Zhisnyakov A.L., Fomin A.A., Privezencev D.G., Baranov
A.A.
Murom Institute of Vladimir
State University
RESEARCHING
OF THE SIGNS SETS BEHAVIOR
ON THE DIGITAL
IMAGES MULTISCALE SEQUENCES
The
problem of images sequence processing occurs often in different fields of
science and engineering. Example of images sequences is video sequences, multiscale
images sequences in the technical vision systems and so on. In spite of
differing forming character of such sequences, many principles of their
processing is same. In this connection, the task of construction of theory
basis, that unifies algorithms of forming, processing and analysis of digital
images sequences describing, is actual.
The
work purpose is constructing of describing and analysis digital images sequences
approach that is based on the insertion of signs inheritance and mutability
terms.
The image conception
Definition 1. Continued image is function
f(x,y), that is definite on the subset P
of plane R2 and having
real values. Function f(x,y) value defines image brighthess in the point (x,y).
Let's consider we can compare each image f to finite signs set X = {X1, X2, …Xn}, that unambiguously determines f into multiple set of another
images given on P.
At that each sign Xi represents finite set
of elements of signs 
place, in example object edges
set that defines the given image.
As a result we can insert set of operators O such that
, 
, … 
, 
(1)
Let for each set X distance between images dX
is defined.
, 
(2)
Images
sequence
Definition 2.
We names images sequence {fn} as ordered
images set f setting index for every one element.
.
As each image from initial set is unambiguously
characterized by signs set X we can
construct the sequence is identical with (3).
{Xn}
= {X0, X1, X2 …}. (4)
Definition 3. We can name vector A sequence limit {Xn} if for each e > 0 there is number n0 = n0(e) that the
inequality 
is true for each real n > n0.
We can
register it as {Xn}®À. (5)
Definition 4. As {fn} is defined unambiguously by {Xn} we can tell images
sequence {fn} converges to a limit g on multiple signs set X
. (6)
We can interpret the vector A as some etalon of signs
vector X of the etalon image.
Theorem 1. On the f we
can forms set of ordered decently sequences {fn}Õ each of ones converges on some signs set X for defined distance dX.
We limit multiformity of viewed images sequences to
case that we can describe sequence forming by some operator.
Definition 5. Let P1
and P2 are two subset of
space P at that Ð1 Í Ð2. Then we can tell
sequence forming operator Ò: Ð2 ® Ð1 is defined on P2
if we can compare each element (i2, j2)Î Ð2 of image f2
to element (i1,j1)Î Ð1 of image f1.
Inheritance
and mutability of images sequence signs
The sequences in which there are some correlation
between images is the most interest for viewing. In practice it means if there
are some images sequence {fn} that is ordered by any signs set X then neighboring images can have some
similarity degree. That is if some sign x
presents on the image fk Î{fn} there is very likely it will appear on neighboring
images fk-1 Î{fn} è fk+1
Î{fn}, k = 1..N-2.
Therefore we can tell about presence of sign
inheritance factor on the images sequences.
At the same time it is obvious, as relationship
between images in the sequence will decrease depending on distance between ones
(increasing image index difference), some signs, that distinctly show on the
one image, can be less visibly on the another images or vanish quite.
We can characterize it as the signs mutability on the
images sequence.
Let is available the sequence 
for which {Xn}®À.
Let’s consider sequence consists of elements {Xn}, that defines same sign
(i.e. sign with index i) for each images of initial sequence.
Oi[{fn}] =
{Oi [f0], Oi [f1], … Oi
[fn]},
{x(i)n}
= {x0(i), x1(i),… xn(i)}. (7)
For describing of one sign behavior on the images
sequence we propose to use mathematical apparatus of fuzzy set. Really same
sign can presents on the several images changing consistently. However as the
result of this changing it can disappears or changes so it will viewed as
another independent sign.
To characterize the presence of sign xj(i) on the
image fi we can insert the attribute operator mi.
Definition 6. We can name the attribute
sign operator as operator that defines entry level of sign x(i) into fuzzy set X(i),
that is conformity degree of one sign realization on the images sequence to the
limit(etalon) value ai, aiÎA, {Xn}®À.
mi : mi[xj(i)]®[0,1]. (8)
Then we
can compare each signs set Xj
of image fjÎ{fn} to
vector m Õj = (m1õ1, m2õ2, .. mkõk )Ò, where k is sings count.
For
each sign we inserts threshold operator
Ã: Ãmi[xj(i)]®{0,1}. (9)
As the
result of appliance (9) to initial vector Xj
that characterized image fj, we get some vector Wj that consists of
ones and zeros.
W = (w1, w2,… wê)Ò, wi Î {0,1}, i = 1,
2,…k. (10)
At this
realization sequences of some sign on the images sequence (7) will agree with
sequence that consists of ones and zeros too.
Q = {q1, q2,… qn}, qj Î {0,1}, j = 1, 2,…n. (11)
Let we
have two neighboring images fm and fm+1 of sequence {fn}.
The vectors Wm and Wm+1 agree with them. We
view elements pairs (wmi, wm+1i), i = 1, 2,..k.
Elements coincidence in this pairs signifies similarity of two images by given
signs set X.
Definition 7. We can think about images sequence mutability as old
signs losing or new signs appearance while transition to each next image in the
sequence.
For mutability characterizing we can use the
expression:
, (12)
Where Å is the inequivalence operator (exclusive disjunction).
Definition 8. Inheriting of images sequence is signs holding
process while transiting to the next image of sequence.
(13)
Theorem 2. Images sequence {fn} is converged if for signs set X
starting with index n0 the expression 
, i = n0 ..n is true,.
Addition
sequences
The mutability that is attended for reflection fk-1
= Tfk brings to losing on
fk-1 of features part that is peculiar for fk. Let fk\fk-1
is image that is result of some operator T*
influencing on fk and holds signs disappeared while fk to
fk-1 transiting.
At this if Tfk
= fk-1 and T*fk
= fk\fk-1 then applying (1), (8) and (9) we can get
O[fk-1] = Xk-1, Ãm[ Xk] = Wk;
O[Òfk] = O[fk-1]
= Xk-1, Ãm[ Xk] = W*k;
O[Ò* fk] = O[fk\
fk-1] = X*k-1, Ãm[ Xk] = Wk-1;
At this Wk = Wk-1 + W*k. (14)
Definition 9. We can name operator T* additional to operator T
if condition (14) is true.
Let gi = fi \ fi-1. (15)
Definition 10. We name {gn} the addition sequence to the
sequence {fn} if there are giÎ{gn} for each fiÎ{fn} that T* fi = gi.
Theorem 3. If the images sequence {fn} is converged
by sings set X to any vector A then addition sequence {gn}
is converged by same signs set X to the
null vector 0 that has same length
that A.
We can insert the operator 
such that
fi/ = Ò (fi
)
Ò*( fi ) (16)
or taking into account (15),
fi/ = fi-1
gi, (17)
At that for each fi/ and fi
the expression 
or 
is true. In other words the operator 
also can not give accurate reconstruction
of image fiÎ{fn} by previous
image fi-1Î{fn} and addition gi-1Î{gn} but
it forms the image fi/ that agree with fi
according to signs set XÎX. Let we have the
sequence {gn}= Ò*{fn}, the operator 
provides reconstruction only
in terms of condition (18). Let we know the image fk Î{fn}.
Definition 11. We name 
estimation of
image fk+mÎ{fn} by the image fkÎ{fn} on
the basis of sequence {gn} with
helping the operator 
. We can determine similarity of two images by specified signs set on the
mutability term (20). We can characterize reconstructing operator precision as
. (19)
The expression (19) defines value of image fk
mutability by attitude to its estimation fk(m) by the
image fk-mÎ{fn}. For characterizing operator 
on the entire sequence we
average (19) by k:
. (20)
The offered approaches
can be applied while developing of new processing methods of digital images. Especially
it is possible to use it while construction of multiscale processing algorithms
of images with variable value of scale.