3D coded aperture imaging, ill-posedness and link with incomplete data radon transform

Jean-François Crouzet

doi:10.3934/ipi.2011.5.341

Article Contents

2011, Volume 5, Issue 2: 341-353. Doi: 10.3934/ipi.2011.5.341

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3D coded aperture imaging, ill-posedness and link with incomplete data radon transform

Jean-François Crouzet^1,

1.
Institut de Mathématiques et de Modélisation de Montpellier, CNRS UMR 5149 Place Eugène Bataillon, 34095 Montpellier

Received: May 31, 2010

Revised: November 30, 2010

Published: April 2011

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Abstract

Coded Aperture Imaging is a cheap imaging process encountered in many fields of research like optics, medical imaging, astronomy, and that has led to several good results for two dimensional reconstruction methods. However, the three dimensional reconstruction problem remains nowadays severely ill-posed, and has not yet furnished satisfactory outcomes.
In the present study, we propose an illustration of the poorness of the data in order to operate a good inversion in the 3D case. In the context of a far-field imaging, an inversion formula is derived when the detector screen can be widely translated. This reformulates the 3D inversion problem of coded aperture imaging in terms of classical Radon transform. In the sequel, we examine more accurately this reconstruction formula, and claim that it is equivalent to solve the limited angle Radon transform problem with very restricted data.
We thus deduce that the performances of any numerical reconstruction will remain shrank, essentially because of the physical nature of the coding process, excepted when a very strong a priori knowledge is given for the 3D source.

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Mathematics Subject Classification: Primary: 44A12, 65R10, 65R30; Secondary: 92C55.

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