Biomedical Engineering Reference
In-Depth Information
Notations
All the scalar variables in this chapter are denoted by the lowercase (
x
), while
the vectors by boldface lowercase alphabets (
x
), and the matrices by the boldface
uppercase alphabets (
X
). As the images are discretized, their spatial support is
Ω
s
=
{
(
x, y, z
):0
≤ x ≤ N
x
−
1
,
0
≤ y ≤ N
y
−
1
,
0
≤ z ≤ N
z
−
1
}
.By
⊂
N
3
O
(
Ω
s
)=
{o
=(
o
xyz
):
Ω
s
→
R
}
, we refer to the possible observable
objects, and we assign the function
h
:
Ω
s
as the microscope Point-spread
function (PSF). The observed intensities are denoted by
i
(
→
R
x
∈ Ω
s
(bounded and
positive), while a 3-D convolution operation between two functions is denoted by
'
x
):
'. However, when the same symbol is used as a superscript over a given function
(
h
∗
(
∗
x
)), it represents the Hermitian adjoint operation.
F
is the forward Fourier
F
−
1
the inverse Fourier transform operations respectively. As the
images are in 3-D, we present them in the 2-D format in this chapter by taking
the maximum of the intensity along the
z
-direction for each pixel in the 2-D XY
plane (or the Maximum Intensity Projection (MIP)) or by taking the maximum along
the
y
-direction for each pixel in the 2-D XZ plane.
The objective lenses of a microscope are defined by their magnification (M),
NA, and the medium in between the lens and the cover slip. For example, a lens of
63
transform and
magnification, 1
.
2NA, and water as medium between the lens and cover slip is
written as '63
×
×/
1
.
2 water immersion'.
4.2
Development of the Auxiliary Computational Lens
4.2.1
Confocality Improves Resolving Power
4.2.1.1
Background Fluorescence Rejection
The euphoria with the developments in the WFM was short lived, because biologists
quickly realized that the illumination excites almost the entire depth of the sample
instead of just the focus plane. Therefore, the intensities detected are not localized
from a single focal plane. For example, in Fig.
4.3
a, we show a
Convallaria
sample
that illustrates the inevitable presence of out-of-focus blur in a WFM image,
that perturbs contrast and image resolution. When imaging biological cells, the
amount of radiation must be limited. When exposed to irradiation, fluorescent
molecules tend to react with molecular oxygen to produce free radicals that can
damage subcellular components and compromise the entire cell. The eventual
photochemical destruction of the fluorophore (or
photobleaching
) usually occurs
after extensive exposure to light [
25
]. This can especially be the case in time-lapse
microscopy. Moreover, live samples are sensitive to the amount of light that they
are exposed to. Even plant cells that are especially attuned to light collection do not
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