LAPÈRDIDA DEL INDIVIDUO EN LO SOCIAL

Analytical image reconstruction methods, such as filtered backprojection (FBP), Fourier reconstruction (FR), and convolution backprojection (CBP), are the reconstruction

Chapter III- Literature Review

Page 67 of 182 methods based on an inversion of the assumed line integral model of the projection data and work well for space-invariant systems [60;86-88;146]. For space variant systems response, these methods ignore any underlying noise distribution and have limited capability to model various image degrading effects and their reconstructed image bear degraded noise and resolution properties [61;104;122;126]. Simple iterative methods, such as ART or SART, try to solve analytical line integral model numerically, as a set, of linear equations without any modeling of the projection process or the object properties, while considering the object as a linear combination of parameters to be estimated, hence, cannot fully exploit imaging physics properly [39;60] and lead to images with lesser than optimized quality [56;79;99]. Methods based on simple modeling of the statistical nature of the data, such as simple likelihood based method or methods based on least squares data modeling; result in noisier images with increasing iteration number due to ill-condition of the mathematical formulation of the reconstruction problem [23;28;93]. These methods try to fit the solution image to the data only, which is noisy in nature, and this noise gets amplified because of very small singular values of the forward model matrix [102, 106].

Several methods based on the concept of restricting or constraining smallest singular values, known as implicit or global regularization methods such as early iteration stopping or post-smoothing of the fully converged MLEM image, are employed to reduce this reconstruction based noise, because these images still have better noise properties as compare to the analytical methods [104;147;148]. Resultant images may have better noise properties, however, their overall resolution and noise characteristics depend on the number of iterations, the sieve or the filter used for post-filtering, with least resolution control and much longer convergence times [24;95]. Modeling various image degrading effects may improve bias and variance of the reconstructed images, however, images may still suffer from reconstruction based noise due to ill-conditioning of the problem. Another feasible approach is to use some explicit regularization technique to overcome this iteration based noise. Methods proposed in the framework of Tikhonov Regularization are Penalized-Likelihood Expectation Maximization (PLEM) reconstruction methods or Maximum-a-Posterior (MAP) reconstruction methods in Bayesian framework [50;111;117]. Explicit regularization allows inclusion of a priori knowledge about the object distribution properties and helps conditioning the reconstruction problem. Arbitrary regularization can be applied to obtain user defined resolution properties of the reconstructed images, which provide extra resolution control [109;129;135]. Regularizing functionals may be added as penalty functions in PLEM framework or the prior distribution functions in Bayesian sense. These regularization functions are also termed as local

Chapter III- Literature Review

Page 68 of 182 regularization methods, because they are generally based on Gibb’s prior distribution functions and work on a small neighborhood vicinity of a pixel of interest [66;109].

Most commonly used penalties or priors are the smoothness priors or the edge preserving priors [24;97;108;148]. Simplest form of smoothing priors, Quadratic prior functions, have been used as conventional regularization choice due to their implementation simplicity; however, they result in images with non-uniform and asymmetric resolution properties across the FOV, even for approximately shift-invariant systems, such as PET near the centre of the scanner, because they lead to an estimator with highly non-uniform and asymmetric response [26]. This is due to the non-uniform smoothing induced by this type of regularization functions, which results from various object dependent, such as non-uniform attenuation distribution, and system dependent factors, such as non-uniform detector response function (DRF) across the field of view in PET [6]. This induced non-uniform smoothing, or consequently non-uniform and asymmetric resolution, properties produce shape deformations in the reconstructed images and affect image quantification and make other processing tasks difficult [29]. A serious attempt to correct for these non-uniformities and asymmetries, in response function, induced by QPs in MAP methods, has been presented in [24;129;138]. A correction scheme for interactions between data term and the prior term has been introduced, which effectively generates locally modified space-variant hyper-parameter values and provides increased uniformity in resolution, based on local certainty with quadratic regularization, however, asymmetry still persists [129]. This behavior of QPs is due to their strong smoothing behavior, which increases with higher intensity differences, especially at the edges. Edge preserving regularization functions provide a way to compensate for this discrepancy and median root priors (MRPs) have a robust ability to preserve edges, though they have theoretical problems in defining their derivatives [107]. Total Variation (TV) regularizing functions are another edge preserving priors with the same theoretical issue, however, their resultant images bear reasonable properties [112;114;116;149;150]. These priors are non-quadratic functionals and bear heavier ill- conditioning as compared to the QPs and take more time to converge.

We have implemented MRPs and TV priors based MAP algorithms with an analysis of their resolution properties as compared to QPs, by an implementation of these functionals in Local Impulse Response (LIR) expression. We have also introduced a correction scheme for non-uniform spatial resolution based on an alternate prior functional, with and without certainty based term. We use an edge-preserving MRPs based non-quadratic regularizing prior, in this scheme, to correct for the above mentioned interactions between data and the prior term [108;110]. Our correction scheme is based on an intrinsic property of MRPs that

Chapter III- Literature Review

Page 69 of 182 they penalize pixel values with respect to the local median, and their behavior is non-linear, which simulates behavior of local certainty type correction method. Also, MRPs have a robust edge preserving characteristic, without any additional parameter required, and they can be run to any number of iterations without adding much noise to the final image. In the following sections, we present a review on tomographic image reconstruction methods, ill- posedness of the image reconstruction, methods to correct for this, non-uniform resolution properties of the reconstruction images and the use of MRPs for the recovery of non- uniform resolution and their activity recovery performance in MAP reconstruction algorithms for histogram and list-mode data.