| The Tikhonov pth order regularization method as a means for spatially invariant and variant smoothing technique to stabilize an ill-posed inverse problem is investigated. The optimal regularization parameter that minimizes mean-square-error (MSE), for unweighted stabilizers, is studied. A deterministic signal model is introduced for this analysis. Using this model, the optimal parameter is found to be the ratio of the total noise variance to the integral of the square of the signal derivatives of the appropriate order. As the regularization parameter increases from small to large, the variance part of MSE decreases but the bias part (often known as blurring) of MSE increases. Therefore, the minimum MSE is a compromise between variance and bias. To further reduce bias calls for a spatially variant method. Weighted regularization is a perfect vehicle for such a method. It is shown that an appropriate number of zero-weights at a spatial location in the weighted stabilizer can eliminate smoothing across that location. Based on this, several weighting schemes are proposed to utilize either available or estimated spatial information to achieve spatially variant regularization. These weighted methods can greatly reduce the bias, thus reduce MSE without the blurring effect. The image smoothing problem is used as an example to evaluate the 2-D version of the proposed methods.;As examples of application of the regularization methods, two problems in nuclear medical imaging are investigated. One is the image reconstruction problem for emission tomography. This is a typical ill-posed inverse problem. Applying the unweighted and weighted regularization methods to the iterative ART algorithm stabilizes it and also improves its convergence rate. The weighted method also provides a good way to incorporate a priori information to improve emission tomographic image reconstruction. The other problem is Compton-scatter correction for nuclear imaging instruments. A deconvolution-fitting method in energy domain using weighted regularization is developed. Good agreement with Monte-Carlo simulation and an existing energy fitting method has been obtained on experimentally measured data. With simulated data, especially in low count level cases, improvement over existing methods has been observed. |
