ESTIMATING THE POSITION OF AN IMAGE WITH UNKNOWN INTENSITY SHAPE

Resume The problem considered is to estimate the image position of a spatially extended object. It is assumed that the shape of the image intensity is a priori unknown, but it can be predicted with some error. In order to synthesize the estimate of the image position, the quasi-likelihood version of the maximum likelihood method is used. Behavior of the signal function in the neighborhood of the real image position is studied. Characteristics of the resulting estimate, such as bias and dispersion, are found by means of the local Markov approximation method. Influence of non-uniformity of the received image intensity upon the estimate accuracy is demonstrated by an example of receiving the rectangular image with linearly varying intensity. ESTIMATING THE POSITION OF AN IMAGE WITH UNKNOWN INTENSITY SHAPE Yury E. Korchagin1, Viacheslav N. Vereshchagin1, Alexander V. Terekhov2, Kirill A. Melnikov2,3,*


Introduction
Systems for estimating (predicting) the position of a spatially extended object by its own image are widely used for security purposes, traffic monitoring and management, in railway transport and in other sectors [1][2][3][4]. However, the information processing algorithms, applied in the specified systems, can often provide the simplest operations only, such as detection of a moving object against a relatively simple background with a subsequent measurement of its speed. Thus, such systems, in fact, are video surveillance systems and not the measuring ones. A more complicated task is when realization of the two-dimensional random field , x y p^h is processed, which, in general case, includes the image of an object with unknown coordinates to be measured, the background and the spatial noise [2][3][4][5]. The useful image is often described by a quasi-deterministic function , s x ŷ h of the two variables determining the dependence of the image intensity upon the coordinates , x ŷ h. One can call , s x ŷ h the intensity distribution or the intensity profile of the image. Depending on both the nature of the image and the resolution of the receiving system, the function , s x ŷ h may be a regular or discontinuous one [5]. Discontinuous functions describe objects with well-defined boundaries.
Some tasks of optimal and suboptimal image processing of the spatially extended objects, with the is necessary to estimate the unknown position 0 m of the image , , s x y 0 m h with the unknown intensity distribution , f x ŷ h.

The estimation algorithm synthesis
If the distribution intensity , f x ŷ h of Equation (6) is known, then it is possible to find the logarithm of the functional of the likelihood ratio (FLR) and then develop the maximum likelihood (ML) estimation algorithm. According to the ML method [11][12], in order to obtain an estimate m t of the position 0 m of Equation (6), it is necessary to form the component of the FLR logarithm depending on the current value λ of the unknown parameter 0 m as follows [13][14] , , , , The ML estimate m m of the image position coincides with the position of the greatest maximum of the decision statistics in Equation (7) If the function , f x ŷ h is a priori known so that then Equation (8) coincides with the FLR logarithm when implementing the optimal reception in Equation (7) and the QL estimate in Equation (9) coincides with the ML estimate m m .

The statistical characteristics of the decision statistics
In order to determine the characteristics of the QL estimate in Equation (9), the statistical properties of the decision statistics in Equation (8) are examined. By substituting Equation (3) into Equation (8), the functional in Equation (8) is presented as the sum of signal and noise components [12]: synthesis and analysis of the QL algorithm for estimating the unknown position of the two-dimensional signal (image) with the unknown intensity distribution are carried out. Influence with a priori ignorance of the image parameters upon the efficiency of the resulting estimate is studied.
Let the realization of the Gaussian random field , , , , , , , be observed within the area Ω. Here 0 m and 0 h are the unknown position parameters and , n x ŷ h is the spatial Gaussian white noise with zero mathematical expectation and the one-side spectral density N0 . In the observation area Ω, one chooses the coordinate system so that the equality [9][10] , , xI x y dxdy yI x y dxdy = X X^ĥ h ## ## (4) is satisfied, that is, the origin is located in the centre of the area , 0 , , , , The abscissa is the value that characterizes the difference between the forms of intensities of the received and the reference images.
The noise component in Equation (17) In the case of the optimal reception, functions in Equations (16) and (20) coincide, that is, for , .  , , , are the signal and noise components, respectively. Similarly to [15], let it be supposed that in the neighborhood of the point 0 m the signal function satisfies the conditions  (9) denote the maximum value of the signal component (12) and denote the dispersion of the noise component in Equation (13). Taking into account the last representations, one can rewrite Equation (11) are the normalized signal component in Equation (16) and noise component in Equation (17), , is the power signal-to-noise ratio (SNR) at the output of the QL receiver, In the case of the optimal reception, the intensity distributions of the received and reference images coincide, that is, Equation (10) The same result has been obtained in [10]. Now consider properties of the correlation function in Equation (20). Taking into account equality in Equation and right parts of the border of the area occupied by the image relative to the line AB are described as X y 1^h and X y 2^h , respectively. Using the introduced notations (see Figure 1), one can rewrite the signal function in Equation (16) If shapes of the received and the reference images coincide (the condition in Equation (10) respectively. It should be noted that these expressions coincide with characteristics of the ML estimate of the image position obtained in [10].
Then, one considers, for example, estimate of the rectangular image position. Let in the observation area of the image be present a rectangle with sides lx and ly , parallel to the axes Ox and Oy, respectively. In this case, one selects the coordinate system so that its origin is located at the intersection of the rectangle diagonals. One assumes that the image intensity is described by a linear function that increases in the direction of the angle θ with the side lx (axis Ox): , are the maximum and minimum intensity values, respectively. For a uniform image that has the same area and shape as the non-uniform one, the slope is q 1 = , so that . Here lx is the length of the image projection on the axis Ox.
In view of the above, one can conclude that the decision statistics in Equation (15) of the QL estimation algorithm in Equation (9) is the Gaussian random process, for which the mathematical expectation and the correlation function, under conditions of high a posteriori accuracy, allow the representations in Equations (22) and (31), respectively. The statistical properties of such a process are studied in details in [10,15]. Taking into account Equation (22), relations in Equation (14) conditioning the consistency of the QL estimate in Equation (9), can be rewritten in a more convenient form: . Thus, the conditions in Equations (14) are satisfied and the QL estimate in Equation (9) is consistent, if for all , y y y min max ! 6 @ the following inequalities hold:  (9) can be found by applying the local Markov approximation method as it is described in [16][17]  axis). From Figure 2 can be seen that the QL estimate is unbiased under q 1 = . If q 1 ! , then, for the same value of q, accuracy of the estimate decreases with increasing i , and that corresponds to the increasing degree of difference between the received and the presupposed images.
In order to characterize the loss in accuracy of the QL estimate, while comparing it to the same kind of loss in case of the ML estimate, one introduces the relation Equations (43) and (44), respectively. In Figure 3, the graphs demonstrate dependence of the loss in the QL estimate accuracy (46) upon the parameter q under 1 c = and the varied values of the parameter i . Here the notations coincide with the ones introduced in Figure 2. From Figure 3, it follows that under q 1 = , there is no loss in the QL estimate accuracy. However, under the fixed q 1 ! , the loss increases with the increasing i and that corresponds to the increasing degree of difference between the intensities of the received and the reference (presupposed) images. When the value q increases and the value i is fixed, the loss increases.

Conclusion
Synthesis and analysis of the rather simple QL algorithm have been carried out for estimating the position of the image with the unknown intensity distribution. As a result, the expansion is found of the signal function of the decision statistics in the neighborhood of the image position real value. Expressions are obtained for the bias and variance of the resulting estimate of the measured image coordinate. The dependence of the estimate characteristics upon the non-uniformity of the image intensity distribution is examined on an example of an image with the set intensity distribution. The relations Taking into account Equation (41), expressions for estimation characteristics in Equations (37) and (38) take the form respectively. Comparing Equations (42) and (43) to Equation (44), one can estimate influence of the non-uniformity of the image brightness distribution upon the accuracy of the QL estimate of the image position. Unlike the ML estimate, the QL estimate is biased as it follows from Equation (42). While the deviation of the parameter q from unity increases, the absolute magnitude of the estimate bias increases too. When the bias is a significant part of the mean-root-square error, it significantly reduces the accuracy of the estimate. Therefore, one introduces the normalized bias that shows what proportion of the mean-root-square is the bias of the QL estimate. Graphs of dependence of the normalized bias in Equation (45) of the QL estimate upon the parameter q under 1 c = and the varied values of the parameter i are presented in Figure 2. Here, the curve 1 corresponds to value 0 i = , the curve 2 -to /16 i r = , the curve 3 -to /8 i r = , the curve 4 -to /6 i r = , the curve 5 -to /4 i r = , the curve 6 -to /3 i r = , the curve 7to /2 i r = (the last curve coincides with the abscissa of the expansion coefficients. The nonparametric estimate of the intensity profile can also be used [18]. These results can be used in design of the measuring systems for object monitoring in the field of transport services, security, process and production control, etc.