Abstract

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.

1. Problem formulation

The reconstruction of the properties of dynamic objects and processes becomes more and more important for the process control and monitoring [1]. 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:

• The reconstruction algorithm structure have to be simple, particularly recursive;
• The algorithm parameters should be conducted beforehand and the outcomes stored in the computer memory;
• The algorithm should be stable in relation with the inaccuracy of the image and observation models.

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.

2. Model equations

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

(1)

where is a S-dimensional vector function, u_k is the regular component, and w_k represents discrete Gaussian white noise with zero expectation and the covariance matrix Q_k . 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

(2)

with the m-dimensional non-linear vector function .

The observation noise v_k+1 is assumed to be discrete Gaussian white noise with zero expectation and the covariance matrix R_k. We suppose that the noise components w_k and v_k are statistically independent and independent on the initial conditions of the equation (1).

We assume, that the non-linear functions a_k(x_k) and h_k(x_k) can be approximated by the following linear relations:

(3)

(4)

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

(5)

(6)

As a result of the linearization the quasi-optimal reconstruction algorithm of a Kalman filter type can be employed [4, 5]

(7)

(8)

(9)

(10)

(11)

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.

3. Statistical linearization approach

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 m_x and the covariance matrix P_x 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

(12)

The problem is to determine the statistical linearization vector m_y and the linearization matrix F . The first item is evaluated by the expression

(13)

giving the expectation of the vector y.

The matrix F is chosen from a requirement that a covariance matrix P_y of the vector y

(14)

is equal to the covariance matrix of the linear approximation FP_xF^T [8, 9]. If we denote the square root of the matrix P by P^½ with the property P^½.(P^½)^T = P , then the above mentioned equality can be noted as follows

with P^½_y = FP^½_x. After substituting (14) the required statistical linearization matrix has the form

(15)

where m_y 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 m_x,k and covariance matrix P_x,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

(16)

(17)

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.

4. Example

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 x_k 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 x_k is presented. For the reconstruction the quasi-optimal algorithm (7)–(11) is employed.

In the considered case the matrix P_k 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 J_k 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 J_k 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.

5. Conclusion

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.

References

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