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Gâteaux derivative

About: Gâteaux derivative is a(n) research topic. Over the lifetime, 219 publication(s) have been published within this topic receiving 3550 citation(s). more


Open accessJournal ArticleDOI: 10.1137/S0036139902408928
Abstract: We consider the problem of segmenting an image through the minimization of an energy criterion involving region and boundary functionals. We show that one can go from one class to the otherby solving Poisson's orHelmholtz's equation with well-chosen boundar y conditions. Using this equivalence, we study the case of a large class of region functionals by standard methods of the calculus of variations and derive the corresponding Euler-Lagrange equations. We revisit this problem using the notion of a shape derivative and show that the same equations can be elegantly derived without going through the unnatural step of converting the region integrals into boundary integrals. We also define a larger class of region functionals based on the estimation and comparison to a prototype of the probability density distribution of image features and show how the shape derivative tool allows us to easily compute the corresponding Gateaux derivatives and Euler-Lagrange equations. Finally we apply this new functional to the problem of regions segmentation in sequences of color images. We briefly describe our numerical scheme and show some experimental results. more

  • Fig. 2. Evolution of the region histogram q̂(α,Ω) of the current frame during the evolution of the active contour.
    Fig. 2. Evolution of the region histogram q̂(α,Ω) of the current frame during the evolution of the active contour.
  • Fig. 1. The reference segmentation of the previous frame (a), the corresponding background reference histogram qc (b), and the corresponding object reference histogram q (c).
    Fig. 1. The reference segmentation of the previous frame (a), the corresponding background reference histogram qc (b), and the corresponding object reference histogram q (c).
Topics: Calculus of variations (58%), Image segmentation (54%), Boundary (topology) (52%) more

283 Citations

Open accessJournal ArticleDOI: 10.1007/BF02391908
Edgar Asplund1Institutions (1)
01 Dec 1968-Acta Mathematica
Abstract: A continuous convex function of one real variable is differentiable, except perhaps at a countable subset of its interval of continuity. The present paper deals with generalizat ions of this e lementary s ta tement to convex functions which are defined on some Banach space E, and continuous in the norm topology, with \"differentiable\" replaced either by \"Frdchet differentiable\" or \"Gateaux differentiable\". Since for E = L ~ ( 0 , 1 ) the very norm funct ion/ (x) = Ilxll for x in E, which is convex and continuous on all of E, is nowhere even G~teaux differentiable (Mazur [13]), this amounts to a classification of the category of all Banach spaces depending upon whether certain differentiability s ta tements hold. Therefore we say tha t a Banach space is a strong di//erentiability space (SDS) if the following theorem holds for it. more

Topics: Subderivative (61%), Danskin's theorem (61%), Convex analysis (59%) more

262 Citations

Open accessBook
01 Jan 1997-
Abstract: Canonical Examples of Weak Asplund Spaces. Properties of Gateaux Differentiability Spaces and Weak Asplund Spaces. Stegall's Classes. Two More Concrete Classes of Banach Spaces that Lie in . Fragmentability. "Long Sequences" of Linear Projections. Vasak Spaces and Gul'ko Compacta. A Characterization of WCG Spaces and of Eberlein Compacta. Main Open Questions and Problems. References. Index. more

Topics: Interpolation space (63%), Compact-open topology (63%), Fréchet space (63%) more

218 Citations

Journal ArticleDOI: 10.1007/BF00940933
Alexander Shapiro1Institutions (1)
Abstract: Various definitions of directional derivatives in topological vector spaces are compared. Directional derivatives in the sense of Gâteaux, Frechet, and Hadamard are singled out from the general framework of σ-directional differentiability. It is pointed out that, in the case of finite-dimensional spaces and locally Lipschitz mappings, all these concepts of directional differentiability are equivalent. The chain rule for directional derivatives of a composite mapping is discussed. more

Topics: Gâteaux derivative (64%), Directional derivative (53%), Lipschitz continuity (53%) more

201 Citations

Journal ArticleDOI: 10.1007/S10092-017-0213-8
Dazhi Zhao1, Maokang Luo1Institutions (1)
10 Jan 2017-Calcolo
Abstract: Fractional calculus is a powerful and effective tool for modelling nonlinear systems. In this paper, we introduce a class of new fractional derivative named general conformable fractional derivative (GCFD) to describe the physical world. The GCFD is generalized from the concept of conformable fractional derivative (CFD) proposed by Khalil. We point out that the term $$t^{1-\alpha }$$ in CFD definition is not essential and it is only a kind of “fractional conformable function”. We also give physical and geometrical interpretations of GCFD which thus indicate potential applications in physics and engineering. It is easy to demonstrate that CFD is a special case of GCFD, then to the authors’ knowledge, so far we first give the physical and geometrical interpretations of CFD. The above work is done by a new framework named Extended Gâteaux derivative and Linear Extended Gâteaux derivative which are natural extensions of Gâteaux derivative. As an application, we discuss a scheme for solving fractional differential equations of GCFD. more

Topics: Gâteaux derivative (64%), Fractional calculus (64%), Generalizations of the derivative (63%) more

150 Citations

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Topic's top 5 most impactful authors

Marián Fabian

5 papers, 105 citations

David Preiss

4 papers, 70 citations

Jonathan M. Borwein

3 papers, 127 citations

Sever S Dragomir

3 papers, 15 citations

A.Y. Aköz

3 papers, 125 citations

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