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RT&A # 02 (29) (Vol.8) 2013, June
IMAGE RECOGNITITON BY MULTIDIMENSIONAL INTERVALS
G. Tsitsiashvili
IAM, FEB RAS, Vladivostok, Russia e-mails: guram@iam.dvo.ru
ABSTRACT
In this paper new algorithm of interval images recognition is suggested. This algorithm gives accuracy solution of considered problem but demands not linear but square complexity by a number of objects. Main motive of such construction is to analyze practically interesting case when there is preliminary silence before predicted events.
1. PRELIMINARIES
In [1] the algorithm of interval images recognition is described. In a case of a single index which characterizes objects of first and second classes a minimal interval contained objects of the first class was constructed. Then an object with an index contained in this interval is considered as the first class object. In a case when each object is characterized by several indexes a one dimensional interval in a recognition procedure is replaced by a multidimensional interval constructed as a direct product of one dimensional intervals. An advantage of the interval images recognition before known algorithms is a linear (by a number of all objects and by a number of all indexes) calculation complexity. This algorithm is successfully used in manifold problems of medical geography and ecology, meteorology and fishing [2] - [9].
The algorithm is sufficiently satisfactory when a number of all objects in a sample is about 20-30 and a number of indexes is larger than 3. But in a problem arisen in a mining an emergence of a rock pressure cannot be predicted using a single interval. It means that there are first class objects which have predecessors and there are first class objects which have not predecessors. In this situation the single interval cannot characterize all first class objects because it goes past first class objects which may be described by a preliminary silence.
In this paper the method of interval recognition is developed in a direction of a consideration of such situation. It is based on a construction of few nonintersecting intervals which contain points characterized different first class objects. So first class objects are divided into some subclasses and their recognitions are realized separately. This algorithm is more complicated and has not linear but square complexity by a number of all objects.
Assume that first class objects are characterized by the set B = , 1 < j <m\ and second
class objects are characterized by the set A = \ai, 1 < i < n\, -roe A, roe A , of real numbers. Suppose
that m is much smaller than n . For real numbers c, d ,c < d, define the interval (c, d) by the
2. MAIN RESULTS
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condition (c, d)={f: c < f < d} if c < d . If c = d then the interval ( c, d ) consists of the single point
c = d . Construct the following rule of a recognition of the object b e B. For each number b e B
contrast two
numbers:
k(b) = max{a e A : a < b}, r(b) = min{a e A : a > b}. As a result we construct for each number b e B the interval (k(b),r(b)).
Theorem 1. If bt,bj eB, then the intervals (k(bi),r(bi)) , (k(bj)r(bj)) coincide or not intersect . Proof. Assume that between the points bt, bj there are not points of the set A . Then by a construction the intervals (k(bi),r(bi)), (k(bi),r(bi)) coincide. Vice versa if between the points bt,bj there are points of the set A then by a construction the intervals (k(bi),r(bi)), (k(bi),r(bi)) not intersect. So the points of the set B are divided into classes of an equivalence by their belonging to coincide intervals.
Suppose now that the set A consists of n objects and its each object i is characterized by l -dimensional vector ai =(a1,...,al). Analogously assume that the set B consists of m objects and its each object j is characterized by l - dimensional vector bi = (b) ,...,bj). Define the interval (k(bj)r(bj)) by the equality
k(bj) = maxja- : aj <bj, 1 < i < n|, r(bj) = minjaj : aj > bj, 1 < i < n|.
Using these intervals construct l - dimensional interval which is its direct product. ®t=1 (k b) r b)).
Theorem 2. If 1 < i * j < m, then l -dimensional intervals ®\=1 (k(bj)r fa)), ®lt=1 (k(bj)r(bj)) coincide or not intersect.
Proof. Indeed, by a construction for any t, 1 < t < l, one dimensional intervals (k(bj),r(bj)) , (k|bj), r(bj))
coincide or not intersect. If these one dimensional intervals for all t, 1 < t < l coincide then their direct products ®\=1 (k(bj),r(bj)), ®l=1 (k|bj),r(bj)) coincide also. In opposite case there is t so that
appropriate intervals not intersect and consequently their direct products not intersect also. Consequently vectors from the set B are divided on subsets (equivalence classes) by their belonging to coincide l -dimension intervals.
Suppose that (l +1) - dimensional vectors arrive in recognition system. The first component equals zero if this vector belongs to the set A and equals one if this vector belongs to the set B . Assume that on the step 0 two (l +1) - dimensional vectors (0,+^,...,+^), (0,-^,...,-^) are introduced into the system. Further on the step n > 0 single vector (an,cn,1,...,cn,l) arrives. Denote n0the first vector for which an = 1. Then the first multidimensional interval containing the vector (cn ,1, . ,c^,/) is constructed.
Further assume that on the step n > n0 the vector (an,cn,1,...,cn,l) arrives in recognition system. Suppose that an = 0. Then if the vector (cn1,..., cnj) does not belong to constructed intervals then the system of these intervals conserves. If (cn1,..., cnj) belongs to one of constructed intervals then this interval is divided onto subintervals by described rule.
Assume that an = 1 then if the vector (cn1,..., cnj) does not belong to constructed intervals then new interval containing this vector is constructed. If (cn1,...,cnj) belongs to one of constructed intervals then the system of intervals does not change.
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REFERENCES
1. Tsitsiashvili G.Sh., Bolotin E.I. 2010. Construction of fast algorithm of images recognition in application to prognoze problems. Informatics and control systems. No. 2. P. 25-27. (In Russian). 2 Bolotin E.I., Tsitsiashvili G.Sh., Golycheva. 2002. Some aspects and perspectives of factor prognoze of epidemics of focus of tick-borne encephalitis foci on base of multidimensional time series. Parazitology. T. 36, No. 2. P. 89-95. (In Russian).
3. Bolotin E.I., Tsitsiashvili G.Sh., Burukhina I.G., Golycheva I.V. 2002. Possibilities of factor prognoze of tick-borne encephalitis foci in Primorye . Parazitology. T. 36, No. 4. P. 280-285. (In Russian).
4. Bolotin E.I., Tsitsiashvili G.Sh. 2003. Space time prognoze of tick-borne encephalitis foci. Bull. of the Far Eastern Branch of Russian Acad. of Sci. 2003. No.1. P. 5-19. (In Russian).
5. Shatilina T.A., Tsitsiashvili G.Sh, Radchenkova T.V. 2006. Experience of application of interval recognition method to prognoze extremal ice-covering of Tatar channel (Japanese sea). Meteorogy and hydrology. No. 10. P. 65-72. (In Russian).
6. Goriainov A.A., Shatilina T.A., Tsitsiashvili G.Sh., Lysenko A.V., Radchenkova T.V. 2007. Clymate causes of decrease of fish resources of Amur salmon in 20-th century. Far eastern region -fishery. No. 1,2 (6,7). P. 94-114. (In Russian).
7. Bolotin E.I., Tsitsiashvili G.Sh., Fedorova S.Yu., Radchenkova T.V. 2009. Factor time prognoze of critical levels of infection illnesses. Human ecology. No. 10. P. 23-29. (In Russian).
8. Bolotin E.I., Tsitsiashvili G.Sh., Fedorova S.Yu. 2010. Theoretical and practical aspects of factor prognoze of infection illnesses. Human ecology. No. 7. P. 42-47. (In Russian).
9. Bolotin E.I., Ananiev V.Yu., Tsitsiashvili G.Sh. 2010. Prognoze of infection illnesses: new
approaches. Population healthy and habitat. No. 5. P. 15-19. (In Russian).
