level set approach to mean curvature flow in arbitrary codimension we develop a level set theory for the mean curvature evolution of surfaces with arbitrary co dimension thus generalizing the previous work 8 15 on hypersurfaces the main idea is to surround the evolving surface of co dimension k in r by a family of hypersurfaces the level sets of a function evolving with normal velocity equal to the sum of the d k smallest principal curvatures the existence and the uniqueness of a weak level set solution is easily established using mainly the results of 8 and the theory of viscosity solutions for second order nonlinear parabolic equations the level set solutions coincide with the classical solutions whenever the latter exist the proof of this connection uses a careful analysis of the squared distance from the surfaces it is also shown that varifold solutions constructed by brakke 7 are included in the level set solutions the idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of de giorgi an introduction to the theory of barriers and his connection to the level set solutions is also provided

consistency and asymptotic normality of some subspace algorithms for systems without observed inputs asymptotic normality for a class of subspace algorithms which estimate the state in a first step is derived expressions for the asymptotic variance are given linear systems with unobserved white noise inputs are considered a class of subspace estimates for the system matrices obtained by estimating the state in the first step is analyzed the main result presented here states asymptotic normality of subspace estimates in addition a consistency result for the system matrix estimates is given an algorithm to compute the asymptotic variances of the estimates is presented in a final section the implications of the result are discussed

subspace identification of circulant systems this article concerns the identification of a class of large scale systems called circulant systems circulant systems have a special property that allows them to be decomposed into simpler subsystems through a state transformation this property has been used in literature for control design and here we show how it can be used for system identification the approach that is proposed here will both reduce the complexity of the problem as well as provide models which have a circulant structure that can be exploited for control design a novel identification algorithm for circulant systems based on subspace identification is presented the algorithm is then tested in simulation on an academic example of circulant system and on a realistic finite element model of a vibrating plate

on the evolution of vector distance functions of closed curves inspired by the work by gomes et al we describe and analyze a vector distance function approach for the implicit evolution of closed curves of codimension larger than one the approach is set up in complete generality and then applied to the evolution of dynamic geometric active contours in mathbb r 4 codimension three case in order to carry this out one needs an explicit expression for the zero level set for which we propose a discrete connectivity method this leads us to make connections with the new theory of cubical homology we provide some explicit simulation results in order to illustrate the methodology

variance analysis of identified linear miso models having spatially correlated inputs with application to parallel hammerstein models this contribution concerns variance analysis of linear multi input single output models when the inputs are temporally white but where different inputs may be correlated an expression is provided for the variance of a linearly parametrized estimate of the frequency response function from one block i e from one input to the output in particular this expression reveals that the variance increases in one block when the number of estimated parameters in another block is increased but levels off when the number of parameters in the other block reaches the number of parameters in the block in question it also quantifies exactly how correlation between inputs affects the resulting accuracy and a graphical representation is provided for this purpose the results are applicable to parallel miso hammerstein models when the nonlinearities are known and generalize an existing variance expression for this type of model

the asymptotic variance of subspace estimates abstract we give new simple general expressions for the asymptotic covariance of the estimated system parameters a b c d in subspace identification the formulas can be applied to a whole class of subspace methods including n4sid moesp cva etc the asymptotic expressions highlight how the conditioning of the estimation problem influences the accuracy of the estimates

maximum entropy vector kernels for mimo system identification abstract recent contributions have framed linear system identification as a nonparametric regularized inverse problem relying on l 2 type regularization which accounts for the stability and smoothness of the impulse response to be estimated these approaches have been shown to be competitive w r t classical parametric methods in this paper adopting maximum entropy arguments we derive a new l 2 penalty to do so we exploit the structure of the hankel matrix thus controlling at the same time complexity measured by the mcmillan degree stability and smoothness of the identified models as a special case we recover the nuclear norm penalty on the squared block hankel matrix in contrast with the previous literature on reweighted nuclear norm penalties our kernel is described by a small number of hyper parameters which are iteratively updated through marginal likelihood maximization constraining the structure of the kernel acts as a hyper regularizer which helps controlling the effective degrees of freedom of our estimator to optimize the marginal likelihood we adapt a scaled gradient projection sgp algorithm which is proved to be significantly computationally cheaper than other first and second order off the shelf optimization methods the paper also contains an extensive comparison with many state of the art methods on several monte carlo studies which confirms the effectiveness of our procedure

order estimation for linear time invariant systems using frequency domain identification methods a two step coarse fine order estimation technique is proposed to determine the order of the numerator and the denominator polynomials of rational transfer function models for single input single output siso linear time invariant systems the coarse order estimation is based on rank detection by verification of the stochastic significance of the singular values of a linearized problem the fine order estimation is based on a statistical analysis of the maximum likelihood cost function the method is tested on measurements of low 4 zeros 6 poles and high 58 poles 58 zeros order systems

subspace model identification part 2 analysis of the elementary output error state space model identification algorithm the elementary moesp algorithm presented in the first part of this series of papers is analysed in this paper this is done in three different ways first we study the asymptotic properties of the estimated state space model when only considering zero mean white noise perturbations on the output sequence it is shown that in this case the moespl implementation yields asymptotically unbiased estimates an important constraint to this result is that the underlying system must have a finite impulse response and subsequently the size of the hankel matrices constructed from the input and output data at the beginning of the computations depends on the number of non zero markov parameters this analysis however leads to a second implementation of the elementary moesp scheme namely moesp2 the latter implementation has the same asymptotic properties without the finite impulse response constraint secondly we compare the moesp2 algorithm with a classical state space model identification scheme the latter

consistent impulse response estimation and system realization from noisy data a novel method for the estimation of a finite part of the impulse response is presented the method is consistent in case the measured output data are prone to colored noise the combination of kung s realization algorithm yields a consistent system identification algorithm that is particularly useful for the identification of systems with uncoupled deterministic and stochastic dynamics this algorithm is called irear a detailed linear sensitivity analysis of the impulse response estimation and kung s realization algorithms yields novel error and covariance formulae for the estimated impulse response and system matrices and the poles and frequency response functions that result from the identified system description

brain structural mapping using a novel hybrid implicit explicit framework based on the level set method this paper presents a novel approach to feature based brain image warping by using a hybrid implicit explicit framework which unifies many prior approaches in a common framework in the first step we develop links between image warping and the level set method and we formulate the fundamental mathematics required for this hybrid implicit explicit approach in the second step we incorporate the large deformation models into these formulations leading to a complete and elegant treatment of anatomical structure matching in this latest approach exact matching of anatomy is achieved by comparing the target to the warped source structure under the forward mapping and the source to the warped target structure under the backward mapping because anatomy is represented nonparametrically a path is constructed linking the source to the target structure without prior knowledge of their point correspondence the final point correspondence is constructed based on the linking path with the minimal energy intensity similarity measures can be naturally incorporated in the same framework as landmark constraints by combining them in the gradient descent body forces we illustrate the approach with two applications 1 tensor based morphometry of the corpus callosum in autistic children and 2 matching cortical surfaces to measure the profile of cortical anatomic variation in summary the new mathematical techniques introduced here contribute fundamentally to the mapping of brain structure and its variation and provide a framework that unites feature and intensity based image registration techniques

a unifying theorem for three subspace system identification algorithms the aim of this paper is to indicate and explore the similarities between three different subspace algorithms for the identification of combined deterministic stochastic systems the similarities between these algorithms have been obscured due to different notations and backgrounds it is shown that all three algorithms are special cases of one unifying theorem the comparison also reveals that the three algorithms use exactly the same subspace to determine the order and the extended observability matrix but that the weighting matrix used to calculate a basis for the column space of the observability matrix is different in the three cases

motion of multiple junctions a level set approach abstract a coupled level set method for the motion of multiple junctions is proposed the new method extends the hamilton jacobi level set formulation of osher and sethian it retains the feature of tracking fronts by following level sets and allows the specification of arbitrary velocities on each front the diffusion equation is shown to generate curvature dependent motion and this is used to develop an algorithm to move multiple junctions with curvature dependent speed systems of reaction diffusion equations are shown to possess inherent properties which prohibit efficient numerical solutions when applied to curvature dependent motion

a subspace algorithm for the identification of discrete time frequency domain power spectra in this paper we present a new subspace algorithm for the identification of multi input multi output linear discrete time systems from measured power spectrum data we show how the state space system matrices can be determined by taking the inverse discrete fourier transform of the given data and applying the result to a new realization algorithm special attention is paid to ensure the positive realness of the identified power spectrum the computational speed is improved by applying a lanczos algorithm the algorithm is illustrated with two practical examples

subspace algorithms for the identification of multivarible dynamic errors in variables models abstract we consider the problem of identifying multivariable finite dimensional linear time invariant systems from noisy input output measurements apart from the fact that both the measured input and output are corrupted by additive white noise the output may also be contaminated by a term which is caused by a white input process noise furthermore all these noise processes are allowed to be correlated with each other we shall develop a solution to this problem in the framework of subspace identification and we shall show that our algorithms give consistent estimates when the system is operating in open or closed loop two realistic simulation studies are presented to demonstrate the practical applicability of the proposed algorithms 1997 elsevier science ltd

identification of the deterministic part of mimo state space models given in innovations form from input output data the problem of linear multivariable state space model identification from input output data can under the presence of process and measurement noise be solved in a non iterative way when incorporating instrumental variables constructed from both input and output sequences in the recently developed class of multivariable output error state space model class of subspace model identification schemes abstract in this paper we describe two algorithms to identify a linear time invariant finite dimensional state space model from input output data the system to be identified is assumed to be excited by a measurable input and an unknown process tloise and the measurements are disturbed by unknown measurement noise both noise sequences are discrete zero mean white noise the first algorithm gives consistent estimates only for the case where the input also is zero mean white noise while the same result is obtained with the second algorithm without this constraint for the special case where the input signal is discrete zero mean white noise it is explicitly shown that this second algorithm is a special case of the recently developed multivariable output error state space moesp class of algorithms based on instrumental variables the usefulness of the presented schemes is highlighted in a realistic simulation study

on line structured subspace identification with application to switched linear systems this article is concerned with the identification of switched linear multiple inputs multiple outputs state space systems in a recursive way first a structured subspace identification scheme for linear systems is presented which turns out to have many attractive features more precisely it does not require any singular value decomposition but is derived using orthogonal projection techniques it allows a computationally appealing implementation and it is closely related to input output models identification second it is shown that this method can be implemented on line to track both the range space of the extended observability matrix and its dimension and thereby the system matrices third by making use of an on line switching times detection strategy this method is applied to blindly identify switched systems and to label the obtained submodels simulation results on noisy data illustrate the abilities and the benefits of the proposed approach

geodesic deformable models for medical image analysis in this paper implicit representations of deformable models for medical image enhancement and segmentation are considered the advantage of implicit models over classical explicit models is that their topology can be naturally adapted to objects in the scene a geodesic formulation of implicit deformable models is especially attractive since it has the energy minimizing properties of classical models the aim of this paper is twofold first a modification to the customary geodesic deformable model approach is introduced by considering all the level sets in the image as energy minimizing contours this approach is used to segment multiple objects simultaneously and for enhancing and segmenting cardiac computed tomography ct and magnetic resonance images second the approach is used to effectively compare implicit and explicit models for specific tasks this shows the complementary character of implicit models since in case of poor contrast boundaries or gaps in boundaries e g due to partial volume effects noise or motion artifacts they do not perform well since the approach is completely data driven

consistency of generalized finite difference schemes for the stochastic hjb equation we analyze a class of numerical schemes for solving the hjb equation for stochastic control problems which enters the framework of markov chain approximations and generalizes the usual finite difference method the latter is known to be monotonic and hence valid only if the scaled covariance matrix is dominant diagonal we generalize this result by given the set of neighboring points allowed to enter the scheme showing how to compute effectively the class of covariance matrices that is consistent with this set of points we perform this computation for several cases in dimensions 2 3 and 4

n4sid subspace algorithms for the identification of combined deterministic stochastic systems abstract recently a great deal of attention has been given to numerical algorithms for subspace state space system identification n4sid in this paper we derive two new n4sid algorithms to identify mixed deterministic stochastic systems both algorithms determine state sequences through the projection of input and output data these state sequences are shown to be outputs of non steady state kalman filter banks from these it is easy to determine the state space system matrices the n4sid algorithms are always convergent non iterative and numerically stable since they only make use of qr and singular value decompositions both n4sid algorithms are similar but the second one trades off accuracy for simplicity these new algorithms are compared with existing subspace algorithms in theory and in practice

region tracking on level sets methods since the work by osher and sethian 1988 on level sets algorithms for numerical shape evolutions this technique has been used for a large number of applications in numerous fields in medical imaging this numerical technique has been successfully used for example in segmentation and cortex unfolding algorithms the migration from a lagrangian implementation to a eulerian one via implicit representations or level sets brought some of the main advantages of the technique i e topology independence and stability this migration means also that the evolution is parametrization free therefore the authors do not know exactly how each part of the shape is deforming and the point wise correspondence is lost in this note they present a technique to numerically track regions on surfaces that are being deformed using the level sets method the basic idea is to represent the region of interest as the intersection of two implicit surfaces and then track its deformation from the deformation of these surfaces this technique then solves one of the main shortcomings of the very useful level sets approach applications include lesion localization in medical images region tracking in functional mri fmri visualization and geometric surface mapping

variational problems and partial differential equations on implicit surfaces a novel framework for solving variational problems and partial differential equations for scalar and vector valued data defined on surfaces is introduced in this paper the key idea is to implicitly represent the surface as the level set of a higher dimensional function and to solve the surface equations in a fixed cartesian coordinate system using this new embedding function the equations are then both intrinsic to the surface and defined in the embedding space this approach thereby eliminates the need for performing complicated and inaccurate computations on triangulated surfaces as is commonly done in the literature we describe the framework and present examples in computer graphics and image processing applications including texture synthesis flow field visualization and image and vector field intrinsic regularization for data defined on 3d surfaces

cortex segmentation a fast variational geometric approach an automatic cortical gray matter segmentation from a three dimensional 3 d brain images magnetic resonance mr or computed tomography is a well known problem in medical image processing in this paper we first formulate it as a geometric variational problem for propagation of two coupled bounding surfaces an efficient numerical scheme is then used to implement the geodesic active surface model experimental results of cortex segmentation on real 3 d mr data are provided

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

brief consistency analysis of subspace identification methods based on a linear regression approach in the literature results can be found which claim consistency for the subspace method under certain quite weak assumptions unfortunately a new result gives a counter example showing inconsistency under these assumptions and then gives new more strict sufficient assumptions which however does not include important model structures such as e g box jenkins based on a simple least squares approach this paper shows the possible inconsistency under the weak assumptions and develops only slightly stricter assumptions sufficient for consistency and which includes any model structure

subspace model identification part 1 the output error state space model identification class of algorithms in this paper we present two novel algorithms to realize a finite dimensional linear time invariant state space model from input output data the algorithms have a number of common features they are classified as one of the subspace model identification schemes in that a major part of the identification problem consists of calculating specially structured subspaces of spaces defined by the input output data this structure is then exploited in the calculation of a realization another common feature is their algorithmic organization an rq factorization followed by a singular value decomposition and the solution of an overdetermined set or sets of equations the schemes assume that the underlying system has an output error structure and that a measurable input sequence is available the latter characteristic indicates that both schemes are versions of the mimo output error state space model identification moesp approach the first algorithm is denoted in particular as the elementary moesp scheme

left ventricular volume estimation for real time three dimensional echocardiography the application of level set techniques to echocardiographic data is presented this method allows semiautomatic segmentation of heart chambers which regularizes the shapes and improves edge fidelity especially in the presence of gaps as is common in ultrasound data the task of the study was to reconstruct left ventricular shape and to evaluate left ventricular volume data were acquired with a real time three dimensional 3 d echocardiographic system the method was applied directly in the three dimensional domain and was based on a geometric driven scheme the numerical scheme for solving the proposed partial differential equation is borrowed from numerical methods for conservation law results refer to in vitro and human in vivo acquired 3 d time echocardiographic data quantitative validation was performed on in vitro balloon phantoms clinical application of this segmentation technique is reported for 20 patient cases providing measures of left ventricular volumes and ejection fraction

asymptotic variance of subspace methods by data orthogonalization and model decoupling a comparative analysis this is a companion of the paper chiuso and picci 2004d where we do asymptotic error analysis of a weighted pi moesp type method and compare accuracy with respect to estimates obtained by customary joint subspace methods the analysis shows that under certain conditions methods based on orthogonal decomposition of the input output data and block decoupled parametrization perform better than traditional joint model based methods in the circumstance of nearly parallel regressors

a class of subspace model identification algorithms to identify periodically and arbitrarily time varying systems abstract subspace model identification algorithms that allow the identification of a linear time varying ltv state space model from an ensemble set of input output measurements are presented in this paper each pair of input and output sequences in this ensemble is recorded when the underlying system to be identified undergoes the same time varying behavior the algorithms operate directly on the available ensemble of input output data and are a generalization of the recently proposed multivariable output error state space moesp class of algorithms to this ensemble type of identification problems a special case is considered in this paper where the repetition of this time varying behavior is intrinsic namely in periodically time varying systems an example of identifying a multirate sampled data system from a recorded input and output sequence demonstrates some of the capabilities of the presented subspace model identification algorithms

subspace based methods for the identification of linear time invariant systems subspace based methods for system identification have attracted much attention during the past few years this interest is due to the ability of providing accurate state space models for multivariable linear systems directly from input output data the methods have their origin in classical state space realization theory as developed in the 1960s the main computational tools are the qr and the singular value decompositions here an overview of existing subspace based techniques for system identification is given the methods are grouped into the classes of realization based and direct techniques similarities between different algorithms are pointed out and their applicability is commented upon we also discuss some recent ideas for improving and extending the methods a simulation example is included for comparing different algorithms the subspace based approach is found to perform competitive with respect to prediction error methods provided the system is properly excited

a novel subspace identification approach with enforced causal models subspace identification methods sims for estimating state space models have been proven to be very useful and numerically efficient they exist in several variants but all have one feature in common as a first step a collection of high order arx models are estimated from vectorized input output data in order not to obtain biased estimates this step must include future outputs however all but one of the submodels include non causal input terms the coefficients of them will be correctly estimated to zero as more data become available they still include extra model parameters which give unnecessarily high variance and also cause bias for closed loop data in this paper a new model formulation is suggested that circumvents the problem within the framework the system matrices a b c d and markov parameters can be estimated separately it is demonstrated through analysis that the new methods generally give smaller variance in the estimate of the observability matrix and it is supported by simulation studies that this gives lower variance also of the system invariants such as the poles

motion of level sets by mean curvature ii we give a new proof of short time existence for the classical motion by mean curvature of a smooth hypersurface our method consists in studying a fully nonlinear uniformly parabolic equation satisfied by the signed distance function to the surface

high performance numerical algorithms and software for subspace based linear multivariable system identification basic algorithmic and numerical issues involved in subspace based linear multivariable discrete time system identification are described a new identification toolbox slident has been developed and incorporated in the freely available subroutine library in control theory slicot reliability efficiency and ability to solve industrial identification problems received a special consideration two algorithmic subspace based approaches moesp and n4sid and their combination and both standard and fast techniques for data compression are provided structure exploiting algorithms and dedicated linear algebra tools enhance the computational efficiency and reliability extensive comparisons with the available computational tools based on subspace techniques show the better efficiency of the slident toolbox at comparable numerical accuracy and its capabilities to solve identification problems with many thousands of samples and hundreds of parameters

morphing active contours a method for deforming curves in a given image to a desired position in a second image is introduced the algorithm is based on deforming the first image toward the second one via a partial differential equation pde while tracking the deformation of the curves of interest in the first image with an additional coupled pde both the images and the curves on the frame slices of interest are used for tracking the technique can be applied to object tracking and sequential segmentation the topology of the deforming curve can change without any special topology handling procedures added to the scheme this permits for example the automatic tracking of scenes where due to occlusions the topology of the objects of interest changes from frame to frame in addition this work introduces the concept of projecting velocities to obtain systems of coupled pdes for image analysis applications we show examples for object tracking and segmentation of electronic microscopy