The article deals with algorithms for signal and image processing in presence of interference from the underlying surface, icker noise, and other types of interference with fractal properties. Models of fractal interference are considered on the basis of the statistical approach. Application of the fractal Brownian motion model with fractional dimension is proved for the statistical description of low-frequency icker noise is proved, and, also, for describing the re ection coe cient of the sensing signal from the background of natural origin under obtaining the radar images. A maximum likelihood algorithm for detecting signals and extended objects, as well at the background of additive fractal noise are developed. The characteristics of detection of extended objects at the background of fractal noise, as well as a low-frequency signal at the background of icker noise are calculated. The statistical modeling of the object detection algorithm on raster and complex images of the earth's surface was carried out and its e ciency was evaluated. It is established that usage of the fractal models allows improving the e ciency of signal and image processing at the background of noise in cases where there are no other di erences between them.
Contemporary studies have made it possible to establish the self-similarity and
fractional measure properties of signals and images obtained by receiving signals
re ected from various objects [
Methods and algorithms for optimal processing of signals and images based
on probabilistic models and the theory of optimal statistical solutions are
wellknown and widely applied. The most general formulation of the problem and the
model of signals and interference are implemented in the
estimation-correlationcompensation approach [4{6]. The statistical approach is also used in processing
the signals and images with fractal properties, for example, fractal Brownian
motion [
Correlation dimension of signals and interferences
The most complete statistical approach is applied to processing the fractal signals
and images. For description of signals and images properties, one of the de
nitions of the fractal measure is used, i.e. the correlation dimension. In [
D = lim lim r!0 n!1 log Cn(r) log r ; where Cn(r) is the correlation integral
Cn(r) =
2 n(n 1) n n X X H(r m=1 j<i kx y kE ); where j:::jE is the norm in the E-dimensional space of embeddings, H(x) = (0; x < 0
is the Heaviside function. The most plausible estimate of the
correla1; x > 0
tion dimension of the proposition in [
For given value of correlation dimension the correlation integral (2) is C(r)
rD that allows one to represent distances between vectors as random value with
the power law of distribution. In the case of distance norming rm=rmax, the
distribution law is F (r) = rD, and multidimension probability density function
is [
M M w(r ; D) = Y w(rm) = Y DrmD 1: (4)
Since the resulting multidimensional density (3) is also a function of unknown
dimension, it can be considered as a likelihood function. The maximum likelihood
estimation of the correlation dimension is obtained as a result of solving the
following extremal problem:
Using logarithm of the likelihood function and extremum condition
ln w(r ; D) =
0, it is possible to calculate the maximum likelihood estimate [
n D =
M M P ln rm m=1 : The above estimate is e ective and asymptotically unbiased. Analysis of the displacement and variance of the estimation error was carried out in [15]. Presence of the fractal noise alters the properties of the observed signals and images, that is re ected in their correlation dimension. Consider the situation when a fractal signal with the dimension DS is observed against the background of additive fractal noise with the dimension DJ . Since in the general case this analysis is extremely complicated, let us consider the case when the intensity of the interference is much greater than the intensity of the useful signal. Considering the signal and interference in the pseudo-phase space and using the Taken's tower, it can be assumed that the presence of a weak signal slightly changes the distances between the vectors of the observed process by a value rm << rm. Under these assumptions, the asymptotic expression for estimating the correlation dimension of the sum of the signal and the interference has the form ^ D1 =
0
M ; where the second term describes the signal presence. The factor D = 1 XM M m=1 rm is a random variable with the asymptotically Gaussian probability distribution when M >> 1. The signal optimal processing against a noise background reduces calculation of a su cient statistics to calculation the logarithm of the likelihood ratio. rm 3
Fractal Brownian motion as a model of fractal signals
and interferences
Fractal Brownian motion is used as a model of fractal interference. Samples of
FBM are formed by one of famous methods [
2 and Hurst exponent H. Dimension of FBM is determined by D = 2 H for
one-dimensional FBM and D = 3 H for two-dimensional FBM. If determined
signals are observed at the background of additive fractal interference in the
form of FBM, then detection and identi cation are complicated. Therefore, one
of the actual problem is synthesis of optimal detection algorithms for signals
at the background of additive fractal interference in the form of FBM. Let the
signal sn is observed at the background of FBM interference xn
yn = sn + xn; n = 1; : : : ; N;
where N is the amount of samples of observed process, xn are the independent
samples of FBM interference. The fractal Brownian motion is a Gaussian
random process; therefore, its properties are completely determined by correlation
matrices for one-dimensional signal [
M f(X(t2)
X(t1)) (X(t4)
X(t3))g = where correlation matrix R depends on Hurst exponent H, and probability density function can be considered as likelihood function w( X =H) = (2 )M=2pdetR(H) exp
X TR 1(H) X : Likelihood ratio of increments of FBM interference and observing determined signal at the background of interference is (6) (7)
Evaluation of determinant and conversion of matrix of increments R are very di cult computational problem in case of randomly given moments of time. Therefore, in several cases, it is useful to consider only non-correlated increments within non-crossing time intervals. In such cases the correlation equals M f Xi; Xj g = ij DX ti2H , where ij is the Kronecker delta, i = 1; : : : ; N 1. In this case, the matrix R is diagonal and its determinant equals detR = DXN QnN=11 t2nH . Multidimensional probability density function (PDF) of the FBM increments is where we can obtain the likelihood ratio of Gaussian signal at the background of fBm interference = s
N 1 2N Q
n=1 (2Dv)(N 1)=2 t2H n exp " N 1 1 X 2 n=1
1 2Dv
x2n 2 t2H n
# x2n : = XM SHmY m +GY0 HmSm m=1 m2H+1 + N0
SHmSm ; (9) (10) (11)
This algorithm is quasioptimal, because it does not consider the increments correlation at the overlapping intervals. But it has signi cant computational advantages, since of absence of matrix inversion operations of high computational cost.
Getting the non-correlated samples of FBM is possible as a result of
transition in the spectral eld. It is well-known [
G0 jXmj2 = m2H+1 + N0; m = 1; : : : ; M = N=2.
Multidimensional probability density function of the FBM spectral components is w(X ) =
M M Q m=1 1
G0 m2H+1 + N0 exp "
M X m=1 jXmj 2
1
G0 m2H+1 + N0 # ;
Thus, in spectral area, the algorithm of likelihood ratio evaluation turnouts simpler because an operation of matrix inversion is excluded. The FBM detection in spectral area against the background of Gaussian noise is made as a result of calculation of statistic (9) and comparing it with a threshold. If a spectrum of signal is of low frequency with harmonics jSmj2 = mG02 , then asymptotic interference immunity of signal processing exists against the background of fractal interference and depends on the Hurst factor: if H < 1=2, then interference immunity is nondecreasing function on signal frequency; if H > 1=2, then the interference decreases with decreasing the signal frequency. 4
It is shown that methods of the theory of optimal statistical solutions can be successfully applied also to processing of the fractal signals and images against the background of additive fractal noise. The basis for the e ectiveness of statistical methods is the irregular character, as well as the relatively large amount of observable data. Under these conditions, the statistical description of fractal signals and images is produced by various methods: the use of a one-dimensional and two-dimensional fractal Brownian motion model, and a statistical description of distances between vectors in a pseudo-phase space. This approach allows us to obtain processing algorithms based on the theory of optimal statistical solutions for solving various problems: detection, discrimination, delineation of boundaries, estimation of parameters, and analysis of the processing e ciency. At the same time, the statistical description is not obtained for all fractal signals and images and their characteristics, and this makes it important to continue research in this direction. 5
The research is supported by the project of the Ministry of education and science of the Russian Federation 8.2810.2017 in the Ryazan State Radio Engineering University.