Inverting the spherical Radon transform for 3D SAR image formation

Redding, N.J.; Payne, T.

doi:10.1109/radar.2003.1278787

Cited by 19 publications

(12 citation statements)

References 9 publications

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“…In fact these applications initiated the authors interest in the problem of recovering a function from spherical means. The inversion from spherical means is, however, is also essential for other technologies such as SONAR (see [2,22]), SAR imaging (see [1,23]), ultrasound tomography (see [17,18]), or seismic imaging (see [3,6]). where ω ⊥ := {y ∈ R n : ω · y = 0} denotes the hyperplane consisting of all vectors orthogonal to ω ∈ S n−1 .…”

Section: Introductionmentioning

confidence: 99%

Universal Inversion Formulas for Recovering a Function from Spherical Means

Haltmeier¹

2014

SIAM J. Math. Anal.

110

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The problem of reconstruction of a function from spherical means is at the heart of several modern imaging modalities and other applications. In this paper we derive universal back-projection-type reconstruction formulas for recovering a function in arbitrary dimension from averages over spheres centered on the boundary of an arbitrarily shaped bounded convex domain with smooth boundary. Provided that the unknown function is supported inside that domain, the derived formulas recover the unknown function up to an explicitly computed integral operator. For elliptical domains the integral operator is shown to vanish and hence we establish exact inversion formulas for recovering a function from spherical means centered on the boundary of elliptical domains in arbitrary dimension.

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Section: Introductionmentioning

confidence: 99%

Universal Inversion Formulas for Recovering a Function from Spherical Means

Haltmeier¹

2014

SIAM J. Math. Anal.

110

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“…One of the obvious areas is the domain of earth surveillance for environmental and meteorological purposes. In this respect it has the ingredients of Synthetic Aperture Radar (SAR) if it uses bursts of microwaves [9]. However SAR can be achieved without collecting surface integral data and some of the problems of SAR are not addressed in this approach of Radon transform on spheres.…”

Section: 3mentioning

confidence: 99%

“…We now provide the inversion formulae for R . The proofs correspond with [9] except for the parametrization of the source which changes the factors. We give then a very short version of the proofs.…”

Section: Inverse Radonmentioning

confidence: 99%

Approximate Image Reconstruction in Landscape Reflection Imaging

Régnier

Rigaud

Nguyen

2015

Mathematical Problems in Engineering

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Simple reflection imaging of landscape (scenery or extended objects) poses the inverse problem of reconstructing the landscape reflectivity function from its integrals on some particular family of spheres. Such data acquisition is encoded in the framework of a Radon transform on this family of spheres. In spite of the existence of an exact inversion formula, the numerical landscape reflectivity function reconstitution is best obtained with an approximate but judiciously chosen reconstruction kernel. We describe the working of this reflection imaging modality and its theoretical handling, introduce an efficient and stable image reconstruction algorithm, and present simulation results to prove the validity of this choice as well as to demonstrate the feasibility of this imaging process.

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“…When using linear of circular integrating detectors in PAT/TAT, these reconstruction problems arise in two spatial dimensions (see [11,19,40,49]). The problems of recovering a function from its spherical means of the wave data also arises in other technologies, such as SONAR [8,42], SAR imaging [5,43,45], ultrasound tomography [35,36], and seismic imaging [9,13].…”

Section: Introductionmentioning

confidence: 99%

The universal back-projection formula for spherical means and the wave equation on certain quadric hypersurfaces

Haltmeier

Pereverzyev

2015

Journal of Mathematical Analysis and Applications

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In many tomographic applications and elsewhere, there arises a need to reconstruct a function from the data on the boundary of some domain given either by the spherical means of the function, or by the corresponding solution of the freespace wave equation. In this paper, we show that the so-called universal backprojection formulas provide exact recovery of the unknown function with compact support for data on any quadric hypersurfaces that can be approximated by elliptic hypersurfaces. These quadric hypersurfaces include elliptic paraboloid as well as parabolic and elliptic cylinders.

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Inverting the spherical Radon transform for 3D SAR image formation

Cited by 19 publications

References 9 publications

Universal Inversion Formulas for Recovering a Function from Spherical Means

Universal Inversion Formulas for Recovering a Function from Spherical Means

Approximate Image Reconstruction in Landscape Reflection Imaging

The universal back-projection formula for spherical means and the wave equation on certain quadric hypersurfaces

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