differential morphology and image processing image processing via mathematical morphology has traditionally used geometry to intuitively understand morphological signal operators and set or lattice algebra to analyze them in the space domain we provide a unified view and analytic tools for morphological image processing that is based on ideas from differential calculus and dynamical systems this includes ideas on using partial differential or difference equations pdes to model distance propagation or nonlinear multiscale processes in images we briefly review some nonlinear difference equations that implement discrete distance transforms and relate them to numerical solutions of the eikonal equation of optics we also review some nonlinear pdes that model the evolution of multiscale morphological operators and use morphological derivatives among the new ideas presented we develop some general 2 d max min sum difference equations that model the space dynamics of 2 d morphological systems including the distance computations and some nonlinear signal transforms called slope transforms that can analyze these systems in a transform domain in ways conceptually similar to the application of fourier transforms to linear systems thus distance transforms are shown to be bandpass slope filters we view the analysis of the multiscale morphological pdes and of the eikonal pde solved via weighted distance transforms as a unified area in nonlinear image processing which we call differential morphology and briefly discuss its potential applications to image processing and computer vision

system structure and singular control the general continuous time linear quadratic control problem is considered it is shown that recently developed linear system theoretic properties and algorithms play an important role in solving this singular control problem

feedback and pole placement in descriptor variable systems the effects and uses of applying linear feedback to continuous time descriptor systems are studied structural changes resulting from feeding back the slow and fast parts of the trajectory separately are characterized it is shown that under certain conditions related to controllability the poles of the slow subsystem may be shifted arbitrarily and the impulsive behaviour of the fast subsystem may be eliminated

linear quadratic control with and without stability subject to general implicit continuous time systems coordinate free interpretations of the optimal costs in terms of dissipation inequality and linear matrix inequality existence and uniqueness of optimal controls and state trajectories abstract we consider linear quadratic control problems with and without stability subject to an arbitrary implicit continuous time system in a simple distributional framework and it is shown that the associated optimal costs if existent are solutions of our dissipation inequality for implicit systems this concept is related to the linear matrix inequality which is expressed in original system coefficients only and the above mentioned optimal costs turn out to be characterizable uniquely by certain solutions of this inequality however these solutions need not be rank minimizing if the underlying system is not standard and we specify why this is the case our statements are valid for regular as well as for singular problems and the possible significance of the algebraic riccati equation is illustrated for both regular and singular problems furthermore we present necessary and sufficient conditions for solvability of our problems and for the existence of optimal controls and associated optimal state trajectories finally we elaborate on the uniqueness of these controls and state trajectories

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

discounted cost linear quadratic gaussian control for descriptor systems we consider the linear quadratic gaussian control problem with a discounted cost functional for descriptor systems on the infinite time horizon based on recent results from the deterministic frame

generalized linear systems geometric and structural approaches abstract this paper presents a new way of introducing invariant subspaces for generalized systems this comes from somewhat known geometric algorithms but in most cases with initial conditions different from those usually seen in this context these definitions are shown to be consistent with other ones directly deduced from classical approaches like matrix pencil tools or discrete algebraic formalism the key starting point for their introduction is the extension of structural descriptions for these generalized systems such as have been available for proper systems many results from the proper case can thus be extended for instance the geometric definition for controllability indices and something similar to the famous morse canonical decomposition of the strictly proper case some links with the inversion algorithm are also sketched

solvability conditions consistency and weak consistency for linear differential algebraic equations and time invariant singular systems the general case we present several solvability concepts for linear differential algebraic equations daes with constant coefficients on the positive time axis as well as for the associated singular systems and investigate under which conditions these concepts are met next we derive necessary and sufficient conditions for global consistency of initial conditions for the dae as well as for the system and generalize these conditions with respect to our concept of weak consistency our distributional approach enables us to generalize results in an earlier paper where singular systems are assumed to have a regular pencil in the sense of gantmacher in particular we establish that global weak consistency in the system sense is equivalent to impulse controllability

projection based polygonality measurement measuring the degree to which a shape resembles a polygon referred to as polygonality is a difficult problem due to the intrinsic diversity in the form and distortion of shapes caused by digitization and similarity transformation this paper proposes a generic approach for this problem by performing the measurement in the projection space where the radon image of some primitive shapes which compose the shape becomes apparent the obtained measures take value in 0 1 with 1 corresponding to analytical polygons they are robust to additive noise and boundary distortion and is invariant to similarity transformation the new framework generalizes existing polygonal measures such as triangularity and quadrangularity in addition the ability to estimate polygon s geometric quantities in the projection space allows approximating a shape by analytical polygons the efficiency of the proposed approach is demonstrated through a number of experiments on both synthetic and real data sets

geometric structure and feedback in singular systems the output nulling a e r b invariant subspaces are defined for singular systems rigorously justifying the name and demonstrating that special cases of these geometric objects are the familiar subspace of admissible conditions and the supremal a e r b invariant subspace a novel singular system structure algorithm is used to compute them by numerically efficient means their importance for describing the possible closed loop geometric structure in terms of the open loop geometric structure is shown an approach to spectrum assignment in singular systems that is based on a generalized lyapunov equation is introduced the equation is used to compute feedback gains to place poles and assign various closed loop invariant subspaces while guaranteeing closed loop regularity

pattern spectrum and multiscale shape representation the results of a study on multiscale shape description smoothing and representation are reported multiscale nonlinear smoothing filters are first developed using morphological opening and closings g matheron 1975 used openings and closings to obtain probabilistic size distributions of euclidean space sets continuous binary images these distributions are used to develop a concept of pattern spectrum a shape size descriptor a pattern spectrum is introduced for continuous graytone images and arbitrary multilevel signals as well as for discrete images by developing a discrete size family of patterns large jumps in the pattern spectrum at a certain scale indicate the existence of major protruding or intruding substructures of the signal at the scale an entropy like shape size complexity measure is also developed based on the pattern spectrum for shape representation a reduced morphological skeleton transform is introduced for discrete binary and graytone images this transform is a sequence of skeleton components sparse images which represent the original shape at various scales it is shown that the partially reconstructed images from the inverse transform on subsequences of skeleton components are the openings of the image at a scale determined by the number of eliminated components in addition two way correspondences are established among the degree of shape smoothing via multiscale openings or closings the pattern spectrum zero values and the elimination or nonexistence of skeleton components at certain scales

classification using geometric level sets a variational level set method is developed for the supervised classification problem nonlinear classifier decision boundaries are obtained by minimizing an energy functional that is composed of an empirical risk term with a margin based loss and a geometric regularization term new to machine learning the surface area of the decision boundary this geometric level set classifier is analyzed in terms of consistency and complexity through the calculation of its e entropy for multicategory classification an efficient scheme is developed using a logarithmic number of decision functions in the number of classes rather than the typical linear number of decision functions geometric level set classification yields performance results on benchmark data sets that are competitive with well established methods

complexity measures of supervised classification problems we studied a number of measures that characterize the difficulty of a classification problem focusing on the geometrical complexity of the class boundary we compared a set of real world problems to random labelings of points and found that real problems contain structures in this measurement space that are significantly different from the random sets distributions of problems in this space show that there exist at least two independent factors affecting a problem s difficulty we suggest using this space to describe a classifier s domain of competence this can guide static and dynamic selection of classifiers for specific problems as well as subproblems formed by confinement projection and transformations of the feature vectors

descriptor variable systems and optimal state regulation linear systems of the form e dot x ax bu with e singular are treated it is desired to find a control which drives the system asymptotically to the origin minimizing a quadratic cost functional no restrictions are placed on initial conditions the cost associated with the impulsive behavior of the system is examined as well as existence and uniqueness of the optimal control through a sequence of coordinate transformations it is proven that the optimal control can be found by solving a reduced order riccati equation

the role of the dissipation matrix in singular optimal control the optimal cost in the regular stationary linear quadratic optimal control problem is given by the maximal solution of an algebraic riccati equation this solution minimizes the rank of a certain matrix which we call the dissipation matrix the rank minimization problem is also meaningful in the singular case does it provide the optimal cost for the singular control problem this question was posed by j c willems in 1971 the answer is yes

a necessary and sufficient condition for solvability of the linear quadratic control problem without stabilty in this short paper a necessary and sufficient condition is given for the existence of a minimizing control for the regular non negative definite linear quadratic control problem without stability

invariant subspaces and invertibility properties for singular systems the general case abstract open loop definitions and properties of several subspaces for general singular systems are characterized by means of a fully algebraic distributional framework simple recursive algorithms for producing these spaces as well as related duality aspects turn out to follow directly from these definitions next we provide definitions and conditions for two notions of left right invertibility of a general singular system in terms of our distributions subspaces and rosenbrock s system matrix and we show which conditions represent the gap between our invertibility concepts finally we prove that in many cases left right invertibility is equivalent to left right invertibility of the system matrix

complexity of shapes embedded in z n with a bias towards squares shape complexity is a hard to quantify quality mainly due to its relative nature biased by euclidean thinking circles are commonly considered as the simplest however their constructions as digital images are only approximations to the ideal form consequently complexity orders computed in reference to circle are unstable unlike circles which lose their circleness in digital images squares retain their qualities hence we consider squares hypercubes in mathbb z n to be the simplest shapes relative to which complexity orders are constructed using the connection between l infty norm and squares we effectively encode squareness adapted simplification through which we obtain multi scale complexity measure where scale determines the level of interest to the boundary the emergent scale above which the effect of a boundary feature appendage disappears is related to the ratio of the contacting width of the appendage to that of the main body we discuss what zero complexity implies in terms of information repetition and constructibility and what kind of shapes in addition to squares have zero complexity

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

a convergent difference scheme for the infinity laplacian construction of absolutely minimizing lipschitz extensions this article considers the problem of building absolutely minimizing lipschitz extensions to a given function these extensions can be characterized as being the solution of a degenerate elliptic partial differential equation the infinity laplacian for which there exist unique viscosity solutions a convergent difference scheme for the infinity laplacian equation is introduced which arises by minimizing the discrete lipschitz constant of the solution at every grid point existence and uniqueness of solutions to the scheme is shown directly solutions are also shown to satisfy a discrete comparison principle solutions are computed using an explicit iterative scheme which is equivalent to solving the parabolic version of the equation

the linear quadratic optimal regulator for descriptor systems in this paper we investigate the linear quadratic optimal regulator problem for the continuous time descriptor system e dot x ax bu where e is in general a singular matrix we solve first a general finite horizon problem by applying the calculus of variations to derive the optimal trajectory of the vector consisting of the concatenated descriptor codescriptor and control vectors from this trajectory the optimal feedback gain relating the control and descriptor variable can be computed by transforming to a coordinate system which can be computed by performing a singular value decomposition of e we derive several riccati differential equations all of which have the same solution this solution gives the optimal cost the steady state optimal feedback gain can be computed by solving an eigenvalue eigenvector problem formulated from the untransformed system parameters in general there does not exist a unique optimal feedback gain but rather the gain is constrained to lie in a linear variety whose dimension is equal to the number of inputs times the rank deficiency of e

sampling curves with finite rate of innovation in this paper we extend the theory of sampling signals with finite rate of innovation fri to a specific class of two dimensional curves which are defined implicitly as the zeros of a mask function here the mask function has a parametric representation as a weighted summation of a finite number of complex exponentials and therefore has finite rate of innovation an associated edge image which is discontinuous on the predefined parametric curve is proved to satisfy a set of linear annihilation equations we show that it is possible to reconstruct the parameters of the curve i e to detect the exact edge positions in the continuous domain based on the annihilation equations robust reconstruction algorithms are also developed to cope with scenarios with model mismatch moreover the annihilation equations that characterize the curve are linear constraints that can be easily exploited in optimization problems for further image processing e g image up sampling we demonstrate one potential application of the annihilation algorithm with examples in edge preserving interpolation experimental results with both synthetic curves as well as edges of natural images clearly show the effectiveness of the annihilation constraint in preserving sharp edges and improving snrs

singular optimal control a geometric approach linear quadratic singular optimal control problem is solved for nonminimum phase and noninvertible systems a state space decomposition is obtained and a reduced order nonsingular subproblem is solved the optimal stabilizing input of the singular problem has been found when there are no transmission zeros on the imaginary axis

the kalman yakubovich popov inequality for differential algebraic systems abstract in this paper we revisit the kalman yakubovich popov lemma for differential algebraic control systems this lemma relates the positive semi definiteness of the popov function on the imaginary axis to the solvability of a linear matrix inequality on a certain subspace further emphasis is placed on the lur e equation whose solution set consists loosely speaking of the rank minimizing solutions of the kalman yakubovich popov inequality we show that there is a correspondence between the solution set of the lur e equation and the deflating subspaces of certain even matrix pencils finally we show that under certain conditions the lur e equation admits stabilizing anti stabilizing and extremal solutions we note that for our results we neither assume impulse controllability nor make any assumptions on the index of the system

higher order continuous time implicit systems consistency and weak consistency impulse controllability geometric concepts and invertibility properties abstract the paper provides a simple and fully algebraic distributional setup for a higher order linear implicit system with arbitrary constant coefficients on the continuous nonnegative time axis the distributional system equation exhibits impulses depending on arbitrary points in the state space and for these points several concepts of weak consistency of increasing degree are introduced in such a way that weak consistency of degree 0 coincides with the standard concept of consistency whereas weak consistency of the highest degree is related to impulse controllability all weakly consistent subspaces are expressed in the consistent subspace and for the latter space a numerically stable algorithm is presented also we derive for every concept of weak consistency a condition in terms of the system coefficients only for the statement that the associated subspace is as large as possible in particular we get a condition for impulse controllability that generalizes the celebrated condition for impulse controllability of a first order system further we state two conditions for control solvability and we specify when distributional state and or input trajectories are unique finally we define and characterize various subspaces for a higher order system in combination with an output equation and link these spaces to two of our four invertibility concepts namely those in the strong sense and we establish that the above composite system is left right invertible in the strong sense if and only if the corresponding system matrix is left right invertible as a rational matrix even if the transfer matrix does not exist

singular lq problem for nonregular descriptor systems in this paper a singular linear quadratic lq problem for non regular descriptor systems is investigated under some general conditions the optimal control and the optimal state of the lq problem are given the optimal control is synthesized as state feedback all the finite eigenvalues of the closed loop system are located on the left half complex plane the state of the closed loop system has the least free entries

singular systems a new approach in the time domain a new approach in the time domain is developed for the study of singular linear systems of the form e dot x ax bu y cx with e singular central to the approach is the fundamental triple alpha e a 1 e alphae a 1 b c with a real number satisfying det alpha e a neq 0 controllability and observability matrices are expressed in terms of the fundamental triple new tests for impulse controllability and impulse observability are also established feedback control problems including pole placement decoupling and disturbance localization are studied by use of a modified proportional and derivative feedback of the state in the form of u f alpha x dot x v

shapes from pixels continuous domain visual signals are usually captured as discrete digital images this operation is not invertible in general in the sense that the continuous domain signal cannot be exactly reconstructed based on the discrete image unless it satisfies certain constraints e g bandlimitedness in this paper we study the problem of recovering shape images with smooth boundaries from a set of samples thus the reconstructed image is constrained to regenerate the same samples consistency as well as forming a shape bilevel image we initially formulate the reconstruction technique by minimizing the shape perimeter over the set of consistent binary shapes next we relax the non convex shape constraint to transform the problem into minimizing the total variation over consistent non negative valued images we also introduce a requirement called reducibility that guarantees equivalence between the two problems we illustrate that the reducibility property effectively sets a requirement on the minimum sampling density we also evaluate the performance of the relaxed alternative in various numerical experiments

principal eigenvalue of a very badly degenerate operator and applications abstract in this paper we define and investigate the properties of the principal eigenvalue of the singular infinity laplace operator u d 2 u d u d u d u d u this operator arises from the optimal lipschitz extension problem and it plays the same fundamental role in the calculus of variations of l functionals as the usual laplacian does in the calculus of variations of l 2 functionals our approach to the eigenvalue problem is based on the maximum principle and follows the outline of the celebrated work of berestycki nirenberg and varadhan h berestycki l nirenberg s r s varadhan the principal eigenvalue and maximum principle for second order elliptic operators in general domains comm pure appl math 47 1 1994 47 92 in the case of uniformly elliptic linear operators as an application we obtain existence and uniqueness results for certain related non homogeneous problems and decay estimates for the solutions of the evolution problem associated to the infinity laplacian

reconstruction of binary shapes from blurred images via hankel structured low rank matrix recovery with the dominance of digital imaging systems we are often dealing with discrete domain samples of an analog image due to physical limitations all imaging devices apply a blurring kernel on the input image before taking samples to form the output pixels in this paper we focus on the reconstruction of binary shape images from few blurred samples this problem has applications in medical imaging shape processing and image segmentation our method relies on representing the analog shape image in a discrete grid much finer than the sampling grid we formulate the problem as the recovery of a rank r matrix that is formed by a hankel structure on the pixels we further propose efficient admm based algorithms to recover the low rank matrix in both noiseless and noisy settings we also analytically investigate the number of required samples for successful recovery in the noiseless case for this purpose we study the problem in the random sampling framework and show that with mathcal o r log 4 n 1 n 2 random samples where the size of the image is assumed to be n 1 times n 2 we can guarantee the perfect reconstruction with high probability under mild conditions we further prove the robustness of the proposed recovery in the noisy setting by showing that the reconstruction error in the noisy case is bounded when the input noise is bounded simulation results confirm that our proposed method outperform the conventional total variation minimization in the noiseless settings

disconnected skeleton shape at its absolute scale we present a new skeletal representation along with a matching framework to address the deformable shape recognition problem the disconnectedness arises as a result of excessive regularization that we use to describe a shape at an attainably coarse scale our motivation is to rely on stable properties the shape instead of inaccurately measured secondary details the new representation does not suffer from the common instability problems of the traditional connected skeletons and the matching process gives quite successful results on a diverse database of 2d shapes an important difference of our approach from the conventional use of skeleton is that we replace the local coordinate frame with a global euclidean frame supported by additional mechanisms to handle articulations and local boundary deformations as a result we can produce descriptions that are sensitive to any combination of changes in scale position orientation and articulation as well as invariant ones

all optimal controls for the singular linear quadratic problem without stability a new interpretation of the optimal cost abstract the singular linear quadratic control problem without stability is solved by means of a generalized dual structure algorithm in order to generate all optimal inputs furthermore it is shown that the optimal cost can be interpreted as the smallest nonnegative rank minimizing solution of a certain matrix inequality the so called dissipation inequality

on a weighted total variation minimization problem abstract the present article is devoted to the study of a constrained weighted total variation minimization problem which may be viewed as a relaxation of a generalized cheeger problem and is motivated by landslide modeling using the fact that the set of minimizers is invariant by a wide class of monotone transformations we prove that level sets of minimizers are generalized cheeger sets and obtain qualitative properties of the minimizers they are all bounded and all achieve their essential supremum on a set of positive measure