Results 1  10
of
164
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 ..."
Abstract

Cited by 116 (24 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 57 (6 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 55 (20 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 36 (14 self)
 Add to MetaCart
(Show Context)
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.
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 v ..."
Abstract

Cited by 32 (0 self)
 Add to MetaCart
(Show Context)
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
TRANSITION SEMIGROUPS OF BANACH SPACE VALUED ORNSTEINUHLENBECK PROCESSES
, 2006
"... We investigate the transition semigroup of the solution to a stochastic evolution equation dX(t) = AX(t) dt + dWH(t), t ≥ 0, where A is the generator of a C0semigroup S on a separable real Banach space E and {WH(t)}t≥0 is cylindrical white noise with values in a real Hilbert space H which is conti ..."
Abstract

Cited by 25 (1 self)
 Add to MetaCart
We investigate the transition semigroup of the solution to a stochastic evolution equation dX(t) = AX(t) dt + dWH(t), t ≥ 0, where A is the generator of a C0semigroup S on a separable real Banach space E and {WH(t)}t≥0 is cylindrical white noise with values in a real Hilbert space H which is continuously embedded in E. Various properties of these semigroups, such as the strong Feller property, the spectral gap property, and analyticity, are characterized in terms of the behaviour of S in H. In particular we investigate the interplay between analyticity of the transition semigroup,
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 ..."
Abstract

Cited by 19 (14 self)
 Add to MetaCart
(Show Context)
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.