IPI
A new exact inversion method for exponential Radon transform using the harmonic analysis of the Euclidean motion group
C E Yarman B Yazıcı
Inverse Problems & Imaging 2007, 1(3): 457-479 doi: 10.3934/ipi.2007.1.457
This paper presents a new method for the exponential Radon transform inversion based on the harmonic analysis of the Euclidean motion group of the plane. The proposed inversion method is based on the observation that the exponential Radon transform can be modified to obtain a new transform, defined as the modified exponential Radon transform, that can be expressed as a convolution on the Euclidean motion group. The convolution representation of the modified exponential Radon transform is block diagonalized in the Euclidean motion group Fourier domain. Further analysis of the block diagonal representation provides a class of relationships between the spherical harmonic decompositions of the Fourier transforms of the function and its exponential Radon transform. These relationships and the block diagonalization lead to three new reconstruction algorithms. The proposed algorithms are implemented using the fast implementation of the Euclidean motion group Fourier transform and their performances are demonstrated in numerical simulations. Our study shows that convolution representation and harmonic analysis over groups motivates novel solutions for the inversion of the exponential Radon transform.
keywords: ray transform Radon transform Euclidean motion group. Exponential Radon transform harmonic analysis convolution representation
IPI
A variational approach to waveform design for synthetic-aperture imaging
T. Varslo C E Yarman M. Cheney B Yazıcı
Inverse Problems & Imaging 2007, 1(3): 577-592 doi: 10.3934/ipi.2007.1.577
We derive an optimal transmit waveform for filtered backprojection-based synthetic-aperture imaging. The waveform is optimal in terms of minimising the mean square error (MSE) in the resulting image. Our optimization is performed in two steps: First, we consider the minimum-mean-square-error (MMSE) for an arbitrary but fixed waveform, and derive the corresponding filter for the filtered backprojection reconstruction. Second, the MMSE is further reduced by finding an optimal transmit waveform. The transmit waveform is derived for stochastic models of the scattering objects of interest (targets), other scattering objects (clutter), and additive noise. We express the waveform in terms of spatial spectra for the random fields associated with target and clutter, and the spectrum for the noise process. This approach results in a constraint that involves only the amplitude of the Fourier transform of the transmit waveform. Therefore, considerable flexibility is left for incorporating additional requirements, such as minimal variation of transmit amplitude and phase-coding.
keywords: Synthetic-aperture filtered backprojection radar imaging. waveform design

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