## 2 Methods
### Regularization of the inverse problem  
A set of admissible images \(\{\mathbf{f}\}\) is convex if for any two elements, \(\mathbf{f}_{a}\) and \(\mathbf{f}_{b}\), all linearly interpolated combinations like \((1-\eta)\mathbf{f}_{a}+\eta\mathbf{f}_{b}\), \(0\leq\eta\leq 1\), also belong to the set. Several deterministic constraints imposed on the solution \(\mathbf{\tilde{f}}\), such as positivity, compact support (i.e., zero value outside a specified region), or specified bounds (i.e., \(\mathbf{f}_{min}\leq\mathbf{f}\leq\mathbf{f}_{max}\)), define convex sets. Constraints can be formulated either in the image space or in the domain of any linear transform of the image \(\mathbf{f}\).  
Both the chi-square surface \(\chi^{2}(\mathbf{f})=constant\) and the entropy surface \(S(\mathbf{f})=constant\) are convex in the image space, so there is only one ME solution consistent with the data and the specified PSF. Consequently, the solution lies at the extremum of:  
\[Q(\lambda,\mathbf{f})=\chi^{2}(\mathbf{f})-\lambda S(\mathbf{f}) \tag{41}\]The first functional, \(\chi^{2}(\mathbf{f})\), measures the goodness or sharpness of fit between the model, \(\mathbf{f}\), and the data, \(\mathbf{g}\). By minimizing \(\chi^{2}(\mathbf{f})\) alone, the agreement becomes unrealistically good, while the solution becomes unstable and oscillating. Thus, \(\chi^{2}(\mathbf{f})\), by itself, defines a highly degenerate minimization problem. Reversely, \(S(\mathbf{f})\) measures the smoothness of the solution with respect to the variations in data. By maximizing \(S(\mathbf{f})\) alone results in a very smooth or stable solution, which is virtually not related with the measured data. Thus, \(S(\mathbf{f})\), by itself, is termed the _stabilizing_ functional or _regularizing_ operator. While \(\chi^{2}(\mathbf{f})\) is related with the _a posteriori_ knowledge of a solution, \(S(\mathbf{f})\) is related with _a priori_ expectation. The adjustable parameter \(\lambda\) is setting the trade-off between the constraints imposed on the two functionals (Figure 5). In addition, \(\lambda\) is required by the necessity of invariance for the minimization principle to the units in which \(\mathbf{f}\) is quantified (e.g., the change from 16-bit to 32-bit sampling) [26].  
This is a typical problem that addresses [PERSON]'s method of undetermined multipliers. The expected output is to settle on a best value of \(\lambda\) based on some criterion, ranging from an objective statistical constraint for \(\chi^{2}(\tilde{\mathbf{f}})\) to entirely subjective ones. As \(\lambda\) varies from 0 to \(\infty\), the solution \(\tilde{\mathbf{f}}\) varies along a trade-off curve between the task of minimizing \(\chi^{2}(\mathbf{f})\) and the task of maximizing \(S(\mathbf{f})\). For any linear combination of \(\chi^{2}(\mathbf{f})\) with a nondegenerate quadratic form \(S(\mathbf{f})\), such as \(\chi^{2}(\mathbf{f})-\lambda S(\mathbf{f})\), the minimization procedure yields a unique solution for \(\mathbf{f}\). That is, when a quadratic minimization principle is combined with a quadratic constraint, and both are positive, only one of the two need to be nondegenerate for the overall problem to be well-posed [26].  
Figure 5: Trade-off between entropy maximization and chi-square minimization.