TY - JOUR

T1 - Windowed Radon transform frames

AU - Shlivinski, Amir

AU - Heyman, Ehud

N1 - Funding Information:
Abbreviations: WRT, windowed Radon transform; WFT, windowed Fourier transform; MS-WRT, multiscale windowed Radon transform; ID, isodiffracting; PB, pulsed beam. * Corresponding author. Fax: +972 3 6423508. E-mail addresses: [email protected] (A. Shlivinski), [email protected] (E. Heyman). 1 This work is supported in part by the Israel Science Foundation (ISF), under Grants Nos. 216/02 and 674/07.

PY - 2009/5/1

Y1 - 2009/5/1

N2 - Two windowed Radon transform (WRT) frame formulations for the decomposition of band limited functions f (x) ∈ L2 (Rℓ), ℓ = 2, 3, are presented. The "basic" formulation consists of two dual frame sets of shifted and rotated windows, one is used to synthesize f and the other to calculate the expansion coefficients as projections of f onto this set. The latter operation is a WRT that samples f at the discrete phase-space lattice of locations and directions. Explicit expressions are derived for a class of isodiffracting (ID) windows, which are matched to the lattice to yield snug frames. The basic formulation is then generalized to multiscales-WRT frames, where the large scales elements are associated with wider windows and sparser (rotation-direction) phase-space lattices that are decimated subsets of the lattice at the smallest scale. The analysis is presented for 3D, with a summary of the modifications for 2D. Finally, we discuss applications to time-dependent wave theory, whereby the source distribution is expanded using a WRT frame. The WRT extracts the local radiation properties of the source, thus describing the radiated field as a sum of collimated isodiffracting pulsed beams (ID-PB) that emerge from the source along the preferred radiation directions.

AB - Two windowed Radon transform (WRT) frame formulations for the decomposition of band limited functions f (x) ∈ L2 (Rℓ), ℓ = 2, 3, are presented. The "basic" formulation consists of two dual frame sets of shifted and rotated windows, one is used to synthesize f and the other to calculate the expansion coefficients as projections of f onto this set. The latter operation is a WRT that samples f at the discrete phase-space lattice of locations and directions. Explicit expressions are derived for a class of isodiffracting (ID) windows, which are matched to the lattice to yield snug frames. The basic formulation is then generalized to multiscales-WRT frames, where the large scales elements are associated with wider windows and sparser (rotation-direction) phase-space lattices that are decimated subsets of the lattice at the smallest scale. The analysis is presented for 3D, with a summary of the modifications for 2D. Finally, we discuss applications to time-dependent wave theory, whereby the source distribution is expanded using a WRT frame. The WRT extracts the local radiation properties of the source, thus describing the radiated field as a sum of collimated isodiffracting pulsed beams (ID-PB) that emerge from the source along the preferred radiation directions.

KW - Frame theory

KW - Multiscale analysis

KW - Pulsed-beams

KW - Wave-theory

KW - Windowed Fourier transform frames

KW - Windowed Radon transform frames

UR - http://www.scopus.com/inward/record.url?scp=62549123715&partnerID=8YFLogxK

U2 - 10.1016/j.acha.2008.07.003

DO - 10.1016/j.acha.2008.07.003

M3 - Article

AN - SCOPUS:62549123715

SN - 1063-5203

VL - 26

SP - 322

EP - 343

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

IS - 3

ER -