Image Processing Reference

In-Depth Information

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

Chapter 9.

ReFeRenCeS

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
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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
,

65-79.

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
,

2593-2605.

Habashy, T. and Wolf, E. 1994. Reconstruction of scattering potentials from

incomplete data.
Journal of Modern Optics
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41
, 1679-1685.

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