Alternate Inverse Methods - Introduction to Imaging from Scattered Fields - page 101

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.
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