{"@context":{"@vocab":"https://cir.nii.ac.jp/schema/1.0/","rdfs":"http://www.w3.org/2000/01/rdf-schema#","dc":"http://purl.org/dc/elements/1.1/","dcterms":"http://purl.org/dc/terms/","foaf":"http://xmlns.com/foaf/0.1/","prism":"http://prismstandard.org/namespaces/basic/2.0/","cinii":"http://ci.nii.ac.jp/ns/1.0/","datacite":"https://schema.datacite.org/meta/kernel-4/","ndl":"http://ndl.go.jp/dcndl/terms/","jpcoar":"https://github.com/JPCOAR/schema/blob/master/2.0/"},"@id":"https://cir.nii.ac.jp/crid/1361137044604754304.json","@type":"Article","productIdentifier":[{"identifier":{"@type":"DOI","@value":"10.1002/cpa.10116"}},{"identifier":{"@type":"URI","@value":"https://api.wiley.com/onlinelibrary/tdm/v1/articles/10.1002%2Fcpa.10116"}},{"identifier":{"@type":"URI","@value":"https://onlinelibrary.wiley.com/doi/pdf/10.1002/cpa.10116"}}],"dc:title":[{"@value":"New tight frames of curvelets and optimal representations of objects with piecewise <i>C</i><sup>2</sup> singularities"}],"description":[{"type":"abstract","notation":[{"@value":"<jats:title>Abstract</jats:title><jats:p>This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise <jats:italic>C</jats:italic><jats:sup>2</jats:sup> edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needle‐shaped elements at fine scales. These elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets. For instance, curvelets obey a parabolic scaling relation which says that at scale 2<jats:sup>−<jats:italic>j</jats:italic></jats:sup>, each element has an envelope that is aligned along a “ridge” of length 2<jats:sup>−<jats:italic>j</jats:italic>/2</jats:sup> and width 2<jats:sup>−<jats:italic>j</jats:italic></jats:sup>.</jats:p><jats:p>We prove that curvelets provide an essentially optimal representation of typical objects <jats:italic>f</jats:italic> that are <jats:italic>C</jats:italic><jats:sup>2</jats:sup> except for discontinuities along piecewise <jats:italic>C</jats:italic><jats:sup>2</jats:sup> curves. Such representations are nearly as sparse as if <jats:italic>f</jats:italic> were not singular and turn out to be far more sparse than the wavelet decomposition of the object. For instance, the <jats:italic>n</jats:italic>‐term partial reconstruction <jats:italic>f</jats:italic><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/tex2gif-stack-1.gif\" xlink:title=\"urn:x-wiley:00103640:media:CPA10116:tex2gif-stack-1\"/> obtained by selecting the <jats:italic>n</jats:italic> largest terms in the curvelet series obeys\n<jats:disp-formula>\n\n</jats:disp-formula>\nThis rate of convergence holds uniformly over a class of functions that are <jats:italic>C</jats:italic><jats:sup>2</jats:sup> except for discontinuities along piecewise <jats:italic>C</jats:italic><jats:sup>2</jats:sup> curves and is essentially optimal. In comparison, the squared error of <jats:italic>n</jats:italic>‐term wavelet approximations only converges as <jats:italic>n</jats:italic><jats:sup>−1</jats:sup> as <jats:italic>n</jats:italic> → ∞, which is considerably worse than the optimal behavior. © 2003 Wiley Periodicals, Inc.</jats:p>"}]}],"creator":[{"@id":"https://cir.nii.ac.jp/crid/1381137044604754304","@type":"Researcher","foaf:name":[{"@value":"Emmanuel J. Candès"}]},{"@id":"https://cir.nii.ac.jp/crid/1381137044604754305","@type":"Researcher","foaf:name":[{"@value":"David L. Donoho"}]}],"publication":{"publicationIdentifier":[{"@type":"PISSN","@value":"00103640"},{"@type":"EISSN","@value":"10970312"}],"prism:publicationName":[{"@value":"Communications on Pure and Applied Mathematics"}],"dc:publisher":[{"@value":"Wiley"}],"prism:publicationDate":"2003-11-14","prism:volume":"57","prism:number":"2","prism:startingPage":"219","prism:endingPage":"266"},"reviewed":"false","dc:rights":["http://onlinelibrary.wiley.com/termsAndConditions#vor"],"url":[{"@id":"https://api.wiley.com/onlinelibrary/tdm/v1/articles/10.1002%2Fcpa.10116"},{"@id":"https://onlinelibrary.wiley.com/doi/pdf/10.1002/cpa.10116"}],"createdAt":"2003-11-18","modifiedAt":"2023-10-13","relatedProduct":[{"@id":"https://cir.nii.ac.jp/crid/1360848660338829440","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@value":"Multiresolution Graph Fourier Transform for Compression of Piecewise Smooth Images"}]},{"@id":"https://cir.nii.ac.jp/crid/1390001205088395008","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@language":"en","@value":"Redundant representation of acoustic signals using curvelet transform and its application to speech denoising"}]},{"@id":"https://cir.nii.ac.jp/crid/1390001205213637888","@type":"Article","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@language":"en","@value":"Image denoising using a multivariate shrinkage function in the curvelet domain"}]},{"@id":"https://cir.nii.ac.jp/crid/1390001205301134464","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@language":"en","@value":"2D tight framelets with orientation selectivity suggested by vision science"}]},{"@id":"https://cir.nii.ac.jp/crid/1390001205424363008","@type":"Article","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@language":"en","@value":"[Paper] Sparse Group Match for Point Cloud Registration"}]},{"@id":"https://cir.nii.ac.jp/crid/1390001206312031872","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@language":"en","@value":"Image Recovery by Decomposition with Component-Wise Regularization"}]},{"@id":"https://cir.nii.ac.jp/crid/1390282680102674432","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@language":"ja","@value":"Multi-scale eFREBAS変換を利用したMR画像の圧縮センシング"},{"@language":"en","@value":"Compressed Sensing of MR Images using Multi-scale eFREBAS Transform"}]},{"@id":"https://cir.nii.ac.jp/crid/1390865718381400704","@type":"Article","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@language":"en","@value":"Constructive Universal Approximation Theorems for Deep Neural Networks: A Group Representation Theoretical Approach"},{"@language":"ja","@value":"深層ニューラルネットの構成的普遍近似定理―群表現論的方法―"}]}],"dataSourceIdentifier":[{"@type":"CROSSREF","@value":"10.1002/cpa.10116"},{"@type":"CROSSREF","@value":"10.3169/mta.6.151_references_DOI_YB6U1wZiPdBJ580GA3qgvUUPKzh"},{"@type":"CROSSREF","@value":"10.1587/transfun.e95.a.2470_references_DOI_YB6U1wZiPdBJ580GA3qgvUUPKzh"},{"@type":"CROSSREF","@value":"10.1587/elex.7.126_references_DOI_YB6U1wZiPdBJ580GA3qgvUUPKzh"},{"@type":"CROSSREF","@value":"10.3902/jnns.31.177_references_DOI_YB6U1wZiPdBJ580GA3qgvUUPKzh"},{"@type":"CROSSREF","@value":"10.3169/itej.70.j118_references_DOI_YB6U1wZiPdBJ580GA3qgvUUPKzh"},{"@type":"CROSSREF","@value":"10.14495/jsiaml.1.9_references_DOI_YB6U1wZiPdBJ580GA3qgvUUPKzh"},{"@type":"CROSSREF","@value":"10.1109/tip.2014.2378055_references_DOI_YB6U1wZiPdBJ580GA3qgvUUPKzh"},{"@type":"CROSSREF","@value":"10.1250/ast.36.457_references_DOI_YB6U1wZiPdBJ580GA3qgvUUPKzh"}]}