Biomedical Engineering Reference
In-Depth Information
Assume that few fluorochromes are used so that each one labels a different entity.
The entities, and therefore also the fluorochromes, may be found either separately
or mixed at every pixel of the image. The purpose of the linear decomposition
algorithm is to find the amount of each fluorochrome at each pixel. In its simplest
form, the algorithm requires measuring and saving the emission spectra of the
distinct fluorochromes prior to the actual analysis with the same experimental
conditions. It also requires that the spectra of the different fluorochromes are
distinguishable from one another and are linearly independent, which means that
none of the spectra can be described as a linear combination of the others.
We describe the algorithm as it is performed on a spectrum at a single pixel, and
it can be repeated for the whole image. The spectrum of each fluorochrome can be
described as I
i
./ where i
D
1;2;:::;N represents the index of the fluorochrome.
The measured spectrum I./ can be viewed as a vector with dimension that is equal
to the number of wavelengths in the spectrum, M .WedefineC
i
as the concentration
of each fluorochrome relative to the concentration of the measured references, and
the measured spectrum can be expressed as I./
D
P
i
C
i
I
i
./. This can be written
in a matrix form
F
where each column corresponds to one of the reference spectra:
2
3
2
3
2
3
2
3
I.
1
/
I.
2
/
:
I.
M
/
I
1
.
1
/I
2
.
1
/ ::: I
N
.
1
/
I
1
.
2
/I
2
.
2
/
:
C
1
C
2
:
C
N
C
1
C
2
:
C
N
4
5
4
5
4
5
4
5
D
D
F
(4.12)
:
:
:
I
1
.
M
/
:::
I
N
.
M
/
:
If M
N (i.e., the number of points in the spectrum is larger than or equal to
the number of fluorochromes), it is possible to find the left-inverse matrix of
F
,
F
LI
,
so that the multiplication
F
LI
F
gives an identity matrix. Then, the values of C
i
are found by
2
3
2
3
C
1
C
2
:
C
N
I.
1
/
I.
2
/
:
I.
N
/
4
5
4
5
D
F
LI
(4.13)
F
LI
depends only on the reference spectra, and therefore, it has to be calculated
only once for the whole image. Because of noise, the calculation can only provide
an estimated solution for the
C
values and the quality of the solution can be tested
by comparing the measured spectrum with the predicted one, I
0
./
D
F
C
.The
vector
C
must have only positive values, and therefore, a constrained algorithm
must be used. One method is to find the vector
C
that minimizes ", the least square
normal,
"
D
ˇ
F
LI
C
I
./
ˇ
2
(4.14)
such that C
i
0 for every i [
62
].
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