Biomedical Engineering Reference
In-Depth Information
4.2.3.1
Total Variation Regularization
We observed a fixed region of the plant
Arabidopsis thaliana
, with a Zeiss LSM
510 confocal, for three different pinhole settings. The gradients of these observed
image volumes were numerically calculated along the
x
-direction as
) and
their histograms are plotted in Fig.
4.9
. We mentioned that as the size of the pinhole
is increased, the observed images have lesser detail due to lowering of contrast and
lesser resolution. From the histograms of the gradients of the observed images,
we notice that the histogram tends to have longer tails when the pinhole sizes
are smaller and smaller tails when the pinhole sizes are larger. We also notice
that there is a large difference between the 1 AU distribution and the 2 AU, but a
negligible difference between the 2 AU and the 5 AU distribution. This is another
reason to confine the working pinhole size in the CLSM to 2-3 AU. Also, we can
say that the observed volumes that have sharper edges tend to have longer tails in
the gradients rather than volumes that are out-of-focus. With this as the basis, we
introduce the following regularization model that can produce restored images with
longer gradient distribution tails.
The object prior distribution in Eq. (
4.19
)is:
∇
x
i
(
x
Pr(
o
)=
Z
−
1
λ
)))
,
with
λ ∈
R
+
,
exp(
−λ
TV(
o
(
x
(4.23)
where
Z
λ
is the
partition function
, TV(
o
)(
x
) is the Total variation (TV) operation
on
o
(
),and
λ
is the regularization parameter described earlier. The TV function
was first described in [
63
] as an iterative denoising method. It was introduced as
a spatial sparsity measure of images by calculating the total amplitude intensity
variations or oscillations in the image. The discrete semi-norm definition of TV that
we will use in this chapter, reads as follows:
x
TV(
o
)=
x
|∇o
(
x
)
|.
(4.24)
Although a large spectrum of regularization functionals exist in literature, in
this chapter, we use an
1
norm based regularization because it is known for
its edge preserving quality and convexity. It also does a non-isotropic smoothing
that acts along the edge direction but not in the direction orthogonal to the
edge. The TV can be used as a constraint as it provides an intuitive and precise
mathematical framework to characterize the piecewise regularity of objects in an
image. Minimizing the TV-norm as introduced by Charbonnier et al. [
17
], Rudin
et al. [
63
], corresponds to constraining the number of discontinuities in an image.
The TV regularization is also well suited for biological images, where the structures
and the background provide very low gradient values, while a finite set of edges
provide high gradient values. Other motivations for using TV are the reduction of
noise and the realization of nearly homogeneous regions with sharp edges.
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