Development and Structure of the First Etalon Dictionary of Fractal ...

PACS №: 47.53+N; 47.52+J

A.A. Potapov, E.M. Il’yin, E.P. Chigin, V.A. German Institute of Radioengineering and Electronics RAS, 11 Mokhovaya av., bild. 7, Moscow, 125009, Russia phone: (095) 2030074; e-mail: [email protected]

Development and Structure of the First Etalon Dictionary of Fractal Properties of Target Classes Contents 1. Introduction

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2. Problem Setting

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3. The Concept of ”Fractal” in Radar

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4. The Fractal Primitives and the Dynamic Text for the Etalon Dictionary of Fractal Properties Development

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5. The Investigation of Space Spectral Signatures for Objects under the Dynamic Test in the Perfect Conditions

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6. The Investigation of Fine Structure in Fractal Cepstrums of Figures at the Dynamic Test

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7. The Computer Estimations for Influence of Geometrical Parameters of Figures under the Dynamic Test in Fractal Recognition of Extensive Rarely Noticeable Objects

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8. The Investigation of Fractal Signature for Dynamic Test Objects under Quality Variation and Noise Absence

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9. The Illustrating Examples of Digital Fractal Generalized Filtering of Targets of Small Contrast

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10.Conclusion

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Abstract The results of development the first etalon dictionary of fractal properties are presented. The investigations are surveyed within the framework of new scientific direction ”Fractal Radiolocation and Fractal Radiophysics”. The perspectives of creation new information technologies offered and developed in IRE RAS.

1.

Introduction

the hardly noticeable and low contrast targets against the land covers it was shown experimentally that the traditional classical algorithms of detection are ineffective and require the large time consumption.

The problem under consideration is of the great scientific and practical interest in the different fields of the radiophysics, electronics and informational control systems. In the current time due to elaboration of the remedy for reduction of the radar and optical visibility of targets and the intensive stealthtechnology development the priority of investigations goes to the new methods of detection and separation of low signals over and under noise level. When detecting

During some years starting at eighties of XX century a part of the scientific investigations in IRE RAS is aimed at the search of the nontraditional algorithms for the processing of the radar signal and images keeping in mind the low-contrast objects detection. It was investigated that one of 107

A.A. Potapov, E.M. Il’yin, E.P. Chigin, V.A. German

the new breaking technology is the development of the textured and fractal methods of the low-contrast processing. The information processing using the textural measures had been mostly presented in [1] and is not considered here. It should be mentioned that one of the most important questions in the fractal geometry now is the correlation of the fractal and textural measures [88]. When composing this work our own publications with our vision of the problem under discussion [1– 124] had been used considerably. It should be noted that in the IRE RAS publications (in contrast to foreign publications) the idea of the fractal processing of the low- contrast images had been used from the very beginning including the fractals on the land fractal background [1–3]. Basing on these investigations the first author has put into operation the idea of the textural and fractal signature, and this scientific direction has been named shortly as the ”Fractal Radiolocation and Fractal Radiophysics” [4– 6,124]. It should be remind that the fractal theory considers the fractional measures instead of integer and is based on the new numerical indexes in a form of the fractional dimension D and the corresponding fractal signatures. Summarizing all mentioned above it should be noted that the conception of the ensemble indications based on the fractional measurement theory and scaling has been introduced into the practical problems of the radar systems for the first time in the publications of IRE RAS. More of it the numerical data for the dynamic test objects has been determined and their availability under the generalized filtering of the low-contrast targets has been demonstrated.

2.

Problem Setting

The general setting of the problem with the radar data processing in our case is based upon the usage of the fractal methods in form recognition. The whole number of the specific aims must be solved at the same time. The question about the etalon dictionary containing the fractal properties of target classes is always central. The following stages are supposed under the classical formalism of the problem [1,3– 5,15,80,87,91,97,99,106,121,124]: 1. the initial classification of targets or their classes a priori, i.e. the construction of the target class alphabet; 2. the determination of the required list of the properties characterizing targets (in our case we are dealing with the fractal properties only); 3. the development of the etalon dictionary of the fractal properties of target or target classes; 108

Fig. 1. Classification and Morphology of Fractal Sets (Signatures). 4. the description of the alphabet for target classes using the language of the fractal property ensemble of etalon dictionary or their combinations; 5. the subdivision of the space of the fractal properties on the regions corresponding to the initial classes of the alphabet; 6. the choice of the metric (deciding rule) or recognition algorithms providing the reference of the target under recognition to one or other target classes. The idea of the fractal, which was introduced into scientific circulation by B. Mandelbrot, has no strict definition for the time being. The fractal classification developed on results [4] and essential for our investigations has been published in [88,98,99,121,124] and is shown in Fig. 1 accordingly. Their attributes are briefly described there with the condition that D0 is the topological space dimension where the fractal with the fractional dimension D is considered. The fractals existing in the real world are called physical and have the limited scale interval. Coming from the data structure at Fig. 1 the following formalized mathematical definition for the fractal had been given in [88].”The fractal is the functional mapping or the set which can be received by the infinite recursive process and which has the following properties: 1. self-similarity or the scale invariance (infinite scaling), i.e. fractals on small scales on the average are looking similar to those on large scales; 2. their dimension (so called Hausdorf’s dimension) is fractional and strictly more than the topological dimension; Электромагнитные Явления, Т.5, №2 (15), 2005 г.

Development and Structure of the First Etalon Dictionary of Fractal Properties of Target Classes

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e) Fig. 2. Illustration of fractal processing (a–d) and RI fractal cepstrum (e). 3. undifferentiation and dealing with the fractional derivatives and integrals”.

The physical definition of the fractal is similar to the definitions introduced by B. Mandelbrot and says [88]: ”The fractals are represented by geometrical objects (lines, surfaces, bodies) with the strongly indented structure and having the attribute of the selfsimilarity in the limited scale”.

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3.

The Concept of ”Fractal” in Radar

The radar image in general form can be represented as the set of elements Xk whose values are proportional to the effective scattering surface (ESS) of the k-th element of radar resolution. Fig. 2a shows the radar image of an area at wavelength 8.6 mm from the helicopter [1–3]. Next to it the radar image of the same area recorded at wavelength 109

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about 30 cm is presented on Fig. 2b. Both images are two- dimensional with the gray level proportional to ESS. Let us suppose the surface (Fig. 2c) has been constructed for each radar image and the heights h are proportional to the gray levels too. Let us assume that the square measurement of the resulted surface is required. For the image corresponding to wavelength 30 cm the square will be less than for the image taken with wavelength 8.6 mm because with the wavelength decreasing the more details of a landscape can be resolved. The sounding electromagnetic wave is similar in such case to the some scale rule. The more and more fine structure of the time-spaced signals or wave fields starts to give notice [1,3,4]. If the radar with the shorter wavelength is available than the image surface will be more and so on. The wavelength decreasing results in the surface square increasing. Then the question arises: what is the real surface square, which was illuminated by the radar? If the surface is covered by the simple objects (rectangle rise, for example) shown at Fig. 2d and its size much more than wavelength, then the radar image surfaces are more or less equal for short and long wavelengths. So one can reply to the question above by calculation the number of resolution elements covering the object. In this case the surface square S will be equal to S ≡ S(λ) = N (λ)δ(λ), (1) where N (λ) is the number of resolution elements required for the object covering; δ(λ) is the resolution element square; λ is the wavelength of the radar. As was mentioned before const. for the simple object (Fig. 2d). For the radar images on Fig. 2a and 2b one can create the dependence S(λ) = f (λ) and with assumption δ(λ) = K(λ) where K is the known function one can get after that the dependence S(λ) = f (δ). It turned out that the measured surface square S is fairly good described by the expression S(λ) = kλ−D .

(2)

So ln(S(λ)) = ln k − D ln λ (k and D are constants) and we can calculate the D value. The dependence ln S(λ) = f (ln δ) fixing the fractal signature of a radar image on Fig. 2a and Fig. 2b is shown on Fig. 2d. It characterizes the space fractal cepstrums of the image (this concept was introduced in 1997 by the first of authors [4–6]). The fractural parameter D is called the Hausdorf-Bezikovitch dimension or the fractal dimension [4]. For the radar images of objects having the simple geometrical shape (rectangulars, circles, smooth curves) this dimension coincides with the topological one, i.e. equal 2 for two-dimensional radar images, and is determined by the angular coefficient between the straight lines (2) in logarithmic scale. However the D values turned out to be more than the 110

topological dimension 2 for the most of the real cover images and meteorological formations. It underlines their complexity and chaotic character. The fractal fractural dimensions and signatures D not only characterize the object topology, but reflect the evolution processes in dynamic systems and are connected with their properties. The fractal theory and non-linearity compose the geometrical chaos. Apparently the fractals are not only the language of the chaos but the language of the nature. By their content the contours of all natural objects are the essence of dynamic processes suddenly coagulates in physical forms thereby uniting the stability and chaos. The software with the algorithms for the fractal processing of data, images and waveforms is developed in IRE RAS, is improving constantly and is presented in [121]. It is important to mark the fact that from the very beginning the developed software package has been directed toward the global problems of recognition the fractal over the fractal according to the value of the fractural dimension D, fractal signature ensemble and the other introduced fractal properties.

4.

The Fractal Primitives and the Dynamic Text for the Etalon Dictionary of Fractal Properties Development

When creating the first etalon dictionary of fractal properties, the latter had been selected by authors from [4,106,124]: 1. the value of the fractal (fractural) topological dimension D; 2. the form of the fractal signature or the fractal cepstrum; 3. the form of the space spectrum and the value of the space frequencies which characterize the image texture. The etalon and really used in practice the working dictionaries of the radar image properties recognition are always utilizing the some signature forms. These signatures are traditionally involving the spatial, temporal, spectral and polarization structures of reflected signals as well as practically any characteristics of reflected signals not connected directly with the coordinates of the radar objects. The final decision about the working dictionary content can not be accepted out of the object classification problem when they are divided on classes (alphabet of classes). The problems of finding the alphabet of classes and the dictionary of properties are directly linked with the implementation of that or Электромагнитные Явления, Т.5, №2 (15), 2005 г.

Development and Structure of the First Etalon Dictionary of Fractal Properties of Target Classes

other decisive regulations: selection and recognition algorithms. The properties based on the fractional dimension theory and the physical scaling had been introduced by us for the first time. The methods developed in IRE RAS let to overcome partially the a priory uncertainty using the information about the geometry (topology) of sampling [80,91,97,106,121,124]. Is it one-dimensional or multidimensional, i.e. in cases of a signal or a image. It should be noted especially that the developed fractal (topological) methods form the independent investigation region and not directly connected nor with the classical probability distribution in the mathematical statistics, nor with the classical theory of ejections, nor with the questions of the statistical topography of random processes and fields. At the same time the topological peculiarities of each specific individual sampling take on the special significance in contrast to averaged realizations having the totally different feature. To take into account especially these peculiarities the term the ”dimensional sclerosis” of the physical signals and signatures [91] had been brought in specially. It should be noted that when describing the non - Markoff processes the special term the ”asymptotical sclerosis” is adopted in the literature to explain the physical sense of the fractional derivatives. In general case it is always possible to talk about the filtration of one-dimensional, two-dimensional,. . . , multi-dimensional signals. So we have to consider the hypothesis to the effect that all typical signatures (properties) must be considered as the components of some generalized spectrum. In this case we can quite easily suggest the new term the ”dictionary of fractal properties” and start with its creation. If under syntax or structural recognition one has to investigate the object structure, its hierarchy and their bonds, under fractal recognition one has to investigate the topology of the object and background, represented in received radar signals both onedimensional and multi-dimensional. Using the fractal approach it is necessary to search, incarnate and use the regulations to which the fractional (complex) topology of the objects under consideration obey. Hence it is absolutely necessary to include into the recognition process the information about the topology or the object structure and then to establish which elements — subimages can be picked out from the object, how much they are elementary and what relations are between them. At the same time some fractal properties can get the different weights indicating the measure of their significance. The idea to detect the regular structural elements inside the recognized images is more productive. Then in a process of recognition (filtering) it is possible to operate with the limited number of primitives and the limited number of rearrangement regulations. Electromagnetic Phenomena, V.5, №2 (15), 2005

When realizing in practice the fractal recognition method we can use the analogy between the creation of a fractal image using the different rearrangement of sub images (or the fractal sub images) and the creation of a sentence in natural language using the words composed of letters. If that’s a case we shall single out the ”fractal primitives” which are the elements of the ”fractal language”. Inevitably the question about the structure of the fractal primitives is arising. The fractal primitives are the fractal symbols which are the least elements of the fractal language. The ensemble of employed fractal symbols we shall name as ”fractal alphabet” or ”fractal dictionary” and identify by the symbol Φ. Using this base one may compose the ”fractal lines”. The fractal line is the finite consequence of symbols composing the alphabet. The line length can be anyone. All possible lines of the fractal alphabet Φ produce the universal multitude of lines or closing Φ. If the multitude of empty lines is denoted as λ, the finite or countable infinite subset of closing fractal alphabet ? is the more precise definition for the concept ”fractal language”. The separate fractal lines composed from the fractal symbols we shall name as ”fractal words”. Later performing some logical operations over the fractal language it is possible to create the new language. The regulations of creating, reforming and interaction of fractal words will be clarified by ”the fractal grammar”. For its composing it is possible to use ideas and methods for the formal grammar developed in the mathematical linguistics. The fractal grammar deduction basing on the given ensemble of teaching objects is essential in practice for its forming. This treatment is in much the same to the problem of teaching in the different methods of recognition. The algorithms of the fractal recognition of appearance are based on utilization of the paradigm ”target topology is formed by its fractal dimension” [4,124]. The space of determined or probability properties a priori can be founded using the dynamic test. The selection and the preparation of the test materials for the experimental verification of the fractal recognition methods and the check-up of principle of the algorithm creation seriously affects on the investigation results reliability. The classical methods of the statistical solution theory are of little use for the radar target recognition in a great deal of problems arising in practice. It is due to the following: strict limitations on the time of analysis and the carrying capacity of information channels; the high level of uncertainty a priori; different noise interference, the large variety of targets features united in the one class, the unknown targets orientation; simultaneous presence of several targets with the different size and orientation. 111

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Fig. 3. Chinese ”Tangram” and 16 figures composed of seven tangram elements.

The best test stuff for the investigation of the specific problems of recognition is the set of the analyzable and etalon images, which can really present the objects corresponding to the given aim of recognition. However, the peculiarities of the each image form in the different recognition aims make difficulties for tracing the general regularities in the recognition process. Therefore the question about the using of the universal test stuff is arising. The space of determined or probability properties a priori can be founded using the dynamic test. The best test stuff for the investigation of the specific problems of recognition is the set of the images, corresponding to the real targets. As fractal primitives the set of figures from ”Tangram” has been used for investigations of the recognition process at the image processing. By the way, the authors are working with this set constantly. In the computer simulations the 16 ”Tangram” figures have been used, representing polygons, artificial construction silhouettes, airplane, ship, human and animals. All figures have the equal area but the number of angles and borders is different as shown on Fig. 3. The ”Tangram” itself consists of seven elements directly (called ”tans”) as a result of cutting the square by the special manner. The cutting diagram is shown at the left upper part on Fig. 3. The possibility of composition the test figures with the different number of angles N and borders M (from 3 to 23) and the different degree of complexity produces convenience in the test image creation close to real. The results of the preliminary ”rough” clasterization are listed in [4,98,99,121,124]. They are approximated by the regressional curves D = f (N, M ) for artificial construction silhouettes and biological objects, where 2 < D < 3. Going back to the general regulations for the elaboration of the dictionary of properties using structural descriptions we should note the following. 112

For the two formal grammars mentioned above the two peculiarities are typical. Firstly, they describe the whole possible results only without the direct indications how to achieve the result for the definite aim. Secondly, all statements in them are formulating in terms of small number well-defined and elementary symbols and operations only. Therefore all formal grammars are simple in respect to their logical formation. The formal grammar may be determined [91] by the generative grammar, i.e. system G =< Ct , Cn , P, A >, which consist of four parts: terminal (basic) vocabulary Ct , non-terminal (auxiliary) vocabulary Cn , ensemble of substitution regulations P , initial symbol or initial axiom A (A ∈ Cn ). Terminal (basic) vocabulary Ct is the set of initial terminal elements or properties, from which the chains generated by the grammar can be constructed. Non-terminal (auxiliary) vocabulary Cn is the set of symbols, which are using for identification of classes of initial elements or initial element chains, as well as some special non-terminal or auxiliary elements. Initial symbol A is the detailed non-terminal symbol, indicating the totality or the class for those lingual objects where their description is made by the given grammar (for example, in the grammar generating sentences the initial symbol is the symbol meaning the sentence and so on). Ensemble of substitution regulations P is the finite set of regulations in a form ϕ → ψ, where ϕ and ψ are the words in dictionary (alphabet) Cn ∪ Ct and ”→” is the symbol not owned by Cn ∪ Ct . For the formal grammar the following relationships are typical: C = Cn ∪ Ct ,

Cn ∩ Ct = 0,

(3)

where C is the dictionary. The process of the language creation starts from the axiom A to which the substitution regulations are applied one after one. As an operation on expressions to get the new ones the conjunction, alternation and negation are used.

5.

The Investigation of Space Spectral Signatures for Objects under the Dynamic Test in the Perfect Conditions

When designing the etalon dictionary of fractal properties for the formation and realization of the fractal detector for the low-contrast targets it was necessary first of all to run the computer experiments investigating the influence of the geometrical parameters for all space figures in dynamic test and their topology. It is aimed to Электромагнитные Явления, Т.5, №2 (15), 2005 г.

Development and Structure of the First Etalon Dictionary of Fractal Properties of Target Classes

the problems of the fractal recognition for the extensive barely noticeable objects. In the experiments mentioned below all ”tans” (from 1-st to 16-th) from Fig. 3 had been investigated in detail for the first time. The results of our computer experiments are produced on Fig. 4 at 100 % image contrast. The unit for the space frequencies measurements F is the value reversed to number of pixels, i.e. [F ] ≡ [pixel−1 ] ≡ [px−1 ]. In that way the invariance to the sizes of the tans image processed is achieved. The ratio of the target area S0 to the scene area S is equal to S0 /S = 25−1 . The space frequency range on Fig. 4 is F = −30 . . . 30 px−1 . The values for the fractal dimensions D averaged along the scene are shown in the working window (Fig. 4) for the fractal space cepstrums. The size values of the current measuring window are listed there too.

6.

The Investigation of Fine Structure in Fractal Cepstrums of Figures at the Dynamic Test

The computer experiments which had been carried out for the first time permit us to compare the fractal features of tans with different configuration. Some specific results are listed below. In the future they will be considered during the design of the fractal non- parametrical detector of radar signals (FNDRS) as well as in the solution of the problems of the fractal recognition for the extensive barely noticeable targets. One can see from Fig. 4 that the fractal space cepstrums for all tans have the quite outstanding individual form. At the same time using the careful comparison of all experimental results we manage to discover the row of regularities. To do this the inflection point in the fractal signature, which corresponds to the appointed size of the measuring window δ, was assigned visually at first. After that the tans with practically the same coordinates of inflection points in the fractal signatures have been compared between each other. As a result some basic deductions about the fine structure in the fractal cepstrums were made (further D(i) stands for the value of the mean fractal dimension of i-th tans), namely: 1. The fractal cepstrums of 5, 6 and 13 tans have the inflection point around the 8-th scale window. In cepstrums of 5 and 13 tans the amplitudes ln S(δ) up to 8-th scale window inclusive exceed the corresponding amplitude for 6 tan. After that when increasing the size of the scale window δ the inversed effect takes place. 2. At the same time having the same cepstrums amplitudes up to 8-th scaling window for 5 and 13 Electromagnetic Phenomena, V.5, №2 (15), 2005

tans, the high frequency cepstrums regions have the absolutely different shape: the convexity up for the fractal cepstrum of 13-th tan and the convexity down for the fractal cepstrum of 5th tan. The more sharp slump of the cepstrums amplitudes after the 8-th scaling window is typical for the fractal cepstrum of 5-th tan. It affects the values of the mean fractal dimension D(5) > D(6) > D(13) . 3. The fractal cepstrums of 3, 9, 12 and 14 tans have the inflection point around the 7-th scale window. The fractal cepstrums of 3 and 14 tans approximating the human body images are close to each other and have the smoothly enough slump. The 9 and 12 tans have about the same behavior between the cepstrums amplitude in low-frequency cepstrums region, and after that the sharp slump in the high-frequency region (especially for the fractal cepstrum of 9 tan) is taking place. These quality differences appear directly in the values of the mean fractal dimension D(9) > D(12) > D(14) > D(3) . 4. The fractal cepstrums of 7 and 15 tans have the barely noticeable inflection point around the 10th scale window. At the same time the slump behavior of the fractal cepstrum amplitude has a pulsate nature. As a result of smooth change of the fractal cepstrum amplitude the mean fractal dimension of these tans are minimal in the whole measured range and comply with the relation D(15) > D(7) . 5. The fractal cepstrums of 1,8 and 9 tans have the inflection point around the 9-th scale window and are quite similar to the 6-th tan behavior. At the same time the general form for fractal cepstrums of 1 and 8 tans have about the same behavior. The mean fractal dimension of these tans comply with the relation D(6) > D(1) > D(8) . 6. The fractal cepstrums of 4,10 and 11 tans have the inflection point around the 12-th scale window. The sharp slump in the high-frequency region for the cepstrum amplitudes of 10-th tan causes the substantial difference in values of the mean fractal dimension D(10) > D(4) > D(11) . 7. The fractal cepstrum of 16 tan has the inflection point around the 13-th scale window. The behavior of the fractal cepstrum of 16 tan is quite similar to the 10-th tan behavior. The measurements have shown D(10) > D(16) . 8. The fractal cepstrum of 2 tan has the inflection point around the 5-th scale window and abruptly differs from the other tans. The fractal cepstrums amplitudes are constant up to the 4-th scale window inclusive and decrease linearly after that. 113

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9. The following comparison of the received tan classes discovered one more peculiarity for the fractal cepstrums of 5,8,10 and 16 tans. Immediately after the inflection point the spasmodic drop of the fractal cepstrums amplitude is taking place. The drop value of the cepstrums amplitude is equal for all tans under consideration. At the same time we have D(5) > D(10) > D(16) > D(8) . 10. Tans 1,2,3,5,6,9,12 and 14 have the fractal cepstrums, amplitudes of which are goes to plateau in case the measuring windows δ have the maximal sizes. At the same time the fractal cepstrums for all tans mentioned above (except 5-th tan) are described by the down concavity of their amplitude after the inflection point. 11. After analysis of experimental data the ranks of the mean fractal dimensions D for all tans have been determined: D(9) > D(5) > D(6) > D(12) > D(10) > D(14) > D(1) > D(16) > D(8) > D(3) > D(13) > D(2) > D(4) > D(11) > D(15) > D(7) . (4) The numerical values of the mean fractal dimensions for all tans under the dynamic test are shown in Table 1.

7.

The Computer Estimations for Influence of Geometrical Parameters of Figures under the Dynamic Test in Fractal Recognition of Extensive Rarely Noticeable Objects

Beneath for the first time one can find the results of the computer investigations based on our experimental data. We are dealing with the space fractal cepstrums in the scene when changing the number of identical tans. The results for two identical figures are presented on Fig. 5 whereas Fig. 6 stands for three identical figures. There is visual resemblance between the space fractal cepstrums for the single and double figures basing on results of our experimental data analysis. All conclusions made before are suitable for the case of the fractal analysis of the scene with two figures. The numerical values of the mean fractal dimensions for the scenes with two of all tans under the dynamic test are shown in Table 2. There is visual resemblance between the space fractal cepstrums for the single, double and triple figures basing on results of our experimental data 114

analysis. All conclusions made before are suitable for the case of the fractal analysis of the scene with three figures. The numerical values of the mean fractal dimensions for the scenes with three of all tans under the dynamic test are shown in Table 3. The anomalous values of the mean fractal dimension D (when D for two or three objects is less then D value for a single object) can be explained by the ”tracking in” of two tans with each other al low distance between them. By other words, at the measuring process two or three tans are imposed upon each other (smoothing procedure). The data from Tables 1–3 permit us to make the conclusion about the sufficient laboriousness in the direct using of the point estimation for the mean fractal dimension D trying to define the number of identical objects at the low ratio between the objects area and the scene area. The procedure of the fractal signature, which is developed and presented in detail below, will be optimal.

8.

The Investigation of Fractal Signature for Dynamic Test Objects under Quality Variation and Noise Absence

The next important step to work out the reference dictionary of the fractal features for the construction and realization of the fractal detector of small contrast targets was the computer experiment on the extensive investigation of fractal signatures and all the dynamic test space figures (thanes) under their amount variation. The results of the numerical experiments carried out first are shown in Fig. 7 bellow in the form of the corresponding fractal signatures ln S = f (δ), where δ is the metering window dimensionality. The mutual contrast of the fractal signature measurement results for each thane partial ensemble (one, two and three ilk figures) allowed to make the following conclusions: 1. The whole thane ensemble of 1–16 dynamical test easily breaks into 5 clusters by the character form of fractal signature variation. Tans 4, 7, 11, 13 can be collected into the first class. At the second one are only two tans 2, 12 (witch are characterizing the aircraft silhouette). At the third cluster 3, 6, 14 tans are got. At the 4-th cluster 1,5,8,9 clusters are got. At the 5-th cluster, the 10-th, 15-th, 16-th tans are got. 2. Clusters segregation has done with respect to the form of fractal signature. The ideally fluent variance of the logarithm of response (amplitude) ln S on the window dimension δ is observed in the first cluster. For the second cluster consisted

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Fig. 4. The fractal space cepstrum (in the middle) and the spectra of the space frequencies (to the right) of tans № 1–16.

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Fig. 5. The space fractal cepstrums for two identical figures of each tan.

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9

10

11

12

13

14

15

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Fig. 7. The measurements of the fractal signatures for all tans under the dynamic test in perfect conditions in the presence of one, two and three targets.  – one figure;  – two figure;  – three figure.

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a)

b)

c)

d)

e)  – S(d);  – dS(d). Fig. 8. The image (a) of the airplane B-2 (Steрlt) and its fractal processing: (b) – the resulting empirical distribution of the fractal dimensionalities, (c) – detailed contour of target, (d) – idealized contour of target, (e) – target cepstrum and its derivation.

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Δ ≈ 0.21 (the thane 6) and Δ ≈ 0.13 (the thane 14) are obtained under the large window dimensions. 6. Two slope ranges also is characterized the fourth cluster. The values Δ ≈ 0.10 and Δ ≈ 0.15 . . . 0.20 are obtained for the thanes 5,8 and 9. The specific form of the fractal signature of the thane 1 (with Δ ≈ 0.26 under window dimension δ = 1 . . . 10 and Δ ≈ 0.80 in the case δ = 10 . . . 15) is determined by the shift of the fraction point δ0 in the pixel neighborhood δ = 10. Therefore, thane 1 is like bridge between the fourth and fifth clusters. Fig. 9. Empirical distributions of local fractal dimensionalities for the image on Fig. 8,a. of airplane figures only, the amplitude decreasing ln S in dependence on window dimension is of undulate character. Withal there is a plateau in the neighborhood of little dimension windows δ: δ = 1, . . . , 4 for the thane 2 and δ = 1, . . . , 5 for the thane 12. The third cluster concentrates the thanes with fractal signatures having linear decline of the amplitude on large and small windows together with their fluent coupling in the middle window dimension. The thanes of the fourth cluster are characterized by sharp fraction in the neighborhood of only one window dimension δ0 . In the fifth cluster, this fraction point (δ0 ) shifts to the large window dimension δ. Withal this fractal signature of the thane 15 is like transitional one between fractal signature of the fourth and fifth clusters. 3. The first cluster concentrates the thanes with fractal signatures characterized by a single slope value Δ. The value Δ is in the range Δ ≈ 0.05 . . . 0.2. 4. For the second cluster (airplane) entries, one can educe three specific sectors of equal slope of fractal signature. Under small window δ = 1, . . . , 4 (the thane 2) and δ = 1, . . . , 5 (the thane 12) the slope is equal to Δ ≈ 0.02 and Δ ≈ 0.04, respectively. It is followed by a middle sector of window dimensions δ = 5 . . . 7 (Δ ≈ 0.5) and δ = 6 . . . 10 (Δ ≈ 0.4) for the thanes 2 and 12, respectively. The slopes have the values Δ ≈ 0.15 (the thane 2) and Δ ≈ 0.14 (the thane 12) on the large window dimension δ = 8 . . . 15 (the thane 2) and δ = 11 . . . 15 (the thane 12). 5. In the third cluster, the fractal signatures have two ranges of slopes. We obtain Δ ≈ 0.35 (the thane 3), Δ ≈ 0.13 (the thane 6) and Δ ≈ 0.20 (the thane 14) under the small window dimensions. The values Δ ≈ 0.12 (the thane 3), Electromagnetic Phenomena, V.5, №2 (15), 2005

7. For the fifth cluster fractal signatures, the slope values Δ ≈ 0.10 . . . 0.15 are observed in large range of window dimensions δ = 1 . . . 12, and after a jump, we have Δ ≈ 0.05 . . . 0.10 on the largest window. 8. All the fractal signatures of the dynamic test allow estimating immediately the number of one type figures in the image processed. The amplitude always increases (in units of ln S) whenever the number of the same figures enlarges. Withal the common form of the fractal signatures does not change. This essential fact experimentally stated allow to bring in the FPDRS future structure the additional capabilities of its adaptation to the radiosituation (abstracting from ”local things”, false ”bright point” or, conversely, the selection of the same type objects, etc.). The Table 4 contains all the values of measured slopes for the fractal signatures of each thane. Here Δ is the signature specific slope in general, Δ is the signature specific slope under small window dimension, Δ is the signature specific slope under the middle window dimension, Δ is the signature specific slope under large window dimension, the line ”—” denotes the absence of this specific parameter for signature.

9.

The Illustrating Examples of Digital Fractal Generalized Filtering of Targets of Small Contrast

The fractal methods considered above were applied to the generalized filtering (processing) of two-dimensional small contrast object images. Some of examples of fractal prosessing of small contrast images were presented earlier in [98,99,104,105,108,110,121,123,124]. All the images of extended targets subjected to fractal 127

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processing are optical with their artificial noising, because there is no such images in the radiowaives diapason. Fig. 8a shows the airplane image B-2 (Steаlth). The measurements of local fractal dimensionalities are made with 4 × 4 pixels of the scale window size and 4 × 4 pixels of metering window. The results are presented on Fig. 8b–8e. Fig 8b shows the form of the obtained normalized empirical fractal distribution W (D)/Wmax (D). Selecting the different sectors on the 128

obtained distribution (parethian), we get the images filtered by the image value D (Fig. 8c). Recently there are not exact relations for determination of the optimal fractal dimensional D diapason. But it is known that this interval is set on the front of the base dependence mode W (D)/Wmax (D), i.e. in this case D ∈ {2.03 . . . 2.50} (Fig. 8b). The values D, which correspond the borders of image (sample) under process, are set behind, after there are the dimensions corresponding the noises, i.e. Электромагнитные Явления, Т.5, №2 (15), 2005 г.

Development and Structure of the First Etalon Dictionary of Fractal Properties of Target Classes

the image small chaotic entries. The obtained filtered image allows constructing the target contour idealized image. On this stage this construction is manually made. The obtained ”idealized” contour (Fig. 8d) is necessary for its cepstrum storage (Fig. 8e). The detection of given target class can be made with fractal cepstrum obtained experimentally. The detection can also be made with fractal cepstrum of dynamic test ”Tangram”. Cepstrum S(D) = f (δ) shown on Fig. 8e is ∂S(D) = f (δ). presented together with its derivation ∂δ In some applications the operation with derivation is more convenience than with fractal cepstrum, for example, in the case of classification of group target. The question on the determination of the empirical distribution sector, which the filtration will be made on, can be solve with the measurement of local fractal dimensions under different scales. The question is that, under the slight smoothing of the initial image, all the small inhomogeneity (interference) merge with large ones and make the significant contribution in obtainind distribution. Otherwise, if the smoothing is strong (e.g., in 10–15 times), the large inhomogeneities appear as separate ”fractals”, and their contribution in distribution is observed on the certain diapason D.

Electromagnetic Phenomena, V.5, №2 (15), 2005

For example, Fig. 9 simultaneously shows three empirical distributions W (D)/Wmax (D) which differ by the parameters of scale and metering windows (the first digit of the diagram right side means scale window, the second one means the metering window). The difference between the scales 15 and 4 is under interest. Under the scale 15, we obtain the sharp decrease of the mean values of dimensionality D. It is explained by decreasing of the background local dimensions contribution under the strong smoothing. That is why one can suppose, that target are on the crossing of these distributions. In Fig. 8 case, the fractal distributions cross in the approximate diapason of dimensionalities 2.03–2.4 (see Fig. 9). This diapason corresponds to the best quality of filtering obtained in experiments (see Fig. 8c). During the experiments it was specified that there is distributed noise with large threshold of dimensionalities in some images. Moreover, this noise was not localized and its presence impairs notably the filtering quality. It was stated that this noise is determined by insufficient resolution of initial images. In many cases, the small quadrate (dimension is up to 5 × 5) were instead of separate points. Their borders formed the steps giving the large diapason of the 129

A.A. Potapov, E.M. Il’yin, E.P. Chigin, V.A. German

a)

b)

c)

d)

Fig. 10. The usage of preliminary noising for the fractal processing of the small contrast images. fractal dimensionalities. Moreover, it was not only under the slight rustication but also the strong one, when these steps performed as periodical structures. The enforcement noising of the initial image was applied for this effect relaxation. It was assumed that high frequency noise of large intensity would destroy the borders of the image small entries. This assumption was proved out. The Fig. 10a shows the initial image the same as on Fig. 8a after noising it by the Gaussian noise with mean square deviation σ ≈ 0.5 to image mean brightness. Fig. 10c shows the result of filtering. In comparison with Fig. 8c, the quality jump is seen. Not only target contours are seen but the target elements such as blowing withdrawer, muzzles, cockpit, rudder. Withal the total amount of interference on filtered image is reduced. Also the insensibility of described filtering algorithm to image contrast should be pointed out. The Fig. 10b 130

shows the initial image, the contrast of which is reduced in 5 times. The processing result (see Fig. 10d) does not differ from the result for the contract image on the Fig. 10c. The result is not changed after the more contrast decreasing, up to the limit when the difference between target brightness and background do not approach to the system resolution. The results of generalized fractal filtration for the military engineering images are presented on Fig. 11– 16. Fig. 11 shows the results of image processing of the fighting machine Su-34 start. The contours were marked out by the heuristic estimates of the resulting distribution of the local fractal image dimensionalities. The obtained results did no allow clustering the image. Nevertheless the target contours can be marked out even in the conditions of noising and low contrast. The airplane muzzle and pulpit can be seen on the resulted images. The analogous method was applied Электромагнитные Явления, Т.5, №2 (15), 2005 г.

Development and Structure of the First Etalon Dictionary of Fractal Properties of Target Classes

a)

b)

c)

d)

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f)

Fig. 11. The image of the airplane Su-34 start, a) – the original image, b) – The result of program processing on the local fractal dimensionalities calculation (the form of obtained empirical distribution D is places on the left), c)–f) the fractal filtering results under different diapasons D.

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Fig. 12. The image of submarine ”Komanch”, a) – the original image, b) – the result of program processing on local fractal dimensionalities calculation (the form of obtained empirical distribution D is places on the left), c)–f) the fractal filtering results under different diapasons.

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a)

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e)

f)

Fig. 13. The result of digital fractal processing of the attacking helicopter image: a) – the original image, b) – the form of obtained empirical distribution D, c)– f) – the results of fractal filtering under the different values of fractal dimension D.

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a)

b)

c) Fig. 14. The results of fractal processing of tank M-1 ”Abrams” image obtained on the inhomogeneous of empirical distribution D: a) – the original image, b)–c) – the results of fractal filtering under the different values of fractal dimension D. for submarine ”Komanch” image (Fig. 12). It should be pointed out that the initial image has very low quality (the low resolution) and low contrast that is about 20 %. Though, as in the previous case, it was answered to educe the submarine specific elements.

Thus, the heuristic approach application for filtering can be used in the problems of analysis and distinction on the bounded set of features (availability of muzzles, hatches, etc.). If the homogenous extended domains exist on the images, their effective clustering is possible by the fractal filtering. Bellow there is the example of attacking helicopter image (Fig. 13). The some domain classes can be educed on the original image, that is ”helicopter”, ”rocket+gas cloud”, ”underlying surface”.

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10.

Conclusion

The carried out investigations showed the prospect of fractal analysis appliance (generalized filtering, in general) for the detection and distinction of radar signals from small contrast targets on the background of heterogeneous intensive disturbances. It was first showed that under the fractal features reference dictionary (or data bank) these features are necessary to be the following: a) the value of fractal topological dimensionality D, b) the form of fractal signatures or fractal cepstrums, c) the form of space spectrum and the values of space frequencies characterizing the image texture. The fractal cepstrum is convenient topological invariant on the one hand namely it does not require the preliminary orienting/scaling. On the second hand, it is the image contrast insensitive. For example, the positions of specific points on fractal cepstrums Электромагнитные Явления, Т.5, №2 (15), 2005 г.

Development and Structure of the First Etalon Dictionary of Fractal Properties of Target Classes

a)

b)

c)

d)

e) Fig. 15. The result of digital fractal processing of the ballistic rocket start image: a) – the original image, b) – the form of obtained empirical distribution D, c)–e) – the results of fractal filtering under the different values of fractal dimension D.

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a)

b)

c)

d) Fig. 16. The rocket start image: (a) – idealized, (b) – in noise, (c) – arbitrary ”star” like image, (d) – consolidated cepstrums of the ballistic rocket start image: 1)  – the cepstrum of reference image (without noise and deformations); 2)  – the real image cepstrum under 3 dB noise; 3)  – the real image cepstrum under geometrical deformations (distention along the one axis); 4) Big  the arbitrary image cepstrum of ”star” type (for the comparison). allow determining by some test the target class, its dimensions and the value of targets. The relative position change allows solving the problem of defined target determination even under low contrast (the minimal contrast is defined by the dynamical diapason of the primary processing device and the properties of analog - digital transformer of this device). The reference and real practical used features dictionaries always use some form of signatures under the radar distinction. The fractal and texture signatures are out in radar practice. It is better to use the stylized set of figures ”Tangram” as fractal primitives for investigation of the image distinction problems. Traditionally authors use the value signal/noise q02 expressed by the quotient of the mean brightness of complicated image (background + target) on the mean square of Gaussian noise. 136

As earlier, it pointed out the important role of fractal distributions (parethians) and their significance for the practical radar problems including the designing of the scheme of fractal parametric detector of radar signals – FPDRS). The tail nodosity of probabilistic distribution of instant fractal dimensionalities of the analyzed signal happens because of serial law influence. In all cases ”Noded Tails” (Heavy Tails or Fat Tails) of the fractal distribution are the result of the feedback increasing the events. The experiment result completely define the specific form of the fractal distribution and the role of the heavy or ”noded” tails which contain all useful under small relation signal/disturber q02 . The results of digital fractal filtering of military ground, air and naval engineering confirm efficiency of fractal processing methods for the radar objects Электромагнитные Явления, Т.5, №2 (15), 2005 г.

Development and Structure of the First Etalon Dictionary of Fractal Properties of Target Classes

masked by intensive noises and disturbances. The scientific results obtained in last decades are the base material for the further development and и underpinning of fractal methods practical application in various fields of radiophysics, radiotechnics, electronics and data-controlling systems. Manuscript received May 12, 2005

References [1] Potapov A.A. Synthesis of the Earth Cover Images in the Optical and Millimeter Wave Bands. Doctoral (Phys.- Math.) Dissertation. – M.: IRE RAS. – 1994. – 436 p. [2] Potapov А.А., Opalenov Yu.V. Long Time Measurement of Scattering Earth Surfaces in Millimeter Band from Board of Helicopter // Elektromagnitnye Volny Elektron. Sist. – 1997. – V. 2, № 3. – P. 71–74. [3] Potapov А.А. Radio Physical Model of Creation Etalon Radar Maps of Earth Surfaces // Elektromagnitnye Volny Elektron. Sist. – 1997. – V. 2, № 4. – P. 31–34. [4] Potapov A.A. Fractaly v Radiofizike i Radiolokatsii (Fractals in Radiophysics and Radar). (First edition). – М.: Logos. – 2002. – 664 p. [5] Potapov А.А. Deterministic Chaos in Radar Images of Earth Surfaces and Fractal Signatures in Problem of Recognition // Proc. of the LII Scientific Session Dedicated to the Day of Radio. Мoscow, May 21–22, 1997. – М.: RNTO RES im. A.S. Popova. – 1997. – V. 1. – P. 169. [6] Potapov А.А. Fractals in Radiophysics and Radar // Proc. XXIII Regional Conf. Radio Wave Propag. St. Petersburg, October 28–29, 1997. – St.P.: Izd-vo SPGU. – 1997. – P. 25. [7] Potapov А.А., German V.А. The Detection of Artificial Objects Based on Fractal Signatures // Proc. III All - Russian - CIS Conf. ”Pattern Recognition and Image Analysis: New Information Technologies” N. Novgorod, December 1–7, 1997. – N. Novgorod: NII PMK at NNGU. – 1997. – Pt. 1. – P. 213–217. Potapov А.А., German V.А. Detection of Artificial Objects with Fractal Signatures // Pattern Recognition and Image Analysis. – 1998. – V. 8, № 2. – P. 226–229. [8] Potapov А.А., Сhekanov R.N. The Wave Scattering on Fractal Surfaces // Proc. of the LII Electromagnetic Phenomena, V.5, №2 (15), 2005

Scientific Session Dedicated to the Day Radio. Мoscow, May 21–22, 1997. – М.: RNTO RES im. A.S. Popova. – 1997. – V. 1. – P. 171–172. [9] Potapov А.А., Sokolov А.V., Chekanov R.N. The Application of Fractal Theory on Radiation Millimeter Wave // Ibid. – P. 167–168. [10] Potapov А.А. Fractals in Radio Physic, Radar and Remote Sensing // Proc. VII All-Russian Workshop ”Physics and Applications of Microwaves”. Krasnovidovo, May 24–30, 1999. – М.: Mos. Gos. Univ. – 1999. – V.2. – P. 140. [11] German V.А., Potapov А.А. About Measurement of Fractal Characteristics Radar and Optical Images // Ibid. – P. 185. [12] Opalenov Yu.V., Potapov А.А. Radar Stochastic Signals and Radon Conversion // Ibid. – P. 186. [13] Potapov А.А., German V.А. Deterministic Chaos in Images of Natural Materials // Ibid. – P. 196–197. [14] Opalenov Yu.V., Potapov А.А. The Digital Radars with Complex Signals and Fractal Processors // Ibid. – P. 199. [15] Potapov А.А., German V.А. The Fractal Signatures in Methods of Detection Low Contrast Radar and Optical Objects // Ibid. – P. 200. [16] Potapov А.А., German V.А. The Fractal Dimensions of Earth Surfaces Images // Proc. Int. Interdisciplinary Symp ”Fractals Applied Characteristics”. Моscow, October 18–21, 1999. – М.: Klassika. – 1999. – P. 51–53. [17] Potapov А.А. About New Applications of Fractal Theory in Radio Physic and Radar // Proc. VII All-Russian Workshop ”Wave Phenomena in Inhomogeneous Media”. Krasnovidovo, May 22– 27, 2000. – М.: Mos. Gos. Univ. – 2000. – V. 2. – P. 57–58. [18] Potapov А.А., German V.А. The Based Principles of Fractal Radar Detection on Radar, Optical and Sintezed Images of Earth Surfaces // Ibid. – P. 58–62. [19] Potapov А.А., German V.А., Chekanov R.N. The Catastrophe Theory and Fractal Theory in Wave Physic // Ibid. – P. 62–66. [20] Potapov А.А. The Fractals in Image Compressing // Ibid. – P. 67–69. [21] Opalenov Yu.V., Potapov А.А., Reutov V.F. Hardware — Methodical Base of a Fractal Multifunction Meter of the Characteristics of Flows and Materials // Ibid. – P. 69–72. 137

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[22] Potapov A.A. Fractals in Remote Sensing // Zarubezh. Radioelectron. Usp. Sovrem. Radioelectron. – 2000. – № 6. – P. 3–65. [23] Potapov A.A., German V.A. Processihg of Optic and Radar Images of the Earth Surface by Fractal Methods // Radiotekh. Elektron. Moscow. – 2000. – V. 45, № 8. – P. 946–953. Journal of Communications Technology and Electronics. – 2000. – V. 45, № 8. – P. 853–860. [24] Potapov А.А. Fractals in Radiophysics and Radar. Elements of the Theory of Fractals: A Review // Radiotekh. Electron. Moscow. – 2000. – V. 45, № 11. – P. 1285–1292; Journal of Communications Technology and Electronics. – 2000. – V. 45, № 11. – P. 1157–1164. [25] Potapov A.A., German V.A., Sokolov А.V. The Radar Extracting of Target from Background by Fractal Method // Radiotekhnika. Moscow. – 2000. № 8. – P. 57–63. [26] Potapov A.A., German V.A. Conceptual Aspects of a Fractal Detection and Discernments of the Targets // Proc. 3 Int. Conf. and Exhibition on Digital Signal Processing and Its Applications. Моscow, November 29–December 1, 2000. – М.: RNTO RES im. A. S. Popova. – 2000. – V. 2. – P. 44–47. [27] Opalenov Yu.V., Potapov A.A., Reutov V.F. Hardvare Methodical Base of a Fractal Multifunction Meter of the Characteristics of Flow and Materials // Ibid. –P. 99–102. [28] Opalenov Yu., Potapov A.A., Reutov V.F. Hardvare - Methodical Base of a Fractal Multifunction Meter of the Characteristics of Flows and Materials // Proc. Second Int. Conf. ”Modern Trends in Computational Physics”. Dubna, July 24–29, 2000. – Dubna: JINR. – 2000. – P. 124. [29] Potapov A.A., German V.A. Fractal Method of Detection of Weak Signals from Complex Images // Ibid. – P. 132. [30] Potapov A.A., German V.A. Chines ”Tangram” in Problems of Fractal Resolution // Ibid. – P. 133. [31] Оpalenov Yu.V., Potapov А.А. Application of Stochastic Signals and the Radon Transform to the Formation of Raster Radar Images by a Microwave Digital Radar with Fractal Data Processing // Radiotekh. Electron. Moscow. – 2000. – V. 45, № 12. – P. 1447–1458. Journal of Communications Technology and Electronics. – 2000. – V. 45, № 12. – P. 1311– 1322. 138

[32] Potapov A.A., German V.A. Fractals, Fractal Target Selection and Fractal Antennas // Proc.I Int. Workshop on Mathematical Modeling of Physical Processes in Inhomogeneous Media. Mexico, Mars 20–22, 2001. - Guanajuato: – 2001. – P. 44–46. [33] Bistrov R.P., Zasovin E.A., Potapov A.A., Sokolov A.V. Millimeter Radar of a High Resolution with the Use of a Fractal Method for Detection // Ibid. – P. 99–101. [34] Оpalenov Yu.V., Potapov А.А. Fractals, Radon Transformation in Radar with Handling on a Carrier Frequency and Prospects of their Application // Zarubezh. radioelektron. Usp. Sovrem. Radioelektron. – 2001. – № 4. – P. 3– 15. [35] Potapov А.А. Fractals in Radio Physics and Radio Location: Fractal Analysis of Signals // Radiotekh. Electron. Moscow. – 2001. – V. 46. № 3. – P. 261–270; Journal of Communications Technology and Electronics. – 2001. – V. 46, № 3. – P. 237–246. [36] Potapov A.A., German V.A. Fractal Selection of Artificial Objects from Radar Images on Inhomogeneous Background // Proc. the Fourth Symp. ”Physics and Engineering of Millimeter and Sub-Millimeter Waves”. Ukraine, Kharkov, June 4–9, 2001. – Kharkov: Kharkov State University. – 2001. – V. 1. – P. 268–270. [37] Bistrov R.P., Potapov A.A., Sokolov A.V., Sokolov S.A. Fractal Methods of Selection of Objects and Millimeter Wave Radar Imaging of the High Resolution // Ibid. – P. 286. [38] Kravchenco V.F., Potapov A.A. Nev Class of Atomic - Fractal Functions and Their Applications // Ibid. – P. 102–107. [39] Kravchenco V.F., Potapov A.A. Atomic - Fractal Antenna Arrays // Proc. URSI Int. Symp. on Electromagnetic Theory. Canada, Victoria, May 13–17, 2001. – Victoria: University of Victoria. – 2001. – P. 660–662. [40] Kravchenko V.F., Potapov A.A., Masiuk V.M. The Atomo - Fraction Function in the Tasks of Antennas Synthesis // Zarubezh. Radioelectron. Usp. Sovrem. Radioelektron. – 2001. – № 6. – P. 4–40. [41] 41. Zasovin E.A., Barmotin V.Ya., Potapov A.A., Sokolov A.V. The Fractal Antennas Based on Serpinsky Fractal // Proc. X Int. Conf. on SpinElectronics and Gyrovector Electrodynamics. Firsanovka, November 16–18, 2001. – М.: МPEI (ТU). – 2001. – P. 208–214. Электромагнитные Явления, Т.5, №2 (15), 2005 г.

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[42] 42. Оpalenov Yu.V., Potapov А.А. Radar Radonometry and Fractal Topology // Ibid. – P. 226–258. [43] 43. Zasovin E.A., Borzov А.B., Bistrov R.P., Ilyasov E.P., Potapov А.А., Sokolov А.V., Тitov А.N. Radioteknicheskie i Radioopticheskie Sistemy. – М.: Кruglyi god. – 2001. – 752 p. [44] German V.A Detection of Low Contrast Radar Target Based on Fractal Properties of Signals: Candidate’s Phys.–Math. Dissertation. – M.: IRE RAS. – 2001. – 130 p. [45] Potapov А.А., German V.A. Fractals in Clustering of Radar Images of Surface Objects and Subsurface Objects // Proc. 3 Int. Conf. and Exhibition on Digital Signal Processing and Its Applications. Моscow, February 27–March 1, 2002. – М.: RNTO RES im. A.S. Popova. – 2002. – V. 1. – P. 217–219. [46] Potapov А.А., Chekanov R.N. Common Fractal Correlator for Radar Scattering // Proc. VIII All-Russian Workshop ”Wave Phenomena in Inhomogeneous Media”. Krasnovidovo, May 26– 31, 2002. – М.: Mos. Gos. Univ. – 2002. – V. 2. – P. 69–70. [47] Оpalenov Yu.V., Potapov А.А. Statistical Structure of Fractal Radar-Radon Measurements // Proc. 2 Int. Congr. ”Nonlinear Dynamic Anal. (NGА - 2)”. Моscow, June 3–8, 2002. – М.: МАI. – 2002. – P. 230. [48] Potapov А.А. Fractals and Fuzzy Sets // Ibid. – P. 234. [49] 49. Potapov А.А., German V.А. Classification and Recognition the Objects by Fractal Methods // Ibid. – P. 235. [50] Potapov А.А. Fractals in Radiophysics and Radar: Fundamental Theory of Wave Scattering by a Fractal Surface // Radiotekh. Electron. Moscow. – 2002. – V. 47. № 5. – P. 517–544. Journal of Communications Technology and Electronics. – 2002. – V. 47, № 5. – P. 461–487. [51] Potapov A.A., German V.A. Effects of Deterministic Chaos and Strange Attractor in the Radar of Dynamics Systems of the Vegetative Cover Type // Pis’ma Zh. Tekh. Fiz. – 2002. – V. 28, № 14. – P. 19–25. [52] Potapov A.A., German V.A. Effects of Deterministic Chaos and Strange Attractor in the Radar of Dynamics Systems of the Vegetative Cover Type // Techn. Phys. Lett. – 2002. – V. 28, № 7. – P. 586–588. Electromagnetic Phenomena, V.5, №2 (15), 2005

[53] Potapov А.А., German V.А. About Applications Theory of Deterministic Chaos in Channel Model Radar and Telecommunications Systems // Proc. IV Int. Sci. Tech. Conf. ”Electron. Informatics– 2002”. Moscow, Zelenograd, November 19–21, 2002. – М.: МIET. – 2002. – V. 2. – P. 219–220. [54] Potapov А.А., Chekanov R.N. Frequency Coherent Function of Fractal Radio Channel // Ibid. – P. 221–222. [55] Potapov А.А. The Topology of Fractal Sets, that Inducted on Fussy Sets // Proc. X Int. Conf. on Spin-Electronics and Gyrovector Electrodynamics. Firsanovka МО, December 20– 22, 2002. – М.: МPEI (ТU). – 2002. – P. 281–297. [56] Potapov А.А., German V.А. The Strange Attractors, Deterministic Chaos and Stability Pareteans as the Elements of New Models Radar Scattering // Ibid. – P. 263–280. [57] Potapov А.А. Fractals on Fuzzy Sets // Elektromagnitnye Volny Elektron. Sist. – 2003. – V. 8, № 1. – P. 26–35. [58] Bunkin B.V., Reutov А.P., Potapov А.А., et al. Aspects of Prospective Radar. / Ed. by А.V. Sokolov. – М.: Radiotekhnika. – 2003. – 512 p. [59] Bistrov R.P., Dmitriev V.G., Potapov A.A., Sokolov A.V., Dmitriev V.G. The Problem of Radar Detection of Low Contrast Targets // Aspects of Prospective Radar./ Ed. by А.V. Sokolov. – М.: Radiotekhnika. – 2003.– P. 20–48. [60] Potapov А.А. The Fussy Sets Formalism and Fuzzy Integrals in Synthesis a New Classes of Fractals and Multifractals in Radar // Ibid. – P. 64–82. [61] Potapov А.А., German V.А. Reconstruction of Strange Attractor in Radar Scattering // Ibid. – P. 83–102. [62] Potapov А.А., German V.А., Sokolov A.V. Fractal Classification Objects on Surfaces and under Surfaces // Ibid. – P. 103–118. [63] Оpalenov Yu.V., Potapov А.А. The Quality of New Methods of Radar based on Radon Transformation // Ibid. – P. 132–147. [64] Оpalenov Yu.V., Potapov А.А. Fractal and Radon Transformations in New Generation Radars with Processing on Carrier Frequency // Ibid. – P. 148–177. [65] Potapov А.А., Sokolov А.V. The Perspective Methods of Radar Signal Processing // Proc. IX All-Russian Workshop ”Physics and Applications of Microwaves”. Zvenigorod, May 26–30, 2003. – М.: Mos. Gos. Univ. – 2003. – V. 1. – P. 71–72. 139

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[66] Potapov А.А., Potapov А.А. (Jr.), Potapov V.А., Sokolov S.А. Radar Detection of Low Contrast Targets // Ibid. – P. 73–74.

[78] Podosenov S.А., Potapov А.А. Fractal Geometry Space — Time of Deterministic Structures // Ibid. – P. 227–229.

[67] 67. Potapov A.A. Fractals, Textural Measures and Dynamic Systems as Basis of Perspective Informational Technologies in Airborne and Ground-based Devices // Proc. V Int. Sci. Tech. Conf. ”Perspective Technology in the Mass Media — PТMM’ 2003”. Vladimir, July 1–4, 2003. – Vladimir: Vladimir. Gos. Univ. – 2003. – P. 230–231.

[79] Bulavkin V.V., Potapov А.А., Vyacheslavova О.F., German V.А. Perspectives of Applications Fractal Theory for Problem of Estimation Quantity of Surfaces // Ibid. – P. 250–253.

[68] Potapov А.А. Fractals, chaos, recursion // Vysshee Obraz. Segodnya. – 2003. – №. 4. – P. 18–26. [69] Potapov А.А., Sokolov А.V. Perspective Methods of Radar Signals Processing Based on Fractal and Textural Measures // Izv. Akad. Nauk. Sеr. Fiz. – 2003. – V. 67. № 12. – P. 1778–1781. [70] Potapov А.А. Fractal, Texture and Dynamic Chaos as Basis of New Radar Technologies: Pt. I.- Results // Proc. I Sci. Tech. Conf. ”Radioopt. Teсhnol. Instrum. Making”. Sochi, September 9– 12, 2003. – М.: МGTU im. N.E. Baumana. – 2003. – P. 58–59. [71] Potapov А.А. Fractal, Texture and Dynamic Chaos as Basis of New Radar Technologies: Pt. II. Perspective Directions // Ibid. – P. 60–61. [72] Podosenov S.А., Potapov А.А. About Fractal Geometry Space - Time of Deterministic Structures // Ibid. – P. 62–63. [73] Zasovin E.А., Potapov А.А. Analysis and Synthesis Lacunars Fractal Antennas // Ibid. – P. 64–65. [74] Potapov А.А., German V.А. Experimental Researching Strange Attractors in Natural Radar Scattering Processes on Millimeter Band // Ibid. – P. 66–67. [75] Potapov А.А. The Methods of Signal Processing Based on Fractal Theory // Proc. I All-Russian Sci. Conf. ”Data Processing Methods and Means”. Моscow, October 1–3, 2003. – М.: Mos. Gos. Univ. – 2003. – P. 559–565. [76] Potapov А.А. Fractal, Texture and Dynamic Chaos as Basis of New Radar Technologies // Proc. Int. Interdisciplinary Symp. ”Fractals Appl. Synergetics”. Моscow, November 17–20, 2003. – М.: МGОU. – 2003. – P. 165–168. [77] Potapov А.А., German V.А. Dependence of Fractal Properties on Topology, Dimensions and Number of Objects // Ibid. – P. 218–221. 140

[80] Potapov А.А. Fractals and Artificial Intelligence // Proc. III Advanced Seminar ”Appl. Methods Artif. Intell. High — Perform. Computations Aerospace Res”. Pereslavl–Zalessky, November 26–27, 2003. – М.: Fizmatlit. – 2003. – P. 76–90. [81] Potapov А.А. New Informational Technologies of Airborne and Ground Based Systems Based on Fractal Theory, Textural Measures and Dynamic Systems // Proc. III Sci. Tech. Conf. ”Prospective Applications of New Technologies and Scientific and Technical Solutions in Khrunichev State Designing Scientific Industrial Center”. Moscow, December 16–18, 2003. – М.: ICS RAS. – 2003. – P. 272–275. [82] Gulyaev Yu.V., Nikitov S.A., Маtveev E.N., Potapov А.А. Mathematic and Physic of Fractals, Models, Results // Proc. XLVI Sci. Conf. ”Modern Problems of Fundamental and Applied Science”. Моscow–Dolgoprudny, November 28– 29, 2003. – М.: МIPT (SU). – 2003. – Pt. V ”Quantum and Physical Electronic”. – P. 103–104. [83] Gulyaev Yu.V., Nikitov S.A., Маtveev E.N., Potapov А.А. Fractals in a New Devices of Communication // Ibid. – P. 105–107. [84] Gulyaev Yu.V., Nikitov S.A., Potapov А.А., Маtveev E.N. Fractals in Photon-Magnetic Crystals // Proc. XII Int. Conf. on SpinElectronics and Gyrovector Electrodynamics. Firsanovka, December 19–21, 2003. – М.: МPEI (ТU). – 2003. – P. 7–42. [85] Zasovin E.А., Potapov А.А. The Researching of Lacunars Fractal Antennas Based on Classical Sets // Ibid. – P. 523–527. [86] Podosenov S.А., Potapov А.А., Sokolov А.А. Impulsnaya Electrodinamika Shirokopolosnyh Radiosistem i Polya Svyazannyh Struktur (Pulse Electrodynamics of Wide-Band Radio Systems). / Ed. by A.A. Potapov. – М.: Radiotekhnika. – 2003. –720 p. [87] Potapov А.А. The Fractal Analysis in Modern Problems of Radars and Radiophysics // Radiotekhnika. Moscow. – 2003. – № 8. – P. 55–66. Электромагнитные Явления, Т.5, №2 (15), 2005 г.

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[88] Potapov А.А. New Information Technology in Radar Detection of Low-Contrast Targets Based on Probabilistic Texture and Fractal Features // Radiotekh. Electron. Moscow. – 2003. – V. 48, № 9. – P. 1101–1119. Journal of Communications Technology and Electronics. – 2003. – V. 48, № 9. – P. 1012–1029. [89] Reutov A.P., Potapov А.А., German V.А. Strange Attractors and Fractals as a Basis of New Dynamics Model of Radar Signals, Scattering by Vegetative Cover // Nelineiny Mir. – 2003. – V. 1, № 1–2. – P. 12–27. [90] Potapov А.А. Brief Historical Essay of the Conception and Formation of the Fractional Calculus // Nelineiny Mir. – 2003. – V. 1, № 1–2. – P. 69–81. [91] Potapov А.А. Topology of Sample // Nelineiny Mir. – 2004. – V. 2, № 1. – P. 4–13. [92] Potapov А.А., Маtveev E.N. Equations of Fractional Orders for Describing Wave Processes in Fractal Media // Proc. IX All-Russian Workshop ”Wave Phenomena in Inhomogeneous Media”. Zvenigorod, May 24–29, 2004. – М.: Mos. Gos. Univ. – 2004. – P. 25–26. [93] 93. Potapov А.А., German V.А. The Distributions of Fractal Dimensions of Synthesis and Natural Images // Ibid. – P. 45–46. [94] Potapov А.А., German V.А. The Parametrical Analysis of Distributions Fractal Dimension with Help Pirson Diagram // Ibid. – P. 47–48. [95] Potapov А.А., German V.А. About Application Fractals for Universe Structures // Proc. Shternberg State Astron. Inst.: ”Horizons of the Universe”. Proc. All-Russia Astron. Conf., Moscow, June 2–10 2004. – М.: GAISh Mos. Gos. Univ. – 2004. – V. 75. – P. 25–26. [96] German V.А., Potapov А.А. The Program for Fractal Counter Detection // Proc. Int. Sci. Conf. ”Smart Data Processing> (Crimea, Aluhta, June 14–19, 2004. – Simferopol: Krymskii Nauch. Tsentr NAN Ukrainy. – 2004. – P. 47. [97] Potapov А.А., German V.А. Fractal Image Recognition // Ibid. – P. 130. [98] Potapov A.A., German V.A. Fractal Radiolocation and Fractal Radiophysics: Stage in the Making, Results, Perspectives // Proc. X Int. Scientific-Research Conf. ”Radiolocation, Navigation, Communication”. Russia, Voronezh, April 13–15, 2004. – Voronezh: NPF Sakvoee. – 2004. – P. 318–341. Electromagnetic Phenomena, V.5, №2 (15), 2005

Voronezh: NPF Sakvoee. – 2004. – V. III. – P. 1869–1896 (in Russian). [99] Potapov A.A. From Editor // Nelineiny Mir. – 2003. – V. 1. № 1–2. – P. 3. – 2004. – V. 2, № 1. – P. 3. [100] Potapov A.A. Fractals, Texture and Deterministic Chaos in New Information Technologies: Synergetic Approaches, Models and Results // Proc. 2nd IEEE Int. Conf. Circuits and Systems for Communications ICCSC–2004. Moscow, June 30–July 2, 2004. – M.: MTUCI. – 2004. – 4 p. (149. pdf). [101] Potapov A.A. Fractals, Textures and Deterministic Chaos in a New Informational Technologies, Synergetic Approaches, Models and Results // Proc. Int. Conf. ”Prospects of Synergetic in the 21th century”. Belgorod, December 3–5, 2003. – Belgorod: BGTU. – 2004. – V. III. – P. 52–58. [102] Potapov A.A. Fractals and Dynamic Chaos in Modern Problems of Remote Sensing // Proc. 2 All-Russian Sci. Conf. ”Remote Sensing of Earth Surfaces and Atmosphere Airspace Devices”. St. Petersburg, June 16–18, 2004. – St. P.: RSHU. – 2004. – V. 3. – P. 51–56. [103] Potapov A.A. Synergetic and Radio Physic Processes in Radar // Proc. 1 Int. Sci. Practical Conf. ”Strategies of Dynamic Development of Russia: Self - Organisation-Administration Union”. Moscow, June 16–18, 2004. – М.: ”Prospekt” RAGC pri Presidente PF. – 2004. – V. 3, Pt. 3. – P. 129–135. [104] Potapov A.A. The Ideas of Synergetic, Fractal Geometry and Deterministic Chaos in Radar Technologies, Fractal Radar and Physic // Proc. Int. Sci. Practical Conf. ”Fundamental Problems of Radioengineering and Device Construction (INTERMATIC–2004)”. Moscow, September 7– 10, 2004. – М.: MIREA-CNII ”Electronica”. – 2004. – Pt. 2. – P. 20–25. [105] Potapov A.A., German V.A. Fractal Signatures: From Nanometers to Parsecs // Ibid. – P. 77–82. [106] Potapov A.A., German V.A. The Creation Fest Dictionary of Fractal Properties // Proc. II Sci. Tech. Conf. ”Radioopt. Teсhnol. Instrum. Making”. Sochi, September 13–17, 2004. – М.: МGTU im. N.E. Baumana. – 2004. – P. 24–25. [107] Оpalenov Yu.V., Potapov А.А., Strukov A.Z. The Experimental Researches of Radar with Full Digital Polar Radiation and Fractal-Radon Filter // Ibid. – 2004. – P. 26–27. 141

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[108] Potapov A.A. Ideas of Synergetic, Fractal Geometry and Deterministic Chaos in Modern Radar Technologies, Fractal Radar and Physic // Proc. 2 All-Russian Sci. Conf. ”The Controlling and Informational Technologies - CIT - 2004”. Pyatigorsk, September 21–24, 2004. – Pyatigorsk: Cpecpechat’. – 2004. – V. 1. – P. 235–242. [109] Potapov A.A. About Theory and Methods of The Digital Fractal Processing of Weak Radar Signals // Proc. 10 Int. Sci. Conf. ”Theory and Technique of Information Transmission, Recrption and Processing”. Tuapse, September 28–October 1, 2004). – Kharkov: KNURE. – 2004. – V. 1. – P. 155–156. [110] Potapov A.A. Geometry of Chaos and Fractal Topology of Samples as Base of Modern System Technologies // Proc. Int. Tech. Conf. ”Applied Synergetic - 2”, and was devoted to Ilya Prigogin. Ufa, October 20–22, 2004. – Ufa: UGNTU. – 2004. – V. 2. – P. 100–106. [111] Potapov A.A. About Stochastic and Fractal in Modern Radar Problem // Proc. Int. Conf. ”Mode Conversion, Coherent Structures and Turbulence - MSS-04” and was devoted to the 75-th anniversary of Professor S.S. Moiseev. Moscow, November 23–25. – M.: IKI RAN. – 2004. – P. 508–513. [112] Potapov A.A. Fractal Radiolocation and Fractal Radiophysics: New Information Technologies // Proc. 13 th Int. Conf. Computational Electromagnetics and Its Applications (ICCEA– 2004). China, November 1–4. - Beijing: Beijing Institute of Technology. – 2004. [113] Potapov A.A., Vyacheslavova О.F. The Estimation of Surfaces Based on Fractal Dimensions and Signatures // Surveys in Applied and Industrial Mathematics. – 2004. – V. 11, № 4. – P. 901–903. [114] Bulavkin V.V., Potapov А.А., Vyacheslavova О.F. Synergetic Approach in the Problem of Qualitative and Quantitative Estimation of the Microrelief of Processed Produce Surface on the Base of Fractal Signature // Nelineiny Mir. – 2005. – V. 3, № 1–2. – P. 128–145. [115] Bulavkin V.V., Potapov А.А., German V.А., Vyacheslavova О.F. The Theory of Fractals in Problem of Estimation Surfaces // Russian Journal of Heavy Machinery ”Tyazheloe mashinostroenie”. – 2005. – № 6. – P. 19– 25. 142

[116] Potapov А.А., Bulavkin V.V., German V.А., Vyacheslavova О.F. The Researching of Micro Structure Surfaces with Methods of Fractal Signatures // Technical Physics. – 2005. – V. 50, № 5. – P. 560–575. [117] Bistrov R.P., Potapov A.A., Sokolov A.V. Millimeter Band Radars with Fractal Processing. – М.: Radiotekhnika. – 2005. – 400 p. [118] Gorbachev A.A., Potapov A.A., Tarakankov S.P. Distant Diagnostic of Dynamical Systems Based on Electromagnetic Waves Nonlinear Scattering // Nelineiny Mir. – 2004. – V. 2, № 5– 6. – P. 310–314. [119] Nonlinear Radar: Collection of Articles. / Ed. by A.A. Gorbachev, A.P. Koldanov, A.A. Potapov, E.P. Chigin (Library of Journal ”Nelineiny Mir”: Scientific Series ”Fractals. Chaos. Probability”). – М.: Radiotekhnika. – 2005. – Pt. 1. – 96 p. [120] Potapov A.A. Fractals Method of Information Transmission // Proc. 3 All-Russian Sci. Conf. ”Irreversible Process in the Sciences and Engineering”. Moscow, January 24–26, 2005. – М.: МGTU im. N.E. Baumana. – 2005. – P. 252– 253. [121] Potapov А.А., German V.А. Methods of Measuring the Fractal Dimension and Fractal Signatures of a Multidimensional Stochastic Signal // Radiotekh. Electron. Moscow. – 2004. – V. 49, № 12. – P. 1468–1491. Journal of Communications Technology and Electronics. – 2004. – V. 49, № 12. – P. 1370– 1391. [122] Potapov A.A. New Information Radiophysical Technologies in Radiolocation: Fractal and Synergetic // Proc. 28th ESA Antenna Workshop on Space Antenna Systems and Technologies. Netherlands, 31 May–3 June. – Noordwijk: ESTEC. – 2005. – Pt. 2. – P. 1047–1050. [123] Potapov A.A., German V.A. Digital Fractal Processing in Signal and Image Analysis: Basises, Methods, Applications // Proc. 2nd IASTED Int. Multi-Conf. ”Automation, Control, and Information Technology – ACIT’ 2005”. Novosibirsk, June 20–24. – Novosibirsk: IAE SB RAS. – 2005. [124] Potapov A.A. Fractals in Radiophysics and Radar: Topology of Sample. – М.: University Book. – 2005. – 848 p. (Second revised and complemented edition).

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