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A New Look at Fundamentals of the Photometric Light Transport and Scattering Theory
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. Естественные науки. 2017. № 5
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probability of transitions became quasi-discontinuous, and the possibility to replace
them by the test number
n
(or
m
,
l
, etc.) appears. In such approach the common
probability of the ''photon'' transition —
p
(
n
,
x
k
|l
,
x
i
) — from
x
i
point to
x
k
point obeys
the general Markov equation [16]:
( ,
, )
( ,
, ) ( ,
, ),
.
k i
k
j
j
i
j
p n x l x
p n x m x p m x l x l m n
(28)
For the homogeneous Markov chain all probabilities of the transitions using ''
s
'' steps
(
s = n – l
) form a matrix of probabilities:
(
, )
( ),
k
i
ik
p x s x
s
p
(29)
herewith, if
N
is a total number of chain's states which are taken into account, the sum
of all matrix elements obeys the closing equation:
1
1.
N
ik
k
p
(30)
The equation (29) shows that probabilities of transitions with the use of any ''
s
'' steps
can be ultimately represented using only one-step probabilities. In fact, it reduces the
solution of the Markov process problem to forming one-step matrixes and making the
formal multiplications of them.
Because of
R
1 in our scattering model, reflectivity
R
can be interpreted as a
probability of the ''photon'' transition from one state to another when it is reflected by
a heterogeneity
r
i
, whilst the probability of the opposite event for the ''photon''
(crossing the heterogeneity
r
i
) will be equal 1
− R
. For the correct usage of the Markov
processes mathematical formalism, it is necessary to enumerate all states of ''photons''.
Consider
N
heterogeneities
r
i
inside Δ
x
. If the simplest enumeration is chosen: before
all
r
i
(before Δ
x
) there is the state number
i
= 1, between
r
1
and
r
2
— the state number
i
= 2, between
r
2
and
r
3
the state number
i
= 3, etc., then one can determine all
one-step probabilities for the ''photon'' migration. Excluding states
i
= 1 and
i
=
N
+1
(after
r
N
), the transition from any
i
-th state to the state
i
=
i +
1 or
i
=
i −
1 has the
probability 1 −
R
. The probability of staying in the state
i
is
R
(the case of back-
reflection in
i
-th state). If ''photon'' goes out of the Δ
x
(
i
= 1 or
i
=
N
+ 1), the
probability of next changing its state falls down to zero. But for this simplest
enumeration approach, the Markovian properties of the process are violated: the
transition from
i
-th state to
i +
1 or
i −
1 states depends on the prehistory of the
''photon'' traveling (was the ''photon'' moved from left to right or from right to left).
Therefore, the problem cannot be resolved using the formalism of Markov processes.
To overcome this difficulty, we offered to consider the much more useful
enumerating approach. The even numbers (
i
= 2, 4, 6, …) we will use for the
enumeration of the ''photon'' states between
r
i
and out of them when ''photon'' travels
from right to left. The uneven (odd) numbers (
i
= 1, 3, 5, …) we will use when a