|NDT.net - December 2002, Vol. 7 No.12|
The reconstruction of the properties of dynamic objects becomes more and more important for process control and monitoring. This task is called process tomography. X-ray tomographic imaging deals with the reconstruction of an object from projection data. In the case discussed here, the corresponding projection model is nonlinear. Both problems, the presence of non- linearity in the image formation process as well as the dynamic behavior of the object properties, yield a non-linear reconstruction problem. A specific mathematical tool and technique is developed to overcome the above restrictions that are based on statistical linearization procedure. A Kalman filter type reconstruction algorithm gives the general principle for the reconstruction. The applicability of the proposal approach is shown by examples.
The reconstruction of the properties of dynamic objects and processes becomes more and more important for the process control and monitoring . The structure of such processes has to be imaged in real time using non-intrusive sensors. X-ray tomographic imaging deals with the reconstruction of an object structure from projection data. The linear projection model case was discussed in [2, 3]. In the case discussed here, the corresponding projection model is non- linear considering the non-linear image and observation properties. A specific mathematical technique has to be developed to overcome the presence of non-linearity in the image formation as well as the dynamic behavior of the object properties. The general principle for the dynamic reconstruction is given by the Bayesian approach leading to a Kalman filter type reconstruction algorithm.
The computational expenditures of high dimensional images reconstruction in real time are usually so large, that they appear unsuitable. It is necessary to use a special reconstruction technique with the following desirable requirements:
Basically these requirements obeys a linear algorithm, therefore they can be formulated as follows: to find quasi-optimal linear reconstruction algorithm for given non-linear image and observation equations. The quasi-optimal non-linear filtering problem is discussed in several publications such as [4, 5]. This technique for the image reconstruction is discussed in [6, 7]. The originality of the presented work consists in employing the statistical linearization procedure [8, 9] for the linear approximation of the equations for the image and the observation model. Such approach permits the solution of the formulated problem.
Let's consider the general case, when the nonlinear model of the image sets is a random Markov sequence and can be described by stochastic difference equation
where is a S-dimensional vector function, uk is the regular component, and wk represents discrete Gaussian white noise with zero expectation and the covariance matrix Qk . The index k counts for the time on a discrete scale. We also suppose the m-dimensional observation equation to be non-linear, given by
with the m-dimensional non-linear vector function .
The observation noise vk+1 is assumed to be discrete Gaussian white noise with zero expectation and the covariance matrix Rk. We suppose that the noise components wk and vk are statistically independent and independent on the initial conditions of the equation (1).
We assume, that the non-linear functions ak(xk) and hk(xk) can be approximated by the following linear relations:
In these expressions the linearization is carried on with respect to the reference process After substituting eqs. (3) and (4) into eqs. (1) and (2) they transform to the following linearized set of equations
As a result of the linearization the quasi-optimal reconstruction algorithm of a Kalman filter type can be employed [4, 5]
There are known series of approaches to determine the linearization vectors a*k(x*k), h*k(x*k) and the linearization matrices A*k(x*k), H*k(x*k). For the formulated problem the statistical linearization approach is used. The basic idea of the statistical linearization approach together with the application to the above problem is discussed in the next section.
The statistical linearization technique of nonlinear functions was simultaneously proposed in [8, 9]. Let's consider an S–dimensional random vector x with the probability density distribution function (PDF) p(x) . The expectation vector mx and the covariance matrix Px are given by the expressions and respectively. The S– dimensional function y= f(x) defines a non-linear transformation. A linear relation of the following aspect approximates this function
The problem is to determine the statistical linearization vector my and the linearization matrix F . The first item is evaluated by the expression
giving the expectation of the vector y.
The matrix F is chosen from a requirement that a covariance matrix Py of the vector y
is equal to the covariance matrix of the linear approximation FPxFT [8, 9]. If we denote the square root of the matrix P by P½ with the property P½.(P½)T = P , then the above mentioned equality can be noted as follows
with P½y = FP½x. After substituting (14) the required statistical linearization matrix has the form
where my is taken from the equation (13).
The linearization vector and the linearization matrix can only be determined from
equations (13) and (15) if the
PDF p(x) is known. Considering the reconstruction problem with the linearized eqs. (5) and (6) the PDF can be approximated by the Gaussian distribution, completely determined by the expectation vector mx,k and covariance matrix Px,k . Here the second index k is introduced to mark the time at which the expectation vector and the covariance matrix is given. As a result the linearization coefficients in eqs. (3) and (4) become functions of these variables. Using these assumptions the coefficients can be represented by the following formulas
These coefficients can be calculated in advance using the prior PDF and eqs. (1)–(2) before being introduced to the reconstruction algorithm (7)–(11). As a result the proposed technique is called prior statistical linearization procedure. This algorithm satisfies the requirements 1–3 made in section 1.
The example is considered for an image model with the dimension S = 16´16, time correlation lasts for 40 time intervals, and the spatial covariance radius is assumed to be 5 pixel lengths. The two-stage image model is chosen such that the generating Gaussian image forms the non-Gaussian image xk by the non-linear transformation identical to all pixels i =1,2,K,S . The image model realizations are shown in figure 1 for different time moments.
Fig 1: Gaussian and non-linear transformed images xk .
Here each realization has a dimension of 2S´1 and consists of two parts. On the left side the Gaussian image is shown with zero expectation, and on the right side the non-Gaussian image xk is presented. For the reconstruction the quasi-optimal algorithm (7)–(11) is employed.
In the considered case the matrix Pk only approximately determines the reconstruction error covariance properties. Therefore the function , defining the quality of reconstruction, also gives only an approximate estimate. The real value of the function Jk is received by the Monte-Carlo simulation method. For each reconstructed image the reconstruction error was determined and averaged over 1000 realizations at time k . As a result analytical values Jk and simulation results are indicated on figure 2. As follows from this figure the theoretical curve gives the underestimated outcomes in comparison to the statistical data tests. This result is expected, since the theoretical results are obtained for the linearized model that does not completely take into account the non-linear properties of the problem.
|Fig 2: The reconstruction relative error: a – analytical; b – Monte-Carlo simulation.|
The problem of dynamic image reconstruction from projections is examined. The image model and the projection data acquisition model are supposed to be given by stochastic non- linear difference equations. The prior statistical linearization technique is employed to obtain the quasi-optimal reconstruction algorithm of the Kalman filter type. Algorithm parameters in term of statistical linearization coefficients can be calculated in advance by using image prior probability density distribution. This technique is demonstrated by an example of non- Gaussian image reconstruction. The comparison of the reconstruction error between the analytical and the Monte-Carlo simulation results is given.
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