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72
ScaleSpace for Discrete Signals
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 1990
"... We address the formulation of a scalespace theory for discrete signals. In one dimension it is possible to characterize the smoothing transformations completely and an exhaustive treatment is given, answering the following two main questions: 1. Which linear transformations remove structure in the ..."
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Cited by 96 (22 self)
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We address the formulation of a scalespace theory for discrete signals. In one dimension it is possible to characterize the smoothing transformations completely and an exhaustive treatment is given, answering the following two main questions: 1. Which linear transformations remove structure in the sense that the number of local extrema (or zerocrossings) in the output signal does not exceed the number of local extrema (or zerocrossings) in the original signal? 2. How should one create a multiresolution family of representations with the property that a signal at a coarser level of scale never contains more structure than a signal at a finer level of scale? We propose that there is only one reasonable way to define a scalespace for 1D discrete signals comprising a continuous scale parameter, namely by (discrete) convolution with the family of kernels T (n; t) = e I n (t), where I n are the modified Bessel functions of integer order. Similar arguments applied in the continuous case uniquely lead to the Gaussian kernel.
Moving coframes. II. Regularization and theoretical foundations
 Acta Appl. Math
, 1999
"... Abstract. The primary goal of this paper is to provide a rigorous theoretical justification of Cartan’s method of moving frames for arbitrary finitedimensional Lie group actions on manifolds. The general theorems are based a new regularized version of the moving frame algorithm, which is of both th ..."
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Cited by 46 (6 self)
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Abstract. The primary goal of this paper is to provide a rigorous theoretical justification of Cartan’s method of moving frames for arbitrary finitedimensional Lie group actions on manifolds. The general theorems are based a new regularized version of the moving frame algorithm, which is of both theoretical and practical use. Applications include a new approach to the construction and classification of differential invariants and invariant differential operators on jet bundles, as well as equivalence, symmetry, and rigidity theorems for submanifolds under general transformation groups. The method also leads to complete classifications of generating systems of differential invariants, explicit commutation formulae for the associated invariant differential operators, and a general classification theorem for syzygies of the higher order differentiated differential invariants. A variety of illustrative examples demonstrate how the method can be directly applied to practical problems arising in geometry, invariant theory, and differential equations.
Stochastic Solutions for Fractional Cauchy Problems
 Calc. Appl. Anal
, 2001
"... Every infinitely divisible law defines a convolution semigroup that solves an abstract Cauchy problem. In the fractional Cauchy problem, we replace the first order time derivative by a fractional derivative. Solutions to fractional Cauchy problems are obtained by subordinating the solution to the or ..."
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Cited by 23 (7 self)
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Every infinitely divisible law defines a convolution semigroup that solves an abstract Cauchy problem. In the fractional Cauchy problem, we replace the first order time derivative by a fractional derivative. Solutions to fractional Cauchy problems are obtained by subordinating the solution to the original Cauchy problem. Fractional Cauchy problems are useful in physics to model anomalous di#usion.
Integrated Semigroups and their Applications to the Abstract Cauchy Problem
 Pacific J. Math
, 1988
"... This paper is concerned with characterizations of those linear, closed, but not necessarily densely defined operators A on a Banach space E with nonempty resolvent set for which the abstract Cauchy problem u'(t) = Au(t), u(0) = x has unique, exponentially bounded solutions for every initial value ..."
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Cited by 15 (0 self)
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This paper is concerned with characterizations of those linear, closed, but not necessarily densely defined operators A on a Banach space E with nonempty resolvent set for which the abstract Cauchy problem u'(t) = Au(t), u(0) = x has unique, exponentially bounded solutions for every initial value x e D(A n). Investigating these operators we are led to the class of "integrated semigroups". Among others, this class contains the classes of strongly continuous semigroups and cosine families and the class of exponentially bounded distribution semigroups. The given characterizations of the generators of these integrated semigroups unify and generalize the classical characterizations of generators of strongly continuous semigroups, cosine families or exponentially bounded distribution semigroups. We indicate how integrated semigroups can be used studying second order Cauchy problems u"{t) — A\u'{t) Aiu(t) = 0, operator valued equations U'(t) = A { U(t) + U(t)A2 and nonautonomous equations u'{t) = A(t)u(t). 1. Introduction. We
Generation theory for semigroups of holomorphic mappings in Banach spaces
 Abstr. Appl. Anal
, 1996
"... Abstract. We study nonlinear semigroups ofholomorphic mappings in Banach spaces and their infinitesimal generators. Using resolvents, we characterize, in particular, bounded holomorphic generators on bounded convex domains and obtain an analog ofthe Hille exponential formula. We then apply our resul ..."
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Cited by 11 (10 self)
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Abstract. We study nonlinear semigroups ofholomorphic mappings in Banach spaces and their infinitesimal generators. Using resolvents, we characterize, in particular, bounded holomorphic generators on bounded convex domains and obtain an analog ofthe Hille exponential formula. We then apply our results to the null point theory ofsemiplus complete vector fields. We study the structure ofnull point sets and the spectral characteristics of null points, as well as their existence and uniqueness. A global version of the implicit function theorem and a discussion of some open problems are also included.
Brownian subordinators and fractional Cauchy problems
, 2007
"... Abstract. A Brownian time process is a Markov process subordinated to the absolute value of an independent onedimensional Brownian motion. Its transition densities solve an initial value problem involving the square of the generator of the original Markov process. An apparently unrelated class of p ..."
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Cited by 8 (4 self)
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Abstract. A Brownian time process is a Markov process subordinated to the absolute value of an independent onedimensional Brownian motion. Its transition densities solve an initial value problem involving the square of the generator of the original Markov process. An apparently unrelated class of processes, emerging as the scaling limits of continuous time random walks, involve subordination to the inverse or hitting time process of a classical stable subordinator. The resulting densities solve fractional Cauchy problems, an extension that involves fractional derivatives in time. In this paper, we will show a close and unexpected connection between these two classes of processes, and consequently, an equivalence between these two families of partial differential equations. 1.
Concavity of eigenvalue sums and the spectral shift function
 J. Funct. Anal
, 2000
"... ABSTRACT. It is well known that the sum of negative (positive) eigenvalues of some finite Hermitian matrix V is concave (convex) with respect to V. Using the theory of the spectral shift function we generalize this property to selfadjoint operators on a separable Hilbert space with an arbitrary spe ..."
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Cited by 7 (2 self)
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ABSTRACT. It is well known that the sum of negative (positive) eigenvalues of some finite Hermitian matrix V is concave (convex) with respect to V. Using the theory of the spectral shift function we generalize this property to selfadjoint operators on a separable Hilbert space with an arbitrary spectrum. More precisely, we prove that the spectral shift function integrated with respect to the spectral parameter from − ∞ to λ (from λ to +∞) is concave (convex) with respect to trace class perturbations. The case of relative trace class perturbations is also considered.
Equivalence of Internal and External Stability for a Class of Distributed Systems
, 1990
"... It is well known that for infinitedimensional systems, exponential stability is not necessarily determined by the location of spectrum. ..."
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Cited by 6 (4 self)
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It is well known that for infinitedimensional systems, exponential stability is not necessarily determined by the location of spectrum.