The paper considers the in uence of the image pixels position in a one-dimensional sequence on the result of correlation dimension evaluation. The sequence formed as a result of reading of the two-dimensional pixel image. Dimension evaluation is performed by maximum likelihood method, using image elements ordering and creating vectors in pseudophase space by Takens theorem, and is used as a texture feature if texture processing and detection of objects. The two di erent scanning methods are considered. There is a problem of optimization of the scan path in order to maintain correlations between pixels in sequence. The rst method is to scan on the criterion of maximum correlation between the adjacent sets of pixels. The second method deals with choice of scan direction by criterion of scalar product maximum of chosen vector and previous one. An estimator of correlation dimension is evaluated.
Problem of detection of objects and its edges, as well as the separation of areas at images is one of the most actual in modern thematic image processing. The distinction between areas and objects from each other is carried out by means of evaluation of certain parameters | textural features. We can make a decision about di erence of objects on the basis of the di erence of characteristic values. One of textural attributes are statistical features, which are determined by taking into account the statistical properties of a sequence of pixels. Among them there is a correlation dimension, which is formally de ned as the probability that the distance between the vectors formed from samples in the time series in pseudophase space is less than a predetermined constant value.
One of the problems of dimension estimation of the image is a violation of
correlations between pixels during ordering them into a one-dimensional sequence
after reading. Radar images usually have a complex structure, which means that
the relationship of pixels is in di erent directions. Conventional methods
perform scanning reading pixels horizontally, vertically or diagonally. In addition,
there are space- lling curves, which allow to read all of the pixels in a selected
area once, sequentially selecting only adjacent pixels. Such a curve is, for
example, the Peano-Hilbert curve [
The aim of investigation is to develop a method of scanning an image, providing more e cient textural processing and evaluation of the correlation dimension by usage of fractal properties of the analysable image fragment as well as algorithms of the fractal dimension estimation that are optimal by the maximum likelihood criterion. An improvement of methods and algorithms is performed by taking into account statistical and geometric dependency of data. An algorithm for data scan within image frame is proposed as well. 2
A maximum likelihood estimation of correlation
dimension
The most precise estimates of correlation dimension are obtained by
maximum likelihood algorithm [
dr
In paper [
m=1
: 0; r 2= [0; 1):
Dependent distances are formed in accordance with the following rule [
rm 2 [rmin k; rmax k); k = 3; ::; N; >> 0; rm 2= [rmin k; rmax k); k = 3; ::; N; > >: m = N + k 3:
rmin k)D 1; w1(r1=D) = 8 N 1 < Q
m=1 : 0; r1 2= [0; 1):
Dr1Dm 1; r1 2 [0; 1); 2) N 2 independent random numbers are generated with the power law distribution of probabilities within the range of values (rmax k; rmin k), k = 3; ::; N ; these numbers set conditionally independent distances from the second vector to other N 2 vectors except the 1st vector. Minimum and maximum values are de ned by the triangle rule: rmin k = jr12
r1kj ; rmax k = r12 + r1k; k = 3; ::; N:
Combined probability density function of these N between vectors is as follows: 2 independent distances (2) (3) (4) As minimum and maximum values rmin k; rmax k depend on distances with numbers 1; ::; N 1, then distances r1; r2 are also statistically dependent and their combined probability density function is derived subject to (2), (3) : w12(r1; r2=D) = w1(r1=D)w2(r2=r1; D): 3) Coordinates of the rst and second vectors in space of embeddings DE = 2 are x1 = 0; y1 = 0; x2 = r12; y2 = 0. Coordinates of other i = 3; ::; N vectors are de ned by geometry of their position using the cosine theorem and distances from i-th vector to the rst and second vectors: xi = r122 + r12i r22i ; yi = pr12i xi2. 2r12 For convenience, sign of coordinates yi is chosen positive.
4) As a result of coordinates evaluation of all vectors we can calculate other N22 52N 3 dependent distances r3 = iri=j=3; p::;(Nxi; j =xji)+2+1;(:y:;iN yj )2 . As distances r3 are absolutely de ned by distances r1; r2, than they do not contain additional information for the correlation dimension estimation.
Optimal estimates of the correlation dimension should take into account statistical and geometrical dependences of vectors, which are represented in the likelihood function. A maximum likelihood estimate is obtained as a result of solving of the optimization problem:
Dest = arg
max 0<D<DE w12(r1; r1jD): (5)
The solving of equation (5) is realized by setting equal to zero of rst order derivative of w12(r1r2jD) with respect to D. A maximum likelihood estimate of correlation dimension is obtained:
D^ = (3N 3) "N 1 2N 3
X ln r1m + X ln(rm m=1 m=N
# rmin k) : 3
Optimal scanning by maximum correlation coe
cient
It is necessary to choose an approximating model for processing of signals and
images as well. The main principle of model choosing is to provide a maximum
conformance of model and real signal parameters. A 2D Fractional Brownian
motion is widely used as a model of a chaotic fractal image. Samples of 2D
fBm are formed X(k; m); k; m = 1; ::; N , by one of famous methods [
M = f(X(k1; m1) 2
X(k3; m3)) 2 (X(k2; m2) 2
X(k4; m4))g = = (6) (7) Two-dimensional Gaussian process X(k; m) is used for representing of image with zero mathematical expectation, dispersion Dx and correlation matrix R = fR(i1; j1; i2; j2) = Dx exp [ ( xji1 i2j + yjj1 j2j)] ; i1; i2; j1; j2 = 1; ::; N g : (8)
In a review paper [
In this paper two method of image scanning is discussed. The rst method deals with choice of scan direction by criterion of correlation coe cient maximum (7), (8) of each subsequent pixel. This method uses image characteristics, averaged by signi cant ensemble of samples, but it doesnt consider properties of particular image. The second method deals with choice of scan direction by criterion of scalar product maximum of chosen vector X and previous one Y. This method doesnt require a priori information about statistical properties of image. Data of particular image are most fully used for construction of optimal scan path. The optimal scanning algorithm consists of a following steps: - rst vector consists of M image pixels from rst line of current frame; - one of Q prede ned directions from last pixel to end of next vector is considered;
- standard scalar product is evaluated for pair of vectors previous X and each of Q considered directions Y | as follows:
RX;Y =
(X; Y) jjXjj jjYjj (9) - maximum value of scalar product de nes a pair of vectors and thereby the direction of scan trajectory; - scan trajectory moves to new pixel, which is last for chosen vector; - the algorithm is repeated until majority pixels are covered.
Figures 3-6 represent optimal scan paths for Gaussian and Fractal Brownian images, obtained by two proposed methods. 4
Dimension evaluation is performed by maximum likelihood method [
Results of analysis of proposed algorithm show, that appliance of maximum likelihood algorithm demands provision of maximum correlations between image pixels in one-dimension sequence. Scanning by maximum correlation coe cient criterion provides bigger di erence between values of correlation dimension estimates and therefore simpli es solution of object detection problem.
Acknowledgments. Investigation was carried out with the Russian Science Foundation support, project number 14-19-01263.