Biomedical Engineering Reference
In-Depth Information
The applicability of super-resolution to PET has been shown, e.g., in [32]
using an approach proposed in [30]. This approach is outlined briefly in the
following to illustrate the general formalism.
Given K shifted LR images L
k
, k 2f1;:::;Kg, K 2N, an initial guess of
the HR image H
(0)
is obtained by averaging the upsampled LR images:
X
1
K
H
(0)
=
t
k
(L
k
" s) ;
(7.33)
k=1
where t
k
is the transformation of the shifted LR images to a common reference
frame and " s is the upsampling of the LR image to the resolution of the HR
image. Based on this initial guess, low resolution images
L
(n)
k
are estimated
by means of
L
(n)
k
= (t
k
(H
(n)
) ? h) # s ;
(7.34)
where ? is the convolution operator, # s denotes downsampling to the resolu-
tion of the LR images, h is a smoothing kernel and (n) denotes the iteration.
The difference between the estimated L
(n
k
and the measured LR images L
k
is used to update the estimate of the HR image:
X
1
K
(((L
k
L
(n)
k
H
(n+1)
= H
(n)
+
t
1
k
) " s) ? p) ;
(7.35)
k=1
where p is a sharpening kernel. Ideally, p is the inverse of the blurring kernel
(if the inverse exists), or the approximate inverse. For details we refer to [30].
The procedure is repeated until either a fixed number of iterations is
reached or the difference between the estimated and measured LR images
is lower than a chosen threshold. In conclusion, the outlined method deter-
mines the HR image by estimating an image which produces the measured
LR images L
k
given the transformations t
k
and the blurring kernel h.
Figure 7.15 gives an artificial example using the outlined method. Fig-
ure 7.15(b) shows one of the eight LR images generated from the original
image in Figure 7.15(a). The smaller gaps in the lines in the LR image
(Figure 7.15(b)) are not clearly resolvable. In the computed HR image (Fig-
ure 7.15(c)) the gaps in the lines are recognizable again.
In [66] improvements regarding SNR (see Definition 14 and 15) for respira-
tory gated phantom data using super-resolution are shown. A concise survey
of super-resolution in medical imaging focusing on PET and MRI is given
in [25].
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