The article considered the way of the fuzzy image representation with using fractal models of images. According to it, the image presented inthe form of a tree range blocks which correspond to image blocks. Everyone range block represents the structure describing similarity of this block to another block of theimage. As a result, it turns out treelike fractal representation of the image describing the distribution of property of self-similarity in the image. The basic possibilities of construction of algorithms of digital processing of the imageswhich based on given representation are listed
Most methods of digital image processing (including defectoscopy images) refer to one of two groups: spatial or frequency. Spatial methods make it possible to identify an object in an image uniquely, and frequency methods allow one to adequately analyse the spectral components of a two-dimensional signal (useful component and noise). In this case, spatial methods, as a rule, are subject to interference and require the source signal to be cleaned of noise, possibly with the loss of useful information. For frequency methods, the main problem is the localisation of objects of interest in the coordinate system of the image.
The use of space-frequency approaches, for example, based on wavelet transforms, allows you to detect objects in images against background noise more accurately. This approach is since the wavelet spectrum contains complete information about the original signal, including the spatial coordinates of the frequency components, and at the same time separates the useful and noise components.
However, the overwhelming majority of digital image processing algorithms based on the algorithms for extracting useful information, which often does not allow the technical vision systems to solve non-trivial problems. In this regard, it is proposed to combine the mathematical theory of fuzzy sets and fuzzy logic with the proven fractal methods of digital image processing.
To develop a system of new features, need to create a new model of a digital image. It is proposed to modify the fractal model developed by the project manager by using a fuzzy distance in it as a measure of similarity of the image areas. This modi cation allows expanding the hierarchy of representations of the original image, thereby increasing the amount of useful information about the original image.
With increasing complexity of the tasks solved, it is expedient to use the specially oriented methods that to the construction of models, methods and algorithms that take into account the incompleteness and inaccuracy of the initial data. It is in such situations that the theory of fuzzy sets and fuzzy logic proves to be the most constructive. Already achieved theoretical results allow to consider fuzzy logic as a promising tool for the analysis of heterogeneous information, in the form of complex qualitative linguistic descriptions and quantitative data. In modern works, we considered methods and algorithms for using fuzzy logic and fuzzy set theory to solve problems of digital image processing.
Analysis of the used fuzzy image processing schemes [
In connection with the preceding, it is proposed to develop a fuzzy mathematical model of a digital image based on its description by the terms of fuzzy sets using fuzzy logic, which is the basis for constructing new algorithms for extracting information about features and image processing algorithms.
The fractal model of the digital image developed and described in [
n f 0 = X k=1
[si w~ Bkdii;li [f ] + oi]: where is an operator of extracting a domain block with the index di from the original image f . The transformation that performs the change in the color characteristics of the domain block and the geometric transformation has the following form (1) (2) (3) (4) (5) (6) D =
Bkdii;li [f ] D0 = si (w[D]) + oi:
M f 0 = X i=1
[D0]: where m,n - images size. For each range block of the image Ri, the domain block Di is selected which best approximates the range block. In other words, there is a pair (Ri; Dj ) for which the metric (6) is minimal.
Ri = D0 = si (w[D]) + oi: For comparison of two images f , h metric used RMS: g(f; h) = X i;j r 1 m n [f (i; j) h(i:j)]2;
To generate a fractal image code, we used a quadtree of range blocks. The root of the tree is the image itself.
Expression 5 can be represented as follows
Ri = D0 + i; i = g(Ri; D0):
A = f(x; A(x)) jx 2 Xg
In other words, the fractal model describes the original image using the quadtree. Each leaf of a tree is a range block of the image, which corresponds to a domain block so that the approximation error is minimal:
A fuzzy set A in a nite set X = x1; x2; ; xn may be represented mathematically as where the function A(x) : X ! [0; 1] is the measure of the degree of belongingness or the membership function of an element x in the nite set X.
An Intuitionistic Fuzzy Set A in a nite set X may be mathematically represented as
A = f(x; A(x); a(x))jx 2 Xg where the functions A(x); A(x) : X ! [0; 1] are, respectively, the membership function and the non-membership function of an element x in a nite set X with the necessary condition
To use all the features of fuzzy sets is necessary to modify the model (5) and bring it to the fuzzy mind. During the construction of the fractal image code, a set of range blocks, domain blocks and approximation error values are generated: Then, using 8, expression 9 can be expressed using a fuzzy set
R = fRig; Ri = fDj ; i;j g: R = fDj ; R(Dj ); 1
R(Dj )jDj 2 Dg:
R(Dj ) = j = max; max = 255=pm n; where max the maximum approximation error, which is de ned as the di erence between absolutely black and absolutely white images: where m; n - image size. (7) (8) (9) (10)
Let the image f is a graph G = (V; U ), where V is the list of range blocks of the image, U is the connections between the range blocks. Then the process of forming a tree of range blocks of the image can be described as follows. There is a range block of zero level R0, which corresponds to the whole image and v1 = R0. It contains range blocks of the rst level R11; R21; R31; R41 2 R0: v2 = R11; v3 = R21; v4 = R31; v5 = R4;
1 u1 = (v1; v2); u2 = (v1; v3); u3 = (v1; v4); u4 = (v1; v5) 2 U : Continuing the formation of the fractal code,a complete tree of range blocks is obtaining, from which the image is constructed. A fragment of the tree of range blocks is shown in Fig. 1. (11) (13)
The operator for the formation of a fractal code called a direct fractal transformation: F (f; D) = ; where is the resulting fractal code of the image f by the operator F . An additional parameter of the operator is the list of domain image blocks D involved in the code generation. There is an inverse fractal transformation that forms the image by the fractal code: f 0 = F ( ; D) = F (F (j; D)): (12) To properly restore an image from its fractal code, it is important that the transformations use the same list of domain blocks D. In the original version, the generated fractal code is full, that means that each vertex of the graph has one parent and four descendants, but the length of the descending chain does not exceed the depth of the quad, which speci ed at the initial stage of the code construction. If the full tree used as the parameter for restoring the image by the operator 12, then the resulting image f 0 is similar to the original image.
For obtaining modi ed images, incomplete trees of range blocks used. To form an incomplete tree, use the operator:
= ( ): The form of the operator depends on the required result. For example, by removing some branches of a fractal tree, you can obtain an image in which the areas for which remote branches respond are smooth, and the remaining areas are clear. Using 10, the threshold operator can be represented as: ( ) = fR0; Dg; R0 = fRijR(Dj ) > g: (14) The operator 16 allows us to use only range blocks whose value of the membership function is greater than a given threshold .
To obtain a fractal representation of the image from the fractal code, need to remove the leaves of the tree. The result is a smoothed, fractal image.
Fig. 2 shows an example of such a decomposition of a test image. The resulting set of images allows for a more detailed analysis of the original image. The transition to other scales allows you to move away from small and random details. It is better to reveal the "internal" structure. It is of interest to consider the structural elements allocated at di erent scales of the representation of the image.
The reason for this approach is the following. Structural elements of the image, such as contours, segments, skeleton, primarily carry all the meaningful information about the original image. At the same time, they have a much more compact representation and are better suited for analysis, both automatic and subjective, than the original halftone image. In the case of automatic analysis, this is determined by the possibility of a formal description of the structural elements, for example, the parameters of the curve (for the contour, the skeleton) or the area geometric object (for segmentation).
Most algorithms for digital image processing using fuzzy systems are carried out in three stages:
the formation of an image fuzzi cation, direct processing, and, if necessary, image recovery from
an image defuzzi cation. At the same time, the quality of image processing by such algorithms
depends more on the correct choice of the fuzzy representation used [
An uncharacteristic area in the image is a section that should not be present in a given image,
for example, a fracture in the image of the microstructure of the metal, a foreign object on the
surface on the aerial photograph, and the like. The task of nding such areas in images is actual
in metallography for the search for defects on the surfaces of materials, in radar to search for
foreign objects, in medicine for the detection of foreign bodies in the blood and other areas in
which image processing is actively applied [
To detect uncharacteristic blocks in the image, we formed a fractal code using a list of speci c blocks instead of domain blocks.
Fractal coding due to the use of approximation implies loss of information, since there is an error in the approximation of rank blocks by domain blocks:
Ri = wi(Dj ) + i: (15)
In other problems this error can be neglected, because the fractal code of the image is used, without the need for decoding. In this problem, we need to obtain a vector f g = f 1; 2; :::; kg; k = 1::NR , then we can obtain a two-dimensional function (x; y): (x; y) =
f i (x; y) 2 Rig where f i (x; y) 2 Rig - the membership function that determines the amount of fuzzy uncharacteristic of the ranking block Ri in the image.
The result is a three-dimensional diagram showing how well each block of the image approximated by the characteristic areas (Fig. 3). It is possible to establish the presence and location of image sections poorly described using speci c areas, by carrying out the threshold processing of the chart
The fractal image code represented in the form of a graph G, in which the vertices of V are range blocks R, and the arcs are the nestedness relations of range blocks. At the same time, the structure of the code remains unchanged. This approach changes only the form of representation and the mathematical apparatus of research.
The described approach to the fractal representation of images can be used to form a set of features re ecting the internal structure of the image since small details discarded. Further on, based on the proposed description, algorithms for isolating contours, image segmentation, skeletonisation can be constructed. This support the usability of the proposed algorithm in practical problems of nondestructive testing.
Also, new fractal attributes of digital images o ered - speci c sites. These signs show which parts of the image are involved in its formation, i.e. according to the self-similarity distribution within the image, it is possible to identify areas using which you can reproduce the image with the highest accuracy. We developed the algorithm for searching for areas of the image that are not characteristic of this class of images. The algorithm based on the fact that uncharacteristic areas of the image poorly approximated by the speci c sections of this class of images. Acknowledgments ThereportedstudywasfundedbyRFBRaccordingtotheresearchproject17-47-330073.