## 2 Methods
### Inverse problems  
Data processing is getting from data space to map space. The theory of the experiment can be regarded as an operator from the map space to data space. The forward problem consists in applying this operator. It is always possible assuming that the experiment can be analyzed. An inverse problem refers to determining the causes for a desired or an observed effect or fitting the parameters of a mathematical model to reproduce observable data. Yet the inverse operator might not exist at all since it usually operates on a space larger than the data space (Figure 1). Typically, inverse problems are ill-posed, in contrast to the well-posed problems, which satisfy [PERSON]'s postulates of well-posedness: existence of solution(s) in the strict sense, uniqueness or continuous dependence on data, and stability [14]. An inverse problem is represented in the context of functional analysis by a mapping between some metric spaces. Most often, inverse problems are formulated in infinite dimensional spaces, whereas limitations imposed by a finite number of practical measurements and model parameters entail recasting inverse problems in a discrete form. Thus, inverse problems turn into ill-posed problems and special numerical methods (regularization) are introduced to relax the conditions imposed on solution(s) in order to circumvent data overfitting. When the stability of solution(s) is violated under data perturbations, regularization to cope with instabilities is needed, too. Several instances of regularized inverse problems may be interpreted in the sense of Bayesian inference [15]. The basic approach to resolving an inverse problem is to consider all solutions (models) in the map space that could give rise to measured data. Once the forward problem has been accomplished, solutions consistent with errors in data measurements and modeling according to some criterion(s) are searched. If all consistent models are very similar, than it is no need to make an option. However, if several distinct solutions exist that comply with the imposed criteria, then a rule for selecting the optimal one in some sense must be introduced in order to solve the inverse problem. In many practical experiments, the observed data are transforms of the quantities of interest. The linear transforms encompass a large class of such experiments. An integral form of a general linear relation between the observable data \(g(s)\) and the unobserved interesting quantity \(f(t)\) has the expression: [17]  
\[g(s)=\int_{D}f(t)r(t,s)\mathrm{d}t+e(s) \tag{1}\]  
where \(r(t,s)\) is the transfer function or the response of the instrument, alternatively termed _blurring function_ or _point spread function_ (PSF) in image processing, which is considered to be perfectly known, while \(e(s)\) stands for the measurement _errors_ or _noise_. It is nevertheless assumed that the observed noise \(e(s)\) is independent of the function \(f(t)\). where \(\mathbf{g}=\{g_{1},g_{2},...,g_{M}\}\) is the data vector, \(\mathbf{f}=\{f_{1},f_{2},...,f_{M}\}\) is the image vector, \(\mathbf{e}=\{e_{1},e_{2},...,e_{M}\}\) is the error vector, and the matrix \(R\) of size \(M\times N\) has the elements:  
\[R_{mn}=r_{m}(t_{n})\left(t_{n+1}-t_{n+2}\right)/2 \tag{5}\]  
Each \(g_{m}\) in the data space may approximate the value of \(f(t)\) at a certain location \(t_{n}\) in the image space, in which case \(r(t_{n},s_{m})=R_{mn}\) should approximately have the form of a narrow instrumental response termed PSF and centered around \(t=t_{n}\). This is the case of components \(g_{m},\,m=1,2,...,M\) located in an entirely different function space from \(f_{n},\,n=1,2,...,N\), such as measuring different Fourier components of \(\mathbf{f}\). Though the measurements have reasonably been reduced to a finite number, the problem remains ill-posed and the matrix \(\mathbf{R}\) is likely to be either _singular_ or very _ill-conditioned_.