Image Processing Reference
prior is chosen such that the object energy lies outside the prior support, then
without any method of regularization, this will lead to spurious oscillations
in the estimate of the object function because the PDFT is data-consistent and
requires a lot of energy in the extrapolated spectral values to accommodate an
underestimated support. To address this problem, the Tikhonov-Miller regu-
larization is used, which helps in removing the ill-conditioning of the P -matrix
by adding a small number to the diagonal elements of the P -matrix (Morris
et al., 1997). This is equivalent to adding a small constant amplitude outside
the prior support. The regularization step essentially reduces the energy in the
estimate arising from the unstable eigenvalues by adding a small value into the
prior function (i.e., not allowing it to be zero outside of the support) to slightly
increase all eigenvalues without altering the eigenvectors.
If no prior knowledge is available then p ( r ) would be a constant, and the
estimator reduces to the DFT of the available Fourier data. Also, when the
regularization constant τ is very large such that 1 + τ ≈ τ, the estimator essen-
tially looks like the DFT. The PDFT estimator is both data-consistent and con-
tinuous, and works in any number of spatial dimensions version (Duchene
et al., 1997). Examples of results using this technique are illustrated later in
Belkebir, K. and Tijhuis, A. G. 2001. Modified gradient method and modified
Born method for solving a two dimensional inverse scattering problem.
Inverse Problems , 17 , 1671-1688.
Byrne, C. L., Fitzgerald, R. L., Fiddy, M. A., Hall, T. J., and Darling, A. M. 1983.
Image restoration and resolution enhancement. Journal of the Optical
Society of America , 73 , 1481-1487.
Chun, Y., Song, L. P., and Liu, Q. H. 2005. Inversion of multi-frequency experi-
mental data for imaging complex objects by DTA-CSA method. Inverse
Problems , 21 , 165-178.
Crocco, L., D'Urso, M., and Isernia, T. 2005. Testing the contrast source extended
Born inversion method against real data: The TM case. Inverse Problems ,
21 (6), S33-S50.
Darling, A. M., Hall, T. J., and Fiddy, M. A. 1983. Stable, noniterative object
reconstruction from incomplete data using a priori knowledge. Journal of
the Optical Society of America , 73 , 1466-1469.
Dubois, A., Belkebir, K., and Saillard, M. 2005. Retrieval of inhomogeneous
targets from experimental frequency diversity data. Inverse Problems , 21 ,
Duchene, B., Lesselier, D., and Kleinman, R. E. 1997. Inversion of the 1996
Ipswich data using binary specialization of modified gradient methods.
IEEE Antennas and Propagation Magazine , 39 (2), 9-12.
Estataico, C., Pastorino, M., and Randazzo, A. 2005. An inexact Newton
method for short range microwave imaging within the second order Born
approximation. IEEE Transactions on Geoscience and Remote Sensing, 43 ,
Habashy, T. and Wolf, E. 1994. Reconstruction of scattering potentials from
incomplete data. Journal of Modern Optics , 41 , 1679-1685.
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