minimum mean squared error transform coding and subband coding knowledge of the power spectrum of a stationary random sequence can be used for quantizing the signal efficiently and with minimum mean squared error a multichannel filter is used to transform the random sequence into an intermediate set of variables that are quantized using independent scalar quantizers and then inverse filtered producing a quantized version of the original sequence equal word length and optimal word length quantization at high bit rates is considered an analytical solution for the filter that minimizes the mean squared quantization error is obtained in terms of its singular value decomposition the performance is characterized by a set of invariants termed second order modes which are derived from the eigenvalue decomposition of the matrix valued power spectrum a more general rank reduced model is used for decreasing distortion by introducing bias the results are specialized to the case when the vector valued time series is obtained from a scalar random sequence which gives rise to a filter bank model for quantization the asymptotic performance of such a subband coder is derived and shown to coincide with the asymptotic bound for transform coding quantization employing a single scalar pre and postfilter traditional transform coding using a square linear transformation and subband coding in filter banks arise as special cases of the structure analyzed here

on the variational approximation of combined second and fourth order geometric evolution equations we present a variational formulation of combined motion by minus the laplacian of curvature and mean curvature flow as well as related flows the proposed scheme covers both the closed curve case and the case of curves that are connected via triple or quadruple junction points or intersect the external boundary on introducing a parametric finite element approximation we prove stability bounds and compare our scheme with existing approaches the presented scheme has very good properties with respect to the equidistribution of mesh points and if applicable area conservation

optimum nonnegative integer bit allocation for wavelet based signal compression and coding in this paper first a new optimum nonnegative integer bit allocation oniba algorithm is proposed for transform coding with lower computational complexity compared to the existing algorithms the algorithm is extended for the cases in which the variances of some sources are zero then an oniba algorithm is developed for wavelet based subband coding finally the algorithm is modified for adaptive subband coding computer simulations demonstrate considerable superiority of the proposed subband coding algorithm over the existing ones

theory of rate distortion optimal constrained filterbanks application to iir and fir biorthogonal designs we design filterbanks that are best matched to input signal statistics in m channel subband coders using a rate distortion criterion previous research has shown that unconstrained length paraunitary filterbanks optimized under various energy compaction criteria are principal component filterbanks that satisfy two fundamental properties total decorrelation and spectral majorization in this paper we first demonstrate that the two properties above are not specific to the paraunitary case but are satisfied for a much broader class of design constraints our results apply to a broad class of rate distortion criteria including the conventional coding gain criterion as a special case a consequence of these properties is that optimal perfect reconstruction pr filterbanks take the form of the cascade of principal component filterbanks and a bank of pre and post conditioning filters the proof uses variational techniques and is applicable to a variety of constrained design problems in the second part of this paper we apply the theory above to practical filterbank design problems we give analytical expressions for optimal iir biorthogonal filterbanks our analysis validates a conjecture by several researchers we then derive the asymptotic limit of optimal fir biorthogonal filterbanks as filter length tends to infinity the performance loss due to fir constraints is quantified theoretically and experimentally the optimal filters are quite different from traditional filters finally a sensitivity analysis is presented

state space solutions to hadamard weighted ℋ and ℋ 2 control problems variations of the standard h and h 2 control problems are presented state space solutions are given for the problems of weighting the individual elements of a closed loop transfer function while i satisfying an h frobenius norm bound or ii minimizing the h 2 norm of the closed loop the optimal h frobenius case is analysed and it is shown that the optimal controller is often unique in contrast to the standard h control problem the solutions of an algebraic riccati equation are examined in the h 2 case to reveal the structure of the h 2 optimal controller

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

on affine plane curve evolution abstract an affine invariant curve evolution process is presented in this work the evolution studied is the affine analogue of the euclidean curve shortening flow evolution equations for both affine and euclidean invariants are developed an affine version of the classical euclidean isoperimetric inequality is proved this inequality is used to show that in the case of affine evolution of convex plane curves the affine isoperimetric ratio is a non decreasing function of time convergence of this affine isoperimetric ratio to the ellipse s value 8 2 as well as convergence in the hausdorff metric of the evolving curve to an ellipse is also proved

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

error analysis of finite element approximations of the inverse mean curvature flow arising from the general relativity this paper proposes and analyzes a finite element method for a nonlinear singular elliptic equation arising from the black hole theory in the general relativity the nonlinear equation which was derived and analyzed by huisken and ilmanen in j diff geom 59 353 437 represents a level set formulation for the inverse mean curvature flow describing the evolution of a hypersurface whose normal velocity equals the reciprocal of its mean curvature we first propose a finite element method for a regularized flow which involves a small parameter a rigorous analysis is presented to study well posedness and convergence of the scheme under certain mesh constraints and optimal rates of convergence are verified we then prove uniform convergence of the finite element solution to the unique weak solution of the nonlinear singular elliptic equation as the mesh size h and the regularization parameter both tend to zero computational results are provided to show the efficiency of the proposed finite element method and to numerically validate the jumping out phenomenon of the weak solution of the inverse mean curvature flow numerical studies are presented to evidence the existence of a polynomial scaling law between the mesh size h and the regularization parameter for optimal convergence of the proposed scheme finally a numerical convergence study for another approach recently proposed by r moser the inverse mean curvature flow and p harmonic functions preprint u bath 2005 for approximating the inverse mean curvature flow via p harmonic functions is also included

rank reduction for modeling stationary signals rank reduction is developed as a general principle for trading off model bias and model variance in the analysis and synthesis of signals the principle is applied to three basic problems stationary time series modeling stationary time series whitening and vector quantization each problem brings its own surprises and insights

fast reaction slow diffusion and curve shortening the reaction diffusion problem u 1 varepsilon delta u varepsilon 1 v n u quad u x 0 varepsilon g x quad partial n u 0 text on partial omega for a vector u x t varepsilon is considered in a domain omega in r m an asymptotic solution is constructed for varepsilon small it shows that at each x u tends quickly to a minimum of v u when v has several minima when u tends to a piecewise constant function boundary layer expansions are constructed around the resulting surfaces of discontinuity or fronts each front is found to move along its normal with a constant velocity determined by the discontinuity v in v across it when v 0 the front s normal velocity is varepsilon kappa where kappa is its mean curvature the motion of fronts in this manner is studied for arcs in the plane which are normal to partial omega at their endpoints and for fronts that are closed curves it is shown a front can shrink to a point in

robust decoupling of discrete systems using spl lscr sup 1 optimization we address a control design problem related to decoupling namely robust decoupling the idea of robust decoupling is defined and shown to be equivalent to robust performance of a hadamard weighted spl lscr sup 1 performance measure a necessary and sufficient condition to determine if a compensator satisfies robust performance of a hadamard weighted spl lscr sup 1 performance measure in the presence of structured uncertainty is given this condition can be used to determine if a compensator robustly decouples a plant finally an iterative scheme for synthesizing a robust decoupling controller is discussed

frequency weighted optimal hankel norm approximation of stable transfer functions abstract this paper shows how to introduce arbitrary frequency weighting into the optimal hankel norm approximation problem for scalar finite dimensional linear time invariant systems the procedure has applications to the closed form design of reduced order controllers

linking anisotropic sharp and diffuse surface motion laws via gradient flows we compare four surface motion laws for sharp surfaces with their diffuse interface counterparts by means of gradient flows on corresponding energy functionals the energy functionals can be defined to give the same dependence on normal direction for the energy of sharp plane surfaces as for their diffuse counterparts the anisotropy of the kinetics can be incorporated into the inner product without affecting the energy functional

on the parametric finite element approximation of evolving hypersurfaces in r 3 we present a variational formulation of motion by minus the laplacian of curvature and mean curvature flow as well as related second and fourth order flows of a closed hypersurface in r 3 on introducing a parametric finite element approximation we prove stability bounds and compare our scheme with existing approaches the presented scheme has very good properties with respect to the distribution of mesh points and if applicable volume conservation

theory and design of optimum fir compaction filters the problem of optimum fir energy compaction filter design for a given number of channels m and a filter order n is considered the special cases where n m and n spl infin have analytical solutions that involve eigenvector decomposition of the autocorrelation matrix and the power spectrum matrix respectively we deal with the more difficult case of m n spl infin for the two channel case and for a restricted but important class of random processes we give an analytical solution for the compaction filter that is characterized by its zeros on the unit circle this also corresponds to the optimal two channel fir filter bank that maximizes the coding gain under the traditional quantization noise assumptions with a minor extension this can also be used to generate optimal wavelets for the arbitrary m channel case we provide a very efficient suboptimal design method called the window method the method involves two stages that are associated with the above two special cases as the order increases the suboptimality becomes negligible and the filter converges to the ideal optimal solution we compare the window method with a previously introduced technique based on linear programming

theory and design of m channel maximally decimated quadrature mirror filters with arbitrary m having the perfect reconstruction property based on the concept of losslessness in digital filter structures this paper derives a general class of maximally decimated m channel quadrature mirror filter banks that lead to perfect reconstruction the perfect reconstruction property guarantees that the reconstructed signal hat x n is a delayed version of the input signal x n i e hat x n x n n 0 it is shown that such a property can be satisfied if the alias component matrix ac matrix for short is unitary on the unit circle of the z plane the number of channels m is arbitrary and when m is two the results reduce to certain recently reported 2 channel perfect reconstruction qmf structures a procedure based on recently reported fir cascaded lattice structures is presented for optimal design of such fir m channel filter banks design examples are included

multiple objective optimal control of linear systems the quadratic norm case control of a linear system to meet requirements embodied in multiple performance criteria is recognized as the typical design problem the authors study the design of control when every criterion of interest takes the form of an h sub 2 norm defined over certain exogenous inputs and controlled outputs of the system the objective is to find all stabilizing controllers that are noninferior with respect to the defined objectives that is to find the set of pareto optimal controllers the novel feature is the use of the youla parametrization to represent the admissible controllers this representation enables the authors to show that each criterion and hence their weighted sum is convex over the class of admissible controllers since the set of admissible controllers is itself convex the pareto optimal controllers can be determined by the standard scalarization technique it is concluded that in general the controller order is higher than the order of the plant

on the use of hadamard weightings in the partial decoupling of h based designs abstract in this paper a solution to the hadamard weighted h hwh 1 block problem is proposed the solution is based on a method by safonov and le 1988 the robust stability of designs based on the hwh method is also discussed

finite element approximation of a phase field model for surface diffusion of voids in a stressed solid we consider a fully practical finite element approximation of the degenerate cahn hilliard equation with elasticity find the conserved order parameter theta x t in 1 1 and the displacement field underline u x t in bbb r 2 such that gamma frac partial theta partial t nabla cdot b theta nabla gamma delta theta gamma 1 psi theta tfrac 12 c theta cal c underline underline cal e underline u underline underline cal e underline u quad nabla cdot c theta cal c underline underline cal e underline u underline 0 subject to an initial condition theta 0 cdot in 1 1 on theta and boundary conditions on both equations here gamma in bbb r 0 is the interfacial parameter psi is a nonsmooth double well potential underline underline cal e is the symmetric strain tensor cal c is the possibly anisotropic elasticity tensor c s c 0 frac 12 1 c 0 1 s with c 0 gamma in bbb r 0 and b s 1 s 2 is the degenerate diffusion mobility in addition to showing stability bounds for our approximation we prove convergence and hence existence of a solution to this nonlinear degenerate parabolic system in two space dimensions finally some numerical experiments are presented

multirate filter banks and transform coding gain an analytical derivation of quantization error in filter banks using additive and signal dependent noise models is presented the coding gain over pcm is derived for both noise models interesting properties of coding gain in optimized filter banks with an ar 1 input signal model asymptotic performance of multirate systems and the analytical constraints required to achieve such performance are discussed connections to the recent developments of optimal subband systems are presented

a structured approximation problem with applications to frequency weighted model reduction the authors consider the problem of finding necessary and sufficient conditions for the existence of a q with no more than k poles in the closed left half plane such that the norm of a certain matrix

capturing the behavior of bubbles and drops using the variational level set approach we reproduce the general behavior of complicated bubble and droplet motions using the variational level set formulation introduced by the authors earlier our approach here ignores inertial effects thus the motion is only correct as an approximation for very viscous problems however the steady states are true equilibrium solutions inertial forces will be added in future work the problems include soap bubbles colliding and merging drops falling or remaining attached to a generally irregular ceiling and liquid penetrating through a funnel in both two and three dimensions each phase is identified with a particular level set function the zero level set of this function is that particular phase boundary the level set functions all evolve in time through a constrained gradient descent procedure so as to minimize an energy functional the functions are coupled through physical constraints and through the requirements that different phases do not overlap and vacuum regions do not develop both boundary conditions and inequality constraints are cast in terms of either local or global equality constraints the gradient projection method leads to a system of perturbed by curvature if surface tension is involved hamilton jacobi equations coupled through a constraint the coupling is enforced using the lagrange multiplier associated with this constraint the numerical implementation requires much of the modern level set technology in particular we achieve a significant speed up by using the fast localization algorithm of h k zhao m kang b merriman d peng and s osher

rate distortion optimal subband coding without perfect reconstruction constraints we investigate the design of subband coders without the traditional perfect reconstruction constraint on the filters the coder uses scalar quantizers and its filters and bit allocation are designed to optimize a rate distortion criterion using convexity analysis we show that optimality can be achieved using filterbanks that are the cascade of a paraunitary principal component filterbank for the input spectral process and a set of pre and postfilters surrounding each quantizer analytical expressions for the pre and postfilters are then derived an algorithm for computing the globally optimal filters and bit allocation is given we also develop closed form solutions for the special case of two channel coders under an exponential rate distortion model finally we investigate a constrained length version of the filter design problem which is applicable to practical coding scenarios while the optimal filterbanks are nearly perfect reconstruction at high rates we demonstrate an apparently surprising advantage of optimal fir filterbanks they significantly outperform optimal perfect reconstruction fir filterbanks at all bit rates

evolution of plane curves driven by a nonlinear function of curvature and anisotropy in this paper we study evolution of plane curves satisfying a geometric equation v beta k nu where v is the normal velocity and k and nu are the curvature and tangential angle of a plane curve gamma we follow the direct approach and we analyze the so called intrinsic heat equation governing the motion of plane curves obeying such a geometric equation the intrinsic heat equation is modified to include an appropriate nontrivial tangential velocity functional alpha we show how the presence of a nontrivial tangential velocity can prevent numerical solutions from forming various instabilities from an analytical point of view we present some new results on short time existence of a regular family of evolving curves in the degenerate case when beta k nu gamma nu k m 0 m le 2 and the governing system of equations includes a nontrivial tangential velocity functional

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

algebraic theory of optimal filterbanks we introduce an optimality theory for finite impulse response fir filterbanks using a general algebraic point of view we consider an admissible set spl lscr of fir filterbanks and use scalability as the main notion based on which performance of the elements in spl lscr are compared we show that quantification of scalability leads naturally to a partial ordering on the set spl lscr an optimal solution is therefore represented by the greatest element in spl lscr it turns out that a greatest element does not necessarily exist in spl lscr hence one has to settle with one of the maximal elements that exist in spl lscr we provide a systematic way of finding a maximal element by embedding the partial ordering at hand in a total ordering this is done by using a special class of order preserving functions known as schur convex there is however a price to pay for achieving a total ordering there are infinitely many possible choices for schur convex functions and the optimal solution specified in spl lscr depends on this subjective choice an interesting aspect of the presented algebraic theory is that the connection between several concepts namely principal component filterbanks pcfbs filterbanks with maximum coding gain and filterbanks with good scalability is clearly revealed we show that these are simply associated with different extremal elements of the partial ordering induced on spl lscr by scalability

on a new method for h_ 2 based decomposition this note proposes a simple linear matrix inequality lmi system for the design of static precompensators to reduce the interactions of a multivariable system the approach is based on minimizing the h2 norm of a modified system and the lmis are derived from the well known technique of pseudodiagonalisation the approach is applied to two complex real life benchmark problems with high levels of interaction it is shown that its performance is significantly better than previously proposed lmi optimization techniques for designing static precompensators

a characterization of all solutions to the four block general distance problem all solutions to the four block general distance problem which arises in h infty optimal control are characterized the procedure is to embed the original problem in an all pass matrix which is constructed it is then shown that part of this all pass matrix acts as a generator of all solutions special attention is given to the characterization of all optimal solutions by invoking a new descriptor characterization of all pass transfer functions as an application necessary and sufficient conditions are found for the existence of an h infty optimal controller following that a descriptor representation of all solutions is derived

a diffuse interface model for alloys with multiple components and phases a nonisothermal phase field model for alloys with multiple phases and components is derived the model allows for arbitrary phase diagrams we relate the model to classical sharp interface models by formally matched asymptotic expansions in addition we discuss several examples and relate our model to the ones already existing