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...esian methods give improved reconstruction quality over traditional methods, see e.g. (Sauer et al., 1994; Hanson and Wecksung, 1983; Bouman and Sauer, 1993; Frese et al., 2002; Yu and Fessler, 2002; =-=Siltanen et al., 2003-=-; Kolehmainen et al., 2003). A major practical difculty in applying Bayesian methods to 3D x-ray imaging is the heavy computational requirements. This is why previous studies on the topic have mostly...

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...y the dilation equations, these integrals can be implemented by a discrete filtering procedure, see [34]. III. THE MEASUREMENT MODEL Here, we describe the pencil beam model for X-ray attenuation (see =-=[7]-=- for more details). Let the object be a compactly supported density f = f(x). Suppose that an X-ray source is placed on one side of the object and the radiation is detected on the other side at a dete...

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...back-projection (FBP) methods, produces reconstructions with notorious artifacts, see Figure 1. In dealing with the ubiquitous limited angle tomography, several approaches have been tested (e.g., see =-=[1, 2, 3, 4, 5]-=- for more recent ones). In terms of artifacts, methods that apply regularization in the image (density) domain show higher degrees of success. Nevertheless, they normally assume piecewise smoothness o...

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...h as FBP, are not well suited for sparse projection data [31, 24]. More promising approaches include algebraic reconstruction (variants of ART) [1, 8, 29], tomosynthesis [11], total variation methods =-=[19, 28, 10, 9]-=-, Bayesian inversion [17, 34, 37, 19, 32], variational methods [20] and deformable models [16, 5, 13, 47, 23]. We introduce a novel variant of the level set method, where the X-ray attenuation coeffic...

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... the Tikhonov functional Tα(f) = ‖Rf − y‖22 + α ‖f‖ 2 2 . (2) The first term in (2) is the data fidelity term, which controls the reconstruction error. The second term is the so-called penalty or prior term, which controls the smoothness of the solution. The disadvantage of Tikhonov regularization is that it tends to oversmooth the solution in some situations. This is not wanted in many medical imaging applications. For this reason, other penalties have been proposed in the literature. Prominent examples are the total variation prior (TV) and the Besov norm prior which both enforce smoothness [5, 10]. Recently, the regularization with sparsity constraints has become popular [3]. Here, one seeks for a solution f∗ which is ‘sparse’ in a given dictionary Ψ = {ψ1, . . . , ψN}, i.e., a solution f∗ = ∑ k xkψk =: Ψx for which only a small fraction of the coefficients x1, . . . , xN is non-zero. It is well-known that `1-norm prefers sparse solutions [3, 1]. Therefore, in sparse regularization one is interested in minimization of the `1-penalized Tikhonov functional Tα(x) = ‖Kx− y‖22 + α ‖x‖1 , (3) where K = RΨ ∈ RM×N . From the theory of compressed sensing it is known that the integration of spar...

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...plications like, e.g., limited angle CT, exploring the full range of Bayesian inversion by also incorporating sample based analysis was, up to now, rather regarded as a theoretical option, see, e.g., =-=[49, 34]-=-. In addition, theoretical questions concerning sparse Bayesian inversion, like, e.g., the ones addressed in [36, 35] can be also be addressed numerically (cf. Section 3.1.4). Acknowledgments We would...

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...more reliable and highly weighted. From a different point of view, if the photon count is large as will be the case here, the Poisson distribution can be approximated by a Gaussian distribution [50], =-=[56]-=-. Furthermore, energy integrating detectors used in commercial CT scanners cause the statistics to deviate from Poisson distribution [57] and their noise characteristics are well approximated by a Gau...

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...rmation is desired, although only a severely limited amount of data can be collected. In intraoral imaging or mammograms, for instance, measurements can be obtained only in a very small angular field =-=[26, 24, 10, 11, 18, 27, 30]-=-. Industrial applications such as systems for material examination or luggage inspection usually are made up of an assembly containing only a few projection units in a restricted projection geometry [...