Image Reconstruction for 3D Lung Imaging - Department of Systems ...

Cheney et al introduced a diagonal matrix, R = diag(H T H), as the prior used in
their NOSER algorithm [35]. This matrix is diagonal, and with SVD decomposition, the
regularized solution can be expressed as
with
ˆx =
f =
M�
i=1
f uTi d
vi
σi
σ 2 i
σ 2 i + λr2 i
f are the “filter factors”, ri are the diagonal elements of R, and σi are the singular values
of R. For small singular values f → 1 and for large singular values f → 0.
3.4.1 Thoughts on Regularization
The use of Tikhonov style regularization techniques is equivalent to introducing a priori
information to the reconstruction process. The fundamental prior information of the conductivity
solution is that it is a positive function, Such methods provide stability but force
solutions to be smooth in some sense thus eliminating the possibility of non-smooth solutions.
This is reasonable given that the undesired, noise dominated, solutions contain
components with high spatial frequencies.
One way in which the generalized Tikhonov regularization method can be understood
to work, is that it draws the solution towards the null space N(R) of the regularization
matrix R [114]. If we use, for example, the first difference matrix, the solution is drawn
toward a constant distribution because a constant solution forms the basis for the null
space of the first difference matrix. Moreover if we have some information on the true
resistivity distribution, the regularization matrix can be constructed in such a way that
the solution is drawn towards the known distribution by the regularization. This can be
implemented by penalizing the difference between the reconstructed conductivity, σ, and
the a priori assumption about the conductivity instead of the just penalizing the solution.
Symbolically, rather than minimize equation 3.7 we minimize
ˆx = arg min
x
�
�Hx − z� 2 + λ 2 �R(x − x ∗ )� 2�
(3.23)
where x ∗ is the a priori assumed distribution. This idea was evaluated by Vauhkonen
et al in [113] and [114]. In [113] the conductivity distribution was approximated as a
linear combination of some pre-selected basis functions that were constructed from prior
information on the structures and conductivities. The method, called the Basis Constraint
Method (BCM), produced good results when the priors were correct but provided misleading
results when the prior was incorrect. Here misleading means that the solutions contained
structural artefacts due to the prior that were not part of the true conductivity. In [114]
Vauhkonen et al used the same kind of idea but rather than forcing the solution to be
in the subspace spanned by pre-selected basis functions they only “draw” the solution
towards the subspace. This method, called the Subspace Regularization Method, provided
an improvement over the BCM in that they could partially avoid misleading results even if
the prior information was not correct.
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