses different approximations than the simultaneous inverse compositional algorithm and handles groupwise geometric and global photometric transformations its name stems from the fact that it uses an inverse compositional update rule for both the geometric and the photometric transformations we demonstrate the proposed algorithm and compare it to previous ones on simulated and real data this shows clear improvements in computational efficiency and in terms of convergence

the pentagram map and y patterns abstract the pentagram map introduced by r schwartz is defined by the following construction given a polygon as input draw all of its shortest diagonals and output the smaller polygon which they cut out we employ the machinery of cluster algebras to obtain explicit formulas for the iterates of the pentagram map

nonmetric calibration of wide angle lenses and polycameras images taken with wide angle cameras tend to have severe distortions which pull points towards the optical center this paper proposes a simple method for recovering the distortion parameters without the use of any calibration objects since distortions cause straight lines in the scene to appear as curves in the image our algorithm seeks to find the distortion parameters that map the image curves to straight lines the user selects a small set of points along the image curves recovery of the distortion parameters is formulated as the minimization of an objective function which is designed to explicitly account for noise in the selected image points experimental results are presented for synthetic data as well as real images we also present the idea of a polycamera which is defined as a tightly packed camera cluster possible configurations are proposed to capture very large fields of view such camera clusters tend to have a nonsingle viewpoint we therefore provide analysis of what we call the minimum working distance for such clusters finally we present results for a polycamera consisting of four wide angle sensors having a minimum working distance of about 4 m on undistorting the acquired images using our proposed technique we create real time high resolution panoramas

true multi image alignment and its application to mosaicing and lens distortion correction multiple images of a scene are related through 2d 3d view transformations and linear and nonlinear camera transformations we present an algorithm for true multi image alignment that does not rely on the measurements of a reference image being distortion free the algorithm is developed to specifically align and mosaic images using parametric transformations in the presence of lens distortion when lens distortion is present none of the images can be assumed to be ideal in our formulation all the images are modeled as intensity measurements represented in their respective coordinate systems each of which is related to an ideal coordinate system through an interior camera transformation and an exterior view transformation the goal of the accompanying algorithm is to compute an image in the ideal coordinate system while solving for the transformations that relate the ideal system with each of the data images key advantages of the technique presented in this paper are i no reliance on one distortion free image ii ability to register images and compute coordinate transformations even when the multiple images are of an extended scene with no overlap between the first and last frame of the sequence and iii ability to handle linear and nonlinear transformations within the same framework results of applying the algorithm are presented for the correction of lens distortion and creation of video mosaics

on almost sure convergence of adaptive algorithms we present an extension of the furstenberg kesten theorem on the convergence of random matrices this extension is applied to the study of almost sure convergence of certain adaptive algorithms in particular we establish that the nlms algorithm is almost surely convergent under extremely weak necessary and sufficient conditions we also discuss the relationship of sufficient conditions that have appeared in the literature with our results