Results 11  20
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711
An introduction to total variation for image analysis
 in Theoretical Foundations and Numerical Methods for Sparse Recovery, De Gruyter
, 2010
"... These notes address various theoretical and practical topics related to Total Variationbased image reconstruction. They focuse first on some theoretical results on functions which minimize the total variation, and in a second part, describe a few standard and less standard algorithms to minimize th ..."
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Cited by 44 (3 self)
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These notes address various theoretical and practical topics related to Total Variationbased image reconstruction. They focuse first on some theoretical results on functions which minimize the total variation, and in a second part, describe a few standard and less standard algorithms to minimize the total variation in a finitedifferences setting, with a series of applications from simple denoising to stereo, or deconvolution issues, and even more exotic uses like the minimization of minimal partition problems.
The spectral action for Moyal planes
 J. Math. Phys
"... Extending a result of D. V. Vassilevich [50], we obtain the asymptotic expansion for the trace of a spatially regularized heat operator LΘ (f)e−t△Θ, where △Θ is a generalized Laplacian defined with Moyal products and LΘ (f) is Moyal left multiplication. The Moyal planes corresponding to any skewsymm ..."
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Cited by 44 (10 self)
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Extending a result of D. V. Vassilevich [50], we obtain the asymptotic expansion for the trace of a spatially regularized heat operator LΘ (f)e−t△Θ, where △Θ is a generalized Laplacian defined with Moyal products and LΘ (f) is Moyal left multiplication. The Moyal planes corresponding to any skewsymmetric matrix Θ being spectral triples [24], the spectral action introduced in noncommutative geometry by A. Chamseddine and A. Connes [6] is computed. This result generalizes the ConnesLott action [15] previously computed by Gayral [23] for symplectic Θ.
Regularization, ScaleSpace, and Edge Detection Filters
 Journal of Mathematical Imaging and Vision
"... . Computational vision often needs to deal with derivatives of digital images. Such derivatives are not intrinsic properties of digital data; a paradigm is required to make them welldefined. Normally, a linear filtering is applied. This can be formulated in terms of scalespace, functional mini ..."
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Cited by 43 (8 self)
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. Computational vision often needs to deal with derivatives of digital images. Such derivatives are not intrinsic properties of digital data; a paradigm is required to make them welldefined. Normally, a linear filtering is applied. This can be formulated in terms of scalespace, functional minimization, or edge detection filters. The main emphasis of this paper is to connect these theories in order to gain insight in their similarities and differences. We take regularization (or functional minimization) as a starting point, and show that it boils down to Gaussian scalespace if we require scale invariance and a semigroup constraint to be satisfied. This regularization implies the minimization of a functional containing terms up to infinite order of differentiation. If the functional is truncated at second order, the CannyDeriche filter arises. 1 Introduction Given a digital signal in one or more dimensions, we want to define its derivatives in a wellposed way. This can...
On Köthe sequence spaces and linear logic
 Mathematical Structures in Computer Science
, 2001
"... We present a category of locally convex topological vector spaces which is a model of propositional classical linear logic, based on the standard concept of Kothe sequence spaces. In this setting, the spaces interpreting the exponential have a quite simple structure of commutative Hopf algebra. The ..."
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Cited by 42 (12 self)
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We present a category of locally convex topological vector spaces which is a model of propositional classical linear logic, based on the standard concept of Kothe sequence spaces. In this setting, the spaces interpreting the exponential have a quite simple structure of commutative Hopf algebra. The coKleisli category of this linear category is a cartesian closed category of entire mappings. This work provides a simple setting where typed calculus and dierential calculus can be combined; we give a few examples of computations. 1
On the spectral decomposition of affine Hecke algebras
 J. Inst. Math. Jussieu
"... Abstract. An affine Hecke algebra H contains a large abelian subalgebra A spanned by the BernsteinZelevinskiLusztig basis elements θx, where x runs over (an extension of) the root lattice. The center Z of H is the subalgebra of Weyl group invariant elements in A. The natural trace (“evaluation at ..."
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Cited by 39 (11 self)
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Abstract. An affine Hecke algebra H contains a large abelian subalgebra A spanned by the BernsteinZelevinskiLusztig basis elements θx, where x runs over (an extension of) the root lattice. The center Z of H is the subalgebra of Weyl group invariant elements in A. The natural trace (“evaluation at the identity”) of the affine Hecke algebra can be written as integral of a certain rational nform (with values in the linear dual of H) over a cycle in the algebraic torus T = spec(A). This cycle is homologous to a union of “local cycles”. We show that this gives rise to a decomposition of the trace as an integral of positive local traces against an explicit probability measure on the spectrum W0\T of Z. From this result we derive the Plancherel formula of the affine Hecke algebra.
Nuclear and Trace Ideals in Tensored *Categories
, 1998
"... We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The comp ..."
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Cited by 39 (12 self)
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We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored categories, all morphisms are nuclear, and in the tensored category of Hilbert spaces, the nuclear morphisms are the HilbertSchmidt maps. We also introduce two new examples of tensored categories, in which integration plays the role of composition. In the first, mor...
Some nonlinear SPDE’s that are second order in time
 ELECTRONIC J. OF PROBABILITY, VOL
, 2003
"... We extend Walsh’s theory of martingale measures in order to deal with hyperbolic stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution ..."
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Cited by 38 (4 self)
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We extend Walsh’s theory of martingale measures in order to deal with hyperbolic stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution can be a distribution in the sense of Schwartz, which appears as an integrand in the reformulation of the s.p.d.e. as a stochastic integral equation. Our approach provides an alternative to the Hilbert space integrals of HilbertSchmidt operators. We give several examples, including the beam equation and the wave equation, with nonlinear multiplicative noise terms.
Jun: A generalization of the HyersUlamRassias stability of Jensens equation
 J. Math. Anal. Appl
, 1999
"... Abstract. We consider the HyersUlamRassias stability problem
2u ◦ A2 u ◦ P1 u ◦ P2 "(jxjp + jyjp); x; y 2 Rn for the Schwartz distributions u, which is a distributional version of the HyersUlamRassias stability problem of the Jensen functional equation2f (x+ y2 f(x) f(y) "(jxj ..."
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Cited by 34 (4 self)
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Abstract. We consider the HyersUlamRassias stability problem
2u ◦ A2 u ◦ P1 u ◦ P2 "(jxjp + jyjp); x; y 2 Rn for the Schwartz distributions u, which is a distributional version of the HyersUlamRassias stability problem of the Jensen functional equation2f (x+ y2 f(x) f(y) "(jxjp + jyjp); x; y 2 Rn for the function f: Rn! C. 1.
Diffraction of Random Tilings: Some Rigorous Results
 J. STAT. PHYS
, 1999
"... The diffraction of stochastic point sets, both Bernoulli and Markov, and of random tilings with crystallographic symmetries is investigated in rigorous terms. In particular, we derive the diffraction spectrum of 1D random tilings, of stochastic product tilings built from cuboids, and of planar rando ..."
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Cited by 30 (17 self)
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The diffraction of stochastic point sets, both Bernoulli and Markov, and of random tilings with crystallographic symmetries is investigated in rigorous terms. In particular, we derive the diffraction spectrum of 1D random tilings, of stochastic product tilings built from cuboids, and of planar random tilings based on solvable dimer models, augmented by a brief outline of the diraction from the classical 2D Ising lattice gas. We also give a summary of the measure theoretic approach to mathematical diraction theory which underlies the unique decomposition of the diffraction spectrum into its pure point, singular continuous and absolutely continuous parts.
Diffractive point sets with entropy
"... Dedicated to HansUde Nissen on the occasion of his 65th birthday After a brief historical survey, the paper introduces the notion of entropic model sets (cut and project sets), and, more generally, the notion of diffractive point sets with entropy. Such sets may be thought of as generalizations of ..."
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Cited by 29 (16 self)
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Dedicated to HansUde Nissen on the occasion of his 65th birthday After a brief historical survey, the paper introduces the notion of entropic model sets (cut and project sets), and, more generally, the notion of diffractive point sets with entropy. Such sets may be thought of as generalizations of lattice gases. We show that taking the site occupation of a model set stochastically results, with probabilistic certainty, in welldefined diffractive properties augmented by a constant diffuse background. We discuss both the case of independent, but identically distributed (i.i.d.) random variables and that of independent, but different (i.e., site dependent) random variables. Several examples are shown.