Marcelo Bertalmıo Vicent Caselles Simon Masnou Guillermo Sapiro
17
pages
English
Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
Découvre YouScribe en t'inscrivant gratuitement
Découvre YouScribe en t'inscrivant gratuitement
17
pages
English
Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
Inpainting Marcelo Bertalmıo, Vicent Caselles, Simon Masnou, Guillermo Sapiro Synonyms – Disocclusion – Completion – Filling-in – Error concealment Related Concepts – Texture synthesis Definition Given an image and a region ? inside it, the inpainting problem consists in modifying the image values of the pixels in ? so that this region does not stand out with respect to its surroundings. The purpose of inpainting might be to restore damaged portions of an image (e.g. an old photograph where folds and scratches have left image gaps) or to remove unwanted elements present in the image (e.g. a microphone appearing in a film frame). See figure 1. The region ? is always given by the user, so the localization of ? is not part of the inpainting problem. Almost all inpainting algorithms treat ? as a hard constraint, whereas some methods allow some relaxing of the boundaries of ?. This definition, given for a single-image problem, extends naturally to the multi-image case therefore this entry covers both image and video inpainting. What is not however considered in this text is surface inpainting (e.g. how to fill holes in 3D scans), although this problem has been addessed in the literature. Fig. 1. The inpainting problem. Left: original image. Middle: inpainting mask ?, in black.
Voir
- problem leading
- patch-based methods
- inpainting
- assisted methods
- image gaps
- convex optimization
- texture synthesis
- problem
- square patch
- methods attempt
THE HEAT KERNEL ON NONCOMPACT SYMMETRIC SPACES
Jean–Philippe Anker & Patrick Ostellari InmemoryofF.I.Karpelevicˇ(1927–2000)
The heat kernel plays a central role in mathematics. It occurs in several fields : analysis, geometry and – last but not least – probability theory. In this survey, we shall focus on its analytic aspects, specifically sharp bounds, in the particular setting of Riemanniansymmetricspacesofnoncompacttype.ItisanaturaltributetoKarpelevicˇ, whose pioneer work [Ka] inspired further study of the geometry of theses spaces and of the analysis of the Laplacian thereon. This survey is based on lectures delivered by the first author in May 2002 at IHP in Paris during the Special Quarter Heat kernels, random walks & analysis on manifolds & graphs . Both authors would like to thank the organizers for their great job, as well as Martine Babillot, Gilles Carron, Sasha Grigor’yan and Jean–Pierre Otal for stimulating discussions.
1. Preliminaries We shall briefly review some basics about noncompact Riemannian symmetric spaces X = GK and we shall otherwise refer to standard texbooks ([GV]; [H1], [H2], [H3]; [Kn]) for their structure and harmonic analysis thereon. Thus G is a semisimple Lie group (real, connected, noncompact, with finite center) or more generally a reductive Lie group in the Harish-Chandra class and K is a maximal compact subgroup. Let θ be the Cartan involution and let g = k ⊕ p be the Cartan decomposition at the Lie algebra level. g is equipped with the inner product (1 1) h X Y i = − B( X θ Y ) where B is the Killing form, appropriately modified if g has a central component. (1.1) enables us to identify g with its dual g ∗ , and likewise for subspaces of g . (1.1) induces the Riemannian structure on X = GK , whose tangent space at the origin 0 = eK is identified with p . Let a be a Cartan subspace of p , let m be the centralizer of a in k and let g = a ⊕ m ⊕ { ⊕ α ∈ Σ g α be the root space decomposition of g with respect to a . Select in a a positive Weyl chamber a + , in Σ the corresponding sets Σ + of positive roots, Σ 0+ of positive indivisible roots, Π of simple roots, and in g the corresponding nilpotent subalgebra n = ⊕ α ∈ Σ + g α . Let ̺ = 21 P α ∈ Σ + m α α be the half sum of positive roots, counted with multiplicities 2000 Mathematics Subject Classification . Primary 22E30, 35B50, 43A85, 58J35; Secondary 22E46, 43A80, 43A90. Key words and phrases. Abel transform, heat kernel, maximum principle, semisimple Lie group, symmetric space, subLaplacian. Both authors partially supported by the European Commission (IHP Network HARP ) Typeset by AMS -TEX 1
