In its discrete form, data are usually assumed to be an ordered vector with 𝑔𝑔 number of components and denoted by 𝑦𝑦 = {𝑦𝑦𝑙𝑙}𝑙𝑙=1𝑔𝑔 ∈ ℝ𝑔𝑔. Object distribution may also be assumed
to consist of an ordered set of multi-indexed numbers with 𝑙𝑙 components denoted by 𝑥𝑥 = �𝑥𝑥𝑗𝑗�𝑗𝑗=1𝑙𝑙 ∈ ℝ𝑙𝑙. Mathematical modeling of the imaging system implies the use of
equations which describe propagation of radiations emitted by the object and detected by a series of detectors [66]. If 𝑓𝑓(𝑥𝑥, 𝑦𝑦) is a function of space variables and denotes properties of the object, a transform mapping this function onto the sampled values of the exact radiations (radiations before detection), denoted by vector 𝑦𝑦 can be expressed as follows,
𝒚𝒚𝑯𝑯 = �𝑯𝑯𝒇𝒇(𝒙𝒙, 𝒚𝒚)�𝑯𝑯+ 𝒃𝒃𝑯𝑯 𝒘𝒘𝑯𝑯𝑯𝑯𝒘𝒘 𝑯𝑯𝑯𝑯𝒊𝒊≥ 𝟎𝟎.
𝒇𝒇𝒇𝒇𝒓𝒓 𝑯𝑯 = 𝑯𝑯, … , 𝑴𝑴 𝑯𝑯𝑳𝑳𝑯𝑯 𝒊𝒊 = 𝑯𝑯, … , 𝑵𝑵 (2.6) Here, 𝐻𝐻 could be a discrete or semi-discrete mapping, sometimes referred to as operator, and 𝑏𝑏 = {𝑏𝑏𝑙𝑙}𝑙𝑙=1𝑔𝑔 is a vector representing background noise. The mapping or
transforming matrix 𝐻𝐻 represents the imaging system and the condition 𝐻𝐻𝑙𝑙𝑗𝑗 ≥ 0 means that
matrix elements are non-negative. This operator is also known as forward model which is towards the noiseless data 𝑦𝑦� = 𝐻𝐻𝑓𝑓 and vector 𝑏𝑏 = 𝑦𝑦 − 𝑦𝑦� is the noise term. It is easier to solve the noiseless forward problem, however, the noise term makes the problem highly ill- conditioned and, to invert it, is very difficult [67].
2.6.1 Tomographic Reconstruction as an Inverse Problem
Almost always, we want to use this forward model 𝐻𝐻 and the observed noisy data 𝑦𝑦 to make inference on the unknown quantity of interest 𝑓𝑓: This is the inversion problem [46;48;49]. Image reconstruction, where we try to estimate source distribution inside the object being imaged using data acquired by an imaging system, is an example of an inverse problem in its mathematical formulation [66;68]. For tomography systems, this matrix is generally very sparse, though of very large size, and its condition number, which is a ratio of the largest to the smallest singular values, is very large [69].
Inverse problems are almost always ill-posed in Hadamard sense, which means that the solution of the problem may not exist, not unique or may not continuously depend on the data [70]. First two conditions are to make sure that inverse of the transform is well defined and whole data space is domain of the inverse transform. Requirement of
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Page 38 of 182 continuous dependence of the solution image on the data is a necessary but not sufficient condition for stability of the solution. Hadamard went on to define a problem to be ill- posed, if it does not satisfy all the above three conditions. So an ill-posed problem is one, where, an inverse does not exist, because, the data is outside the range of 𝐻𝐻, or the inverse is not unique because more than one image is mapped on to the same data, or because an arbitrarily small change in the data can cause an arbitrarily large change in the image [70]. Simplest way to solve this problem is to mathematically invert it.
2.6.1.1 Direct Matrix Inversion
The ill-conditioning of the reconstruction problem is due to very small singular values of 𝐻𝐻, which can be observed if we replace 𝐻𝐻 by its Singular Value Decomposition (SVD); or 𝐻𝐻 = 𝑈𝑈𝑈𝑈𝑉𝑉𝑡𝑡, where 𝑈𝑈 and 𝑉𝑉 are unitary matrices and 𝑈𝑈 is a diagonal matrix composed of
singular values. Least square solution, for noiseless data will become 𝑓𝑓̌𝐿𝐿𝐿𝐿𝐿𝐿 = 𝑉𝑉𝑈𝑈−1𝑈𝑈𝑡𝑡𝑦𝑦� =
∑ 𝜎𝜎1 𝑙𝑙𝑢𝑢𝑙𝑙
𝑡𝑡𝑦𝑦�𝑣𝑣 𝑙𝑙 𝑁𝑁
𝑙𝑙=1 . Here, 𝑢𝑢𝑙𝑙 and 𝑣𝑣𝑙𝑙 are the 𝑙𝑙𝑡𝑡ℎ column of matrices 𝑈𝑈 and 𝑉𝑉. Clearly, for very small
singular values 𝜎𝜎𝑙𝑙, the solution will become unstable [67]. For a brief analysis we may
observe that in case of a well-posed problem, relative error propagation from the data to the solution is controlled by the condition number. If ∆𝑦𝑦 is a variation in 𝑦𝑦 and ∆𝑓𝑓, the corresponding variation in 𝑓𝑓 then,
‖∆𝒇𝒇‖
‖𝒇𝒇‖ ≤ 𝒄𝒄𝒇𝒇𝑳𝑳𝑯𝑯(𝑯𝑯) ‖∆𝒚𝒚‖
‖𝒚𝒚‖ , (2.7)
and, for linear forward problems,
𝒄𝒄𝒇𝒇𝑳𝑳𝑯𝑯 (𝑯𝑯) = ‖𝑯𝑯‖�𝑯𝑯−𝑯𝑯�
Since, the fractional error in 𝑓𝑓 equals the condition number multiplied by the fractional error in 𝑦𝑦; smaller values of condition number for matrix 𝐻𝐻 are desirable [67;71;72]. If 𝑠𝑠𝑔𝑔𝑙𝑙𝑎𝑎(𝐻𝐻) is not too large, the problem is considered to be well-conditioned and the solution is stable with respect to small variations in the data. Otherwise, the problem is said to be ill- conditioned. It should be noted that the above relation is only valid for square matrices for the purposes to calculate matrix inverse.
Direct inversion is not very popular due to ill-conditioned reconstruction problem because it is not straight forward to include regularization explicitly. Even very small ill- conditioned matrix will require regularization for stable solutions. Similarly, for very large
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Page 39 of 182 matrices inversion is not very efficient and requires large memory space to hold the matrices involved. Hence, factorization is used to reduce the problem size and its complexity in iterative manner, though large memory requirements may still slow down or hinder the numerical evaluations [48;67;69;72].
2.6.2 Solving the Inverse Problem - Reconstruction Methods
A concept of well-conditioned problems, as discussed above, is lesser distinct than the concept of well-posed problems. Most correctly stated, inverse problems turn out to be ill- posed or at least ill-conditioned due to very high condition number of the inverse operator; any attempt to solve it by simple analytical inversion methods fails, though there are some simplified inversion techniques available such as DFR or FBP. However, meaningful information can be gained from ill-posed inverse problems, even though they cannot be strictly inverted. Mathematical theory of regularization or theory of generalized inverses, which is an extension of the theory of the Moore-Penrose inverse of a matrix, is usually applied to investigate problems that are not well posed [73]. Classical Tikhonov regularization is a well developed theory for linear ill-posed problems with a key idea to approximate a discontinuous operator with its continuous approximate functionals [71;74]. For linear ill-posed inverse problems, with compact linear forward transformation, Singular Value Decomposition (SVD) can be used to find the inverse transformation. However, for ill-posed problems, solution gets unstable for very small eigenvalues as discussed above. Inverse transformation can be regularized by limiting the smallest singular values. A filter factor, where filtering is introduced as a type of implicit regularization, can be included to set very small singular values equal to zero. Various types of filter factors such as Truncated SVD Filter, Tikhonov Filter or Exponential Filter have been proposed [67]. Variation regularization and regularization by defining some prior function in MAP estimators are different options to define explicit regularization functionals. Generally available mathematical tools for such inverse problems are either the deterministic regularization theory or the probabilistic Bayesian inference and estimation theory [60;74-77]. Filter factor has the least control on the reconstructed resolution which mainly depends on the properties of the filter chosen and it is also very difficult to choose a cut-off value.
Analytical reconstruction of images, from tomographic data, means to obtain function 𝑓𝑓(𝑥𝑥, 𝑦𝑦) by inverting analytical equation (2.2) assuming continuous variables given in the relation. These types of methods are known as direct or Analytical Reconstruction Methods [78]. They are considerably faster as compared to the iterative reconstruction methods,
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Page 40 of 182 where a discrete representation of the object is being estimated using some estimation algorithm from a discrete data vector using some underlying forward model. However, some of the iterative algorithms involve evaluation of certain analytic inversion formulae, too [56;79].