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## Linguistic Description of Curves, Contours and Images. Similarities and Differences

(Kazan State Technical University)

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In present time field of soft computing have taken a significant place in mathematic. Soft computing aim is to adapt to imprecision, uncertainty and partial truth of real word for achieving tractability, robustness, low solution cost, and better rapport with reality [1].

The fundamental direction of this field is fuzzy sets theory, which goal is mathematical formalization of fuzzy, qualitative and linguistic terms such as "tall", "quick", "small", "large" and to allow using it in describing of various objects.

In field of soft computing a high attention is given to subjects, connected to linguistic approximation of different functions [1,2]. This attention is caused by proving of theorem FAT (Fuzzy Approximation Theorem) by Bart Kosko in 1992. This theorem claims that any continuous function f:X(r)Y (X,YÌR) specified on compact set, may be approximated with any accuracy by some additive fuzzy system F, which rules are defined in linguistic form [3].

 picture 1

For example function on picture 1 may by described in linguistic form as
IF x is small THEN y is large
IF x is medium THEN y is small
IF x is large THEN y is large

To achieve a more precise linguistic approximation one must do more detail granulation of abscissa axis i.e. use more linguistic terms such as "least", "very small", "nearly small" etc.

Linguistic approximation of simple functions can be done fairly easy. But linguistic approximation with good accuracy of some complicated functions may induce some problems because of necessity of using too big number of rules, necessity of partitioning abscissa axis on multitude granules. Bart Kosko suggests to use the process of learning designed for searching optimal rules to decide this problem.

Our proposition for solving this problem consists in usage of linguistic approximation in common with different types of structural methods (for example, in common with generative grammars).
Denote by P the set of primitive curves, which can be simply described on linguistic level. Denote by R the set of relations between primitives from P. Let pair (P,R) is complete for description of all curves of some data domain À in the range of used structural method. Next, any complicated curve Á from À can be described by combining some primitives of set P and some relations between them into certain structure

 Á =L(P' , R '), P' Ì P,R 'ÌR (1)

Describing all primitives from P in linguistic form and defining mapping for every relation between primitives to some linguistic term we can map description (1) to linguistic description of complicated curve Á
FAT may be easy transformed from functions to 2D-contours. One can get linguistic description of 2D-contours by using next methods.

1. Transformation 2D-contour to 1D-function, which is unique defined contour (for example to curvature function). Next, one can construct linguistic description for this function.
2. Constructing linguistic approximation for mapping , which is unique defined contour. There are two ways may be used in this method.
2.1. Using not 1D-linguistic terms but 2D-linguistic terms which are describing 2D-planar area, for example "large rectangle", "small neighborhood" etc.
2.2. Using 1D-linguistic terms as antecedents in additive fuzzy systems in FAT but defining area using logical connectives AND, OR.

One may use these methods for linguistic description of contour images in any data domain À. In this case one can determinate set of closed primitive contours P that are easy described on linguistic level and set of relations R between them. (pair (P,R) must be complete in À). Thus everything said about level of functions may be transformed to level of images through level of contours, any contour image may be described on linguistic level.

Synthesizing linguistic description of any function f may be carrying out by next scheme.

Stage 1
Decomposition of f into parts which hypothetical are primitives (this decomposition may be not unique). This stage may be named as recognition of structure relations between primitives.

Stage 2
Recognition of extracted parts of f as primitives in P. This stage is released by enumeration of all linguistic description of primitives in P and chosing those of them which are the best approximation for this parts. This stage may be named as recognition of primitives.

Stage 3
Calculation of linguistic approximation precision e i for function f. This precision depends on approximation precision of any decomposed parts.

Stage 4
Taking other decomposition way of function f and returning to stage 1. If there is no more decomposition ways in presence then go to stage 5.

Stage 5
Choosing this decomposition way that leads to approximation precision not more than any cutoff E and to least description complicated.

In this scheme stage of recognition of structure relations between primitives (function decomposition) is primary to stage of recognition of primitives (linguistic approximation decomposing parts of f). This scheme may be named as "from top to bottom" scheme of synthesizing linguistic description.

On the contour images level in addition to synthesizing of its linguistic descriptions one must often solve the task of localization any object K on image I where K is described in linguistic form. This task may also be solved also by "from top to bottom" scheme. However in this case methods of primarily recognition of relations between primitives are very difficult in sense of computing (for example, Hough transform). But if linguistic description of localization of object K is defined then primitives in set P' , which are presented in structure of K, and set of relations between them R'are defined too. Thus one may use next scheme for synthesizing linguistic description of contour images.

Stage 1
Extracting on I set of closed contours P" that are contained in set P'. This stage may be named as recognition of primitives.

Stage 2
Trying to connect primitives of P" by relations in R'. This connection may be not unique. This stage may be named as recognition of structural relations between primitives. The result of this stage is the set of relations R" which connects set of primitives .

Stage 3
Synthesizing linguistic description of pair () and computing precision e i of fitting this description to linguistic description of localization of object K.

Stage 4
Returning to stage 2 and trying other connections of primitives P" in by relations in R'. If there are no more possible connections present then go to stage 5.

Stage 5
Choosing that pair from (), which leads to best precision of approximation of localization of object K.

In this scheme stage of primitives recognition is primary to stage of recognition of structure relations between primitives. This scheme may be named as "from bottom to top" scheme of synthesizing of linguistic description. This scheme has advantage in less computational difficulties in comparison to "from top to bottom" scheme.

However this scheme has next drawback. "From top to bottom" scheme is not too sensitivity to high deformations of some primitives, it balances faulty estimation by best recognition of other decomposition parts. Opposed to this "from bottom to top" scheme does not consider primitives which not satisfy required description and discards them on stage 1. Thus "from bottom to top" scheme may be used to solve tasks of localization of object K on contour image I only if deformations of primitives are not generative (2).

Both "from top to bottom" and "from bottom to top" schemes have been implemented in software system for ultrasound images recognition for automated fetometry of fetus in mother's womb [3]. We have picked out set of ultrasound images of fetus, which hold demand (2).

1. L.A. Zadeh. Fuzzy computing = computing with words // IEEE Transactions on FS, Vol. 4, No. 2, 1996.
2. B. Kosko. Fuzzy systems as universal approximators // IEEE Transactions on Computers, Vol. 43, No. 11, 1994.
3. Anikin I.V. Software system for automated fetometry of fetus in mother's womb // In "Computer Technologies in Science, Projecting and Production", N. Novgorod, 2000. (in Russian).

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