A discussion of emission tomography physics, with reference to PET and SPECT has been presented, in this chapter, for various types of data acquisition and system models. Various image degrading effects have been discussed which ultimately degrade final image quality and consequently make reconstructed resolution non-stationary and cause a signal loss in form of PVE due to limited resolution capabilities of these systems [6;11;14;19;23;28;41;61;93;94]. Asymmetry of these effects across the FOV and noise in the data with its incomplete nature make the mathematical image reconstruction problem ill-posed. We discussed different approaches, for example implicit or explicit regularization techniques, to solve the ill-posed reconstruction problems in the context of their capabilities to control final reconstructed resolution and noise [46;109;134;135]. We argued that explicit regularization methods, such as MAP or PLEM reconstruction methods
Chapter II - Background
Page 64 of 182 have better control over the final reconstructed resolution through a use of penalty or prior distribution functions as compared to the other methods [66;95;105;111;125]. Different prior functions have been proposed to reconstruct images bearing specific characteristics depending on the selected priors [77;105;121;128;129]. A brief description of the requirements and capabilities of different reconstruction approaches is given in Table 2.1. This table indicates that analytical reconstruction methods do not take care of the ill- conditioning of the problem due to noisy and incomplete data whereas they are much faster as compared to the iterative and statistical reconstruction methods.
Table 2.1: Comparison of different reconstruction methods for their resolution control and other characteristics such as edge preservation, ill-posedness and additional parameters required.
Reconstruction Method Posedness Ill- Speed Conv. Preservation Edge Prior Used Resolution Uniformity Parameters Required Analytical Reconstruction
Direct Inversion Yes Fast Smoothed No Uniform No
Fourier Reconstruction Yes Fast Smoothed No Uniform No
Filtered Backprojection Yes Fast Smoothed No Uniform No
Iterative Reconstruction
Least Square Yes Fast Smoothed No No No
Algebraic Reconstruction Yes Fast Smoothed No No One
Conjugate Gradient Yes Slow Smoothed No No One
Maximum Likelihood Yes Slow Smoothed No No No
Reg. Least Square Less Slow Smoothed Yes No One
Statistical Iterative
MAP-QPs Less Mod Smoothed Yes Controlled One
MAP-NonQPs Less Mod Preserved Yes Controlled More than One
MAP-MRPs Less Mod Robust Yes Controlled One
MAP-TV Less Mod Robust Yes Controlled More than one
Similarly, analytical methods produce images with almost uniform resolution for space-invariant systems, however, they do not consider any underlying noise distribution and their ability to include various system models is limited [84;86;86;88;122;136]. Simple
Chapter II - Background
Page 65 of 182 iterative methods, without regularization, cannot compensate for ill-conditioning of the reconstruction problem and produce reconstruction based noise because they use information embedded in the data only and do not use any available priori information about the object [105;111;125]. Their convergence is slow and low frequency content is superimposed by high frequency noise if large number of iterations is used, whereas their resolution control is poor [98;100;120;134;137].
Statistical reconstruction methods, with explicit regularization included in form of QPs, require only one parameter to be tuned and have better control on the reconstructed resolution, however, their reconstructed images have anisotropic smoothing characteristics with blurred and de-shaped regions. Non-quadratic regularization techniques need extra empirical parameters to define edges and like TV priors produce patchy artifacts even in uniform activity regions [26;30;60;127;138].
Regularized image reconstruction MAP methods, based on MRPs, require only one empirical hyper-parameter; have robust edge preservation, better resolution control, lower blurring nature and moderate convergence speed [30;107;110;139;140]. They have a problem of analytical description of their convergence properties like that of TV priors, however heuristic approaches are available for the solution. Hence, we described, in this chapter, a brief derivation and modeling of the priors based on quadratic, TV and MRPs based priors in the context of OSL-MAP reconstruction algorithms for image reconstruction [107;112]. We have also presented a brief derivation of the LIR expression, including these priors, used for the valuation and comparison of resolution properties of the images reconstructed by these priors. In rest of the thesis, an exhaustive analysis and comparison of resolution properties of these priors has been presented along with a comparison of their activity recovery performance. Our objective is to find the best available prior distribution function in MAP reconstruction algorithms if uniform resolution characteristics across the whole reconstructed image are desirable.
Chapter III- Literature Review
Page 66 of 182
CHAPTER 3
Literature Review
In PET, SPECT or CT, final reconstructed images are expected to depict underlying activity concentration (in PET, SPECT) or attenuation distribution (in CT) inside the object being imaged [4;141-143]. Clinical community assesses these images in two different ways, namely, qualitatively and quantitatively. Different figures-of-merit (FoMs), depending on the task and requirements of the study being carried out to measure quality of the images, are used [27;144]. The most commonly used metrics include spatial resolution, contrast recovery coefficients, absolute activity values and standard deviation (noise) images etc. Spatial resolution expresses the ability of an imaging system to distinguish between two smallest separable structures, inside the object being imaged, and is an important evaluation criterion for the comparison of different images [22;50;65;145]. Images can only be compared to each other, or various image processing tasks such as segmentation or registration or feature recognition can only be performed accurately, if they have matched reconstructed resolution characteristics [24;29]. Finite reconstructed resolution of these systems depends on many physical or estimator dependent, image degrading effects and it is very difficult to obtain images bearing uniform and matched resolution characteristics [6;10;60;141]. Also due to many noise processes present in emission and detection processes, a compromise is always needed for a trade-off between resolution and noise which ultimately depends on the reconstruction method or model used. In this chapter, we reviewed image formation process, causes of image quality degradation and non-uniform reconstructed resolution, various image reconstruction methods, their resolution properties and methods to compensate for resulting non-uniform resolution properties of images reconstructed by different methods.