Image Processing Reference
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reduce the computational effort, the type 3 basis (a piecewise-linear approximation
of the Gaussian) is the most suitable. The type-3 basis is 0 almost everywhere except
(
2
pixels. Consequently the number of multiplications required in equation
(1.3) reduces from 2
N to only
2
+
)
2
r
1
2
.
The pre-computing of RB and its storage allows to avoid calculations of
B
i
,
j
for
each new measurement position as the counter automaton advances. In the case of
a fixed radius (e.g. choosing
r
(
+
)
2
r
1
2, for a similar reconstruction error as in [4]) the
number of multiplications in the recovery algorithm is of the order of 10
6
=
=
20000 (easily computed in less than 0.1 second on actual computers). For reference,
in [2] where an improved, highly effective reconstruction algorithm is considered,
more than 100 seconds are reported for a similar number of measurements. This
comparison indicates a significant speed-up of our image scan method with several
orders of magnitude, not only in the measurement but also in the reconstruction
phase, when compared to compressive sensing.
for
K
The Influence of Radius
r
:
As expected, the radius
r
has a significant role on the
quality of the reconstructed image. It corresponds to
L
in the naïve reconstruction
scheme discussed in Fig. 1.4. As seen in Figure 1.10, for a given
r
the PSNR im-
proves almost linearly up to a value
K
, then it saturates. In order to avoid saturation
r
must be smaller. But on the other hand, a small
r
is in contrast with a small
K
because the sliding basis cannot provide good reconstruction of all pixels located in
the neighborhood of the measurement. The reconstructed image will look noisy (as
it happens for
r
3. Consequently, it
follows that
r
must adapt to the value of
K
. Based on experiments we propose the
following formula:
r
=
1
.
5 in Fig. 1.10) but will improve when
r
=
. Such a value may be used as a fixed
one in the recovery scheme but an adaptive radius is also possible, for instance by
updating the radius value any time
K
becomes a power of 2. Experiments with both
schemes revealed no major quality differences in the reconstructed image, except
that the adaptive radius scheme would lead to more computational effort (recalcu-
lating the sliding basis each time the radius value changes).
Note that in the above experiments
K
(
K
)=
log
2
(
N
)
−
log
2
(
K
)
5000 is about 7.6% of all pixels in the
original image corresponding thus to a compression rate of about 13 times or a rate
of 0.61 bpp (bits per pixel) assuming a coding of each pixel with 8 bits. In order
to provide a comparison with compressive sensing approaches, the same medical
image as in [4] is considered. While our method leads to a slight degradation in the
PSNR when compared to the above mentioned compressive sensing method (PSNR
= 24,1 dB for ours, instead of 26.5 dB in [4] for
K
= 10000 measurements) the
visual quality of reconstructed images is rather similar (Figure 1.11). Such a slight
degradation is acceptable given the important reduction in the overall complexity,
particularly the complexity of the sensing device.
Further improvements (i.e. obtaining a better PSNR for a given
K
) are expected,
at the expense of computational complexity, by introducing novel reconstruction
sliding bases, tailored to the nature of the images to be compressed. Though,
given the very low complexity of the compression stage such a method is suit-
able for embedding in various smart sensing devices with low power consumption
=
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