http: // shulga.tripod.com/formaved

                  ABSRTUCT

Interference-defended - joint solving of segmentation and approximation problems, based on usage of bell-like (fuzzy) membership function. – Bell-approximation.
Approximating segments with maximal functional accentuation - loading in center, with graduate step-by-step falling down of it to the ends. – Segments with fuzzy ends - contiguous segments. – As much as possible strict – as much as possible lengthy - universal linear units. – Geometrical ground of primary approach – delineating ("spanning-capture") of arbitrary - curvilinear contour forms.
Unison of positional hypotheses as interference-defended indicator of simultaneous detection and recognition of defined configurations. Possibility of operation under partial overlapping of sought objects.

Keywords:

data modeling, regression analysis, interpolation, approximation, segmentation, piecewise-smooth curves, splines, fuzzy sets; shape analysis, image analysis, pattern recognition, image recognition, template matching, Hough transform; computer vision, machine vision, industrial vision, robot vision, drawing recognition, documents analysis; analysis of schemes, maps, graphs, trajectories and signal processing as a whole; applied geometry.

INTRODUCTION. Conceptual content.

The present work deals with the problem of jam-resistant representation of contour forms as sets of elementary line segments. It elaborates the problem of following positional detection (detection with finding the position and orientation) among these sets of specified configurations. These configurations are determined through their samples (templates) which, herein, themselves represent the individual images of piecewise-smooth objects. Origin materials were contour preparation of actual images. 

Line - one of primary geometrical concepts by means of which the man describe of environmental actuality. An apparent ease - "evidence" from which the man operates with his own abstract representations in this field, makes false impression about simplicity of algorithmization (transposition on a computer) such operation. One of key tasks here - a partition of registered linear shapes on individual components - the elementary segments. Difficulties start already in the elementary case - a case "angular" - polylines. It appears, rather uneasy to pick out an angle in a actuality when lines are set discretely as a set of the individual points scattered around of true positions.

In the first section of the present work it is arising here difficulties systematically understand. The physical sense of minimization of mean-square deviation - base procedure of the analysis of actual data is considered. Necessity of usage of bell-like (fuzzy) function of a membership of contour points to individual segment is obviously shown. Tasks of partition of contour configuration on individual segments and their approximation are reduced together - to searching local maxima of a "fuzzy" measure of likeness constructed on the basis of such function.

In the second section it is considered the essence more complicated problem, namely, -  if we should divide into units a lines having the smooth shapes, - when generally there are no precise angles, - one segment gradualy - smoothly pass into another. The concept of segment with fuzzy ends is developed. It is suggested to use a membership function of bell-like shape also in the longitudinal (axial) direction. 

Such - bell-like function enables to rest on the length of the contour lines for a maximum degree when they are approximated with the curves of some standard type, mainly in the center, with gradual weakening in the periphery direction, according to their gradual divergence.
The obtained approximating segments are named as contiguous segment because their non-sharp (fuzzy) ends are only adjacent, in the general case, to contours under approximation.

In the third section the problem of practical usage of contiguous segments in the task of positional detection of arbitrary formed contour objects under their exemplars is considered. The offered solution is based on a procedure of registration of a unison of parametric (positional) hypotheses about probable positins of sought configuration which are independently generated by separate primary fiatures - fragments. As such indications - fragments contiguous segments here appear. Common ontological (essential) cscheme of such procedure is offered. Practical implementation of this approach leads to procedure of detection similar to procedures in which basis known Hough transform. The offered summarizing enables to see their depth generality - to simplify logic of their development and the further development.

Moreover, and what is above all, the developed procedure is based on an application of the universal primery elements, which limit the sets of the hypotheses about the global (positional) parameters of the search objects in the maximum. It is applied the described above tangent-contiguous segments with fuzzy ends. It was an use of these initial units that enabled us to draw the carried out search the hypothesis unison principle up to a possibility of the constructive application in the problem of positional detection of arbitrary form objects. In the other hand, it was the use of such way of detection that enable a principal way to "cut up" the "sliding uncertainty" of contiguous segments that have its place in general. It may be told about uncial unity of this conceptual pair.

At this base, the sequence of procedures was created for efficient solving the problem of segmentation-approximation and following positional detection (detection with definition of the position and orientation) of arbitrary form flat contour figures with respect to their samples in the case when one figure may partially overlap another.

All these may be essentially advanced and widely applied as the whole and individually.

The paper is a otherwise accented variant of the following work with some specifications in the terminology:

Shulga V.I. "Complex (Collection) of Interference Defended Procedures of Contour Boundary Approximation and Recognition of Objects on Contour Images (research work)" / Glushkov Institute of Cybernetics, Academy of Sciences of the Ukraine, Kiev, 1992, 75p. (In Russian) / Deposited in the All-Union Institute of Scientific and Engineering Information (Moscow), 04.01.92, N 12-B92.
https://shulga.tripod.com/GEOLIN/glnfl1.htm
In its turn, this work summarizes the author's investigations, the first publications are dated 1981 [2] and 1982 [19] years. In the scientific literature, there is lot of papers, thematics of which intersects with the subjects of the present investigation. In Internet there is a wide set of sites, which is searched under enquiries:  «weighed least squares», «fuzzy lines», «generalized Hough transform» … Definition of relationship degree  of these materials with the present work and determination here authoring superiority – is a subject for different  examination. To not passing virtues of the present work, it is possible to refer that instead of scrappy hints and sacral appeals to higher mathematics, semantic - physical explanations are given, which are a basis of any science. Besides all procedures is united in the unique technological line-up, showing a place of each concepts in the common problem of positional detection - recognition as a whole.