NDT.net - December 2002, Vol. 7 No.12 |

(Federal Institute for Materials Research and Testing (BAM),

Unter den Eichen 87, 12205 Berlin, Germany)

(Gerd-Ruediger.Tillack@bam.de)

V. M. Artemiev and A. O. Naumov

(Institute of Applied Physics, Akademicheskaya str. 16, 220072,

Minsk, Belarus) (artemiev@iaph.bas-net.by)

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 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.

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:

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

(1) |

where is a *S*-dimensional vector function, * u_{k}* is the
regular component, and

(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

We assume, that the non-linear functions * a_{k}*(

(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}*(

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)

(12) |

The problem is to determine the statistical linearization vector * m_{y}* and the linearization matrix

(13) |

giving the expectation of the vector *y*.

The matrix * F* is chosen from a requirement that a covariance matrix

(14) |

is equal to the covariance matrix of the linear approximation * FP_{x}F^{T}* [8, 9]. If we denote the square root of the matrix

with **P**^{½}* _{y}* =

(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

(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.

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

Fig 1: Gaussian and non-linear transformed images x .
_{k} |

Here each realization has a dimension of 2*S*´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

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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