Adaptive Image Reconstruction with Predictive Model

Prior knowledge concerning information about the image and noise properties strongly influence the performance of image reconstruction from projections in computerized tomography. The authors propose an adaptive recursive 2D image reconstruction under uncertain conditions for the image and statistical noise properties. The Reconstruction is considered as adaptive estimation problem on the basis of empirical data generated by a predictive image model. The projection model is introduced by the vector y(n) given by the components \({{y}_{m}}(n) = \sum\nolimits_{{{{r}_{1}} = 1}}^{R} {\sum\nolimits_{{{{r}_{2}} = 1}}^{R} {{{x}_{{mr}}}} } (n){{a}_{r}} + {{\xi }_{m}}(n), r = {{r}_{1}},{{r}_{2}}\) with the random image values a r on a rectangular grid of size R× R, the number of current projection n(n =1, 2, 3, ⋯), the detector number m= 1, 2, ⋯ M, the elements of the M× R 2 projection matrix x mr (n), and the noise component ξ m (n). For the reconstruction step nonly the new information provided by the projection y m (n) (m= 1, 2, ⋯, M) and the previous estimations â r (k) (k= n− 1, n − 2, ⋯, n−1 − n 0) are used with predictive image model algorithm a* (n) = Φ [â (k), k = n− 1, ⋯, n− 1− n0] = Φ (â, n0), where n 0 gives the order of the prediction model. For the solution of this problem the cost function F(â, a, n) with the constraints Q m (â, y, n) = 0 is employed. Because the prior FDD function for the image parameters and the noise properties are unknown, the empirical PDD is used which is determined from predicted image data. Therefore the adaptive, empirical data-based estimation criteria has the form. The proposed reconstruction algorithm is applied to simulated and experimental projection data, and compared with standard CT algorithms.