RENACIMIENTO DEL VERDAERO INDIVIDUO

MLEM type methods are expected to produce images with uniform resolution characteristics, if allowed to fully converge with complete system matrix modeling, however, they need large number of iterations before convergence can be achieved [104;147;161]. Unfortunately, simple ML problem, in ECT, is highly ill-conditioned, mathematically, and reconstructed images get noisier as we continue increasing the iteration number due to incomplete data and noise. Images exhibit high spatial variance and image features are overcome by the induced noise [17;72;164]. In Hadamard sense, solution image is not unique or may discontinuously depend on the data. This is because of very small singular values of highly ill-conditioned system matrix. Several implicit regularization techniques have been proposed, such as stopping the reconstruction before convergence, post-filtering the fully converged MLEM images or methods of Sieves, which in essence try to get rid of these very small singular values [95;147]. However, these methods are generally space-invariant and induce bias towards a uniform starting guess image [29]. These methods have either no or very least control over reconstructed image resolution. Additional constraints are needed to limit the set of solution images. These constraints are generally implemented in form of an additional function, describing object

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Page 74 of 182 properties in form an object distribution function, and are generally referred to as explicit regularization methods.

3.3.6 Explicit Regularization Techniques

Explicit regularization simply means to include object distribution properties, along with data distribution function, as part of a cost function which is optimized to find the final reconstructed image. In Tikhonov regularization framework, key idea of explicit regularization is to introduce a continuous approximation of a non-continuous operator and is known as Penalized-Likelihood Expectation Maximization (PLEM) reconstruction method in penalty function framework [71;75]. This approach is equivalent to add a penalty term to the likelihood function and then to maximize this modified objective function [96;118]. An explicit form of regularization, to overcome the problem of ill- conditioning, can be implied in form of a prior distribution probability along with the likelihood in Bayesian framework and is, generally, known as Maximum-a-Posterior (MAP) reconstruction algorithm [77;111;117]. This prior function is used to satisfy the data and to include any prior knowledge available about the object being reconstructed and may be a generalized image description based function or an anatomical image of the object being imaged [93;111;122]. A very popular choice of theoretical image description is based on a property of images derived from morkov-random-fields (MRF) as Gibbs distribution function. This function works on a very small local neighborhood of image pixels assuming that images are locally smooth [111;125]. Hence, most favored regularization choice is that of smoothness priors or penalty functions based on piece-wise pair differences of pixels in local neighborhoods of an image [24;26;138].

3.3.7 Prior Distributions

Smoothness priors work on a basic image description that images are locally smooth and are generally applied in form of Gibbs distribution priors. Quadratic regularization priors, based on pair-wise pixel differences, have been most advocated and analyzed as smoothness priors in MAP reconstruction methods due to their simple implementation and good smoothing properties to reduce noise. However, these priors cause an over smoothing at regional edges and high count areas in the image, because their smoothing behavior is spatially non-uniform and depends on various factors such as activity concentration, system geometry, and non-uniform attenuation. Fessler (1996), has shown that QPs produce over smoothing in high count regions and consequently, images bear spatially-

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Page 75 of 182 variant resolution properties with poor resolution in these areas [24;26;129;138;148]. Difficulty in hyper-parameter tuning is another drawback of MAP reconstruction methods, because, this parameter unlike cut-off frequency in FBP, does not have any units and is difficult to evaluate. Smoothing heavily depends on this parameter and several methods have been proposed to tune this parameter much like a tabulation of cut-off frequency against resolution [26;126]. It has also been shown that effective parameter value for QPs is spatially non-uniform and depends on local certainty, though, impulse response is independent of the object intensity values, because Hessian of QPs becomes independent of the object.

Interestingly, non-uniform resolution is not specific to QPs only and, for non-quadratic priors; local resolution may not only depend on the smoothing parameter but also on object through Hessian of the prior term. However, this dependence, somehow, is also desirable such as in edge preservation. Non-quadratic priors, in general, require more than one parameter to be tuned which is a tedious problem [107;111;125]. The quadratic and non- quadratic (excluding MRPs) priors work on a basic description of the images that they are locally smooth and try to penalize pixel differences without any respect for basic image features, such as edges, where pixel differences are highest [77;121]. This results in an estimator form which drags the final image towards its locally smooth version. Hence, these estimators result in images with non-uniform and asymmetric resolution properties.

3.3.8 Edge Preserving Priors – MRPs and TV

Quadratic regularization blurs edges and induces low resolution in high count regions, whereas, to avoid this problem an edge preserving non-quadratic regularization may be used [77]. Several edge preserving techniques have been proposed. However, either these methods use non-quadratic regularization (other than Median Root Priors - MRPs) such as Huber and Lang’s penalty functions or they are computationally intensive and use complex edge defining techniques such as deterministic annealing or the method of level sets [77;120;127]. Also, these methods have only been implied to improve edge preservation without addressing the problem of non-uniform resolution and, generally, use more than one empirical parameter to define edges [134;135].

Median Root base Priors (MRPs) have a robust property to preserve edges [107;128]. This is mainly due to the fact that MRPs assume that images are locally monotonic and median follows the edge. Alenius et. al., (1998) documented several advantages of MRPs, such as lesser quantification errors because of noise suppression and automatic edge preservation without additional parameters required [128]. MRPs do not induce noise with

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Page 76 of 182 proceeding iteration number and virtually can be iterated to any feasible number of iterations. Our results have shown that reconstructed resolution with MRPs is less sensitive to the hyper-parameter value, which is an advantage in MAP reconstruction methods [30].

We gained motivation for our analysis and design to use MRPs by Alenius et. al. (1998). They proposed non-quadratic regularization based on these priors and assumed a Gaussian-like distribution for their prior function about the median taken as its mean in a local neighborhood [128;165]. Their proposed reconstruction algorithm may be thought of as a generalization of one-step-late (OSL) variation of PLEM methods by Green [105]. There are certain drawbacks of using MRPs for example there are no straight forward analytical derivatives available for MRPs due to their nonlinear dependence in a local neighborhood and one has to resort to an empirical definition, because response of the system in MAP methods depends on Hessian of the prior function [107;112].

Hsiao et. al., (2003), have discussed analytical behavior of MRPs in detail and propose their own prior function based on an approximation to absolute function, because optimized absolute function in a local neighborhood turns out to be the local median [110]. They defined an auxiliary field of variables, in registration with local medians, for minimization of their suggestive objective function and claim that their posterior is a true joint estimation due its analytical derivability. Interestingly, even though the convergence properties of MRPs are not very well known, still they pointed out that, all the images they tried with MRPs converged. Their method results in images that bear same properties as images reconstructed by MRPs and the same has been observed by us [30]. Hence, in our point of view, it is more a matter of theoretical description and we used MRPs in our analysis borrowing heuristic derivatives while leaving their convergence properties as an open question. Another class of prior functions, having same problem with their analytical derivatives, is the TV regularizing functional [112;149]. Even with theoretical issues, these priors have been vastly used in image restoration problems with acceptable results. We have also compared TV regularizing functionals in our studies and compared them with QPs and MRPs.