Optimally Sparse Representations of Cartoon-Like Cylindrical Data

Easley, Glenn R.; Guo, Kanghui; Labate, Demetrio; Pahari, Basanta R.

doi:10.1007/s12220-020-00493-0

Cited by 2 publications

(8 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We start by defining the class of 4-dimensional cylindrical cartoon-like functions associated with our data model. This definition extends a similar definition in the 3-dimensional setting that was introduced by some of the authors [19] as a modification of the better known class of cartoon-like functions, originally proposed by Donoho [21] to provide a simplified model of natural images.…”

Section: Sparse Cylindrical Shearlets Approximationsmentioning

confidence: 66%

“…As we discuss below, the cylindrical shearlets we consider in this paper are derived from a modified construction that handles spatial and temporal coordinates with different geometric sensitivities. As already indicated by some of the authors in [19] and further argued in this paper, this construction entails distinct mathematical properties with respect to conventional shearlets and significant potential advantages in the context of spatio-temporal data.…”

Section: Introductionmentioning

confidence: 77%

“…In Sec. 2, we introduce a new construction of 4-dimensional cylindrical shearlets by generalizing the 3dimensional construction in [19]. In Sec.…”

Section: Outlinementioning

confidence: 99%

“…Cylindrical shearlets were recently introduced by some of the authors [19] as a variant of the shearlet construction in the 3-dimensional setting. As indicated above, this construction is motivated by applications in the analysis of data where the main discontinuities are perpendicular to one of the coordinate axes so that it is useful to employ representations that are direction-sensitive with respect to one hyperspace.…”

Section: Cylindrical Shearletsmentioning

confidence: 99%

“…2.1. Smooth Parseval frame of cylindrical shearlets on L 2 pR 4 q Our construction below adapts the main ideas from the standard shearlet construction [20] and extends the 3-dimensional cylindrical shearlet construction [19].…”

Section: Cylindrical Shearletsmentioning

confidence: 99%

See 4 more Smart Citations

Efficient representation of spatio-temporal data using cylindrical shearlets

Bubba¹,

Easley²,

Heikkilä³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

Efficient representations of multivariate functions are critical for the design of state-of-the-art methods of data restoration and feature extraction. In this work, we consider the representation of spatio-temporal data such as temporal sequences (videos) of 2-and 3-dimensional images, where conventional separable representations are usually very inefficient, due to their limitations in handling the geometry of the data. To address this challenge, we consider an idealized class EpAq Ă L 2 pR 4 q of functions of 4 variables dominated by hypersurface singularities in the first three coordinates that we apply to model 4-dimensional data corresponding to temporal sequences (videos) of 3-dimensional objects. To provide an efficient representation for this type of data, we introduce a new multiscale directional system of functions based on cylindrical shearlets and prove that this new approach achieves superior approximation properties with respect to conventional multiscale representations. To further illustrate the advantages of our approach, we apply a discrete implementation of the new representation to problems of dynamic tomography using synthetic data.

show abstract

Section: Sparse Cylindrical Shearlets Approximationsmentioning

confidence: 66%

Section: Introductionmentioning

confidence: 77%

“…In Sec. 2, we introduce a new construction of 4-dimensional cylindrical shearlets by generalizing the 3dimensional construction in [19]. In Sec.…”

Section: Outlinementioning

confidence: 99%

Section: Cylindrical Shearletsmentioning

confidence: 99%

Section: Cylindrical Shearletsmentioning

confidence: 99%