RU2484427C1 - Method for cartographic display of two-dimensional distributions given in digital form - Google Patents

Abstract

FIELD: physics.

SUBSTANCE: images of discrete graphic distributions are converted to a continuous half-tone form with further display thereof in form of isolines. During optical simulation, digital feature values are encoded at the given point of the board by optical symbols - different-sized spots with optical density which is proportional to the feature value. Topographical relief is constructed by interpolating height and/or depth points in form of two-dimensional irregular rational fundamental splines. A two-dimensions spline function is constructed, which is defined as a tensor product of one-dimensional splines. When constructing the topographical relief, iterated functions and wavelets are determined in order to present fractal relief by forming, for a piecewise linear surface, Kronrod-Reeb graph and Morse-Smale complexes. The piecewise linear surface is simplified using structures of the Kronrod-Reeb graph and Morse-Smale complexes obtained for it. Fractal relief parameters are estimated based on the given structures of the Reeb graph and Morse-Smale complexes.

EFFECT: broader functional capabilities, high reliability of cartographic display of two-dimensional distributions given in digital form.

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Description

The invention relates to the field of geodesy and cartography, in particular to cartographic modeling in structural tectonic, geophysical, geochemical, etc. research, exploration, engineering and geological surveys, etc.

There is an optical method for constructing density distribution maps of graphically defined elements (spots, dots and lines) on the area under study, which includes converting images of discrete graphic distributions into a continuous grayscale form with their further presentation in the form of contour lines (copyright certificate SU No. 365562, 1970 [1 ]). However, this method does not provide for the processing of data specified in digital form.

There is also a method of cartographic display of two-dimensional distributions defined in digital form, which includes converting images of discrete graphic distributions into a continuous grayscale form with their further representation in the form of isolines — an excitance, in which, in order to increase accuracy in optical modeling, digital sign values are encoded at a given point on the tablet optical symbols - different-sized spots with an optical density proportional to the size of the attribute (copyright certificate S U No. 640113, 1978 [2]), which provides a display of two-dimensional distributions given in digital form.

However, in most cases of cartographic mapping, it is necessary to build a three-dimensional elevation model defined in an analytical form when displaying the results of the ecological state of the regions (patent RU No. 2079891, application No.92007530 dated 11.23.1992 [3]), the results of situational monitoring of economic objects, for example, offshore oil and gas deposits and terminals, including underwater exploration with a mapping of the terrain, which is not provided by known methods.

There is also a method of constructing a three-dimensional relief model in the form of piecewise spline functions of two variables. The source information is a typographic map of the area. The method is implemented through a geospatial information system (GIS) (Berlyant A.M. Cartography. - M.: Aspect Press, 2002. - 336 p. [4]). The effectiveness of analytical data processing in geospatial information systems largely depends on the capabilities provided by the digital elevation model (DEM), defined as a set of elevations taken at the nodes of some regular or irregular network of points with given coordinates [4]. In automated systems, DEM serves as the basis for obtaining direct and indirect information about the terrain. For example, obtaining information on morphometric data, including the calculation of slope angles and exposure of slopes; visibility / invisibility analysis; construction of three-dimensional images; cross-sectional profiles; assessment of the shape of the slopes through the curvature of their cross and longitudinal sections; calculation of positive and negative volumes; generation of lines of the thalweg network and watersheds forming the wireframe network of the relief, its structural lines and other special points of the relief: local minima (depressions) and local maxima (vertices), saddles, edges, cliff lines and other violations of the “smoothness” of the surface, etc. .

The source data for creating a DEM, for example, land, are topographic maps, aerial photographs, satellite images, altimetry measurements, marine navigation charts, and hydrographic surveying data. At the same time, the following technology was adopted for constructing the DEM (Suvorov S.G., Dvoretsky E.M., Kovalenko S.A. Methodology for creating digital elevation models of increased accuracy // Information and Space. No. 1, 2005 - p.52-54 [5 ]). All available information is digitized. The data obtained from various sources are combined into a single set of coordinates of points and heights in them. This set is triangulated (usually by the Delaunay method). The triangulation procedure gives a system of disjoint triangles covering the considered region of the earth's surface (TIN model). As a result, the relief appears as a multifaceted (elementary face - a triangle) surface with elevations (depth marks) at the nodes of the triangular network. Each face of this surface is described either by a linear function (polyhedral model) or by a polynomial surface whose coefficients are determined by the values at the vertices of the triangle faces. This technology in various versions is implemented in all GIS used in practice.

At the same time, the goal of building a DEM — obtaining adequate direct and indirect terrain information in automated systems — is not achieved. The source of all the disadvantages of this technology is the triangulation stage. In this case, the relief is represented as a continuous function, but with discontinuities already in the corresponding function of the first differential on the edges of the triangulation (i.e., a nonsmooth function). This contradicts the relief model, which is accepted when constructing topographic or navigation maps, where the relief surface appears to be a smooth function. In addition, the true purpose of triangulation is to set the order (network) according to the degree of proximity and relative position on a set of points in the plane, therefore, the relationship of heights (depths) between points is not taken into account, which leads to a distortion of the spatial direction and a shift in the location of structural relief lines. The main types of structural lines of relief include ridge and keel lines, lines of convex and concave bends. By ridge lines, or watersheds, we mean lines of planned correlation of points with maximum heights. Keel lines (thalwegs, beds) connect the points with minimum heights. In addition, the result of triangulation will change dramatically and unpredictably with a change in the initial set of points, i.e. when deleting, adding points (points) or when changing coordinates in the original array of points. This property of triangulation does not allow “controlling” (editing) the construction of a local landform. In addition, if the DEM was constructed using triangulation, the results of the calculation of relief differentials of various orders are not reliable. It can be stated that in this area of geoinformatics there is a problem situation, which is expressed in the fact that the technology of DEM construction using the triangulation procedure does not allow achieving the desired goal. The resolution of the current problematic situation can be achieved by using such DTM construction tools that do not use the triangulation procedure and which lead to the construction of a smooth surface everywhere, which is implemented in the well-known technical solution (patent RU No. 2415381 C1, 08.28.2011 [6]).

In the known method for cartographic display of two-dimensional distributions, set in digital form, including the conversion of images of discrete graphic distributions into a continuous grayscale form with their further presentation in the form of isolines, an extinction, in which, in optical modeling, the digital values of the attribute are encoded at the given point of the tablet with optical symbols - different spots with optical density proportional to the size of the feature, the construction of the terrain, in contrast to analogues [1- 5], the construction of the terrain is performed by interpolating the points of heights (depths) in the form of two-dimensional irregular rational fundamental splines, by constructing a two-dimensional spline function, defined as the tensor product of one-dimensional splines [6].

The known method [6] involves the interpolation of height points (depths) using two-dimensional spline functions, and specifically in the form of two-dimensional irregular rational fundamental splines (NURBS) (Golovanov N.N. Geometric modeling. - M.: Fizmatlit, 2002. - 472 s . [7]). The advantage of this method is the interpolation of elevation points in the form of two-dimensional rational two-dimensional spline-functions NURBS, which allows you to build a smooth surface for any shape of the relief, even for cliffs with a negative angle of inclination. Secondly, the surface of the relief is determined by the analytical dependence, i.e. a finite set of parameters of a fixed set of functions (polynomial splines). The analytical form of the terrain assignment, i.e. in the form of a superposition of analytical functions of two variables, it allows you to use the entire apparatus of differential geometry to describe the morphometric properties of the relief, for example, calculating the value of a function (height, depth) or its differential (slope) at any point (points) of the domain of the function. Third, NURBS provides the ability to locally edit surface shapes. In addition, for the same area of the earth, the DEM data array using NUBRS will be at least an order of magnitude smaller than with the traditional point representation. The use of NURBS improves the efficiency of automated geospatial systems by reducing processing time and the required amount of memory. The use of NURBS in computing has long been a fact - in the graphics packages of all operating systems, NURBS processing and visualization algorithms are integrated, for example, in low-level graphics packages: DirectX and OpenGL for Windows. However, when constructing a DTM, there are obstacles associated with the effect of the occurrence in some situations of a violation of monotony in surface changes — the local appearance of false oscillations. In the known method, this obstacle is removed either by adding new points to the array for interpolation, or by using methods of isogeometric approximation by splines (Kvasov B.I. Methods of isogeometric approximation by splines. - M. - Izhevsk: Research Center “Regular and chaotic dynamics”, Institute of Computer Research, 2006. - 416 p. [8]). In the first case, the solution to the problem is associated with an increase in the expert’s work in the iterative procedure for constructing NURBS, and in the second, with a significant complication of the mathematical algorithms of the technology.

In the known method, the construction technology of DEM based on NURBS is implemented in the form of an iterative expert automated procedure. The programming language used is the MatLab language. In this system, the quality of DEM construction is determined by expert comparison of the position of contours calculated according to NURBS with the position of the corresponding isohypses (isobaths) on the original map.

In a specific implementation of the known method, raster maps serve as a source of terrain information.

In the general case, when approximating the relief profile with one-dimensional splines, the values of the first two derivatives at the end points of the section are set. However, such information is unknown, and it is impossible to obtain it in practice. Therefore, the simplest cubic spline with zero boundary derivatives was used as the base spline for approximating the relief profile along the section. The coordination of the first two DPC differentials for adjacent rectangular map sections is ensured by overlapping task areas of adjacent NURBS.

The technology for building DTMs in an analytical form based on NURBS allows you to exclude the triangulation stage and thereby eliminate the disadvantages of existing technologies.

However, when compiling a DEM in support of conducting geological and seismic underwater studies, it is necessary to perform more detailed detailing of the seabed topography to select the installation sites for the bottom measuring equipment.

The lack of differentiability of fractal functions leads to the need to use a specific apparatus to represent them. Two types of instrumental techniques can be attributed to such an apparatus: Iterated Function Systems (IFS) and wavelets. These tools allow you to represent a fractal function as a pre-fractal only as an average (weighted average value) on a discrete set of cells (points), usually regular. The area of the cells corresponds to a certain multiple of the power of two. For IFS, there are no fundamental restrictions on scale values at all. For wavelets, all scales are possible, smaller than some maximum, determined only by the density of the source data.

Methods for representing terrain using IFS and wavelets allow implementation (Sahr K., White D., Kimerling A. Geodesic Discrete Global Grid Systems // Cartography and Geographic Information Science, Vol. 30, No. 2, 2003, pp. 121-134 . White. D. Global Grids From Recursive Diamond Subdivisions of The Surface of an Octahedron or Icosahedron. // Environmental Monitoring and Assessment, 4 (1), 2000, pp. 93-103. Bartholdi. J., Goldsman P. Continuous Indexing of Hierarchical Subdivisions of the Globe. // Int. J. Geographical Information Science, 15 (6), 2001, pp. 489-522. Goodchild MF, Yang S. A Hierarchical Data Structure for Global Geographic Information Systems. // Computer Vision and Geographic Image Processing, 54 (1), 1992, pp. 31-44. Matos P. SMOS L1 Processor Discrete Global Grids Document - DEIMOS, Engenharia, 2003 .-- 64 pp. Lessig C. Orthogonal and Symmetric Haar Wavelets on the Sphere - University of Toronto, 2007. - 169 pp Peter Schroder and Wim Sweldens. Spherical wavelets: Efficiently representing functions on the sphere. Computer Graphics Proceedings (SIGGRAPH 95), 1995, pages 161-172):

- binary generalization of the surface given on a regular grid of points;

- imitation of increasing the scale of resolution of the source information;

- identification of sudden changes in function values;

- calculation of fractal parameters of the function;

- filtering "noise".

In this case, the following operations can be implemented:

- relief reconstruction by discrete measurements;

- solutions of the inverse IFS problem;

- finding the optimal continuous and discrete wavelet families to represent the relief.

These operations can be implemented on the methods of describing the relief using Morse functions, Kronrod-Rieb graphs and Morse-Smale complexes, which will provide the opportunity to:

- topological coding of relief forms;

- cartographic generalization;

- recognition of geomorphological objects;

- formal classification of geomorphological objects;

- tiling the surface of the relief with a family of parametrized (polynomial) functions defined on Morse-Smale cells;

- hierarchically assess the degree of similarity of two terrain maps representing the same area, at the same or different scales;

- assessment of the reliability of the selected relief forms, taking into account the error and power of the source information;

- assessing the degree of sufficiency of a set of point measurements to restore the relief with a given detail.

The objective of the proposed technical solution is to expand the functionality while increasing the reliability of the method of cartographic display of two-dimensional distributions, set in digital form based on algorithms of relief reconstruction from discrete measurements using topological maps; the formation of the Kronrod-Reeb graph for a piecewise linear surface; the formation of Morse-Smale complexes for a piecewise linear surface; simplification of a piecewise linear surface using the structures of the Kronrod-Rieb graph and Morse-Smale complexes obtained for it; estimates of fractal relief parameters based on given structures of the Kronrod-Reeb graph and Morse-Smale complexes.

The problem is solved due to the fact that in the method of cartographic display of two-dimensional distributions defined in digital form, which includes converting images of discrete graphic distributions into a continuous grayscale form with their further representation in the form of isolines, an excidence, in which optical values are encoded in optical modeling the given point of the tablet with optical symbols - different-sized spots with an optical density proportional to the value of the feature, and the terrain is performed by interpolating the height and / or depth points in the form of two-dimensional irregular rational fundamental splines, by constructing a two-dimensional spline function, defined as the tensor product of one-dimensional splines, in which, in contrast to the prototype, iterating functions and wavelets are determined when constructing the terrain to represent the fractal relief, by forming a Kronrod-Rieb graph and Morse-Smale complexes for a piecewise linear surface, while piecewise linear simplifications are performed in surfaces using the structures of the Kronrod-Rieb graph and Morse-Smale complexes obtained for it; estimation of fractal relief parameters based on the given structures of the Kronrod-Reeb graph and Morse-Smale complexes.

The essence of the proposed method is illustrated by drawings.

Figure 1. Illustration of the sequence of the first six pre-fractals, in the procedure for constructing a fractal line (“Sierpinski carpet”) with a branch index equal to or greater than three for internal points (deleted triangles are indicated in white).

Figure 2. The graphical result of the recursive use of matrix expressions.

Figure 3. Illustration of determining the distance between the point x and the subset B, as well as the distance from the subset B, and from B to A.

Figure 4. Graphic illustration of an IFS action. The positions denote: the original image 1, three subsequent iterations 2, 3, 4, respectively.

Figure 5. Initial steps for generating a fractal surface using IFS. Generation sequences 5, 6, 7, 8.

6. Approximation of IFS prefractal 9 with another prefractal 10.

7. The time-frequency diagram for the Fourier transform (Fig. 7a) and the time-scale diagram for the wavelet transform (Fig. 7b). Frequency 11, amplitude 12, scale 13.

Fig. 8. Scheme of two-dimensional wavelet decomposition. Figa - Chain of calculations, Figb - large-scale correspondence of levels of decomposition. Initial data 14, the first level 15 decomposition, the second level 16 decomposition, the third level 17 decomposition.

Fig.9. Examples of imitation of relief by fractals.

Figure 10. The behavior of the function f (x, y) near the critical point with coordinates (0,0). Minimum - a, saddle - s, maximum - d.

11. Plot of the surface near the point with coordinates (0,0) for a degenerate critical point of the “monkey saddle” type.

Fig. 12. Count of Kronrod-Riba. Figa, b - contour image of the function for which the Kronrod-Riba graph is constructed, Fig.12c - the Kronrod-Riba graph, Fig.12g - embedding of the Conrod-Riba graph in the plane of the contour drawing of the function: maxima - points I, J, G , minima - points D, A, B, L, saddles - points H, F, E, C, K.

Fig.13. Illustration of the transition from a smooth surface to a simplicial complex. Smooth surface 18, simplicial complex 19.

Fig.14. Simplexes of small dimension.

New distinctive features of the proposed method, which consists in the fact that when constructing the terrain, iterative functions and wavelets are determined to represent the fractal relief by forming the Kronrod-Rieb graph and Morse-Smale complexes for the piecewise-linear surface, while simplifying the piecewise-linear surface using the structures of the Kronrod-Rib graph and Morse-Smale complexes obtained for it; estimation of fractal relief parameters based on the given structures of the Kronrod-Reeb graph and Morse-Smale complexes are based on the following postulates.

The first postulate (continuity) - the relief of the Earth is a continuous surface.

The second postulate (topological) - the relief of the Earth is a closed two-dimensional surface in three-dimensional space, topologically equivalent to a two-dimensional sphere.

These two obvious postulates determine fairly general geometric properties of the relief. For a constructive mathematical description, it is necessary to have some way of arithmeticizing geometric objects. Therefore, it is necessary to introduce an auxiliary postulate that uses some numerical system that is consistent in its properties with the first two postulates. As such a numerical system, we choose the most familiar system of real numbers, then the postulate of arithmetizing the surface of the relief can be written in the following form.

гомеоморфное двумерной сфере (S²), где R - множество вещественных чисел, R∈R₊, R₊ - положительные вещественные числа, h₀∈R₊ - наибольшее возможное отклонение от R, 2h₀/R<<1. Для определенности положим, что вещественные координаты точки (x₁, x₂, x₃) соответствуют декартовой прямоугольной правой системе координат с начальной точкой в центре сферы.Third postulate (arithmetization): topography is a topological space with a Euclidean metric $$\left\{\left({\text{x}}_{\text{one}}{\text{, <figure-callout id="2" label="x" filenames="00000027.png,00000028.png" state="{{state}}">x</figure-callout>}}_{\text{2}}{\text{, x}}_{\text{3}}\right)\in {\text{<figure-callout id="3" label="R" filenames="00000028.png,00000030.png" state="{{state}}">R</figure-callout>}}^{\text{3}}|{\left|{\text{x}}_{\text{one}}^{\text{2}}+{\text{x}}_{\text{2}}^{\text{one}}+{\text{x}}_{\text{3}}^{\text{2}}\right|}^{\text{1/2}}-\text{R}<{\text{h}}_{\text{0}}\right\}$$

homeomorphic to the two-dimensional sphere (S ² ), where R is the set of real numbers, R∈R ₊ , R ₊ are positive real numbers, h ₀ ∈R ₊ is the greatest possible deviation from R, 2h ₀ / R << 1. For definiteness, we assume that the real coordinates of the point (x ₁ , x ₂ , x ₃ ) correspond to a Cartesian rectangular right coordinate system with a starting point in the center of the sphere.

The last postulate allows us to move from a purely geometric representation of a relief to its representation in the form of a point set.

In addition, suppose that the terrain has no "sheer and slopes with negative angles." This condition is expressed as the following postulate.

Fourth postulate (technical): every ray emerging from the center of the Earth crosses the surface of the relief at a single point.

The set of mathematical surfaces s satisfying the above postulates is denoted by Ξ (Ξ = {s} or s∈Ξ). Note that using rotations around the center, some surfaces s from the set Ξ can be aligned pointwise. In addition, we will not distinguish between surfaces that are identical up to a scale factor. Such subsets of surfaces form equivalence classes.

Therefore, as Ξ we will consider a set consisting only of representatives of the equivalence classes s.

) и класс всюду не дифференцируемых - негладких ({'s}) поверхностей

So, a certain set of mathematical surfaces Ξ is extremely extensive. By definition, all continuous surfaces are included. All continuous surfaces are divided into two disjoint classes: the class of differentiable - smooth (we denote this set by the symbol

) and the class of everywhere non-differentiable non-smooth ({'s}) surfaces

In cartography and geoinformatics, without argument, it is implicitly assumed that the topography of the Earth is a smooth function. Indeed, the cartographic image of the relief operates with smooth isobath and isohypsum lines, which is possible only for smooth functions. The fifth postulate for cartographic mapping of relief: the cartographic relief of the Earth appears to be a smooth surface.

The work (Mandelbrot B. Fractal geometry of nature. - M .: Institute for Computer Research, 2002. - 655 pp.) Provides a detailed justification for the assertion that the Earth's relief is a fractal surface.

Fractals can be considered as sets of points embedded in space. For example, the set of points forming a line in ordinary Euclidean space has a topological dimension D _T = 1 and a Hausdorff-Besikovich dimension D = 1. The Euclidean dimension of space is equal to E = 3. Since for the line D = D _T , the line, according to the definition of Mandelbrot, is not fractal, which confirms the reasonableness of the definition. Similarly, the set of points forming a surface in a space with E = 3 has the topological dimension D _T = 2 and D = 2. We see that a normal surface is not fractal, no matter how complex it is. Finally, the ball, or full sphere, has D = 3 and D _T = 3.

The central place in the definition of the Hausdorff-Besikovich dimension and, therefore, the fractal dimension D is occupied by the concept of the distance between points in space. In the general case, as δ → 0, the measure M _d = ∑h (δ) is equal to zero or infinity, depending on the choice of d, the measure dimension. The Hausdorff-Besikovich dimension D∈R _{+ of the} set ℑ is the critical dimension at which the measure M _d changes its value from zero to infinity:

$$M_d={\displaystyle \sum \gamma \left(d\right){\delta }^d=\gamma \left(d\right)N\left(\delta \right){\delta }^d\underset{\delta \to 0}{\to }\{\begin{array}{c}0d>D\\ \infty ,d<D\end{array}.\left(\mathrm{one}\right)}$$

The value of M _d for d = D is often finite, but may be zero or infinity. It is important to consider at what value of d the value of M _d changes stepwise. Note that in the above definition, the Hausdorff-Besikovich dimension appears as a local property, since it characterizes the properties of a set of points with an infinitely small diameter, or the size δ of the test function used to cover the set. Therefore, the fractal dimension D can also be a local characteristic of the set.

In accordance with (1) for the fractal surface, the logarithm of the normalized power spectrum of elevations as a function of the logarithm of the roughness wavelength λ should be a linear function.

An informal interpretation of fractals is that the border of a fractal object looks the same, no matter what scale they are observed. This is a manifestation of the property of scale invariance.

Fractals can be considered as sets of points embedded in space. For example, the set of points forming a line in ordinary Euclidean space has the topological dimension D _T = 1 and the Hausdorff-Besikovich dimension D = 1. The Euclidean dimension of space is equal to E = 3. Since for the line D = D _T , the line, according to the definition of Mandelbrot (Mandelbrot B. Fractal geometry of nature. - M .: Institute for Computer Research, 2002. - 655 p.), Is not fractal, which confirms the reasonableness of the definition. Similarly, the set of points forming a surface in a space with E = 3 has the topological dimension D _T = 2 and D = 2.

We see that a normal surface is not fractal, no matter how complex it is. Finally, the ball, or full sphere, has D = 3 and D _T = 3.

The central place in the definition of the Hausdorff-Besikovich dimension and, therefore, the fractal dimension D is occupied by the concept of the distance between points in space. In the general case, as δ → 0, the measure M _d = ∑h (δ) is equal to zero or infinity, depending on the choice of d, the dimension of the measure. The Hausdorff-Besikovich dimension D∈R _{+ of the} set ℑ is the critical dimension at which the measure M _d changes its value from zero to infinity.

The value of M _d for d = D is often finite, but may be zero or infinity.

It is important to consider at what value of d the value of M _d changes stepwise. Note that in the above definition, the Hausdorff-Besikovich dimension appears as a local property, since it characterizes the properties of a set of points with an infinitely small diameter, or the size δ of the test function used to cover the set. Therefore, the fractal dimension D can also be a local characteristic of the set. In fact, there are several nuances worth considering. In particular, the definition of the Hausdorff-Besikovich dimension allows one to cover the set with “balls” of not necessarily the same size, provided that the diameters of all balls are less than δ. In this case, the d - measure is the lower bound, i.e. the minimum value obtained with all possible coatings.

Then the sixth postulate for the Earth's relief (the a posteriori postulate of the physical relief): the physical relief of the Earth is adequately represented by an everywhere undifferentiated surface - a fractal.

.Thus, the physical surface of the Earth's relief is a mathematical object from the set Ξ = {'s}, while the cartographic representation of the relief refers to smooth functions $$\text{Ξ}=\left\{\tilde{s}\right\}$$

.

The most important consequence is that any section of a fractal surface will also be a fractal line. To solve applied problems, it is important to note a significant difference between the "vertical" sections of the surface 's and the "horizontal". The intersection line of 's with the plane passing through the center of the' s ("vertical" section) is always a fractal closed one-dimensional simple line. Such a line is topologically equivalent to a circle. Another thing is the "horizontal" section. This section forms the set of intersection points 's with some sphere with center corresponding to' s and with radius R + h, where | h | <h ₀ .

Many points of this intersection are also a fractal, but this fractal has a much more complex structure. Obviously, such a section can consist of many "closed" connected pieces isolated from each other - the "coastlines of the islands." But this is not the main difference from the "vertical" section. The “horizontal” section leads to lines that are much more complex from a topological point of view. Each connected piece of a “horizontal” section is topologically not equivalent to a circle. The fact is that near any intersection point of a fractal surface 's with a secant sphere, the number of intersection points is infinite. Therefore, near each intersection point, “splitting” of a simple line can occur. Thus, the “horizontal” sections give some set of disconnected lines having a branch index equal to or greater than three. Therefore, the fractal line cannot be graphically depicted. However, graphically, a fractal can be indirectly represented as some infinite iterative procedure. At each step of this procedure, you can depict an intermediate state in the construction of the fractal. Such an intermediate state is called a prefractal. The procedure for constructing a line with a branch index equal to or greater than three for internal points is illustrated by an example of constructing a line C (called a triangular "Sierpinski carpet") with a branch index equal to or greater than three for internal points, which is constructed as follows: in an equilateral triangle with the side equal to unity draws the middle lines, and the internal points of the triangle bounded by the middle lines are ejected from it. The remaining set consists of triangles of the first rank. The same operation is performed with each of the triangles of the first rank: the middle lines are drawn in it and the internal points of the triangle bounded by them are ejected. Similarly, they act with each of the nine resulting triangles of the second rank and come to 27 triangles of the third rank. Proceeding in the same way further, for each natural number n, a set consisting of 3 ’triangles of the nth rank is obtained (FIG. 1).

The set C remaining after performing all these operations is a continuum. In the general case, the “horizontal” section of an everywhere non-differentiable surface is a thin network with cells of various sizes. Note that the Sierpinski carpet has a unique property: any line embedded in the Euclidean plane is also embedded in the Sierpinski carpet. This property allows us to consider the "Sierpinski carpet" as a universal line on the plane (Boltyansky V.G., Efremovich V.A. Visual topology. - M .: Nauka, 1983. - 160 p.).

, но для фрактальной кривой производная не определена, следовательно, не определена и длина. Очевидно, что понятие длины не определено и для более сложно устроенной линии "горизонтального" сечения 's.One of the consequences of the fractality of the line of the "vertical" section 's, that is, the lack of a derivative at any point of it, is its non-rectifiability - the inapplicability of the concept of length to it. Indeed, the length L of a smooth plane curve (rectifiable) given by the equation y = ƒ {x) is determined by the integral of the derivative: $$\text{L}={\displaystyle {\int }_{\text{a}}^{\text{b}}\sqrt{\text{one}+{ƒ}^{\text{'2}}\left(\text{x}\right)\text{dx}}}$$

, but for the fractal curve the derivative is not defined, therefore, the length is not defined either. Obviously, the concept of length is not defined for a more complicated line of the "horizontal" section.

A similar statement is true regarding surface area for a simply connected region or the entire 's - the area for fractal surfaces is not defined. Indeed, for piecewise smooth functions of surfaces with a piecewise smooth edge (or without edge), the surface area is usually determined using the following construction. The surface is divided into small parts with piecewise-smooth borders: in each part, select a point at which the tangent plane exists, and project the part under consideration orthogonally onto the tangent plane of the surface at the selected point; the area of the obtained flat projections is summarized; finally, they go over to the limit with ever smaller partitions (such that the largest of the diameters of the parts of the partition tends to zero). On the indicated class of surfaces, this limit always exists, and if the surface is given by a parametrically piecewise smooth function z = ƒ (x, y) over the domain D on the plane (x, y), then the area S is expressed by the double integral

$$\text{S}={\displaystyle \underset{\text{D}}{\iint }\sqrt{\text{one}+{\left(\frac{\text{d}ƒ}{\text{dx}}\right)}^{\text{2}}+{\left(\frac{\text{d}ƒ}{\text{dy}}\right)}^{\text{2}}}\text{dxdy}\text{.}\left(\text{2}\right)}$$

However, by definition, the surface 's is not differentiable everywhere; therefore, the integral for the area does not exist.

In a first approximation, fractal can be understood as a geometric figure that has the property of self-similarity, i.e. composed of several parts, each of which is similar to the whole figure. A small part of the fractal contains information about the entire fractal. Fractals are similar to themselves, they are similar to themselves on everyone at any scale - no matter how large they look at fractal objects, they will still be fragmented and broken. This property of self-similarity leads to the fact that in order to describe fractal measures and relationships, it is necessary to use the parameter of the fractal consideration scale. The power dependence on the scale of the form (1) is valid for:

- measures of one-dimensional and two-dimensional fractals;

- average values of maximum deviations from the line connecting the two points of the fractal;

- the ratio of the perimeter of the closed fractal curve and the area covered by it;

- distribution of closed areas with a "horizontal" section's area.

The property of fractality is manifested in cartography in various aspects. The main aspect is that a long line of map scales is used to display the terrain. At each scale, it is assumed (implicitly) that the relief is a smooth surface. This follows from the fact that isohypses (isobaths) are smooth curves. However, the relief of the same area on maps of different scales is different in detail. Moreover, with increasing scale, there is no convergence in the position of isohypsum (isobath), which indicates the absence of convergence to some smooth surface. It is this circumstance that explains the need for a cartographic generalization procedure in the technology for constructing maps of various scales. The generalization procedure is carried out by an expert. In geoinformatics, there are no acceptable automated algorithms for cartographic generalization. The fractality property is manifested in the mathematical dependence (power function) of the course of the intersection curve and is used to describe the roughness of the relief surface. The algorithm for constructing the intersection curve is as follows. On a map of a certain scale, pairs of points are taken that are uniformly distributed over the area and are spaced apart by a distance r. The height difference h of each pair of points is determined, and the arithmetic mean of the absolute values of these differences is found. Then the distance doubles, triples, etc .; pairs of points are again selected on the map, height differences are measured and average values corresponding to the increased distance between points are found. These average values are plotted on a graph in which the distance between points is plotted along the abscissa axis, and the average values of height differences are plotted along the ordinate axis. The curve near which points on the graph fit is the intersection curve h = ƒ (r). This dependence is approximated by a power function, which is characteristic of fractal surfaces.

The most transparent property of undifferentiated relief is manifested in estimates of the length of coastlines from maps of various scales. The graphic image of coastlines on maps of various scales is essentially a pre-fractal of various levels of detail, displaying a “horizontal” section corresponding to zero sea level. When measuring the coastline at a constantly enlarging scale, ever smaller bends fall into consideration, and each new part increases the total length of the coast. In the case of relief fractality, the observed length should increase unboundedly. Estimates of the coastline for various sections of the coast showed that length estimates increase with scale according to the power law of the form (1). At the same time, the graph for a smooth curve should tend to a constant value.

To date, it has been established that fractal objects are the lengths of rivers, the structure and boundaries of catchments and ravine-gully networks, the distribution of islands by area (Melnik M.A. Fractal analysis of morphologically homogeneous sections of rivers (using the example of the Tomsk region) // Materials of the XIII Scientific Meeting geographers of Siberia and the Far East, Irkutsk: IG SORAN Publishing House, 2007. Vol. 1. P.165-167 Tarboton DG Fractal river networks, Horton's laws and Tokunaga cyclicity // Journal of Hydrology. 187. (1996). 105. -117. (New2 / 10.1.1.80.2625.pdf). Ivanov A.V., Koronovsky A.A., Minyukhin I.N., Yashkov I.A. Determination of fractal size Roughnesses of the ravine-girder network of the city of Saratov // University proceedings. Applied nonlinear dynamics. 2006. v.14. No. 2. pp. 64-73. Dodds PS, Rothman DH SCALING, UNIVERSALITY, AND GEOMORPHOLOGY // Annu. Rev. Earth Planet. Sci. 2000.28: 571-610 (new2 / AnnRev.28.1.571.pdf) Turcotte DL Self-organized complexity in geomorphology: Observations and models // Geomorphology 91 (2007) 302-310).

The lack of differentiability of fractal functions leads to the need to use a specific apparatus to represent them. Such a device can include: Iterated Function Systems (IFS) and wavelets. These mathematical tools make it possible to represent a fractal function as a pre-fractal only as an average (weighted average value) on a discrete set of cells (points), usually regular. The area of the cells corresponds to a certain multiple of the power of two. For IFS, there are no fundamental restrictions on scale values at all. For wavelets, all scales are possible, smaller than some maximum, determined only by the density of the source data. The basic principle of IFS is to present a self-similar fractal as a composition of many “smallest” copies of itself. The relief may also appear to be more complex fractal structures than a self-similar fractal (Gagnon J., Lovejoy S., Schertzer D. Multifractal earth topography // Nonlin. Processes Geophys., 13, 2006, 541-570). IFS is a relatively cumbersome set-theoretic construct; therefore, we restrict ourselves to a general schematic description. Let us give an example of a general IFS expression for the R ² plane:

$$\text{W}\left(\text{x}\right)={\text{w}}_{\text{i}}\left[\begin{array}{c}{\text{x}}_{\text{one}}\\ {\text{x}}_{\text{2}}\end{array}\right]=\left[\begin{array}{cc}{\text{a}}_{\text{i}}& {\text{b}}_{\text{i}}\\ {\text{c}}_{\text{i}}& {\text{d}}_{\text{i}}\end{array}\right]\left[\begin{array}{c}{\text{x}}_{\text{one}}\\ {\text{x}}_{\text{2}}\end{array}\right]+\left[\begin{array}{c}{\text{e}}_{\text{i}}\\ {ƒ}_{\text{i}}\end{array}\right]={\text{A}}_{\text{i}}\text{x}+{\text{t}}_{\text{i}}\text{.}\left(\text{3}\right)$$

From here, the IFS form for the Sierpinski triangular carpet will be:

$${\text{w}}_{\text{i}}=\left[\begin{array}{c}{\text{x}}_{\text{one}}\\ {\text{x}}_{\text{2}}\end{array}\right]=\left[\begin{array}{cc}\text{0}\text{.5}& \text{0}\\ \text{0}& \text{0}\text{.5}\end{array}\right]\left[\begin{array}{c}{\text{x}}_{\text{one}}\\ {\text{x}}_{\text{2}}\end{array}\right]+\left[\begin{array}{c}\text{one}\\ \text{one}\end{array}\right]\text{,}\left(\text{four}\right)$$

$${\text{w}}_{\text{i}}\left[\begin{array}{c}{\text{x}}_{\text{one}}\\ {\text{x}}_{\text{2}}\end{array}\right]=\left[\begin{array}{cc}\text{0}\text{.5}& \text{0}\\ \text{0}& \text{0}\text{.5}\end{array}\right]\left[\begin{array}{c}{\text{x}}_{\text{one}}\\ {\text{x}}_{\text{2}}\end{array}\right]+\left[\begin{array}{c}\text{one}\\ \text{fifty}\end{array}\right]\text{,}\left(\text{5}\right)$$

$${\text{w}}_{\text{i}}\left[\begin{array}{c}{\text{x}}_{\text{one}}\\ {\text{x}}_{\text{2}}\end{array}\right]=\left[\begin{array}{cc}\text{0}\text{.5}& \text{0}\\ \text{0}& \text{0}\text{.5}\end{array}\right]\left[\begin{array}{c}{\text{x}}_{\text{one}}\\ {\text{x}}_{\text{2}}\end{array}\right]+\left[\begin{array}{c}\text{fifty}\\ \text{fifty}\end{array}\right]\text{.}\left(\text{6}\right)$$

The graphical result of the recursive application of these matrix expressions is shown in FIG. 2.

The metric space (X, d) is the set X on which the real distance function d: X × X → R is defined with the following properties:

d (a, b) ≥0 for all a, b∈X;

d (a, b) = 0 if and only if a = b for all a, b∈X;

d (a, b) = d (b, a) for all a, b∈X;

d (a, c) ≤ d (a, b) + d (b, c) for all a, b, c∈X.

Such functions are called metrics. This means that the metric space is a combination of two elements: the set of points from X∈R ⁿ and the distances between two points of the set d (a, b).

Let (X, d) be a complete metric space. Denote by ℜ (X) the space whose points are nonempty compact subsets of X.

The distance between a point and a subset is defined as

d (x, B) = min {d {x, y): y∈B},

where B∈ℜ (X), x∈X. This definition is illustrated in FIG.

The distance between two subsets is defined as d (A, B) = max {d (x, B): x∈A}.

Here the metric is not symmetrical, as shown in Fig. 1.8, d (A, B) ≠ d (B, A}. To overcome the asymmetric problem of this metric, the Hausdorff metric is introduced. The Hausdorff metric on ℜ (X) is defined as

$$\text{h}\left(\text{A}\text{, B}\right)=\text{max}\left(\text{d}\left(\text{Ab}\right)\text{, d}\left(\text{B}\text{, A}\right)\right)\text{.}\left(\text{7}\right)$$

The Hausdorff metric allows you to determine the convergence of IFS by calculating the distance between map points for each iteration. We give some more necessary definitions guaranteeing the compressibility of mappings:

. $$\text{W}\left(\text{S}\right)={\displaystyle {\cup }_{\text{i}=\text{one}}^{\text{n}}{\text{w}}_{\text{i}}\left(\text{S}\right)}$$

.

A transformation W: X → X on a metric space (X, d) is called a contraction map (or contraction) if there exists a number s, 0 <s <1, such that

d (W (x), W (y)) ≤s · d (x, y}, ∀x, y∈X.

The number s is called the compression ratio.

, сжимающее отображение. Тогда существует единственная точка x_ƒ∈Х, такая, что для любой точки x∈ХThe main results of the theory of contracting mappings are connected with fixed points of such mappings. Let X be a complete metric space and $$\text{W: x}\to \text{X | W}={\cup }_{\text{i}=\text{one}}^{\text{n}}{\text{w}}_{\text{i}}$$

compressive mapping. Then there exists a unique point x _ƒ ∈X such that for any point x∈X

$${\text{x}}_{ƒ}=\text{W}\left({\text{x}}_{ƒ}\right)={\displaystyle \underset{\text{i}=\text{one}}{\overset{\text{n}}{\cup }}{\text{w}}_{\text{i}}\left({\text{x}}_{ƒ}\right)}=\underset{\text{n}\to \infty }{\text{lim}}{\text{W}}^{\circ \text{n}}\left(\text{x}\right)\text{.}\left(\text{8}\right)$$

A point x _ƒ is called a fixed point or an attractor of the map W.

This definition implies that any contraction mapping applied to any initialization set will always result in a fixed finite set. A system of iterative mappings describes a set of contraction functions that act on a set (in the space R ⁿ ) to obtain another set in the same space or a space of smaller dimension. IFS shared record is

$$\text{W}(\cdot )={\displaystyle \underset{\text{i}=\text{one}}{\overset{\text{n}}{\cup }}{\text{w}}_{\text{i}}(\cdot )}\text{,}\left(\text{9}\right)$$

where (·) corresponds to the space parameter R ⁿ , w _i - corresponds to each individual compressive map {w _i : R ⁿ → R ⁿ | i = 1, ..., n}. Graphically, this can be considered as shown in FIG. 4. 5 and 6 are illustrations of the effects of IFS on a surface.

Wavelets as a mathematical tool for the hierarchical representation of functions allow us to describe an arbitrary function in terms of a rough average approximation and details of various scales.

Even for a schematic formal description of a two-dimensional wavelet transform, a huge number of definitions are required. This is not necessary, since there is currently a large volume of literature on wavelets, for example (Malla S. Wavelets in signal processing. - M .: Mir, 2005. - 671 p. Blatter K. Wavelet analysis. Fundamentals of the theory. - M .: Technosphere, 2004. - 280 pp. I. Dobeshi Ten lectures on wavelets. - Izhevsk, SRC "Regular and chaotic dynamics", 2001. - 464 pp. Chui Ch. Introduction to wavelets. - M .: Mir, 2001. - 412 p.). We give only an informal justification of discrete wavelets. A classic tool for data analysis is Fourier analysis, which can be used to convert observational data at a point into a form more convenient for frequency analysis. However, when using the Fourier methods, one difficulty arises: each Fourier coefficient contains complete information about the behavior of a series at only one frequency and no information about its behavior at other frequencies. In addition, Fourier methods are difficult to adapt to many situations of practical importance. For example, most of the time series encountered in practice are finite and aperiodic, while the discrete Fourier transform can be applied only to periodic functions. The disadvantages of the Fourier methods and the need to provide a hierarchical representation of functions, which is called multi-scale analysis, have led to the development of methods based on which the function under study is represented by a set of coefficients, each of which carries limited information about the position and frequency of the function (Fig. 7).

Despite the existence of a wide variety of methods for hierarchical representation of functions, the developed wavelet theory contains an arsenal of extremely useful tools that allow hierarchical decomposition of functions in an effective and theoretically justified way. Generally speaking, the wavelet representation of functions consists of a general rough approximation and refinement coefficients that allow working with the function at various scales.

Two-dimensional discrete wavelets are a family of four tensor products (HH, HL, LH, LL) of two filters with some special properties: low-frequency L and high-frequency H, parameterized by scale and shear coefficients. These two filters are one-dimensional discrete wavelets. The initial data for a two-dimensional wavelet decomposition are scalar functions defined on a regular grid of points (matrices).

The process of two-dimensional wavelet decomposition is carried out according to the scheme shown in Fig. 8.

Wavelets are widely used to represent the relief defined on a regular grid of points at various scales (Keitt, T. N., Urban DL Scale-Specific Inference Using Wavelets // Ecology, 86 (9), 2005, pp. 2497-2504. Marchuk AG , Simonov KV Detecting possible impact craters on Earth's surface using the DEM data proscessing. // Bull. Nov. Comp. Center, Math. Model. In Geoph., 10 (2005), 59-66. Swai CJ, Kirby JF An effective elastic thickness map of Australia from wavelet transforms of gravity and topography using Forsyth's method // GEOPHYSICAL RESEARCH LETTERS, VOL. 33 2006 78-82. Audet P. Directional wavelet analysis on the sphere: Application to gravity and topography of the terrestrial planets // JOURNAL OF GEOPHYSICAL RESEARCH, Vol. 116, 2011, 111-127. Tassara A., Swain C., Hackney R., Kirby J. Elastic thickness structure of South America estimated using wavelets and satellit e-derived gravity data // Earth and Planetary Science Letters 253 (2007) 17-36). Using wavelets, sharp differences in the values of the function are detected, fractal parameters are estimated. It is the relationship between the coefficients of the wavelet decomposition by fractal parameters that is the basis for the use of wavelet transform in the description of fractal functions (Berkner K. F Wavelet-based Solution to the Inverse Problem for Fractal Interpolation Functions // Fractal in Engineering '97, Springer-Verlag, 1997. - 123-135 pp. Struzik ZR, Dooijes EH, Groen FCA The solution of the inverse fractal problem with the help of wavelet decomposition. In MM Novak, editor. Fractals reviews in the natural and applied sciences, pages 332-343. Chapman and Hall, February 1995). Currently, the practical application of fractal representation of the relief lies in the field of imitation. A large number of algorithms have been developed that simulate the relief by fractal surfaces (Franceschetti G., Riccio D. Scattering, Natural Surfaces and Fractals - Academic Press is an imprint of Elsevier, 2007. - 307 pp.). Figure 9 presents two examples of simulating fractal relief.

Representation of the relief in the form of a fractal function leads to the practical impossibility of fully (in all details and details) to describe and display the real relief in any way. The fractality of the relief is manifested in the fact that the surface of the relief is a combination of various spatial forms (irregularities) of an infinite set of spatial scales. However, in this variety of landforms, a regularity is traced in the form of a similarity of variability at various scales.

The main feature of terrain mapping is that terrain on maps is always presented as a smooth function. The presence of smooth contours on maps (isogypsum, isobath) is evidence of the smoothness of the surface of the relief. In cartography, the fractality of the relief is manifested indirectly, through cartographic generalization. Traditionally, the relief is presented in the form of maps of various scales. Large scale maps display smaller relief forms. The transition to a smaller scale requires the application of the laws of cartographic generalization — smoothing, exaggerating, and eliminating the relief forms of small sizes.

Geomorphology studies various forms of relief at various scales. In geomorphology, it is customary to divide relief forms of various sizes into orders. Currently, up to 12 orders of the earth's surface are distinguished (Simonov Yu.G., Bolysov S.I. Methods of geomorphological research: Methodology. - M.: Aspect Press, 2002. - 191 p.).

For example, the first category includes continental ledges and oceanic depressions (the area of the forms is 10 ⁷ km ² ), the second category includes parts of the first with a size of 10 ⁶ km ² , and the third category includes parts of the relief elements of the second category with dimensions of about 10 ⁵ km ² . The division is carried out further, and each subsequent category occupies an area an order of magnitude smaller than the previous one. It is important to note here that the objects of each category have their own spatial scales of the map at the scales that these objects can be observed. At other scales, these objects are not observable. In addition, objects of different categories that are similar in shape are given different names, for example, a hill, a hill, a hill, a mountain, a mountainous area, a ridge.

These features of the geomorphological classification left their mark on the use of mathematics in geomorphological research. For almost every registered landform, its own methods and methods of mathematical description were developed. The proposed technical solution uses a more abstract mathematical apparatus - topology, in terms of elements of the theory of differential and algebraic (combinatorial) topology. The use of this apparatus allows one to mathematically describe all relief forms irrespective of the scale and specific forms for displaying on a map or display.

Algebraic topology provides a connection between geometry and algebra, between a continuous and discrete description of the relief, a description of the structural features of the relief: minima, maxima, network lines of talwegs and watersheds. Differential topology provides a theoretical basis for identifying and reconciling the global properties of a relief surface with a set of its local structural features. All this allows the implementation of constructive algorithms for computers.

In addition, the apparatus of differential and algebraic topology makes it possible to approximately reconstruct the surface from a set of point data, to compare the degree of proximity of two representations of the relief surface of a fixed region, both for one scale and for different scales. To get a logically reasoned way to simplify the terrain, both for generalization purposes and for removing measurement noise.

The possibility of an approximate representation of a fractal surface using smooth functions is guaranteed by the Weierstrass theorem (G. Fikhtengolts, Differential and Integral Calculus Course, Vol. 3 - M .: Fizmatlit, 2001. - 662 pp.), Which states that the continuous function (we emphasize only that enough of continuity, smoothness requirements no function at all) several variables ƒ (x _1, ..., x _n) on a closed bounded set Q can be uniformly approximated polynomial sequence: for any ε> 0, there exists a polynomial P ( _1, ..., x _n), that its maximum deviation from ƒ (x _1, ..., x _n) to Q does not exceed a given ε:

$$\underset{\text{Q}}{\text{max}}\text{|}ƒ\left({\text{x}}_{\text{one}}\text{,}...{\text{, x}}_{\text{n}}\right)-\text{P}\left({\text{x}}_{\text{one}}\text{,}...{\text{, x}}_{\text{n}}\right)\text{|}<\text{ε}\text{.}\left(\text{2}\text{.one}\right)$$

There is a generalization of Weierstrass theorem - Stone's theorem (Stone MN The Generalized Weierstrass Approximation Theorem // Mach. Mag. 1948. Vol. 21. P.167-183, 237-254), which proves that not only the set of polynomials in coordinate functions, but generally a ring of polynomials from any set of functions separating points. Therefore, for example, the set of trigonometric polynomials is dense.

Among the variety of possible smooth surfaces, one should single out a class of surfaces that will represent all other smooth functions. It is advisable to use the class of Morse functions as such a class. We can assume that surfaces belonging to the class of Morse functions are most “simply” constructed from a mathematical point of view. The fact is that Morse functions have the smallest possible set of types of critical points, namely: points of local maxima, minima, and simple saddle points.

Usually consider two-dimensional functions defined on the plane. In our case, the relief surface is defined on the sphere S ² . This manifold is closed and bounded, therefore compact. Smooth functions defined on an arbitrary manifold are investigated by Morse theory (Milnor J. Morse Theory. - M.: Publishing House of LCI, 2011. - 184 p.). This mathematical theory explores the relationship between the topology of the manifold M (for us, M is a sphere) and the critical points of the scalar function ƒ: M → R defined on the manifold M.

Consider a smooth function ƒ: M → R on a smooth manifold M, and let x ₁ , ..., x _n be smooth regular coordinates in a neighborhood of the point p∈M. A point p is called critical for the function ƒ if the differential

$$\text{d}ƒ={\displaystyle \sum \frac{\text{d}ƒ}{{\text{dx}}_{\text{i}}}{\text{dx}}_{\text{i}}\left(\text{10}\right)}$$

vanishes at the point p. This is equivalent to the condition that all partial derivatives of the function vanish at a given point. A critical point is called non-degenerate if the second differential

$${\text{<figure-callout id="2" label="d" filenames="00000027.png,00000028.png" state="{{state}}">d</figure-callout>}}^{\text{2}}ƒ={\displaystyle \sum \frac{{\text{<figure-callout id="2" label="d" filenames="00000027.png,00000028.png" state="{{state}}">d</figure-callout>}}^{\text{2}}ƒ}{{\text{dx}}_{\text{i}}{\text{dx}}_{\text{j}}}{\text{dx}}_{\text{i}}{\text{dx}}_{\text{j}}\left(\text{eleven}\right)}$$

non-degenerate at this point. This is equivalent to the condition that the matrix of second partial derivatives has a non-zero determinant. According to the well-known Morse lemma, in the vicinity of each non-degenerate critical point, one can always choose such local coordinates in which the function is written in the form of a quadratic form:

$$ƒ\left(\text{p}\right)=-{\text{x}}_{\text{one}}^{\text{2}}-{\text{x}}_{\text{2}}^{\text{2}}-...-{\text{<figure-callout id="2" label="x λ" filenames="00000027.png,00000028.png" state="{{state}}">x </figure-callout>}}_{\text{λ}}^{}+{\text{<figure-callout id="2" label="x λ" filenames="00000027.png,00000028.png" state="{{state}}">x </figure-callout>}}_{\text{λ}}^{}+...+{\text{x}}_{\text{<figure-callout id="2" label="n" filenames="00000027.png,00000028.png" state="{{state}}">n</figure-callout>}}^{\text{2}}\text{.}\left(\text{12}\right)$$

For each non-degenerate critical point, the number λ is uniquely determined and is called its index.

A smooth function ƒ: M → R is a Morse function if all its critical points are non-degenerate. A remarkable fact in the theory of critical points is that for Morse functions on S ^{2 the} complete classification of critical points and the geometry in their neighborhood are known. It turns out that near the critical point p∈S ² there is always the possibility of choosing such local coordinates x and y that a function near this point can be written as

$$ƒ\left(\text{x}\text{, y}\right)=\pm {\text{x}}^{\text{2}}\pm {\text{<figure-callout id="2" label="y" filenames="00000027.png,00000028.png" state="{{state}}">y</figure-callout>}}^{\text{2}}\text{,}\left(\text{13}\right)$$

and all these critical points are isolated (Morse lemma).

Figure 10 presents the surface graphs for all four possible critical points near the point with coordinates (0,0).

If a smooth function on a sphere is not a Morse function, then it can have not only non-degenerate critical points of type (13), but also degenerate types of critical points other than (13), for example, a degenerate critical point of the “monkey saddle” type shown in FIG. .eleven.

However, each bounded smooth function ƒ: M → R can be uniformly approximated by a smooth function g that does not have degenerate critical points. Moreover, g can be chosen so that the ith derivatives of g on the compact set K uniformly approximate the corresponding derivatives ƒ for i≤k. Morse functions are everywhere dense in the space of all smooth functions on a smooth manifold. In other words, any smooth function with an arbitrarily small stir can be turned into a Morse function. Moreover, complex degenerate critical points are scattered into the union of a certain number of Morse points, i.e. non-degenerate, features. This is because non-degenerate critical points of type (13) are stable with respect to small “movements” of the graphs (Fig. 10), while degenerate critical points with small “movements” of the corresponding graphs either disappear or go to the surface with some set of non-degenerate critical points type (13).

Of great importance is the concept of the index i (p) of the function ƒ at the critical point p∈M, which is defined as the number of minuses in the corresponding equation (13). The critical point of index 0, 1 or 2 is called the minimum saddle and maximum, respectively.

Thus, it is always possible to represent any smooth function on the sphere as a Morse function. In other words, any surface of the relief represented on the map can be unambiguously correlated with some Morse function. Further, for the sake of simplicity of presentation, we assume that the Morse functions under consideration belong to the class - C ^{∞ of} differentiable functions. The program-mathematical apparatus for implementing the method includes a mathematical apparatus for describing the structural properties of smooth functions based on algebraic topology.

Algebraic topology provides a set of tools that allow you to capture and describe the shape of the surface, to determine in which surfaces coincide or differ. In addition, in algebraic topology, there are classical tools, such as Morse theory, homotopy, and homology, which are suitable for solving a number of problems related to the surface shape.

The goal of algebraic topology will provide means for solving a typical problem: recognition (distinguishing) of objects belonging to the category G ₁ (in our case, smooth surfaces). Algebraic topology, using the functor F: G ₁ → G _2, replaces this problem with a similar task of recognizing objects belonging to the category G ₂ . The meaning of the replacement is that this task in the category G ₂ may turn out to be easier. It should be noted that when moving from category G ₁ to category G _2, part of the information about objects of category G ₁ is usually lost. However, functors of algebraic topology provide “good” properties of such a replacement, namely: the parameters used are relatively easily computable on a computer; there are simple ways to clarify the difference or isomorphism of objects; when moving from object X to object F (X), a lot of information is not lost.

Moreover, the functions of two variables are considered not just as functions of the two variables x and y, but as functions defined at the points of the two-dimensional domain - S ² .

Algebraic topology reveals a sharp splitting of the properties of functions of two variables. Some of them turn out to be close to the properties of functions of one variable. Other properties of functions of two variables, on the contrary, are sharply “two-dimensional”. In addition, it turns out that there are concepts that depend on both “one-dimensional” and “two-dimensional” properties of functions.

The “one-dimensional” properties of the function of two variables are described using the Kronrod-Reeb graph, and the properties of higher dimensions are described using Morse-Smale complexes.

Morse Theory (Milnor J. Morse Theory. - M.: LKI Publishing House, 2011. - 184 pp.) Establishes the basis for describing the set of critical points of a smooth function given on a manifold. Using Morse's theory, we can determine a method for describing the shape of a surface based on the evolution of the surface of level isolines that displays a function. This method, which consists in comparing the levels of critical points on the surface, is usually considered as one of the simplest ways to describe the geometry of the surface. Each one-dimensional continuum, its one-dimensional tree, is associated with each function. The study of a number of properties of the function itself is reduced to the study of the properties of the corresponding function on a one-dimensional tree. The separation of the properties of the function of two variables into “one-dimensional” and “two-dimensional” seems to be a fundamental fact. From this point of view, the introduction of a one-dimensional tree is just essential: with its help, one-dimensional properties of a two-dimensional function are particularly clearly distinguished.

In what follows, by ƒ ^-1 (r) we denote the complete inverse image of the value r of the scalar function ƒ defined on S ² (ƒ: S ² → R). By a we denote the regular values of the function, i.e. such values, in the prototype of which there is not a single critical point. In this case, ƒ ^-1 (a) is always a smooth submanifold in S ^{2 by} virtue of the well-known implicit function theorem. Let c denote the critical values of the function, i.e. such values in the prototype of which there is at least one critical point.

Let ƒ be the Morse function on a compact smooth manifold S ² . Consider an arbitrary level surface ƒ ^-1 (a) and its connected components, which we call layers. As a result, the variety is divided into a union of layers, a foliation with features is obtained. We emphasize that each layer is connected by definition. Declaring each layer as one point and introducing the natural factor topology in the space of layers Γ, we obtain a certain quotient space. It can be considered as the base of this foliation. For the Morse function, the space Γ is a graph.

Graph G is called the Kronrod-Reeb graph (A. Kronrod, On functions of two variables // UMN, 5: 1 (35) (1950), 24-134. Reeb G. Sur les points singuliers d'une forme de pfaff compl ' etement int'egrable ou d'une fonction num'erique. Comptes Rendus de L'Acad'emie ses S'eances, Paris, 222: 847-849, 1946) for the Morse function ƒ on the manifold S ² . The vertex of the Kronrod-Reeb graph is the point corresponding to the special layer of the function ƒ, i.e. a connected component of the level containing the critical point of the function. A vertex of the Kronrod-Reeb graph is called terminal if it is the end of exactly one edge of the graph. All other vertices are called internal.

The end vertices of the Kronrod-Reeb graph correspond uniquely to the local minima and maxima of the function. The inner vertices of the Kronrod-Reeb graph correspond one-to-one to special layers of the function containing saddle critical points (Fig. 12).

For simplicity of the image of two-dimensional surfaces, everywhere further in this section it is assumed that the functions depicted behave like - r far from the origin, continuing them near the point ∞ so that it is a local minimum point.

If it is known in advance that the surface under study is orientable or non-orientable, then the Kronrod-Rieb graph of an arbitrary simple function on it allows one to restore the surface topology. Kronrod-Reeb graphs are considered up to isomorphism of oriented graphs. Two Morse functions on an orientable surface are layerwise equivalent if and only if their Kronrod-Reeb graphs are isomorphic.

The generalized mechanism for constructing the Kronrod-Reeb graph is as follows. Let a Morse function be given on a manifold. Two critical points are connected by an edge on the Kronrod-Reeb graph if and only if there exists a monotone smooth path on the manifold that connects these points and does not intersect the critical layers (except for its ends). A monotonous path is a path along which a given function increases. Two saddles are connected by two edges on the Kronrod-Reeb graph if and only if there are two monotone smooth paths on the manifold that connect these saddles and do not intersect critical layers (except for their ends), and these paths cannot be connected in a constant way on the manifold ( that is, the internal points of the paths belong to different layers).

Thus, the one-dimensional tree of the Morse function consists of a set of end points, plus no more than a countable number of simple arcs that pairwise intersect at no more than one point, which is also a branch point.

It should be noted that for a degenerate Morse function (the presence of a degenerate critical point), an arbitrarily small stirring of the Morse function can be achieved so that at each critical level c (i.e., on the set of points p for which ƒ (p) = c) lies exactly one critical point. In other words, critical points that fall on one level can be divided into close levels. Morse functions having exactly one critical point at each critical level are called simple. For a non-degenerate Morse function, the Kronrod-Riba graph is a tree with T triple branch points, K = T + 2 end points and P = 2T + 1 edge connecting K + T = 2T + 2 vertices of the graph (Arnol VI Smooth functions statistics // Abdus Salam International Center for Theoretical Physics, ICTR. 2006. IC / 2006/012. 9pp.). Note that not every finite (or infinite) graph will be a Kronrod-Reeb graph for some proper smooth function with isolated critical points on the surface (Sharko V.V. Smooth functions on non-compact surfaces. // Zbirnik prats Institute of Mathematics, National Academy of Sciences of Ukraine 2006 , vol. 3, No. 3, 443-473). There is a condition that guarantees that the graph will be the Kronrod-Reeb graph.

Let ν be the vertex of the graph G. The graph Г _ν = Г - ν is obtained from the graph Г by removing the vertex ν and all edges incident to it. A vertex v is called breaking if the graph Γ is a disconnected set. Let Ω denote the set of vertices of order 1 in the graph G.

A graph Γ satisfies condition (W) if:

a) D is a connected set;

, то $${\text{Г}}_{\text{ν}}^{\text{i}}{\displaystyle \cap \text{Ω}\ne \varnothing }$$

;b) For each dividing vertex ν of Γ such that if among the connected components of Γ _ν there are finite graphs $${\text{G}}_{\text{ν}}^{\text{i}}$$

then $${\text{G}}_{\text{ν}}^{\text{i}}{\displaystyle \cap \text{Ω}\ne \varnothing }$$

;

c) If the graph Γ is finite, then Ω consists of at least two vertices.

Let Γ be a graph with condition (W). Then on Γ there is an increasing (decreasing) function. Each graph with condition (W) is a Kronrod-Reeb graph of a smooth function with isolated singularities defined on the surface.

Suppose that a graph Γ satisfies condition (W), and a function of height g is given on it, then there exists an oriented surface M and a smooth function ƒ with isolated singularities on it, which is a function of height for which the Kronrod-Reeb graph is isomorphic to G.

The Kronrod-Riba graph of any eigenfunction smooth with isolated singularities defined on the surface (compact or non-compact) satisfies condition (W).

For a graph to be a Kronrod-Reeb graph a smooth function ƒ with isolated singularities on some surface, it is necessary and sufficient that it satisfies condition (W).

The Kronrod-Reeb graphs allow one to calculate the number of possible topologically different Morse functions on a sphere (Nicolaescu LI Counting Morse functions on the 2-sphere // Compositio Math. 144 (2008) 1081-1106), establish the topological equivalence of two Morse functions, defines this surface uniquely up to diffeomorphism.

For a Morse function, the topology of many levels is associated with critical points and the gradient field of the function. This relationship provides an opportunity for a formal description of the surface in algebraic form. Unlike other topology methods based, for example, on Kronrod-Reeb trees, the use of the Morse-Smale complex provides a description of two-dimensional and multidimensional surface properties, which allows one to obtain a representation of the local topology of smooth functions and segment the surface into regions with a "uniform" gradient field . In other words, the geometry of a smooth surface is mapped into simple geometric images (simplicial complexes, Fig. 13), the analysis of which allows us to describe the structural features of the original surface, of different dimensions: zero, one and two-dimensional. These structural features are easily interpreted in geomorphological terms, for example, peaks (peaks) and troughs (pits) correspond to zero-dimensional objects, one-dimensional lines of a network of thalwegs, watersheds, two-dimensional - monotonous slopes. This ensures the connection of geometric properties through structural features with geomorphological semantics.

More recently, Morse theory has been extended to piecewise linear functions (Gyulassy A.G. Combinatorial Construction of Morse-Smale Complexes. Dissertation Submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy. University of California. 2008). Such an extension to piecewise linear functions and discrete grids allows the use of Morse theory to solve practical problems with real data sets.

The critical points of the Morse function are those points on a two-dimensional surface where the function is stationary. In order to fully describe the Morse function, we must highlight its structural features. To do this, you need to define a vector field called a gradient.

The gradient of the Morse function is a vector field on S ² . We integrate this vector field in order to decompose S ² into regions with homogeneous flows.

для всех t∈R.The curve l (t) is called the integral line ƒ if - $$\frac{\text{d}}{\text{ds}}\text{l}\left(\text{t}\right)=\text{d}ƒ\left(\text{l}\left(\text{t}\right)\right)$$

for all t∈R.

In other words, the integral line is a path for which the tangent vector is parallel to the gradient at each point of the path.

Integral lines represent the flow along the gradient between critical points. At any point where the gradient is not equal to zero, the integral line passing through this point can be found by tracing forward and backward along the gradient vector field. The integral lines are open 1-manifolds, and since S ^{2 is} compact, restrictions exist at both ends. The following definition defines the upper and lower limit of the points of the integral line.

называется источником интегральной линии l(t) и обозначается org(l).Limit $$\underset{\text{s}\to -\infty }{\text{lim}}\text{l}\left(\text{t}\right)$$

is called the source of the integral line l (t) and is denoted by org (l).

называется приемником интегральной линии l(t) и обозначается dest(l).Limit $$\underset{\text{s}\to +\infty }{\text{lim}}\text{l}\left(\text{t}\right)$$

is called the receiver of the integral line l (t) and is denoted by dest (l).

Integral lines on smooth functions have the following properties:

1) two integral lines either intersect or coincide;

2) integral lines cover all S ² ,

3) sources and receivers of integrated lines are critical points ƒ.

Integral lines are monotonic and, therefore, org (l) ≠ dest (l). These properties provide the condition that each point S ² has exactly one integral line passing through it. All points on S ² can be classified as sources or receivers.

The listed properties follow from the standard differential calculus.

The integral lines that connect the maximum and the saddle, or the minimum and the saddle, are called separatrix lines. In geomorphology, the lines of separatrices that connect the minima and saddles are usually called gullies, or the lines of the valleys, and those that connect the saddles and maxima are called the lines of the ridges.

Stable / unstable manifolds. Let p be some critical point of the function ƒ: S ² → R. An unstable manifold for a point p is a set of points that belong to an integral line for which the source is p, U (p) = {p} U {x∈S ² | x∈im (l), org (l) = p}. A stable manifold for a point p is the set of points that belong to the integral line for which p is the receiver, S (p) = {p} U {x∈S ² | x∈im (l), dest (l) = p}. Here im (l} is the map of the curve l∈S ² .

The stable manifold S (p) of the critical point p with index i = i (p) is an open cell of dimension dim (S (p)) = i.

Note that the unstable manifolds of the function ƒ are stable manifolds of the function - ƒ, since d (-ƒ) = - dƒ. Thus, two types of manifolds have the same structural properties. Therefore, unstable manifolds of the function ƒ are also open cells, but with dimension dim (U (p)) = 2-i, where i is the index of the critical point.

The Morse function is called the Morse-Smale function if stable and unstable manifolds intersect only transversally. In two dimensions, this means that the stable and unstable 1-manifolds intersect at angles close to straight lines. Their intersection point is necessarily a saddle; their intersection at a regular point would be contrary to property 1) for integral lines.

The connected components U (p) ∩s (q) for all critical points p, q∈S ² are called Morse-Smale cells. We are talking about cells measuring 0, 1 and 2, as vertices, arcs and regions, respectively.

A set of Morse-Smale cells forms a Morse-Smale complex. The one-dimensional skeleton of the Morse – Smale complex consists of critical points and separatrix lines. This skeleton is called a critical network.

Since U (p) ∩ S (p) = {p}, and if p ≠ q, then U (p) ∩s (q) is the set of regular points r∈S ² that lie on the integral lines l with org ( l) = p and dest (l) = q. It is possible that the intersection of stable and unstable manifolds consists of more than one component.

As a result of the intersection of the stable and unstable manifolds mapped, each vertex of the Morse-Smale complex is a critical point, each arc is half of the stable or unstable 1-manifold of the saddle, and each region is one of the components of the intersection of the stable 2-manifold of the maximum, and the unstable 2-manifold minimum.

We note that each cell of the Morse – Smale complex has a simple geometry — an almost monotonic function that can be well approximated by a polynomial equation.

The cells of the Morse – Smale complex are triangulated to go over to simplicial complexes — the combinatorial basis of algebraic topology (L. Pontryagin, Fundamentals of combinatorial topology. - M.: Nauka, 1976. - 136 pp.). If a linearly independent collection of k + 1-st points {e ₀ , ..., e _k } ∈R ^{k is given} , then the convex hull constructed on these points is called the k-th simplex. Points from a set of points are called vertices. On Fig shows all simplexes of small dimension for 0≤k≤3.

The proposed method is as follows.

The method of cartographic display of two-dimensional distributions defined in digital form includes converting images of discrete graphic distributions into a continuous grayscale form with their further presentation in the form of isolines - excidence. In optical modeling, digital values of a sign are encoded at a given point on the tablet with optical symbols - different-sized spots with an optical density proportional to the value of the sign. The construction of the terrain is performed by interpolating the points of heights and / or depths in the form of two-dimensional irregular rational fundamental splines, by constructing a two-dimensional spline function, defined as the tensor product of one-dimensional splines.

When constructing the terrain, iterative functions and wavelets are determined to represent the fractal terrain, by forming the Kronrod-Rieb graph and Morse-Smale complexes for the piecewise-linear surface, while simplifying the piecewise-linear surface using the Kronrod-Rieb graph structures obtained for it and Morse-Smale complexes; estimation of fractal relief parameters based on the given structures of the Kronrod-Reeb graph and Morse-Smale complexes.

Representation of the relief using IFS and wavelets allows you to:

binary surface generalization defined on a regular grid of points;

imitation of increasing the scale of resolution of the source information;

identification of sudden changes in function values;

calculation of fractal parameters of a function;

noise filtering.

The basic algorithms for software and mathematics allow:

restore relief from discrete measurements;

solve the inverse IFS problem;

choose the optimal continuous and discrete wavelet families to represent the relief.

Methods for describing the relief using Morse functions, Kronrod-Reeb graphs, and Morse-Smale complexes provide the ability to:

topological coding of relief forms;

cartographic generalization;

recognition of geomorphological objects;

formal classification of geomorphological objects;

tiling the surface of the relief with a family of parametrized (polynomial) functions defined on Morse-Smale cells;

hierarchically assess the degree of similarity of two elevation maps representing the same area, at the same or different scales;

assessment of the reliability of the selected relief forms, taking into account the error and power of the source information;

assess the degree of sufficiency of a set of point measurements to restore the relief with a given detail.

To implement these features, the following list of basic algorithms is developed:

relief reconstruction from discrete measurements using topological;

the formation of the Kronrod-Reeb graph for a piecewise linear surface;

the formation of Morse-Smale complexes for a piecewise linear surface;

simplification of a piecewise linear surface using the structures of the Kronrod-Reeb graph and Morse-Smale complexes obtained for it;

estimation of fractal relief parameters based on the given structures of the Kronrod-Reeb graph and Morse-Smale complexes.

The selected postulate formulations describe the simplest surface structure, but this is not a limitation. It is quite possible to move on to more complex structures, for example, to allow the existence of "caves", "tunnels", etc. The apparatus of topology remains quite adequate in this case, it is only necessary to apply a more complex technique of algebraic topology - the theory of homology.

The proposed method is implemented on the elemental basis of microprocessor technology, video plotters and video plates, information connected with the measuring equipment of the parameters of the relief of the seabed, etc.

For example, the control unit is made on the basis of microprocessors of the ATMES AVR family with special software that allows input / output of information and conversion of signals from several navigation sensors (instruments), for example, an echo sounder, which is a multi-beam echo sounder of the R2Sonic 2022 type with a central beam 1 degree, a high-frequency profilograph, which is a Chirp-type linear frequency modulation profilograph, side-scan sonars, which are batim an insulating sonar type «Benthos C3D» with a swath width of 1.2 km, and the unit conversion map information is based on a microprocessor DSP - processor running embedded «UCLinux» operating system.

A microprocessor based on a DSP processor is a device that provides software and hardware integration of the individual units that make up the hardware. The microprocessor allows you to perform operations on 32-bit numbers in a floating point format, which ensures the accuracy of calculations sufficient to solve most control and data conversion tasks. The processor clock speed is 400 MHz. In addition to the processor, the cartographic information conversion unit includes SDRAM memory microcircuits, flash memory microcircuits, and input-output interface microcircuits. Such a system construction allows real-time complex computational tasks to be solved; a large amount of system RAM allows the implementation of resource-intensive algorithms.

Information sources

1. Copyright certificate SU No. 365562, 1970.

2. Copyright certificate SU No. 640113, 1978.

3. Patent RU No. 2079891.

4. Berlyant A.M. Cartography. - M.: Aspect Press, 2002 .-- 336 p.

5. Suvorov S. G., Dvoretsky E. M., Kovalenko S. A. The technique of creating digital elevation models of high accuracy // Information and space. No. 1, 2005. - S.52-54.

6. Patent No. 2415381.

7. Golovanov N.N. Geometric modeling. - M .: Fizmatlit, 2002 .-- 472 p.

8. Kvasov B.I. Methods of isogeometric approximation by splines. - M.-Izhevsk: Research Center “Regular and Chaotic Dynamics”, Institute for Computer Research, 2006. - 416 p.

Claims (1)

A method for cartographic display of two-dimensional distributions defined in digital form, which includes converting images of discrete graphic distributions into a continuous grayscale form with their further representation in the form of isolines - excidence, in which optical modeling encodes digital values of a sign at a given point on the tablet with optical symbols - different-sized spots with optical density proportional to the value of the feature, the construction of the terrain is performed by interpolating points in heights and / or depths in the form of two-dimensional irregular rational fundamental splines by constructing a two-dimensional spline function, defined as the tensor product of one-dimensional splines, characterized in that when constructing the terrain, iterative functions and wavelets are defined to represent the fractal relief by forming a piecewise-linear surface the Kronrod-Reeb graph and Morse-Smale complexes, while simplifying a piecewise linear surface using the graph structures obtained for it ronroda Riba and complexes Morse-Smale; The fractal parameters of the relief are estimated based on the given structures of the Kronrod-Reeb graph and Morse-Smale complexes.

Images