**Figure 5.**

*(a) The appearance of non-physical oscillations in the reconstructed spectrum; (b) Peak smoothing due to over regularization.*

*Electron Beam Processing of Biological Objects and Materials DOI: http://dx.doi.org/10.5772/intechopen.112699*

example of regularization of the second type is the imposition of a non-negativity condition. Such regularization is used in problems in which the non-negativity of an unknown quantity is guaranteed by its physical nature, for example, mass, spectrum, energy, etc. The first method usually consists of the introduction of regularizing operators. One of the most popular methods is *L*<sup>2</sup> regularization, better known as Tikhonov regularization [58–60]:

$$L\_2 = \sum\_i \left( D^i - \sum\_i d^{ij} a^j \right)^2 + a \bullet \theta \left[ a^i \right], \tag{8}$$

where *α* is the regularization parameter, *θ aj* � � is the regularizing operator.

There is a wide range of regularizing operators [61, 62] and values of regularization parameters. In general, the higher the value of the regularization parameter, the less the regularized problem will be related to the nonregularized problem, the smaller the value, the less noticeable the regularization effect will be. In the simplest case, the regularizing operator is a Euclidean norm *L*<sup>2</sup> . From practice, it is known [63] that for the simplest regularizing operator, the best results are obtained by the residual method, which prescribes to choose such values of the regularization parameter that:

$$\sum\_{i} \left( D^{i} - \sum\_{i} d^{ij} d^{i} \right)^{2} \approx \sigma^{2} \left\| D^{i} \right\|^{2},\tag{9}$$

where *σ* is the relative error of *Di* .

It is worth noting that the Tikhonov regularization with the simplest regularizing operator leads to a smoothing of the peaks of the spectrum (**Figure 5b**), which may be undesirable. This can be addressed by modifying the regularizing operator, for example [64]:

$$\theta\left[\mathfrak{a}^{j}\right] = \sum\_{j} \log\left(\mathfrak{a}^{j}\right)^{2}.\tag{10}$$

Also, one can decompose the spectrum into a regular and singular part [64]:

$$f(E) = f\_s(E) + f\_r(E),\tag{11}$$

where *fr*ð Þ *E* is regular and *fs* ð Þ *E* is singular component:

$$f\_s(E) = \beta \mathfrak{e}^{-\frac{(E-\mu)^2}{\mathfrak{a}^2}}.\tag{12}$$

Such decomposition allows to treat two parts of spectrum separately and can lead to improved results.
