 2. SEGMENTS WITH FUZZY ENDS.

Problem of structural segmentation (approximation) of arbitrary shaped curves is solved.
2.1. - It is noted a non-adequacy to the actuality of such the concepts as sharpen beginnings and ends of segments in a gemeral case of smooth contour lines, when one segment gradually passes into another one. 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.

2.2. - Some particularities of program procedure elaborated on this base are demonstrated. The results of experiments are discussed.
2.3. - Some improvements are supposed.
2.4. - Longitudinal fixation of the contiguous segments.

2.5.
A general phenomenon of sliding of approximating segments is pointed out.

2.1.  Statement of the Problem.

The concept of bell-like (fuzzy) membership function has enabled in principle new approach to the problem of subdivision and approximation of contour boundaries of arbitrary - curvilinear forms. Principal complexity of representation of such contours as a set of separate segments consists in that such terms as legible ends of a segment here are not applicable. Here one segment gradually (smoothly) passes in another. In this case, function, estimating degree of a membership of points to a separate segment should fall down gradually to its ends, i.e. to have the bell-like form and in the longitudinal direction. The provisional view of such function is shown on fig.7. Here parameter B corresponds a half-length of an approximating segment, is similar to, how parameter D - to a half-width. Fig.7

To the conclusion about expediency of use of such membership function leads also the fact that there is no limited collection of curves of any standard type, capable to exhaust all variety of curvilinear segments of curvilinear-smooth forms. Such segments are described by curves-standards only approximately. Here it is also natural to demand a maximum approximation of an approximating (reference) segment on the center with a possibility of gradual slacking of this approximation to the periphery.

These reasons also have underlain a suggested method of the description of contour boundaries in terms of segments with fuzzy ends. In the carried out researches as axial lines of approximating segments curves of the simplest analytical type, namely - the central pieces of parabolas were used. And each branch could have own curvature. For fine length tuning of branches the mechanism similar to fine tuning of width of segments was used. The conformity measure builted according to these reasons, has the following mathematical expression: Where

 b_i, d_i - the parameters defining a position of a contour point in own frame of an approximating segment (fig.8); these parameters are defined on coordinates of the point x_i, y_i with the help of transformations b_i =   (x_i - X)*cos(A) + (y_i - Y)*sin(A), d_i = - (x_i - X)*sin(A) + (y_i - Y)*cos(A); Fig.8
 X,Y - coordinates of the center of the segment; A   - the angle of declination of the segment; P   = P1, if b_i >= 0 (curvature of the right branch), P2, if b_i <  0 (curvature of the left branch); B   = B1, if b_i >= 0 (length of the right branch), B2, if b_i  < 0 (length of the left branch); D   - a half-width of the membership function.

A separate approximating segment defined by such measure of conformity is symbolically shown on fig.8. Here decrease of thickness of the segment reflects decrease of its clearness, i.e. degrees of correspondence to an approximated contour.

2.2. PROCEDURAL PARTICULARITIES. RESULTS OF THE EXPERIMENTS.

The proposed approximation method has been realized in the language FORTRAN (a small part - on the autocode). Experiments with this program were carried out on a computer by efficiency of 1 million operations from a floating point.

For searching of local maximums of the conformity measure of contiguous segments a gradient procedure with a governed step was used. Components of a gradient were calculated on the basis of analytical expressions for partial derivatives.

The contour image on which basic serviceability of this procedure was checked is shown on fig.10a. Fig.10a

The size of an initial half-tone picture is equal - 290x196 points. For detection of contour boundaries the known Hueckel's operator [5,6] was used. In total, 800 contour points were chosen approximately. Initial positions of contiguous segments were defined by nodes of a regular grid with 12 raster points step (reasons on the basis of which this step was set, are stated in Appendix_2.2.A). Values of remaining parameters were accepted as the following: B1 = B2 = 5, D = 2.5, A = 0, P1 = P2 = 0.

Initial approximations had the most various positions concerning approximated pieces. Nevertheless, in supressing majority, from 384 starts having here a place, procedure of gradient "fine tuning" founded the maximum of the bell-conformity measure defining an actual contiguous segment (fig.10b). Fig.10b

No more than 10 starts led to an erroneous approximation: or to a capture of casual accumulation of the contour points not similar to a segment of lines (stain), or to a capture of the several close placed segments. Such cases were defined by explicit violation of the search process, i.e. too excessive increase of length (width) of an obtained segment or number of steps, or by criterion of filling ratio of a resulting segment by contour points [refered below].

The experiments carried out in this way have shown that the maximums corresponding to the segments in search have pure slopes. Really, on these slopes there are no ejections (they are swallowed by the basic extremums) in which gradient procedure could "lose the way". These experiments have proved a basic possibility of deriving of the description of contour figures in terms of contiguous (tangents) of the segments corresponding to our intuitive representations.

At the same time, the analysis of behaviour of the gradient procedure originally used for search of local maximums of a bell-conformity measure , has shown, that in neighbourhoods of the maximums (i.e. at the final stages of search) the conformity function looks like the long, narrow ridges prolated along axial lines of contour boundaries. Thus, approximating segments can long oscillate in the cross direction to these lines, with very slow movement in the longitudinal direction. For fast transition of these ridges in  it was offered to displace during search the central point of an approximating segment not on axes of the image coordinate system but along and cross of flowing direction of its axial line. As the axial line of an approximating segment is gradually arranged under the axial line of the represantation piece of contour boundary, longitudinal moving of the segment is carried out directly along the axial line of the mentioned ridges. Such organization of the search procedure of local maximums of the conformity measure enabled rather fast and legiblly to pass the above mentioned ridges of this measure. More in detail about the choice of value of steps it is said in Appendix_2.2.B.

Further. As initial approximations in the subsequent experiments, the most legible elements of contour boundaries selected by the Hueckel's operator were accepted. Besides, sparsed representation of contour boundaries was used. As contour points the central points of contour-boundary (edge) patterns of the Hueckel's operator were assumed, instead of all raster points placed under them as it was in the first series of experiments were accepted.

According to sparsed representation, the criterion of filling ratio of contiguous segments [Appendix 2.2.C] was modified also.

A contour image on which serviceability of the advanced procedure of representation of contour boundaries was checked by contiguous segments is shown on fig.13. A size of the initial half-tone picture - 216x216. In this case, diameter of the entering piece of the Hueckel's operator was equal to seven (7) points of the raster. Overlapping at tracking of the contours was equal to a half of diameter of this piece. Hence, a step between "sparsed" contour points was 3.5. Contours of the exibited image contain 215 points. Fig.13

On the same picture the outcomes of work of new procedure (resulting contiguous segments) are shown.

The proposed organization of the procedure of gradient searching, sparsing of contour points and choice of the initial approximations defined on the best pieces of contour boundaries enabled to increase essentially stability of the process of the description of these boundaries in terms of contiguous segments. Necessary accounting time was sharply reduced also. Representation of a contour picture (fig.13) with contiguous segments has demanded 50 sec of computation time, and a picture of the same order of the complexity shown on fig.10 (the first series of experiments ), with the help of initial variant of this procedure - over 20 minutes.

The carried out experiments confirmed a possibility of practical realization of the concept of contiguous segments - segments with fuzzy ends.

In Appendix_2.2.D the some tactical problems of the organization of the search process of local maximums of the bell-conformity measure are considered.

2.3. THE POSSIBLE IMPROVEMENTS.

Program realization of the suggested approximation mode was intended for examination of its principal serviceability. Operational organization of the program can be improved considerably.

Further researches on size optimization of steps of gradient procedure seems to be expedient. In the considered variant the choice of their value was carried out on the base of the most common reasons.

Now for detection of false situations (in particular, at capture of several segments),the final, rather bulky property criterion of filling ratio of contiguous segments is used only. For more operative estimation of the situations arising during the bell-approximation process, a use of the information on behaviour of the gradient procedure - character and number of steps - is possible. For the same purposes it is possible to use a mutual coherence (confirmation) of the approximating segments related to the same piece of contour boundaries. There is that natural reason, that if one contiguous segment confirms by its branch the cross position and orientation of the central piece of other segment (and vice versa), then, most likely, there is the set of contour points under these approximating segments corresponding to our concept about an uniform line segment.

The applied mode of preliminary choice of contour points is deintended for work in the conditions of full absence of information on topological connections between individual elementary contours. However, as an operational experience shows for actual images, contour boundaries are traced through the help of local operators on rather extended pieces. In this connection treatment of the connected pieces of contour boundaries instead of acnodes from some area is prospective. By the way, it can facilitate approximation of close placed contours too.

Use of the Hueckel's operator for search of contour boundaries of analysed images is caused by a necessity of simplification of experiments. Most likely, the suggested approximation method will be efficient at use of more simple operators too, for example at the use of the simplified Hueckel's operator , the Sobel's operator and possibly of the Roberts simplest operator . Besides, a possibility is supposed to build-up of an operator of similar outcomes quality (and of operation principle) to the Hueckel's operator, but more fast-acting.

The basic share of computation time in the procedure of detection of contiguous segments is expended on a sequential evaluation of contributions of contour points in a flowing value of the coincidence measure gradient. The operating time of procedure can be reduced essentially if to calculate these contributions in the parallel way using a specialized computer.

All evaluations were carried out in format of numbers with floating point. There are no principal difficulties in their transformation into integer form. Thus byte (or slightly greater) representations would be enough in most cases. Obvious replacement of direct analytical evaluations by the table representation of bell-like (one-dimensional) functions and their derivatives here will allow to reduce sharply calculation time. Necessary for its products can also be determined employing tables. Though the volume of such tables is great, but it does not exceed engineering possibilities of used computing tools (for storage of these tables memory about several tens kilobyte is required).

As a whole, the increase of operational speed of the representation procedure of contour boundaries by contiguous segments on several orders is quite real.

2.4. LONGITUDINAL FIXATION OF THE CONTIGUOUS SEGMENTS.

The concept of contiguous segments was developed as a base for the following recognition-matching of contour fragments of arbitrary form as a whole. From this position, the description of contour boundaries of contiguous segments which would correspond to the pieces of minimum curvature, looks more preferably. Matching of contours on such pieces can be executed more precisely than on curved. On the other hand, such description naturally (if to start with our intuitive representations) divides contour boundaries on pieces with unit contributions to weight of the search object.

In this connection an experimental examination of serviceability of the representation procedure of contour boundaries by contiguous segments has revealed the following problem. As appeared, parabolic contiguous segments tend to run into smooth inflections of contour boundaries. The example of representation of contour boundaries when movement of approximating segments in a longitudinal direction is carried out in a direction of corresponding component of conformity measure gradient is shown on fig.14. Fig.14

And such arrival is caused not by width of memberships functions as it was in the case of root-mean-square approximation of contour boundaries by rectilinear segments. In order to prevent a possible disclosure of structure delicacies of the conformity function caused by sparsing of contour boundaries, the minimum width of memberships functions was limited to a size approximately equal to a step between sparsed contour points (D = 1.5). Experiments have shown noncriticality of the obtained outcomes to a choice of this size (trial experiments were carried out at D_min = 0.5 - 2.5).

The analysis of this situation has allowed to draw a conclusion that arrival here is defined by the form of approximating curves. Curvature of parabolas is maximum in the center and decreases to the periphery, therefore parabolas are easily entered in such pieces of contour boundaries.

In work  for restriction of the running tendency of parabolas it was offered to move aside branches of smaller curvature. However, as appeared, thus counteraction is required to slipping of contiguous segments to the ends of representation pieces at fuzzy expressed linearity maximums (fig.15). To solve this problem correctly was not possible, - it was not possible to select univalently any strategy of switching of longitudinal moving force which would enable to avoid simultaneously slipping of approximating segments and arrivals on smooth inflections. Fig.15

In last version of the procedure of approximation with contiguous segments a soft restriction of width of rectilinear corridors, which these segments can occupy, was suggested. Mathematically, such restriction expresses that at an evaluation of individual components of a gradient of the conformity measure the contour points are taken into account with weight computed under the formula Where

 d_i - deviation of a contour point from a straight line tangenting the contiguous segment in its central point, i.e. from the longitudinal axis of its own frame; U   - a half-width of a rectilinear corridor occupied by the segment (fig.16). Fig.16

The second series of the experiments described in the previous section (their outcomes are shown on fig.13) carried out with the similar count of contour points, i.e. contributions of these points to values of components of the conformity measure gradient were multiplied on their weights with the count of width of an admissible corridor. Thus for an evaluation of the component corresponding to longitudinal movement, the corridor with a half-width equal 2.5, and for all remaining components - a corridor with a half-width equal 5 was used.

Smooth (gradual) restriction of width of the rectilinear corridors occupied by contiguous segments, has allowed to avoid their explicit arrivals on smooth inflections and slipping to the ends of represantation pieces.

At the same time, here again a legible solution of the problem of longitudinal fixing of approximating segments to obtain it was not possible. At use of boundaring rectilinear corridors the separate approximating segment can lose sensitivity ends of a represented contour line. It can happen, if this line has a sufficient expansion and curvature and consequently can leave for longitudinal boundaries of the boundaring corridor. To increase width of the corridor it is impossible, - the tendency of a parabola to run into smooth inflections begins to affect. An example of such contour lines are contour boundaries of insoles corresponding to its heel (fig.13), and also to lateral faces of an iron when contours of this object are submitted full enough (fig.17). Fig.17

In similar cases the approximating segments should be fixed arbitrarily-compulsorily. The mechanism of such fixing consisted in that longitudinal movement of the approximating segment in one direction was authorized only on half of the distance between two points in which there was a change of movement direction (in the beginning this distance was set equal to a diagonal of the analysed image).

It is necessary to pay attention that width restriction of the rectilinear corridors occupied by approximating segments leads also to respective breakage of these segments (on cross corridorboundaries). In outcome the contour line which without cross restrictions could be submitted by one segment, will be submitted by several ones (see, for example, segments 4, 15 on fig.13).

Probably, to resolve the problems of longitudinal fixing of approximating segments is possible through the use of such curves (as the standards of the simplest segments) which curvature will increase to their ends and is minimum in the center (for example, curves of elliptic type). Apparently, approximation of such type is capable to give objective (in our intuitive understanding) subdivision of contour boundaries into pieces of minimum curvature, but this problem demands the further researches.

The expediency of further investigation in this direction as a whole is obvious. In particular, legible longitudinal fixing of approximating segments essentially would facilitate a solution of the subsequent problem of integral recognition-matching. Obviously, the more precisely such fixing, the less final verifications of standards of search objects with analysis pictures is required.

2.5. SLIDING OF LINEAR ELEMENTS.

Despite of the some successes, a final solution of the problem of longitudinal fixing of approximating segments to obtain it was not possible. It was not possible to fix, for example, these segments on lateral faces of an iron, and also on a heel of an insole (see fig.13). Certainly it is possible to consider, that these examples concern to some degenerated case - too smooth curves. Nevertheless, in the exhibited examples basic unsteadiness of the edge permitting to distinguish legibly a case when the contour should be represented with an integral arc and when - composite one (namely as the approximating segments should place in a true case on the contour) is shown. Depending on initial approximations, random interferences, and also not perfecteness of the form of objects themselves can obtain various representations - subdivisions - of the contour lines corresponding to identical elements of observation objects.

This situation is aggravated also with that in practice a full detection of the contour boundaries meets rather seldom. Only scraps of lines are selected. The blacked out pieces (or superlighted) are badly reflected. Some pieces can be absent basically, - owing to the possible partial overlapping by other objects. The last circumstance represents one of the most characteristic peculiarities of a solution of the problem of object recognition in actual conditions. Generally, the approximating segments are not fixed on the defined places basically.

All these circumstances induce to do a conclusion about necessity to build-up the procedures of integral pose matching in the supposition of a possibility of sliding of approximating segments on the contour boundaries of the search objects. Speculations in this occasion as the whole lead to a conclusion that the separate segment of the simplest form (with feebly marked changes of curvature), inherently, legibly defines the passing contour only in the cross direction. Often to understand finally the longitudinal fixing it is possible only proceeding from integral models of search contour figures.

Fig.18 illustrates this position for the simplest cases, particularly, - an incomplete detection of rectilinear contours (fig.18a). Fig.18a,b,c,d

Here the chosen segments correspond to an angle of some figure. This angle can slide concerning each of the chosen segments (fig.18b,c), and only the combination of such segments catches this angle legibly (fig.18d). The similar example for a curvilinear angle (iron), which for example, is fixed at least by two approximating (contiguous) segments (and by that defines these segments as the defined parts of a contour of the search object), is represented on fig.18e. Fig.18e

Naturally, at the so poorly fixing - fuzzy - primary-elementary matching of contours the solution of the integral positional matching problem represents a nontrivial problem. In this case, the separate segment generates as possible all those positions of the search object, at which its standard contour is only juxtaposed with this segment in the cross direction. To choose a search position among so extensive set generated by each segment of an observed picture is rather uneasy.

The following stage of the given work also has been devoted to a solution of this problem. At this stage the method  interpreting the positional detection problem as accumulation of hypotheses , which in the case of objects defined - fixed - forms allows to solve costructively the positional detection problem in conditions of sliding primary-elementary matching of contours has been developed.